TOPICS IN TOPOLOGY AND HOMOTOPY THEORY
Garth Warner
Department of Mathematics
University of Washington
PREFACE
This book is addressed to those readers who have been through Rotmany (or i*
*ts
equivalent), possess a wellthumbed copy of Spanierz, and have a good background*
* in
algebra and general topology.
Granted these prerequisites, my intention is to provide at the core a state*
* of the art
treatment of the homotopical foundations of algebraic topology. The depth of co*
*verage is
substantial and I have made a point to include material which is ordinarily not*
* included,
for instance, an account of algebraic Ktheory in the sense of Waldhausen. Ther*
*e is also
a systematic treatment of ANR theory (but, reluctantly, the connections with mo*
*dern
geometric topology have been omitted). However, truly advanced topics are not c*
*onsidered
(e.g., equivariant stable homotopy theory, surgery, infinite dimensional topolo*
*gy,etale K
theory, : : :). Still, one should not get the impression that what remains is e*
*asy: There
are numerous difficult technical results that have to be brought to heel.
Instead of laying out a synopsis of each chapter, here is a sample of some *
*of what is
taken up.
(1) Nilpotency and its role in homotopy theory.
(2) Bousfield's theory of the localization of spaces and spectra.
(3) Homotopy limits and colimits and their applications.
(4) The James construction, symmetric products, and the DoldThom theo*
*rem.
(5) Brown and Adams representability in the setting of triangulated ca*
*tegories.
(6) Operads and the MayThomason theorem on the uniqueness of infinite*
* loop
space machines.
(7) The plus construction and theorems A and B of Quillen.
(8) Hopkins' global picture of stable homotopy theory.
(9) Model categories, cofibration categories, and Waldhausen categorie*
*s.
(10) The Dugundji extension theorem and its consequences.
A book of this type is not meant to be read linearly. For example, a reader*
* wishing
to study stable homotopy theory could start by perusing x12 and x15 and then pr*
*oceed
to x16 and x17 or a reader who wants to learn the theory of dimension could imm*
*ediately
turn to x19 and x20. One could also base a second year course in algebraic topo*
*logy on
x3x11. Many other combinations are possible.
_________________________
yAn Introduction to Algebraic Topology, Springer Verlag (1988).
zAlgebraic Topology, Springer Verlag (1989).
Structurally, each xhas its own set of references (both books and articles)*
*. No attempt
has been made to append remarks of a historical nature but for this, the reader*
* can do no
better than turn to Dieudonney. Finally, numerous exercises and problems (in th*
*e form of
"examples" and "facts") are scattered throughout the text, most with partial or*
* complete
solutions.
_________________________
yA History of Algebraic and Differential Topology 19001960, Birkh"auser (19*
*89); see also, Adams,
Proc. Sympos. Pure Math. 22 (1971), 122 and Whitehead, Bull. Amer. Math. S*
*oc. 8 (1983), 129.
01
x0. CATEGORIES AND FUNCTORS
In addition to establishing notation and fixing terminology, background mat*
*erial from
the theory relevant to the work as a whole is collected below and will be refer*
*red to as the
need arises.
Given a category C , denote by Ob C its class of objects and by Mor C its*
* class of
morphisms. If X, Y 2 Ob C is an ordered pair of objects, then Mor (X; Y ) is t*
*he set of
morphisms (or arrows) from X to Y . An element f 2 Mor (X; Y ) is said to have *
*domain_
X and codomain_Y . One writes f : X ! Y or X f!Y . Functors preserve the arrows*
*, while
cofunctors reverse the arrows, i.e., a cofunctor is a functor on C OP, the cate*
*gory opposite
to C .
Here is a list of frequently occurring categories.
(1) SET , the category of sets, and SET *, the category of pointed s*
*ets. If
X; Y 2 Ob SET , then Mor (X; Y ) = F (X; Y ), the functions from X to Y , and i*
*f (X; x0),
(Y; y0) 2 Ob SET *, then Mor (X; x0); (Y; y0) = F (X; x0; Y; y0), the base po*
*int preserving
functions from X to Y .
(2) TOP , the category of topological spaces, and TOP *, the categor*
*y of pointed
topological spaces. If X; Y 2 Ob TOP , then Mor (X; Y ) = C(X; Y ), the cont*
*inuous
functions from X to Y , and if (X; x0), (Y; y0) 2 Ob TOP *, then Mor (X; x0);*
* (Y; y0) =
C(X; x0; Y; y0), the base point preserving continuous functions from X to Y .
(3) SET 2, the category of pairs of sets, and SET 2*, the category o*
*f pointed
pairs of sets. If (X; A), (Y; B) 2 Ob SET 2, then Mor (X; A); (Y; B) = F (X; *
*A; Y; B), the
functions from X to Y that take A to B, and if (X; A; x0); (Y; B; y0) 2 Ob SET*
* 2*, then
Mor (X; A; x0); (Y; B; y0) = F (X; A; x0; Y; B; y0), the base point preservin*
*g functions
from X to Y that take A to B.
(4) TOP 2, the category of pairs of topological spaces, and TOP 2*, *
*the category of
pointed pairs of topological spaces. If (X; A), (Y; B) 2 Ob TOP 2, then Mor (*
*X; A); (Y; B)
= C(X; A; Y; B), the continuous functions from X to Y that take A to B, and if *
*(X; A; x0),
(Y; B; y0) 2 Ob TOP 2*, then Mor (X; A; x0); (Y; B; y0) = C(X; A; x0; Y; B; *
*y0), the base
point preserving continuous functions from X to Y that take A to B.
(5) HTOP , the homotopy category of topological spaces, and HTOP **
*, the ho
motopy category of pointed topological spaces. If X; Y 2 Ob HTOP , then Mor (*
*X; Y ) =
[X; Y ], the homotopy classes in C(X; Y ), and if (X; x0); (Y; y0) 2 Ob HTOP *
* *, then
Mor (X; x0); (Y; y0) = [X; x0; Y; y0], the homotopy classes in C(X; x0; Y; y0*
*).
(6) HTOP 2, the homotopy category of pairs of topological spaces, an*
*d HTOP 2*,
the homotopy category of pointed pairs of topological spaces. If (X; A); (Y; B)*
*2Ob HTOP 2,
02
then Mor (X; A); (Y; B) = [X; A; Y; B], the homotopy classes in C(X; A; Y; B)*
*, and if
(X; A; x0), (Y; B; y0) 2 Ob HTOP 2*, then Mor (X; A; x0); (Y; B; y0) = [X; *
*A; x0; Y; B; y0],
the homotopy classes in C(X; A; x0; Y; B; y0).
(7) HAUS , the full subcategory of TOP whose objects are the Hausdor*
*ff spaces
and CPTHAUS, the full subcategory of HAUS whose objects are the compact space*
*s.
(8) X, the fundamental groupoid of a topological space X.
(9) GR, AB, RG (AMOD or MODA ), the category of groups, abelian
groups, rings with unit (left or right Amodules, A 2 Ob RG ).
(10) 0, the category with no objects and no arrows. 1, the category w*
*ith one
object and one arrow. 2, the category with two objects and one arrow not the id*
*entity.
A category is said to be discrete_if all its morphisms are identities. Ever*
*y class is the
class of objects of a discrete category.
[Note: A category is small_if its class of objects is a set; otherwise it *
*is large_. A
category is finite_(countable)_if its class of morphisms is a finite (countable*
*) set.]
In this book, the foundation for category theory is the "one universe" appr*
*oach taken by Herrlich
Strecker and Osborne (referenced at the end of the x). The key words are "set",*
* "class", and "conglomer
ate". Thus the issue is not only one of size but also of membership (every set *
*is a class and every class is
a conglomerate). Example: {Ob SET } is a conglomerate, not a class (the members*
* of a class are sets).
[Note: A functor F : C ! D is a function from MorC to MorD that preserves i*
*dentities and respects
composition. In particular: F is a class, hence {F} is a conglomerate.]
A metacategory_is defined in the same way as a category except that the obj*
*ects and the morphisms
are allowed to be conglomerates and the requirement that the conglomerate of mo*
*rphisms between two
objects be a set is dropped. While there are exceptions, most categorical conce*
*pts have metacategorical
analogs or interpretations. Example: The "category of categories" is a metacate*
*gory.
[Note: Every category is a metacategory. On the other hand, it can happen t*
*hat a metacategory
is isomorphic to a category but is not itself a category. Still, the convention*
* is to overlook this technical
nicety and treat such a metacategory as a category.]
ae
Given categories A ; B; C and functors TS::AB!!CC, the comma_category_ T*
*; S is
ae
the category whose objects are the triples (X; f; Y ) : XY22ObOAbB& f 2 Mor (*
*T X; SY )
ae *
* 0
and whose morphisms (X; f; Y ) ! (X0; f0; Y 0) are the pairs (OE; ) : OE 22M*
*orM(X;oXr)(Y;fYo0)r
03
f
T?X ! SY?
which the square TOEy yS commutes. Composition is defined component*
*wise
T X0 !f0 SY 0
and the identity attached to (X; f; Y ) is (idX ; idY).
(A\C ) Let A 2 Ob C and write KA for the constant functor 1 ! C with v*
*alue
A_then A\C KA ; idC is the category of objects_under_A_.
(C =B) Let B 2 Ob C and write KB for the constant functor 1 ! C with v*
*alue
B_then C =B idC; KB  is the category of objects_over_B_.
Putting together A\C & C =B leads to the category of objects_under_A_and_o*
*ver_B_:
A\C =B. The notation is incomplete since it fails to reflect the choice of the*
* structural
morphism A ! B. Examples: (1) ;\TOP =* = TOP ; (2) *\TOP =* = TOP *; (3)
A\TOP =* = A\TOP ; (4) ;\TOP =B = TOP =B; (5) B\TOP =B = TOP (B), the
"exspaces" of James (with structural morphism idB).
The arrow_category_C(!) of C is the comma category idC; idC. Examples: *
* (1)
TOP 2 is a subcategory of TOP (!); (2) TOP 2*is a subcategory of TOP *(!).
[Note: There are obvious notions of homotopy in TOP (!) or TOP *(!), from*
* which
HTOP (!) or HTOP *(!).]
The comma category KA; KB  is Mor(A; B) viewed as a discrete category.
A morphism f : X ! Y in a category C is said to be an isomorphism_if there *
*exists
a morphism g : Y ! X such that g O f = idX and f O g = idY. If g exists, then*
* g is
unique. It is called the inverse_of f and is denoted by f1 . Objects X; Y 2 Ob*
* C are said
to be isomorphic_, written X Y , provided that there is an isomorphism f : X !*
* Y . The
relation "isomorphic to" is an equivalence relation on Ob C .
The isomorphisms in SET are the bijective maps, in TOP the homeomorphisms*
*, in HTOP the
homotopy equivalences. The isomorphisms in any full subcategory of TOP are the*
* homeomorphisms.
ae
Let FG::CC!!DD be functors_then a natural_transformation_ from F to G is a
function that assigns to each X 2 Ob C an element X 2 Mor (F X; GX) such that
F?X X! GX?
for every f 2 Mor (X; Y ) the square Ff y y Gf commutes, being termed*
* a
F Y ! GY
Y
natural_isomorphism_if all the X are isomorphisms, in which case F and G are sa*
*id to be
naturally_isomorphic_, written F G.
04
ae
Given categories CD, the functor_category_[C ; D] is the metacategory who*
*se ob
jects are the functors F : C ! D and whose morphisms are the natural transforma*
*tions
Nat (F; G) from F to G. In general, [C ; D] need not be isomorphic to a categor*
*y, although
this will be true if C is small.
[Note: The isomorphismsaine[C ; D]aareethe natural isomorphisms.] ae
Given categories CDand functors KL::AD!!CB , there are functors [K;[D*
*]C:;[CL;]D]: [C ; D]
*
* ae
! [A ; D] precomposition *
* K
! [C ; B]definedabyepostcomposition. If 2 Mor ([C ; D]), then we shall write*
* L
in place of [K;[D]C;,Ls]o L(K) = (L)K.
There is a simple calculus that governs these operations:
ae ae
(K O K0) = (K)K0 and (L0O L) = L0(L) :
(0O )K = (0K) O (K) L(0O ) = (L0) O (L)
A functor F : C ! D is said to be faithful_(full_) if for any ordered pair *
*X; Y 2 Ob C ,
the map Mor (X; Y ) ! Mor (F X; F Y ) is injective (surjective). If F is full *
*and faithful,
then F is conservative_, i.e., f is an isomorphism iff F f is an isomorphism.
A category C is said to be concrete_if there exists a faithful functor U : *
*C ! SET . Example: TOP
is concrete but HTOP is not.
[Note: A category is concrete iff it is isomorphic to a subcategory of SET *
*.]
Associated with any object X in a category C is the functor Mor (X; _ ) 2 O*
*b [C ; SET ]
and the cofunctor Mor (_ ; X) 2 Ob [C OP; SET ]. If F 2 Ob [C ; SET ] is a fun*
*ctor or if
F 2 Ob [C OP; SET ] is a cofunctor, then the Yoneda lemma establishes a biject*
*ion X
between Nat (Mor (X; _ ); Fa)eor Nat (Mor (_ ; X); F ) and F X, viz.aeX () = X *
*(idX ).
OP ! [C ; *
*SET ]
Therefore the assignments XX!!MorM(X;o_r)(_l;eX)ad to functors CC! [C OP; S*
*ET ] that
are full, faithful, and injective on objects, the Yoneda_embeddings_. One says*
* that F is
representable_(by X) if F is naturally isomorphic to Mor (X; _ ) or Mor (_ ; X)*
*. Repre
senting objects are isomorphic.
The forgetful functors TOP ! SET , GR ! SET , RG ! SET are representable.*
* The power set
cofunctor SET ! SET is representable.
05
A functor F : C ! D is said to be an isomorphism_if there exists a functor *
*G : D ! C
such that G O F = idC and F O G = idD. A functor is an isomorphism iff it is fu*
*ll, faithful,
and bijective on objects. Categories C and D are said to be isomorphic_provi*
*ded that
there is an isomorphism F : C ! D .
[Note: An isomorphism between categories is the same as an isomorphism in *
*the
"category of categories".]
The full subcategory of TOP whose objects are the A spaces is isomorphic to*
* the category of ordered
sets and order preserving maps (reflexive + transitive = order_).
[Note: An A_space_is a topological space X in which the intersection of eve*
*ry collection of open sets
is open. Each x 2 X is contained in a minimal open set Ux and the relation x y*
* iff x 2 Uy is an order
on X. On the other hand, if is an order on a set X, then X becomes an A space *
*by taking as a basis
the sets Ux = {y : y x} (x 2 X).]
A functor F : C ! D is said to be an equivalence_if there exists a functor *
*G : D ! C
such that G O F idC and F O G idD. A functor is an equivalence iff it is full*
*, faithful,
and has a representative_image_, i.e., for any Y 2 Ob D there exists an X 2 Ob *
*C such that
F X is isomorphic to Y . Categories C and D are said to be equivalent_provided *
*that there
is an equivalence F : C ! D . The object isomorphism types of equivalent catego*
*ries are
in a onetoone correspondence.
[Note: If F and G are injective on objects, then C and D are isomorphic (ca*
*tegorical
"SchroederBernstein").]
The functor from the category of metric spaces and continuous functions to *
*the category of metrizable
spaces and continuous functions which assigns to a pair (X; d) the pair (X; od)*
*, od the topology on X
determined by d, is an equivalence but not an isomorphism.
[Note: The category of metric spaces and continuous functions is not a subc*
*ategory of TOP .]
A category is skeletal_if isomorphic objects are equal. Given a category C *
*, a skeleton_
__ __
of C is a full, skeletal subcategory C for which the inclusion C ! C has a repr*
*esentative
image (hence is an equivalence). Every category has a skeleton and any two skel*
*etons of a
category are isomorphic. A category is skeletally_small_if it has a small skele*
*ton.
The full subcategory of SET whose objects are the cardinal numbers is a ske*
*leton of SET .
A morphism f : X ! Y in a category C is said to be a monomorphism__ if it*
* is left
cancellable with respect to composition, i.e., for any pair of morphisms u; v :*
* Z ! X such
that f O u = f O v, there follows u = v.
06
A morphism f : X ! Y in a category C is said to be an epimorphism_if it is*
* right
cancellable with respect to composition, i.e., for any pair of morphisms u; v :*
* Y ! Z such
that u O f = v O f, there follows u = v.
A morphism is said to be a bimorphism_if it is both a monomorphism and an e*
*pimor
phism. Every isomorphism is a bimorphism. A category is said to be balanced_i*
*f every
bimorphism is an isomorphism. The categories SET , GR, and AB are balanced but *
*the
category TOP is not.
In SET , GR, and AB, a morphism is a monomorphism (epimorphism) iff it is i*
*njective (surjective).
In any full subcategory of TOP , a morphism is a monomorphism iff it is injecti*
*ve. In the full subcategory
of TOP * whose objects are the connected spaces, there are monomorphisms that a*
*re not injective on the
underlying sets (covering projections in this category are monomorphisms). In T*
*OP , a morphism is an
epimorphism iff it is surjective but in HAUS, a morphism is an epimorphism iff *
*it has a dense range. The
homotopy class of a monomorphism (epimorphism) in TOP need not be a monomorphi*
*sm (epimorphism)
in HTOP .
Given a category C and an object X in C , let M(X) be the class of all pair*
*s (Y; f),
where f : Y ! X is a monomorphism. Two elements (Y; f) and (Z; g) of M(X) are
deemed equivalent if there exists an isomorphism OE : Y ! Z such that f = g O *
*OE. A
representative_class_of_monomorphisms_in M(X) is a subclass of M(X) that is a s*
*ystem
of representatives for this equivalence relation. C is said to be wellpowered_*
*provided that
each of its objects has a representative class of monomorphisms which is a set.
Given a category C and an object X in C , let E(X) be the class of all pair*
*s (Y; f),
where f : X ! Y is an epimorphism. Two elements (Y; f) and (Z; g) of E(X) a*
*re
deemed equivalent if there exists an isomorphism OE : Y ! Z such that g = OE O*
* f. A
representative_class_of_epimorphisms_in E(X) is a subclass of E(X) that is a sy*
*stem of
representatives for this equivalence relation. C is said to be cowellpowered_p*
*rovided that
each of its objects has a representative class of epimorphisms which is a set.
SET, GR, AB, TOP (or HAUS ) are wellpowered and cowellpowered. The categ*
*ory of ordinal
numbers is wellpowered but not cowellpowered.
A monomorphism f : X ! Y in a category C is said to be extremal_provided th*
*at in
any factorization f = h O g, if g is an epimorphism, then g is an isomorphism.
An epimorphism f : X ! Y in a category C is said to be extremal_provided th*
*at in
any factorization f = h O g, if h is a monomorphism, then h is an isomorphism.
07
In a balanced category, every monomorphism (epimorphism) is extremal. In a*
*ny
category, a morphism is an isomorphism iff it is both a monomorphism and an ext*
*remal
epimorphism iff it is both an extremal monomorphism and an epimorphism.
In TOP , a monomorphism is extremal iff it is an embedding but in HAUS, a m*
*onomorphism is
extremal iff it is a closed embedding. In TOP or HAUS, an epimorphism is extre*
*mal iff it is a quotient
map.
A source_in a category C is a collection of morphisms fi: X ! Xi indexed by*
* a set I
and having a common domain. An nsource_is a source for which #(I) = n.
A sink_in a category C is a collection of morphisms fi : Xi ! X indexed by *
*a set I
and having a common codomain. An nsink_is a sink for which #(I) = n.
A diagram_in a category C is a functor : I ! C , where I is a small catego*
*ry, the
indexing_category_. To facilitate the introduction of sources and sinks associa*
*ted with ,
we shall write i for the image in Ob C of i 2 Ob I.
(lim) Let : I! C be a diagram_then a source {fi : X ! i} is said to be
natural_if for each ffi 2 Mor I, say i ffi!j, ffi O fi = fj. A limit_of is a *
*natural source
{`i : L ! i} with the property that if {fi : X ! i} is a natural source, then t*
*here
exists a unique morphism OE : X ! L such that fi = `iO OE for all i 2 Ob I. Li*
*mits are
essentially unique. Notation: L = limI (or lim).
(colim) Let : I! C be a diagram_then a sink {fi : i ! X} is said to be
natural_if for each ffi 2 Mor I, say i ffi!j, fi = fj O ffi. A colimit_of is *
*a natural sink
{`i: i! L} with the property that if {fi: i! X} is a natural sink, then there e*
*xists a
unique morphism OE : L ! X such that fi= OE O `i for all i 2 Ob I. Colimits are*
* essentially
unique. Notation: L = colimI (or colim).
There are a number of basic constructions that can be viewed as a limit or *
*colimit of
a suitable diagram.
Let I be a set; let I be the discrete category with Ob I = I. Given a coll*
*ection
{Xi: i 2 I} of objects in C , define a diagram : I! C by i= Xi (i 2 I).
(Products) A limit {`i : L ! i} of is said to be a product_of the Xi.
Q Q
Notation: L = Xi (or XI if Xi = X for all i), `i = pri, the projection_from *
* Xi to
i i
Xi. Briefly put: Products are limits of diagrams with discrete indexing categ*
*ories. In
particular, the limit of a diagram having 0 for its indexing category is a fina*
*l object in C .
[Note: An object X in a category C is said to be final_if for each object *
*Y there is
exactly one morphism from Y to X.]
08
(Coproducts) A colimit {`i : i ! L} of is said to be a coproduct_of t*
*he
`
Xi. Notation: L = Xi (or I . X if Xi = X for all i), `i = ini, the injection_*
*from Xi to
` i
Xi. Briefly put: Coproducts are colimits of diagrams with discrete indexing *
*categories.
i
In particular, the colimit of a diagram having 0 for its indexing category is a*
*n initial object
in C .
[Note: An object X in a category C is said to be initial_if for each object*
* Y there is
exactly one morphism from X to Y .]
In the full subcategory of TOP whose objects are the locally connected spa*
*ces, the product is the
product in SET equipped with the coarsest locally connected topology that is f*
*iner than the product
topology. In the full subcategory of TOP whose objects are the compact Hausdor*
*ff spaces, the coproduct
is the StoneCech compactification of the coproduct in TOP .
a
Let Ibe the category 1 o !!o 2. Given a pair of morphisms u; v : X ! Y in C*
* , define
ae b ae
a diagram : I! C by 1 = X & a = u .
2 = Y b = v
(Equalizers) An equalizer_in a category C of a pair of morphisms u; v *
*: X ! Y
is a morphism f : Z ! X with u O f = v O f such that for any morphism f0 : Z0 !*
* X
with u O f0 = v O f0 there exists a unique morphism OE : Z0 ! Z such that f0 = *
*f O OE. The
2source X f Z uOf!Y is a limit of iff Z f!X is an equalizer of u; v : X ! Y *
*. Notation:
Z = eq(u; v).
[Note: Every equalizer is a monomorphism. A monomorphism is regular_if it *
*is an
equalizer. A regular monomorphism is extremal. In SET , GR , AB , TOP (or HA*
*US ),
an extremal monomorphism is regular.]
(Coequalizers) A coequalizer_in a category C of a pair of morphisms u;*
* v : X ! Y
is a morphism f : Y ! Z with f O u = f O v such that for any morphism f0 : Y ! *
*Z0
with f0 O u = f0 O v there exists a unique morphism OE : Z ! Z0 such that f0 = *
*OE O f. The
2sink Y !fZ fOuX is a colimit of iff Y !fZ is a coequalizer of u; v : X ! Y *
*. Notation:
Z = coeq(u; v).
[Note: Every coequalizer is an epimorphism. An epimorphism is regular_if *
*it is a
coequalizer. A regular epimorphism is extremal. In SET , GR , AB , TOP (or H*
*AUS ),
an extremal epimorphism is regular.]
There are two aspects to the notion of equalizer or coequalizer, namely: (1*
*) Existence of f and
(2) Uniqueness of OE. Given (1), (2) is equivalent to requiring that f be a mon*
*omorphism or an epimor
phism. If (1) is retained and (2) is abandoned, then the terminology is weak_eq*
*ualizer_or weak_coequalizer_.
09
For example, HTOP *has neither equalizers nor coequalizers but does have weak *
*equalizers and weak
coequalizers.
ae
Let I be the category 1 o a!o3b o 2. Given morphisms fg::XY!!ZZ in C , de*
*fine a
8
< 1 = X aea = f
diagram : I! C by : 2 = Y & . j
3 = Z b = g P? ! *
* Y?
(Pullbacks) Given a 2sink X f!Z g Y , a commutative diagram y *
* yg is
X !f*
* Z
0 j0
said to be a pullback_square_if for any 2source X P 0! Y with f O0= gOj0ther*
*e exists a
unique morphism OE : P 0! P such that 0= O OE and j0= j O OE. The 2source X *
* P j!Y
is called a pullback_of the 2sink X f!Z g Y . Notation: P = X xZ Y . Limits of*
* are
pullback squares and conversely. ae
Let I be the category 1 o a o3b!o 2. Given morphisms fg::ZZ!!XY in C , de*
*fine a
8
< 1 = X aea = f
diagram : I! C by : 2 = Y & . g
3 = Z b = g Z? ! *
* Y?
(Pushouts) Given a 2source X f Z g!Y , a commutative diagram yf *
* yj
X ! *
* P
0 j0
is said to be a pushout_square_if for any 2sink X ! P 0 Y with 0O f = j0O g t*
*here exists
a unique morphism OE : P ! P 0such that 0= OE O and j0= OE O j. The 2sink X !*
* P j Y
is called a pushout_of the 2source X f Z g!Y . Notation: P = X tZY . Colimits *
*of are
pushout squares and conversely.
The result of dropping uniqueness in OE is weak_pullback_or weak_pushout_. *
*Examples are the com
mutative squares that define fibration and cofibration in TOP .
Let I be a small category, : IOP x I! C a diagram.
(Ends) A source {fi : X ! i;i} is said to be dinatural_if for each ffi*
* 2 Mor I,
say i ffi!j, (id; ffi) O fi = (ffi; id) O fj. An end_of is a dinatural source *
*{ei : E ! i;i}
with the property that if {fi: X ! i;i} is a dinatural source, then there exist*
*s a unique
morphism OE : X ! E such that fi=ZeiO OE forZall i 2 Ob I. Every end is a limit*
* (and every
limit is an end). Notation: E = i;i(or .
i I
(Coends) A sink {fi : i;i! X} is said to be dinatural_if for each ffi *
*2 Mor I,
say i ffi!j; fiO (ffi; id) = fj O (id; ffi). A coend_of is a dinatural sink {e*
*i : i;i! E}
with the property that if {fi : i;i! X} is a dinatural sink, then there exists *
*a unique
010
morphism OE : E ! X such that fi = OE O eiZfor all i 2 Ob I. Every coend is a c*
*olimit (and
i Z I
every colimit is a coend). Notation: E = i;i(or ).
ae
Let F : I! C be functors_then the assignment (i; j) ! Mor(Fi; Gj) defines *
*a diagram IOPxI !
G : I! C Z
SET and Nat(F; G) is the end Mor(Fi; Gi).
i
INTEGRAL YONEDA LEMMA Let I be a smallZcategory,iC a completeZand cocomple*
*te
category_then for every F in [IOP; C], Mor(_ ; i) . Fi F FiMor(i;_.)
i
Let I6= 0 be a small category_then I is said to be filtered_if
ae (F 1) Given any pair of objects i; j in I, there exists an object k an*
*d morphisms
i ! k
j ! k ;
(F 2) Given any pair of morphisms a; b : i ! j in I, there exists an o*
*bject k and
a morphism c : j ! k such that c O a = c O b.
Every nonempty directed set (I; ) can be viewed as a filtered category I, w*
*here
Ob I = I and Mor (i; j) is a one element set when i j but is empty otherwise.
Example: Let [N ] be the filtered category associated with the directed se*
*t of non
negative integers. Given a category C , denote by FIL (C ) the functor category*
* [[N ]; C]_
then an object (X ; f) in FIL (C ) is a sequence {Xn; fn}, where Xn 2 Ob C & f*
*n 2
Mor (Xn; Xn+1), and a morphism OE : (X ; f) ! (Y ; g) in FIL (C ) is a sequence*
* {OEn}, where
OEn 2 Mor (Xn; Yn) & gn O OEn = OEn+1 O fn.
(Filtered Colimits) A filtered_colimit_in C is the colimit of a diagra*
*m : I! C ,
where I is filtered.
(Cofiltered Limits) A cofiltered_limit_in C is the limit of a diagram *
* : I! C ,
where I is cofiltered.
[Note: A small category I6= 0 is said to be cofiltered_provided that IOP is*
* filtered.]
A Hausdorff space is compactly generated iff it is the filtered colimit in *
*TOP of its compact subspaces.
Every compact Hausdorff space is the cofiltered limit in TOP of compact metriz*
*able spaces.
Given a small category C , a path_in C is a diagram oe of the form X0 ! X1 *
* . .!.
X2n1 X2n (n 0). One says that oe begins_at X0 and ends_at X2n. The quotient*
* of
Ob C with respect to the equivalence relation obtained by declaring that X0 ~ *
*X00iff there
011
exists a path in C which begins at X0 and ends at X00is the set ss0(C ) of comp*
*onents_of
C , C being called connected_when the cardinality of ss0(C ) is one. The full s*
*ubcategory of
C determined by a component is connected and is maximal with respect to this p*
*roperty.
If C has an initial object or a final object, then C is connected.
[Note: The concept of "path" makes sense in any category.]
Let I6= 0be a small category_then Iis saidatoebe pseudofiltered_if
(PF 1) Given any pair of morphisms a : i !ijn I, there exists an obj*
*ect ` and morphisms
ae b : i ! k
c : j ! `such that c O a = d O b;
d : k ! `
(PF 2) Given any pair of morphisms a; b : i ! j in I, there exists a m*
*orphism c : j ! k such
that c O a = c O b.
I is filtered iff Iis connected and pseudofiltered. Iis pseudofiltered iff *
*its components are filtered.
ae
Given small categories IJ, a functor r : J ! Iis said to be final_provide*
*d that for
every i 2 Ob I, the comma category Ki; r is nonempty and connected. If J is *
*filtered
and r : J ! Iis final, then I is filtered.
[Note: A subcategory of a small category is final_if the inclusion is a fin*
*al functor.]
Let r : J ! I be final. Suppose that : I! C is a diagram for which colim O*
* r
exists_then colim exists and the arrow colim O r ! colim is an isomorphism.
Corollary: If i is a final object in I, then colim i.
[Note: Analogous considerations apply to limits so long as "final" is repla*
*ced through
out by "initial".]
Let Ibe a filtered category_then there exists a directed set (J; ) and a fi*
*nal functor r : J! I.
Limits commute with limits. In other words, if : I x J ! C is a diagram,*
* then
under the obvious assumptions
limIlimJ limIxJ limJxI limJlimI:
Likewise, colimits commute with colimits. In general, limits do not commute*
* with co
limits. However, if : Ix J ! SET and if I is finite and J is filtered, then *
*the arrow
colimJlimI ! limIcolimJ is a bijection, so that in SET filtered colimits commu*
*te
with finite limits.
[Note: In GR , AB or RG , filtered colimits commute with finite limits. *
* But, e.g.,
filtered colimits do not commute with finite limits in SET OP .]
In AB (or any Grothendieck category), pseudofiltered colimits commute with *
*finite limits.
012
A category C is said to be complete_(cocomplete_) if for each small categor*
*y I, every
2 Ob [I; C] has a limit (colimit). The following are equivalent.
(1) C is complete (cocomplete).
(2) C has products and equalizers (coproducts and coequalizers).
(3) C has products and pullbacks (coproducts and pushouts).
(4) C has a final object and multiple pullbacks (initial object and mu*
*ltiple
pushouts).
[Note: A source {i : P ! Xi} (sink {i : Xi ! P }) is said to be a multiple_*
*pullback_
(multiple_pushout_) of a sink {fi : Xia!eX} (source {fi : X ! Xi}) provided th*
*at
fi O i = fj O j (i O fi = j O fj) 8 ij and if for any source {0i: P 0! Xi} (s*
*ink
ae
{0i: Xi ! P 0}) with fi O 0i= fj O 0j(0iO fi = 0jO fj) 8 ij, there exists a u*
*nique
morphism OE : P 0! P (OE : P ! P 0) such that 8 i, 0i= iO OE (0i= OE O i). Ever*
*y multiple
pullback (multiple pushout) is a limit (colimit).]
The categories SET , GR , and AB are both complete and cocomplete. The same*
* is true of TOP
and TOP * but not of HTOP and HTOP *.
[Note: HAUS is complete; it is also cocomplete, being epireflective in TOP*
* .]
A category C is said to be finitely_complete_(finitely_cocomplete_) if for*
* each finite
category I, every 2 Ob [I; C] has a limit (colimit). The following are equival*
*ent.
(1) C is finitely complete (finitely cocomplete).
(2) C has finite products and equalizers (finite coproducts and coequa*
*lizers).
(3) C has finite products and pullbacks (finite coproducts and pushout*
*s).
(4) C has a final object and pullbacks (initial object and pushouts).
The full subcategory of TOP whose objects are the finite topological space*
*s is finitely complete and
finitely cocomplete but neither complete nor cocomplete. A nontrivial group, co*
*nsidered as a category,
has multiple pullbacks but fails to have finite products.
If C is small and D is finitely complete and wellpowered (finitely cocomp*
*lete and
cowellpowered), then [C ; D] is wellpowered (cowellpowered).
SET (!); GR (!); AB(!); TOP (!) (or HAUS (!)) are wellpowered and cowellpow*
*ered.
[Note: The arrow category C(!) of any category C is isomorphic to [2; C].]
Let F : C ! D be a functor.
013
(a) F is said to preserve a limit {`i : L ! i} (colimit {`i : i ! L}) *
*of a
diagram : I! C if {F `i : F L ! F i} ({F `i : F i ! F L}) is a limit (colimit)*
* of the
diagram F O : I! D .
(b) F is said to preserve limits (colimits) over an indexing category *
*I if F pre
serves all limits (colimits) of diagrams : I! C .
(c) F is said to preserve limits (colimits) if F preserves limits (col*
*imits) over all
indexing categories I.
The forgetful functor TOP ! SET preserves limits and colimits. The forgetfu*
*l functor GR ! SET
preserves limits and filtered colimits but not coproducts. The inclusion HAUS *
*! TOP preserves limits
and coproducts but not coequalizers. The inclusion AB ! GR preserves limits but*
* not colimits.
ae
There are two rules that determine the behavior of MorM(X;o_r)(_w;iX)th r*
*espect to
limits and colimits.
(1) The functor Mor (X; _ ) : C ! SET preserves limits. Symbolically,*
* there
fore, Mor (X; lim) lim(Mor (X; _ ) O ).
(2) The cofunctor Mor (_ ; X) : C ! SET converts colimits into limits*
*. Sym
bolically, therefore, Mor (colim; X) lim(Mor (_ ; X) O ).
REPRESENTABLE FUNCTOR THEOREM Given a complete category C , a functor
F : C ! SET is representable iff F preserves limits and satisfies the solution*
*_set_condition_:
There exists a set {Xi} of objects in C such that for each X 2 Ob C and each y *
*2 F X,
there is an i, a yi2 F Xi, and an f : Xi! X such that y = (F f)yi.
Take for C the category opposite to the category of ordinal numbers_then th*
*e functor C ! SET
defined by ff ! * has a complete domain and preserves limits but is not represe*
*ntable.
Limits and colimits in functor categories are computed "object by object". *
*So, if C is
a small category, then D (finitely) complete ) [C ; D] (finitely) complete and *
*D (finitely)
cocomplete ) [C ; D] (finitely) cocomplete.
Given a small category C , put bC= [C OP; SET ]_then bCis complete and coco*
*mplete.
The Yoneda embedding YC : C ! bCpreserves limits; it need not, however, preserv*
*e finite
colimits. The image of C is "colimit dense" in bC, i.e., every cofunctor C !*
* SET is a
colimit of representable cofunctors.
An indobject_in a small category C is a diagram : I ! C , where I is filt*
*ered.
Corresponding to an indobject , is the object L in bCdefined by L = colim(YC *
*O ).
014
The indcategory_IND(C ) of C is the category whose objects are the indobjects a*
*nd whose
morphisms are the sets Mor (0; 00) = Nat(L0 ; L00 ). The functor L : IND (C )*
* ! bC
that sends to L is full and faithful (although in general not injective on ob*
*jects), hence
establishes an equivalence between IND (C ) and the full subcategory of bCwhose*
* objects
are the cofunctors C ! SET which are filtered colimits of representable cofunc*
*tors. The
category IND (C ) has filtered colimits; they are preserved by L, as are all li*
*mits. Moreover,
in IND (C ), filtered colimits commute with finite limits. If C is finitely coc*
*omplete, then
IND (C ) is complete and cocomplete. The functor K : C ! IND (C ) that send*
*s X
to KX , where KX : 1 ! C is the constant functor with value X, is full, faith*
*ful, and
injective on objects. In addition, K preserves limits and finite colimits. The *
*composition
C !K IND (C ) L!bC is the Yoneda embedding YC . A cofunctor F 2 Ob Cb is said *
*to be
indrepresentable_if it is naturally isomorphic to a functor of the form L , 2*
* Ob IND (C ).
An indrepresentable cofunctor converts finite colimits into finite limits and c*
*onversely,
provided that C is finitely cocomplete.
[Note: The procategory_PRO (C ) is by definition IND (C OP)OP . Its object*
*s are the
proobjects_in C , i.e., the diagrams defined on cofiltering categories.]
The full subcategory of SET whose objects are the finite sets is equivalent*
* to a small category. Its
indcategory is equivalent to SET and its procategory is equivalent to the full *
*subcategory of TOP whose
objects are the totally disconnected compact Hausdorff spaces.
[Note: There is no small category C for which PRO (C ) is equivalent to SET*
* . This is because in
SET , cofiltered limits do not commute with finite colimits.]
ae ae
Given categories CD, functors FG::CD!!DC are said to be an adjoint_pair*
*_if the func
ae OP
tors MorMOo(FrO (ixdidD) from C OP x D to SET are naturally isomorphic, i.e.*
*, if it is
COP x G) ae
possible to assign to each ordered pair XY22ObOCbDa bijective map X;Y : Mor (*
*F X; Y ) !
Mor (X; GY ) which is functorial in X and Y . When this is so, F is a left_adj*
*oint_for G
and G is a right_adjoint_for F . Any two left (right) adjoints for G (F ) are *
*naturally
isomorphic. Left adjoints preserve colimits; right adjoints preserve limits. *
*In order that
(F; G) beaaneadjoint pair, it is necessary andasufficientethat there exist natu*
*ral transfor
mations 22Nat(idC;NGaOtF()FsOuG;bidject to (G) O (G) = idG . The data (F;*
* G; ; ) is
D ) (F ) O (F ) = idF ae
referred to as an adjoint_situation_, the natural transformations ::idCF!OGG*
*O!Fidbeing
*
* D
the arrows_of_adjunction_.
(UN) Suppose that G has a left adjoint F _then for each X 2 Ob C , each
015
Y 2 Ob D , and each f : X ! GY , there exists a unique g : F X ! Y such that f *
*= GgOX .
[Note: When reformulated, this property is characteristic.]
The forgetful functor TOP ! SET has a left adjoint that sends a set X to t*
*he pair (X; o), where o
is the discrete topology, and a right adjoint that sends a set X to the pair (X*
*; o), where o is the indiscrete
topology.
Let I be a small category, C a complete and cocomplete category. Examples:*
* (1) The constant
diagram functor K : C ! [I; C] has a left adjoint, viz. colim: [I; C] ! C, and*
* a right adjoint, viz.
lim: [I; C] ! C; (2) The functor C ! [IOP x I; C] that sends X to (i; j) ! Mor(*
*i; j) . X is a left adjoint
for end and the functor that sends X to (i; j) ! XMor(j;i)is a right adjoint fo*
*r coend.
GENERAL ADJOINT FUNCTOR THEOREM Given a complete category D , a func
tor G : D ! C has a left adjoint iff G preserves limits and satisfies the sol*
*ution_set_
condition_: For each X 2 Ob C , there exists a source {fi : X ! GYi} such that *
*for every
f : X ! GY , there is an i and a g : Yi! Y such that f = Gg O fi.
The general adjoint functor theorem implies that a small category is comple*
*te iff it is cocomplete.
ae
KAN EXTENSION THEOREM Given small categories CD, a complete (cocomplete)
category S, and a functor K : C ! D , the functor [K; S] : [D ; S] ! [C ; S] h*
*as a right
(left) adjoint ran (lan) and preserves limits and colimits.
[Note: If K is full and faithful, then ran (lan) is full and faithful.]
Suppose that S is complete. Let T 2 Ob [C ; S]_then ranTZis called the rig*
*ht_Kan_
extension_of T along K. In terms of ends, (ran T )Y = T XMor (Y;KX). The*
*re is a
X
"universal" arrow (ran T ) O K ! T . It is a natural isomorphism if K is full a*
*nd faithful.
Suppose that S is cocomplete. Let T 2 Ob [C ; S]_thenZlanT is called the le*
*ft_Kan_
X
extension_of T along K. In terms of coends, (lanT )Y = Mor (KX; Y ) . T X. T*
*here is
a "universal" arrow T ! (lanT ) O K. It is a natural isomorphism if K is full a*
*nd faithful.
Application: If C and D are small categories and if F : C ! D is a functor,*
* then the
precomposition functor bD! bChas a left adjoint bF: bC! bDand bFO YC YD O F .
[Note: One can always arrange that bFO YC = YD O F .]
The construction of the right (left) adjoint of [K; S] does not use the ass*
*umption that
D is small, its role being to ensure that [D ; S] is a category. For example, *
*if C is small
016
and S is cocomplete, then taking K = YC , the functor [YC ; S] : [Cb; S] ! [C ;*
* S] has a left
adjoint that sendsZT 2 Ob [C ; S] to T 2 Ob [Cb; S], where T O YC = T . On an *
*object
X Z X
F 2 bC, T F = Nat(YC X; F ) . T X = F X . T X. T is the realization_fun*
*ctor_; it
is a left adjoint for the singular_functor_ST , the composite of the Yoneda emb*
*edding S !
[S OP; SET ] and the precomposition functor [S OP; SET ] ! [C OP; SET ], thus (*
*ST Y )X =
Mor (T X; Y ).
[Note: The arrow of adjunction T O ST ! idSis a natural isomorphism iff ST *
*is full
and faithful.]
CAT is the category whose objects are the small categories and whose morp*
*hisms
are the functors between them: C ; D 2 Ob CAT ) Mor (C ; D) = Ob [C ; D]. CA*
*T is
concrete and complete and cocomplete. 0 is an initial object in CAT and 1 is*
* a final
object in CAT .
Let ss0 : CAT ! SET be the functor that sends C to ss0(C ), the set of co*
*mponents of C; let
dis: SET ! CAT be the functor that sends X to disX, the discrete category on X;*
* let ob: CAT ! SET
be the functor that sends C to ObC , the set of objects in C; let grd: SET ! CA*
*T be the functor that
sends X to grdX, the category whose objects are the elements of X and whose mor*
*phisms are the elements
of X x X_then ss0 is a left adjoint for dis, disis a left adjoint for ob, and o*
*bis a left adjoint for grd.
[Note: ss0 preserves finite products; it need not preserve arbitrary produc*
*ts.]
GRD is the full subcategory of CAT whose objects are the groupoids, i.e.*
*, the small
categoriesaein which every morphism is invertible. Example: The assi*
*gnment
: TOPX !!GRDX is a functor.
Let iso: CAT ! GRD be the functor that sends C to isoC, the groupoid whose*
* objects are those
of C and whose morphisms are the invertible morphisms in C_then isois a right a*
*djoint for the inclusion
GRD ! CAT . Let ss1 : CAT ! GRD be the functor that sends C to ss1(C ), the f*
*undamental_groupoid_
of C, i.e., the localization of C at MorC _then ss1 is a left adjoint for the i*
*nclusion GRD ! CAT .
is the category whose objects are the ordered sets [n] {0; 1; : :;:n} (*
*n 0)
and whose morphisms are the order preserving maps. In , every morphism can be
written as an epimorphism followed by a monomorphism and a morphism is a monomo*
*r
phism (epimorphism) iff it is injective (surjective). The face_operators_are th*
*e monomor
phisms ffini: [n  1] ! [n] (n > 0; 0 i n) defined by omitting the value i. *
*The
degeneracy_operators_are the epimorphisms oeni: [n + 1] ! [n] (n 0; 0 i n) d*
*e
017
fined by repeating the value i. Suppressing superscripts, if ff 2 Mor ([m]; [n*
*]) is not the
identity, then ff has a unique factorization ff = (ffii1O . .O.ffiip) O (oej1O *
*. .O.oejq), where
n i1 > . .>.ip 0, 0 j1 < . .<.jq < m, and m + p = n + q. Each ff 2 Mor ([m];*
* [n])
determines a linear transformation R m+1 ! R n+1 which restricts to a map ff: m*
* !
n. Thus there is a functor ? : ! TOP that sends [n] to n and ff to ff. Since*
* the
objects of are themselves small categories, there is also an inclusion : !*
* CAT .
Given a category C , write SIC for the functor category [ OP; C] and COSIC*
* for the
functor category [ ; C]_then by definition, a simplicial_object_in C is an obj*
*ect in SIC
and a cosimplicial_object_in C is an object in COSIC . Example: Y is a cosi*
*mplicial
object in b .
Specialize to C = SET _then an object in SISET is called a simplicial_s*
*et_and a
morphism in SISET is called a simplicial_map_.aGivenea simplicial set X, put X*
*n = X([n]),
so for ff : [m] ! [n], Xff : Xn ! Xm . If di=sXffii, then di and si are conne*
*cted by the
i= Xoei
simplicial_identities_:
ae 8 j + 1)
The simplicial_standard_nsimplex_is the simplicial set [n] = Mor (_ ; [n]), i.*
*e., [n] is
the result of applying to [n], so for ff : [m] ! [n], [ff] : [m] ! [n]. Owing *
*to the
Yoneda lemma, if X is a simplicial set and if x 2 Xn, then there exists one and*
* only one
simplicial map x : [n] ! X that takes id[n]to x. SISET is complete and cocomp*
*lete,
wellpowered and cowellpowered.
S
Let X be a simplicial set_then one writes x 2 X when one means x 2 Xn. Wi*
*th
n
this understanding, an x 2 X is said to be degenerate_if there exists an epimor*
*phism
ff 6= idand a y 2 X such that x = (Xff)y; otherwise, x 2 X is said to be nondeg*
*enerate_.
The elements of X0 (= the vertexes_of X) are nondegenerate. Every x 2 X admits*
* a
unique representation x = (Xff)y, where ff is an epimorphism and y is nondegene*
*rate.
The nondegenerate elements in [n] are the monomorphisms ff : [m] ! [n] (m n).
A simplicial_subset_of a simplicial set X is a simplicial set Y such that Y*
* is a subfunctor
of X, i.e., Yn Xn for all n and the inclusion Y ! X is a simplicial map. Not*
*ation:
Y X. The nskeleton_of a simplicial set X is the simplicial subset X(n) (n *
*0) of
X defined by stipulating that X(n)pis the set of all x 2 Xp for which there exi*
*sts an
epimorphism ff : [p] ! [q] (q n) and a y 2 Xq such that x = (Xff)y. Therefore
X(n)p= Xp (p n); furthermore, X(0) X(1) . . .and X = colimX(n). A proper
simplicial subset of [n] is contained in [n](n1), the frontier__[n] of [n]. Of*
* course,
018
_[0] = ;. X(0)is isomorphic to X0 . [0]. In general, let X#n be the set of non*
*degenerate
elements of Xn. Fix a collection {[n]x : x 2 X#n} of simplicial standard nsim*
*plexes
indexed by X#n_then the simplicial maps x : [n] ! X (x 2 X#n) determine an arrow
X#n.?_[n] ! X(n1)?
X#n. [n] ! X(n) and the commutative diagram y y is a pushout
X#n. [n] ! X(n)
square. Note too that _ [n] is a coequalizer: Consider the diagram
a u a
[n  2]i;j!! [n  1]i;
0i 1 such that every elem*
*ent of C(XE ; E) is a
constant.]
A morphism f : A ! B and an object X in a category C are said to be orthogo*
*nal_
(f?X) if the precomposition arrow f* : Mor (B; X) ! Mor (A; X) is bijective. G*
*iven a
class S Mor C , S? is the class of objects orthogonal to each f 2 S and given*
* a class
D Ob C , D? is the class of morphisms orthogonal to each X 2 D. One then says *
*that
a pair (S; D) is an orthogonal_pair_provided that S = D? and D = S? . Example: *
*Since
???=?, for any S, (S?? ; S? ) is an orthogonal pair, and for any D, (D? ; D?? )*
* is an
orthogonal pair.
[Note: Suppose that (S; D) is an orthogonal pair_then (1) S contains the is*
*omor
phisms of C ; (2) S is closed under composition; (3) S is cancellable_, i.e., g*
* O f 2 S &
A? ! A0?
f 2 S ) g 2 S and g O f 2 S & g 2 S ) f 2 S. In addition, if fy yf0 is*
* a
B ! B0
pushout square, then f 2 S ) f0 2 S, and if 2 Nat(; 0), where , 0: I! C , then
i2 S (8 i) ) colim 2 S (if colim, colim0 exist).]
Every reflective subcategory D of C generates an orthogonal pair. Thus, *
*with R :
C ! D the reflector, put T = O R, where : D ! C is the inclusion, and de*
*note
by ffl : idC ! T the associated natural transformation. Take for S Mor C the*
* class
consisting of those f such that T f is an isomorphism and take for D Ob C the *
*object
class of D , i.e., the class consisting of those X such that fflX is an isomorp*
*hism_then (S; D)
is an orthogonal pair.
A full, isomorphism closed subcategory D of a category C is said to be an o*
*rthogonal_subcategory
023
of C if ObD = S? for some class S MorC . If D is reflective, then D is orthogo*
*nal but the converse is
false (even in TOP ).
[Note: Let (S; D) be an orthogonal pair. Suppose that for each X 2 ObC ther*
*e exists a morphism
fflX : X ! TX in S, where TX 2 D_then for every f : A ! B in S and for every g *
*: A ! X there exists
a uniqueftf:lB ! TX such that fflX O g = t O f. So, for any arrow X ! Y , there*
* is a commutative diagram
X ! TX X
?y ?y
, thus T defines a functor C ! C and ffl : idC! T is a natural tra*
*nsformation. Since
Y !fflTY
Y
fflT = Tffl is a natural isomorphism, it follows that S? = D is the object clas*
*s of a reflective subcategory of
C .]
(DEF) Fix a regular cardinal _then an object X in a cocomplete categ*
*ory
C is said to be definite_provided that 8 regular cardinal 0 ; Mor (X; _ ) pr*
*eserves
colimits over [0; 0[, so for every diagram : [0; 0[! C , the arrow colimMor (X*
*; ff) !
Mor (X; colimff) is bijective.
Given a group G, there is a for which G is definite and all finitely pres*
*ented groups are !definite.
REFLECTIVE SUBCATEGORY THEOREM Let C be a cocomplete category. Sup
pose that S0 Mor C is a set with the property that for some , the domain and c*
*odomain
of each f 2 S0 are definite_then S?0is the object class of a reflective subcat*
*egory of C .
(P Localization) Let P be a set of primes. LetaSPe= {1} [ {n > 1 : p*
* 2 P )
p=n}_then a group G is said to be P_local_if the map Gg!!Ggn is bijective 8*
* n 2 SP .
GR P, the full subcategory of GR whose objects are the P local groups, is a*
* reflective
subcategoryaofeGR . In fact, Ob GR P = S?P, whereanoweSP stands for the set o*
*f homo
morphisms Z1!!Zn (n 2 SP ). The reflector LP : GRG !!GRGP is called P_lo*
*calization_.
P
Plocalization need not preserve short exact sequences. For example, 1 ! A3*
* ! S3 ! S3=A3 ! 1,
when localized at P = {3}, gives 1 ! A3 ! 1 ! 1 ! 1.
A category C with finite products is said to be cartesian_closed_provided t*
*hat each of
the functors _ xY : C ! C has a right adjoint Z ! ZY , so Mor (XxY; Z) Mor (X;*
* ZY ).
The object ZY is called an exponential_object_. The evaluation_morphism_evY;Z*
* is the
morphism ZY x Y ! Z such that for every f : X x Y ! Z there is a unique g : X !*
* ZY
such that f = evY;ZO (g x idY).
*
* 024
In a cartesian closed cat*
*egory:
*
* q Yi Q
(1) XY xZ (XY )*
*Z ; (3) X i (XYi);
Q Q *
* `i `
(2) ( Xi)Y *
*(XYi); (4) X x ( Yi) (X x Yi):
i i *
* i i
SET is cartesian closed *
*but SET OP is not cartesian closed. TOP is not cartesian closed but does
have full, cartesian closed s*
*ubcategories, e.g., the category of compactly generated Hausdorff spaces.
[Note: If C is cartesian *
*closed and has a zero object, then C is equivalent to 1. Therefore neither
SET * nor TOP * is cartesian *
*closed.]
CAT is cartesian closed:*
* Mor (C x D; E) Mor(C ; ED), where ED = [D ; E]. SISET is cartesian
closed: Nat(X x Y; Z) Nat(X;*
* ZY ), where ZY ([n]) = Nat(Y x [n]; Z).
[Note: The functor ner: C*
*AT ! SISET preserves exponential objects.]
A monoidal_category_is a *
*category C equipped with a functor : C x C ! C (the
multiplication_)aandean objec*
*t e 2 Ob C (the unit_), together with natural isomorphisms R,
L, and A, where RXL: X e !*
* X and AX;Y;Z : X (Y Z) ! (X Y ) Z, subject
X : e X*
* ! X
to the following assumptions.
(MC 1) The diagram
A *
* A
X Y (Z W?) !(*
*X Y ) (Z W ) ! (X Y ) Zx W
idA ?y *
* ?? Aid
X (Y Z) W *
*! X (Y Z) W
*
* A
commutes.
(MC 2) The diagram
*
* A
X *
*(e? Y ) !(X e)? Y
idL *
* ?y ?yRid
X *
* Y =======================X Y
commutes.
[Note: The "coherency" pr*
*inciple then asserts that "all" diagrams built up from in
stances of R, L, A (or their *
*inverses), and id by repeated application of necessarily
commute. In particular, the d*
*iagrams
A *
* A
e ( X Y ) !(e X)*
* Y X (Y e) !(X Y ) e
?? *
*? ? ?
L idR y *
*?yLid ?y ?yR
X Y ================*
*=======X Y X Y =======================X Y
*
* 025
commute and Le = Re : e e ! *
*e.]
Any category with finite *
*products (coproducts) is monoidal: Take X Y to be X Y (X q Y ) and
let e be a final (initial) ob*
*ject. The category AB is monoidal: Take X Y to be the tensor product and
let e be Z. The category SET *
**is monoidal: Take X Y to be the smash product X#Y and let e be the
two point set.
A symmetry_ for a monoida*
*l category C is a natural isomorphism >, where >X;Y :
X Y ! Y X, such that >Y;X O>X*
*;Y : X Y ! X Y is the identity, RX = LX O>X;e,
and the diagram
*
*A >
X ( Y Z) *
*!(X Y ) Z ! Z (X Y )
?? *
* ?
id> y *
* ?yA
X ( Z Y ) *
*!(X Z) Y ! (Z X) Y
*
* A > id
commutes. A symmetric_monoid*
*al_category_is a monoidal category C endowed with a
symmetry >. A monoidal catego*
*ry can have more than one symmetry (or none at all).
[Note: The "coherency" pr*
*inciple then asserts that "all" diagrams built up from in
stances of R, L, A, > (or the*
*ir inverses), and idby repeated application of necessarily
commute.]
Let C be the category of *
*chain complexes of abelian groups; let D be the full subcategory of C whose
objects are the graded abelia*
*n groups. C and D are both monoidal: Take X Y to beatheetensor product
and let e = {en} be the chain*
* complex defined by e0 = Z and en = 0 (n 6= 0). If X = {Xp} and if
ae *
* ae Y = {Yq}
x 2 Xp, then the assignment*
* X Y ! Y X is a symmetry for C and there are no others.
y 2 Yq *
* x y ! (1)pq(y x) ae
By contrast, D admits a secon*
*d symmetry, namely the assignment X Y ! Y X .
*
* x y ! y x
A closed_category_is a sy*
*mmetric monoidal category C with the property that each
of the functors _ Y : C ! C *
*has a right adjoint Z ! hom (Y; Z), so Mor (X Y; Z)
*
* OP
Mor X; hom(Y; Z) . The funct*
*or hom : C x C ! C is called an internal_hom_functor_.
The evaluation_morphism_evY;Z*
*is the morphism hom (Y; Z) Y ! Z such that for every
f : X Y ! Z there is a uniqu*
*e g : X ! hom (Y; Z) such that f = evY;ZO (g idY).
Agreeing to write Ue for the *
*functor Mor (e; _ ) (which need not be faithful), one has
UeOhom Mor . Consequently, *
*X hom (e; X) and hom (XY; Z) hom X; hom(Y; Z) .
*
* 026
A cartesian closed catego*
*ry is a closed category. AB is a closed category but is not
cartesian closed.
TOP admits, to within is*
*omorphism, exactly one structure of a closed category. For let X and Y
be topological spaces_then th*
*eir productaXe Y is the cartesian product X x Y supplied with the final
topology determined by the in*
*clusions {x} x Y ! X x Y(x 2 X; y 2 Y ), the unit being the one point
*
* X x {y} ! X x Y
space. The associated interna*
*l hom functor hom(X; Y ) sends (X; Y ) to C(X; Y ), where C(X; Y ) carries
the topology of pointwise con*
*vergence.
Given a monoidal category*
* C , a monoid_in C is an object X 2 Ob C together with
morphisms m : X X ! X and ff*
*l : e ! X subject to the following assumptions.
(MO 1) The diagram
*
* A mid
X ( X X) *
*!(X X) X ! X X
?? *
* ?
idm y *
* ?ym
X X *
*!mX
commutes.
(MO 2) The diagrams
fflid *
* idffl
e X ! X *
*X X X  X e
?? *
* ? ? ?
L m y *
* ?ym ?y ?yR
X=============*
*==========X X=======================X
commute.
MON C is the category w*
*hose objects are the monoids in C and whose morphisms
(X; m; ffl) ! (X0; m0; ffl0) *
*are the arrows f : X ! X0 such that f O m = m0O (f f) and
f O ffl = ffl0.
MON SET is the category *
*of semigroups with unit. MON AB is the category of rings with unit.
Given a monoidal category*
* C, a left_action_of a monoid X in C on an object Y 2 Ob C
is a morphism l : X Y ! Y su*
*ch that the diagram
A *
* mid fflid
X (X Y ) ! (*
*X X) Y ! X Y  e Y
?? *
* ? ?
idl y *
* ?yl ?yL
X Y *
*! Y =======================Y
*
* l
*
* 027
commutes.
[Note: The definition of *
*a right_action_is analogous.]
LACT X is the category *
*whose objects are the left actions of X and whose morphisms
(Y; l) ! (Y 0; l0) are the ar*
*rows f : Y ! Y 0such that f O l = l0O (id f).
If X is a monoid in SET ,*
* then LACT X is isomorphic to the functor category [X ; SET], X the
category having a single obje*
*ct * with Mor(*; *) = X.
A triple_T= (T;am;effl) i*
*n a category C consists of a functor T : C ! C and natural
transformations mf2fNat(TlO*
*2T;NTa)t(idsubject to the following assumptions.
*
* C; T )
(T 1) The diagram
*
* mT
*
* T O?T O T ! T?O T
Tm *
* ?y ?ym
*
* T O T !mT
commutes.
(T 2) The diagrams
fflT *
* Tffl
T ! *
*T O T T O T  T
?? *
* ? ?? ??
id m y *
* ?ym y yid
T =======*
*====T T ============ T
commute.
[Note: Formally, the func*
*tor category [C ; C] is a monoidal category: Take F G to
be F O G and let e be idC. Th*
*erefore a triple in C is a monoid in [C ; C] (and a cotriple_in
C is a monoid in [C ; C]OP )*
*, a morphism of triples being a morphism in the metacategory
MON [C ;C.]]
Given a triple T = (T; m;*
* ffl) in C , a T_algebra_is an object X in C and a morphism
: T X ! X subject to the fol*
*lowing assumptions.
(TA 1) The diagram
*
* T
*
* T (T X) !T X
*
* ?? ?
mX *
* y ?y
*
* T X ! X
*
* 028
commutes.
(TA 2) The diagram
*
*fflX
X *
*! T X
?? *
* ?
id y *
* ?y
X =*
*========X
commutes.
T ALG is the category whose obj*
*ects are the T algebras and whose morphisms
(X; ) ! (Y; j) are the arrows f : X ! *
*Y such that f O = j O T f.
[Note: If T = (T; m; ffl) is a cot*
*riple in C , then the relevant notion is T_coalgebra_and
the relevant category is T COALG .]
TakeaCe= AB . Let A 2 ObRG . Defi*
*ne T : AB ! AB by TXa=eA X, m 2 Nat(T O T; T) by
mX : A (A X) ! A X , ffl 2 Nat(i*
*dAB; T) by fflX : X ! A X _then TALG is isomorphic
a (b x) ! ab x *
* x ! 1 x
to AMOD .
Every adjoint situation (F; G; ; )*
* determines a triple in C , viz. (G O F; GF; ) (and
a cotriple in D , viz. (F O G; F G; ))*
*. Different adjoint situations can determine the same
triple. Conversely, every triple is de*
*termined by at least one adjoint situation, in general by
many. One realization is the construct*
*ion of EilenbergMoore: Given a triple T = (T; m; ffl)
in C , call FT the functor C ! TALG *
* that sends X f!Y to (T X; mX ) Tf!(T Y; mY ), call
GT the functor TALG ! C that send*
*s (X; ) f!(Y; j) to X f!Y , put X = fflX , and
(X;) = _then FT is a left adjoint for *
*GT and this adjoint situation determines T .
Suppose that C = SET , D = MON SE*
*T. Let F : C ! D be the functor that sends X to the
free semigroup with unit on X_then F i*
*s a left adjoint for the forgetful functor G : D ! C. The triple
determined1by this adjoint situation i*
*s T = (T; m; ffl), where T : SET ! SET assigns to each X the set
S
TX = Xn, mX : T(TX) ! TX is defined *
*by concatenation and fflX : X ! TX by inclusion. The
0
corresponding category of Talgebras i*
*s isomorphic to MON SET.
Let (F; G; ; ) be an adjoint situa*
*tion. If T = (G O F; GF; ) is the associated
triple in C , then the comparison_func*
*tor_ is the functor D ! T ALG that sends Y to
(GY; GY ) and g to Gg. It is the only *
*functor D ! T ALG for which O F = FT and
GT O = G.
Consider the adjoint situation pro*
*duced by the forgetful functor TOP ! SET _then TALG =
SET and the comparison functor TOP !*
* SET is the forgetful functor.
029
ae
Given categories CD, a functor G : D ! C is said to be monadic_(strictly_*
*monadic_)
provided that G has a left adjoint F : C ! D and the comparison functor : D ! *
*TALG
is an equivalence (isomorphism) of categories.
In order that G be monadic, it is necessary that G be conservative. So, e.g*
*., the forgetful functor
TOP ! SET is not monadic. If D is the category of Banach spaces and linear c*
*ontractions and if
G : D ! SET is the "unit ball" functor, then G has a left adjoint and is conser*
*vative, but not monadic.
Theorems due to Beck, Duskin and others lay down conditions that are necessary *
*and sufficient for a
functor to be monadic or strictly monadic. In particular, these results imply *
*that if D is a "finitary
category of algebraic structures", then the forgetful functor D ! SET is strict*
*ly monadic. Therefore the
forgetful functor from GR , RG , : :,:to SET is strictly monadic.
[Note: No functor from CAT to SET can be monadic.]
Among the possibilities of determining a triple T = (T; m; ffl) in C by a*
*n adjoint
situation, the construction of EilenbergMoore is "maximal". The "minimal" cons*
*truction
is that of Kleisli: KL (T ) is the category whose objects are those of C , the*
* morphisms
from X to Y being(Mor (X; T Y ) with fflX 2 Mor (X; T X) serving as the identit*
*y. Here, the
f!T Y
composition of XY ! T Z in KL (T ) is mZ OT gOf (calculated in C). If KT : C *
*! KL (T )
g
is the functor that sends X f!Y to X fflYOf!T Y and if LT : KL (T ) ! C is the *
*functor that
sends X f!T Y to T X mYOTf!T Y , then KT is a left adjoint for LT with arrows o*
*f adjunction
fflX ; idTX and this adjoint situation determines T .
[Note: Let G : D ! C be a functor_then the shape_of G is the metacategory
S G whose objects are those of C , the morphisms from X to Y being the conglom*
*erate
Nat (Mor (Y; G_ ); Mor (X; G_ )). While ad hoc arguments can sometimes be used *
*to show
that SG is isomorphic to a category, the situation is optimal when G has a lef*
*t adjoint
F : C ! D since in this case SG is isomorphic to KL (T ), T the triple in C det*
*ermined
by F and G.]
Consider the adjoint situation produced by the forgetful functor GR ! SET *
*_then KL (T ) is
isomorphic to the full subcategory of GR whose objects are the free groups.
A triple T = (T; m; ffl) in C is said to be idempotent_provided that m is*
* a natural
isomorphism (hence fflT = m1 = T ffl). If T is idempotent, then the comparison*
* functor
KL (T ) ! T ALG is an equivalence of categories. Moreover, GT : T ALG ! C *
*is full,
faithful, and injective on objects. Its image is a reflective subcategory of C *
*, the objects
030
being those X such that fflX : X ! T X is an isomorphism. On the other hand, *
*every
reflective subcategory of C generates an idempotent triple. Agreeing that two i*
*dempotent
triples T and T 0are equivalent if there exists a natural isomorphism o : T ! T*
* 0such that
ffl0 = o O ffl (thus also o O m = m0O oT 0O T o), the conclusion is that the co*
*nglomerate of
reflective subcategories of C is in a onetoone correspondence with the congl*
*omerate of
idempotent triples in C modulo equivalence.
[Note: An idempotent triple T = (T; m; ffl) determines an orthogonal pair (*
*S; D). Let
f : X ! Y be a morphism_then f is said to be T_localizing_if there is an isomo*
*rphism
OE : T X ! Y such that f = OE O fflX . For this to be the case, it is necessary*
* and sufficient
that f 2 S and Y 2 D. If C 0is a full subcategory of C and if T 0= (T 0; m0;*
* ffl0) is an
idempotent triple in C 0, then T (or T ) is said to extend_T0 (or T 0) provided*
* that S0 S
and D0 D (in general, (S0)? D (D0)?? , where orthogonality is meant in C ).]
Let (F; G; ; ) be an adjoint situation_then the following conditions are eq*
*uivalent: (1) (G O
F; GF; ) is an idempotent triple; (2) G is a natural isomorphism; (3) (F O G; F*
*G; ) is an idem
potent cotriple; (4) F is a natural isomorphism. And: (1), : :,:(4) imply that *
*the full subcategory C of
C whose objects are the X such that X is an isomorphism is a reflective subcate*
*gory of C and the full
subcategory D of D whose objects are the Y such that Y is an isomorphism is a *
*coreflective subcategory
of D.
[Note: C and D are equivalent categories.]
Given a category C and a class S Mor C , a localization_of_C_at_Sis a pair*
* (S1 C,
LS), where S1 C is a metacategory and LS : C ! S1 C is a functor such that 8 *
*s 2 S,
LSs is an isomorphism, (S1 C; LS) being initial among all pairs having this pr*
*operty,
i.e., for any metacategory D and for any functor F : C ! D such that 8 s 2 S*
*, F s is
an isomorphism, there exists a unique functor F 0: S1 C ! D such that F = F 0*
*O LS.
S1 C exists, is unique up to isomorphism, and there is a representative that h*
*as the same
objects as C itself. Example: Take C = TOP and let S Mor C be the class of ho*
*motopy
equivalences_then S1 C = HTOP .
__
[Note: If S is the class of all morphisms rendered invertible by LS (the sa*
*turation_of
__1
S), then the arrow S1 C ! S C is an isomorphism.]
Fix a class I which is not a set. Let C be the category whose objects are X*
*, Y , and {Zi: i 2 I} and
whose morphisms, apart from identities, are fi: X ! Zi and gi: Y ! Zi. Take S =*
* {gi: i 2 I}_then
S1C is a metacategory that is not isomorphic to a category.
[Note: The localization of a small category at a set of morphisms is again *
*small.]
031
Let C be a category and let S Mor C be a class containing the identities o*
*f C and
closed with respect to composition_then S is said to admit a calculus_of_left_f*
*ractions_if
(LF 1) Given a 2source X0 s X f!Y (s 2 S), there exists a commutativ*
*e square
X? f! Y
ys ?yt, where t 2 S;
X0 !f0Y 0
(LF 2) Given f; g : X ! Y and s : X0 ! X (s 2 S) such that f O s = g*
* O s,
there exists t : Y ! Y 0(t 2 S) such that t O f = t O g.
[Note: Reverse the arrows to define "calculus of right fractions".]
Let S MorC be a class containing the identities of C and closed with respe*
*ct to composition such
that 8 (s; t) : t O s 2 S & s 2 S ) t 2 S_then S admits a calculus of left frac*
*tions if every 2source
X? f! Y?
X0 sX f!Y (s 2 S) can be completed to a weak pushout square ys yt, where*
* t 2 S. For an
X0 !f0 Y 0
illustration, take C = HTOP and consider the class of homotopy classes of homo*
*logy equivalences.
Let C be a category and let S Mor C be a class admitting a calculus of lef*
*t fractions.
Given X; Y 2 Ob S1 C; Mor (X; Ya)eis the conglomerate of equivalence classes *
*of pairs
(s; f) : X f!Y 0s Y , two pairs (s;(f)t;bg)eing equivalent iff there exist u;*
* v 2 Mor C :
ae
u O s 1
v O t 2 S, with u O s = v O t and u O f = v O g. Every morphism in S C can*
* be
represented in the form (LSs)1LSf and if LSf = LSg, then there is an s 2 S suc*
*h that
s O f = s O g.
[Note: S1 C is a metacategory. To guarantee that S1 C is isomorphic to a *
*category,
it suffices to impose a solution_set_condition_: For each X 2 Ob C , there exis*
*ts a source
{si : X ! X0i} (si 2 S) such that for every s : X ! X0 (s 2 S), there is an i a*
*nd a
u : X0 ! X0isuch that u O s = si. This, of course, is automatic provided that X*
*\S, the
full subcategory of X\C whose objects are the s : X ! X0 (s 2 S), has a final *
*object.]
If C is the full subcategory of HTOP *whose objects are the pointed connec*
*ted CW complexes and
if S is the class of pointed homotopy classes of pointed nequivalences, then S*
* admits a calculus of left
fractions and satisfies the solution set condition.
Let (F; G; ; ) be an adjoint situation. Assume: G is full and faithful or, *
*equivalently,
that is a natural isomorphism. Take for S Mor C the class consisting of those*
* s such
that F s is an isomorphism (so F = F 0O LS)_then {X } S and S admits a calculus
032
of left fractions. Moreover, S is saturated and satisfies the solution set con*
*dition (in
fact, 8 X 2 Ob C , X\S has a final object, viz. X : X ! GF X). Therefore S1*
* C is
isomorphic to a category and LS : C ! S1 C has a right adjoint that is full an*
*d faithful,
while F 0: S1 C ! D is an equivalence.
[Note: Suppose that T = (T; m; ffl) is an idempotent triple in C . Let D*
* be the
corresponding reflective subcategory of C with reflector R : C ! D , so T = O *
*R, where
: D ! C is the inclusion. Take for S Mor C the class consisting of those *
*f such
that T f is an isomorphism_then S is the class consisting of those f such that *
*Rf is an
isomorphism, hence S admits a calculus of left fractions, is saturated, and sat*
*isfies the
solution set condition. The Kleisli category of T is isomorphic to S1 C and T *
*factors as
C ! S1 C ! D ! C , the arrow S1 C ! D being an equivalence.]
ae ae 1
Let (F; G; ; ) be an adjoint situation. Put S = {X } MorC _then S C *
* are isomor
ae ae T = {Y } MorD ae T1 D
0 : S1C ! T1 D G0O F0*
* id1
phic to categories and F induce functors F such that *
* S C , thus
ae G G0: T1 D ! S1C F0O G0*
* idT1D
S1C are equivalent. In particular, when G is full and faithful, S1C is equ*
*ivalent to D (the saturation
T1 D *
* __
of S being the class consisting of those s such that Fs is an isomorphism, i.e.*
*, S is the "S" considered
above).
Given a category C , a set U of objects in C is said to be a separating_set*
*_if for every
f
pair X !!Y of distinct morphisms, there exists a U 2 U and a morphism oe : U ! *
*X such
g
that f Ooe 6= g Ooe. An object U in C is said to be a separator_if {U} is a sep*
*arating set, i.e.,
if the functor Mor (U; _ ) : C ! SET is faithful. If C is balanced, finitely c*
*omplete, and
has a separating set, then C is wellpowered. Every cocomplete cowellpowered c*
*ategory
with a separator is wellpowered and complete. If C has coproducts, then a U 2 O*
*b C is a
`
separator iff each X 2 Ob C admits an epimorphism U ! X.
[Note: Suppose that C is small_then the representable functors are a separa*
*ting set
for [C ; SET ].]
Every nonempty set is a separator for SET . SET xSET has no separators but*
* the set {(;; {0}); ({0},
;)} is a separating set. Every nonempty discrete topological space is a separat*
*or for TOP (or HAUS ).
Z is a separator for GR and AB , while Z[t] is a separator for RG . In AMOD *
*, A (as a left Amodule)
is a separator and in MODA , A (as a right Amodule) is a separator.
Given a category C , a set U of objects in C is said to be a coseparating_*
*set_if for
033
f
every pair X !!Y of distinct morphisms, there exists a U 2 U and a morphism oe *
*: Y !
g
U such that oe O f 6= oe O g. An object U in C is said to be a coseparator_i*
*f {U} is
a coseparating set, i.e., if the cofunctor Mor (_ ; U) : C ! SET is faithful*
*. If C is
balanced, finitely cocomplete, and has a coseparating set, then C is cowellpowe*
*red. Every
complete wellpowered category with a coseparator is cowellpowered and cocomplet*
*e. If C
has products, then a U 2 Ob C is a coseparator iff each X 2 Ob C admits a monom*
*orphism
Q
X ! U.
Every set with at least two elements is a coseparator for SET . Every indis*
*crete topological space
with at least two elements is a coseparator for TOP . Q=Z is a coseparator for *
*AB . None of the categories
GR , RG , HAUS has a coseparating set.
SPECIAL ADJOINT FUNCTOR THEOREM Given a complete wellpowered category
D which has a coseparating set, a functor G : D ! C has a left adjoint iff G*
* preserves
limits.
A functor from SET ; AB or TOP to a category C has a left adjoint iff it p*
*reserves limits and a
right adjoint iff it preserves colimits.
Given a category C, an object P in C is said to be projective_if the functo*
*r Mor (P; _ ) :
C ! SET preserves epimorphisms. In other words: P is projective iff for each *
*epimor
phism f : X ! Y and each morphism OE : P ! Y , there exists a morphism g : P !*
* X
such that f O g = OE. A coproduct of projective objects is projective.
A category C is said to have enough_projectives_provided that for any X 2 *
*Ob C
there is an epimorphism P ! X, with P projective. If a category has enough proj*
*ectives
and a separator, then it has a projective separator. If a category has coprodu*
*cts and a
projective separator, then it has enough projectives.
The projective objects in the category of compact Hausdorff spaces are the *
*extremally disconnected
spaces. The projective objects in AB or GR are the free groups. The full subc*
*ategory of AB whose
objects are the torsion groups has no projective objects other than the initial*
* objects. In AMOD or
MODA , an object is projective iff it is a direct summand of a free module (a*
*nd every free module is a
projective separator).
Given a category C, an object Q in C is said to be injective_if the cofunct*
*or Mor (_ ; Q) :
C ! SET converts monomorphisms into epimorphisms. In other words: Q is injec*
*tive
034
iff for each monomorphism f : X ! Y and each morphism OE : X ! Q, there exists*
* a
morphism g : Y ! Q such that g O f = OE. A product of injective objects is inje*
*ctive.
A category C is said to have enough_injectives_provided that for any X 2 Ob*
* C , there
is a monomorphism X ! Q, with Q injective. If a category has enough injectives*
* and
a coseparator, then it has an injective coseparator. If a category has product*
*s and an
injective coseparator, then it has enough injectives.
The injective objects in the category of compact Hausdorff spaces are the r*
*etracts of products
[0; 1]. The injective objects in the category of Banach spaces and linear co*
*ntractions are, up to iso
morphism, the C(X), where X is an extremally disconnected compact Hausdorff spa*
*ce. In AB , the
injective objects are the divisible abelian groups (and Q=Z is an injective cos*
*eparator) but the only injec
tive objects in GR or RG are the final objects. The module Hom Z(A; Q=Z) is a*
*n injective coseparator
in AMOD or MODA .
A zero_object_in a category C is an object which is both initial and final*
*. The cat
egories TOP *, GR , and AB have zero objects. If C has a zero object 0C (o*
*r 0), then
for any ordered pair X; Y 2 Ob C there exists a unique morphism X ! 0C ! Y ,*
* the
zero_morphism_0XY (or 0) in Mor (X; Y ). It does not depend on the choice of a*
* zero ob
ject in C . An equalizer (coequalizer) of an f 2 Mor (X; Y ) and 0XY is said t*
*o be a kernel_
(cokernel_) of f. Notation: kerf (cokerf).
[Note: Suppose that C has a zero object. Let {Xi: i 2 I} be a collectionaof*
*eobjects in
Q ` idX (i *
*= j)
C for which Xiand Xiexist. The morphisms ffiij: Xi! Xj defined by i
i i ` Q 0XiXj(i*
* 6= j)
then determine a morphism t : Xi ! Xi such that prjO t O ini= ffiij. Exampl*
*e: Take
i i
#(I) = 2_then this morphism can be a monomorphism (in TOP *), an epimorphism (*
*in
GR ), or an isomorphism (in AB ).]
A pointed_category_is a category with a zero object.
Let C be a category with a zero object. Assume that C has kernels and co*
*kernels.
Given a morphism f : X ! Y , an image_(coimage_) of f is a kernel of a cokernel*
* (cokernel
of a kernel) for f. Notation: imf (coim f). There is a commutative diagram
f
kerf! X ! Y ! cokerf
?? x
y ??
coimf !_imf;
f
__
where f is the morphism parallel_to f. If parallel morphisms are isomorphisms, *
*then C is
said to be an exact_category_.
035
__ *
*__
[Note: In general, f need be neither a monomorphism nor an epimorphism and *
*f can
be a bimorphism without being an isomorphism.]
A category C that has a zero object is exact iff every monomorphism is the*
* kernel
of a morphism, every epimorphism is the cokernel of a morphism, and every morph*
*ism
admits a factorization: f = g O h (g a monomorphism, h an epimorphism). Such a*
* fac
torization is essentially unique. An exact category is balanced; it is wellpowe*
*red iff it is
cowellpowered. Every exact category with a separator or a coseparator is wellpo*
*wered and
cowellpowered. If an exact category has finite products (finite coproducts), t*
*hen it has
equalizers (coequalizers), hence is finitely complete (finitely cocomplete).
AB is an exact category but the full subcategory of AB whose objects are t*
*he torsion free abelian
groups is not exact. Neither GR nor TOP * is exact.
Let C be an exact category.
(EX) A sequence . .!.Xn1 dn1!Xn dn!Xn+1 ! . .i.s said to be exact_pr*
*ovided
that imdn1 kerdn for all n.
[Note: A short_exact_sequence_is an exact sequence of the form 0 ! X0 ! X !*
* X00!
0.]
(KerCoker Lemma) Suppose that the diagram
X1 _______wX2 _______wX3 _______w0
f1 f2 f3
u u u
0 ________wY1 ________wY2 ________wY3
is commutative and has exact rows_then there is a morphism ffi : kerf3 ! cokerf*
*1, the
connecting_morphism_, such that the sequence
kerf1 ! kerf2 ! kerf3 ffi!cokerf1 ! cokerf2 ! cokerf3
is exact. Moreover, if X1 ! X2 (Y2 ! Y3) is a monomorphism (epimorphism), then
kerf1 ! kerf2 (cokerf2 ! cokerf3) is a monomorphism (epimorphism).
(Five Lemma) Suppose that the diagram
X1 _______wX2 _______wX3 _______wX4 _______wX5
f1 f2 f3 f4 f5
u u u u u
Y1 ________wY2 ________wY3 ________wY4 ________wY5
is commutative and has exact rows.
036
(1) If f2 and f4 are epimorphisms and f5 is a monomorphism, then f3 is*
* an
epimorphism.
(2) If f2 and f4 are monomorphisms and f1 is an epimorphism, then f3 i*
*s a
monomorphism.
(Nine Lemma) Suppose that the diagram
0 0 0
  
u u u
0 _______wX0 _______wX _______wX00_______w0
  
u u u
0 ________wY 0________wY ________wY 00_______w0
  
u u u
0 ________wZ0 ________wZ ________wZ00________w0
  
u u u
0 0 0
is commutative, has exact columns, and an exact middle row_then the bottom row *
*is
exact iff the top row is exact.
In anaexactecategory C, there are two short exact sequences associated with*
* each morphism f : X !
Y , viz. 0 ! kerf ! X ! coimf ! 0.
0 ! imf ! Y ! cokerf ! 0
An additive_category_is a category C that has a zero object and which is eq*
*uipped with
a function + that assigns to each ordered pair f; g 2 Mor C having common domai*
*n and
codomain, a morphism f + g with the same domain and codomain satisfying the fol*
*lowing
conditions.
(ADD 1) On each morphism set Mor (X; Y ), + induces the structure of *
*an abelian
group. ae
(ADD 2) Composition is distributive over + : f(Og(g++hh))=O(fkO=g)(*
*+g(fOOkh)).+ (h O k)
(ADD 3) The zero morphisms are identities with respect to + : 0+f = f*
* +0 = f.
An additive category has finite products iff it has finite coproducts and w*
*hen this is
so, finite coproducts are finite products.
[Note: If C is small and D is additive, then [C ; D] is additive.]
037
AB is an additive category but GR is not. Any ring with unit can be view*
*ed as an additive
category having exactly one object (and conversely). The category of Banach spa*
*ces and continuous linear
transformations is additive but not exact.
An abelian_category_is an exact category C that has finite products and fi*
*nite co
products. Every abelian category is additive, finitely complete, and finitely *
*cocomplete.
A category C that has a zero object is abelian iff it has pullbacks, pushouts,*
* and ev
ery monomorphism (epimorphism) is the kernel (cokernel) of a morphism. In an ab*
*elian
`n Qn
category, t : Xi! Xi is an isomorphism.
i=1 i=1
[Note: If C is small and D is abelian, then [C ; D] is abelian.]
AB is an abelian category, as is its full subcategory whose objects are th*
*e finite abelian groups but
there are full subcategories of AB which are exact and additive, yet not abelia*
*n.
A Grothendieck_category_is a cocomplete abelian category C in which filtere*
*d colimits
commute with finite limits or, equivalently, in which filtered colimits of exac*
*t sequences
are exact. Every Grothendieck category with a separator is complete and has an *
*injective
coseparator, hence has enough injectives (however there exist wellpowered Groth*
*endieck
categories that do not have enough injectives). In a Grothendieck category, ev*
*ery fil
tered colimit of monomorphisms is a monomorphism, coproducts of monomorphisms a*
*re
` Q
monomorphisms, and t : Xi! Xi is a monomorphism.
i i
[Note: If C is small and D is Grothendieck, then [C ; D] is Grothendieck.]
AB is a Grothendieck category but its full subcategory whose objects are t*
*he finitely generated
abelian groups, while abelian, is not Grothendieck. If A is a ring with unit, t*
*hen AMOD and MODA
are Grothendieck categories.
ae
Given exact categories CD, a functor F : C ! D is said to be left_exact_(*
*right_exact_)
if it preserves kernels (cokernels) and exact_if it is both right and left exac*
*t. F is left exact
(right exact) iff for every short exact sequence 0 ! X0 ! X ! X00! 0 in C , the*
* sequence
0 ! F X0 ! F X ! F X00(F X0 ! F X ! F X00! 0) is exact in D . Therefore F is ex*
*act
iff F preserves short exact sequences or still, iff F preserves arbitrary exact*
* sequences.
[Note: F is said to be half_exact_if for every short exact sequence 0 ! X0 *
*! X !
X00! 0 in C , the sequence F X0 ! F X ! F X00is exact in D .]
The projective (injective) objects in an abelian category are those for whi*
*ch Mor(X; _)(Mor (_ ; X))
is exact. In AB , X _ is exact iff X is flat or here, torsion free. If Iis sma*
*ll and filtered and if C is
Grothendieck, then colim: [I; C] ! C is exact.
038
ae
Given additive categories CD, a functor F : C ! D is said to be additive_*
*if for all
X; Y 2 Ob C , the map Mor (X; Y ) ! Mor (F X; F Y ) is a homomorphism of abelia*
*n groups.
Every half exact functor between abelian categories is additive. An additive fu*
*nctor be
tween abelian categories is left exact (right exact) iff it preserves finite li*
*mits (finite co
limits). The additive_functor_category_[C ; D]+ is the full submetacategoryaofe*
*[C ; D] whose
OP ! [C ; AB*
* ]+
objects are the additive functors. There are Yoneda embeddings CC! [C OP; AB *
*]+. If
C and D are abelian categories with C small, if K : C ! D is additive, and*
* if S is a
complete (cocomplete)aabelianecategory, then there is an additive version of Ka*
*n extension
+
applicable to [C[;DS];.S]+The functors produced need not agree with those obt*
*ained by
forgetting the additive structure.
Let A be a ring with unit viewed as an additive category having exactly one*
* object_then AMOD
is isomorphic to [A; AB]+ and MODA is isomorphic to [AOP ; AB]+.
[Note: A right AmoduleZX and a left Amodule Y define a diagram AOP xA ! A*
*B (tensor product
A
over Z) and the coend X Y is X A Y , the tensor product over A.]
If C is small and additive and if D is additive, then
(1) D finitely complete and wellpowered (finitely cocomplete and cowel*
*lpowered)
) [C ; D]+ wellpowered (cowellpowered);
(2) D (finitely) complete ) [C ; D]+ (finitely) complete and D (finite*
*ly) cocom
plete ) [C ; D]+ (finitely) cocomplete;
(3) D abelian (Grothendieck) ) [C ; D]+ abelian (Grothendieck).
[Note: Suppose that C is small. If C is additive, then [C ; AB ]+ is a*
* complete
Grothendieck category and if C is exact and additive, then [C ; AB ]+ has a s*
*eparator
which as a functor C ! AB is left exact.]
Given a small abelian category C and an abelian category D , write LEX (C ;*
* D) for the
full, isomorphism closed subcategory of [C ; D]+ whose objects are the left exa*
*ct functors.
DERIVED FUNCTOR THEOREM If C is a small abelian category and if D is
a wellpowered Grothendieck category, then LEX (C ; D) is a reflective subcateg*
*ory of
[C ; D]+ . As such, it is Grothendieck. Moreover, the reflector is an exact fun*
*ctor.
[Note: The reflector sends F to its zeroth_right_derived_functor_R0F .]
If C is a small abelian category, then LEX (C ; AB ) is a Grothendieck cat*
*egory with
a separator. Therefore LEX (C ; AB ) has enough injectives. Every injective *
*object in
039
LEX (C ; AB ) is an exact functor. The Yoneda embedding C OP ! [C ; AB ]+ is l*
*eft exact.
It factors through LEX (C ; AB ) and is then exact.
[Note: Since C is abelian, every object in [C ; AB ]+ is a colimit of re*
*presentable
functors and every object in LEX(C, AB) is a filtered colimit of representable *
*functors.
Thus LEX(C, AB) is equivalent to IND(COP ) and so LEX(C, AB)OP is equivalent to
PRO(C).]
The full subcategory of AB whose objects are the finite abelian groups is e*
*quivalent to a small
category. Its procategory is equivalent to the opposite of the full subcategory*
* of AB whose objects are
the torsion abelian groups.
Given an abelian category C , a nonempty class C Ob C is said to be a Serr*
*e_class_
providedathatefor any short exact sequence 0 ! X0 ! X ! X00! 0 in C , Xa2eC iff
X0 0 00 X0
X00 2 C or, equivalently, for any exact sequence X ! X ! X in C , X00 2 C*
* )
X 2 C.
[Note: Since C is nonempty, C contains the zero objects of C .]
Given an abelian category C with a separator and a Serre class C, let SC *
*Mor C
be the class consisting of those s such that kers 2 C and cokers 2 C_then SC ad*
*mits a
__
calculus of left and right fractions and SC = SC, i.e., SC is saturated. The me*
*tacategory
S1CC is isomorphic to a category. As such, it is abelian and LSC : C ! S1CC i*
*s exact
and additive. An object X in C belongs to C iff LSCX is a zero object. Moreover*
*, if D is
an abelian category and F : C ! D is an exact functor, then F can be factored t*
*hrough
LSC iff all the objects of C are sent to zero objects by F .
[Note: Suppose that C is a Grothendieck category with a separator U_then fo*
*r any
Serre class C, LSC : C ! S1CC has a right adjoint iff C is closed under copro*
*ducts, in
which case S1CC is again Grothendieck and has LSCU as a separator.]
Take C = AB and let C be the class of torsion abelian groups_then C is a Se*
*rre class and S1CCis
equivalent to the category of torsion free divisible abelian groups or still, t*
*o the category of vector spaces
over Q.
Given a Grothendieck category C with a separator, a reflective subcategory *
*D of C
is said to be a Giraud_subcategory_provided that the reflector R : C ! D is exa*
*ct. Every
Giraud subcategory of C is Grothendieck and has a separator. There is a onet*
*oone
correspondence between the Serre classes in C which are closed under coproducts*
* and the
Giraud subcategories of C .
040
[Note: The GabrielPopescu theorem says that every Grothendieck category wi*
*th a
separator is equivalent to a Giraud subcategory of AMOD for some A.]
Attached to a topological space X is the category OP (X) whose objects are *
*the open subsets of X
and whose morphisms are the inclusions. The functor category [OP (X)OP ; AB] is*
* the category of abelian
presheaves on X. It is Grothendieck and has a separator. The full subcategory o*
*f [OP (X)OP ; AB] whose
objects are the abelian sheaves on X is a Giraud subcategory.
Fix a symmetric monoidal category V _then a V_category_M consists of a cl*
*ass
O (the objects_) and a function that assigns to each ordered pair X; Y 2 O an *
*object
HOM (X; Y ) in V plus morphisms CX;Y;Z : HOM (X; Y ) HOM (Y; Z) ! HOM (X;*
* Z),
IX : e ! HOM (X; X) satisfying the following conditions.
(V cat1) The diagram
HOM (X; Y ) (HOM (Y; Z) HOM (Z; W )) _______widCHOM(X; Y ) HOM (Y; *
*W )

 
A 
u 
(HOM (X; Y ) HOM (Y; Z)) HOM (Z; W ) C

 
Cid  
u u
HOM (X; Z) HOM (Z; W ) _____________________wCHOM(X; W )
commutes.
(V cat2) The diagram
e HOM (X; Y ) _____________wLHOM(X;Y)u_________HOM_R(X; Y ) e
 
  
Iid   idI
u  u
HOM (X; X) HOM (X; Y ) _______wCHOM(X; Y )u_____HOM_C(X; Y ) HOM (Y; Y )
commutes.
[Note: The opposite of a V category is a V category and the product of t*
*wo V 
categories is a V category.]
The underlying_category_UMof a Vcategory M has for its class of objects th*
*e class O,
Mor (X; Y ) being the set Mor (e; HOM (X; Y )). Composition Mor (X; Y ) x Mor *
*(Y; Z) !
Mor (X; Z) is calculated from e e e fg!HOM (X; Y ) HOM (Y; Z) ! HOM (X; *
*Z),
while IX serves as the identity in Mor (X; X).
[Note: A closed category V can be regarded as a V category (take HOM (X*
*; Y ) =
hom (X; Y )) and UV is isomorphic to V .]
041
Every category is a SET category and every additive category is an AB cat*
*egory.
A morphism F : V ! W of symmetric monoidal categories is a functor F : V !*
* W , a morphism
ffl : e ! Fe, and morphisms TX;Y : FX FY ! F(X Y ) natural in X, Y such that *
*the diagrams
Feu FX _______wTF(e X) FX uFe _______wTF(X e)
fflid  idffl 
 FL  FR
 u  u
e FX __________wLFX FX e __________wRFX
FX (FY FZ) _______wA(FX FY ) FZ
idT  Tid
 
u u
FX F(Y Z) F(X Y ) FZ
 
T T
u u
F(X (Y Z)) _________wFAF((X Y ) Z)
commute with F>X;Y O TX;Y = TY;XO >FX;FY .
Example: Given a symmetric monoidal category V, the representable functor M*
*or(e; _) determines
a morphism V ! SET of symmetric monoidal categories.
Let F : V ! W be a morphism of symmetric monoidal categories. Suppose that *
*M is a Vcategory.
Definition: F*M is the W category whose object class is O, the rest of the da*
*ta being FHOM (X; Y ),
FHOM (X; Y )FHOM (Y; Z) T!F(HOM (X; Y )HOM (Y; Z)) FC!FHOM (X; Z), e ffl!Fe*
* FI!FHOM (X; X).
[Note: Take W = SET and F = Mor(e; _) to recover UM .]
Fix a symmetric monoidal category V . Suppose given V categories M ; N_th*
*en a
V_functor_F : M ! N is the specification of a rule that assigns to each objec*
*t X in M an
object F X in N and the specification of a rule that assigns to each ordered pa*
*ir X, Y 2 O
a morphism FX;Y : HOM (X; Y ) ! HOM (F X; F Y ) in V such that the diagram
HOM (X; Y ) HOM (Y; Z) ____________wCHOM(X; Z)
F F  F
X;Y Y;Z   X;Z
u u
HOM (F X; F Y ) HOM (F Y; F Z) _______wCHOM(F X; F Z)
commutes with FX;X O IX = IFX .
[Note: The underlying_functor_UF : UM ! UN sends X to F X and f : e !
HOM (X; Y ) to FX;Y O f.]
Example: HOM : M OP x M ! V is a V functor if V is closed.
042
A V category is small_if its class of objects is a set; otherwise it is la*
*rge_. VCAT ,
the category of small V categories and V functors, is a symmetric monoidal ca*
*tegory.
Take V = AB _then an additive functor between additive categories "is" a V*
*functor.
Fix a symmetric monoidal category V . Suppose given V categories M , N a*
*nd V 
functors F; G : M ! N _then a V_natural_transformation_ from F to G is a clas*
*s of
morphisms X : e ! HOM (F X; GX) for which the diagram
e HOM u(X; Y ) _________wXHGX;YOM(F X; GX) HOM (GX; GY )
1 
L  C
 u
HOM (X; Y ) HOM (F X; GY )u
1 
R  C
u 
HOM (X; Y ) e _________wFX;YYHOM(F X; F Y ) HOM (F Y; GY )
commutes.
Assume that V is complete and closed. Let M , N be Vcategories with M smal*
*l_then the category
V [M ; N] whose objects are the VfunctorsZM ! N and whose morphisms are the V*
*natural transfor
Q
mations is a V category if HOM (F; G) = HOM (FX; GX), the equalizer of *
*HOM (FX; GX)!!
Q X X2O
hom (HOM (X0; X00); HOM (FX0; GX00)).
X0;X002O
Let C be a category with pullbacks_then an internal_category_(or a category*
*_object_)
in C consists of an object M, an object O, and morphisms s : M ! O, t : M ! O,
e : O ! M, c : M xO M ! M satisfying the usual category theoretic relations (he*
*re,
M xO?M ! M
y ?yt). Notation: M = (M; O; s; t; e; c).
M !s O
[Note: There are obvious notions of internal_functor_and internal_natural_t*
*ransforma_
tion_.]
An internal category in SET is a small category. An internal category in *
*SISET is a simplicial
object in CAT .
An internal category in CAT is a (small) double_category_.
[Note: Spelled out, such an entity consists of objects X; Y; : :,:horizonta*
*l morphisms f; g; : :,:ver
tical morphisms OE; ; : :,:and bimorphisms (represented diagramatically by squ*
*ares). The objects and
043
h
the horizontal morphisms form a category with identities X X!X. The objects a*
*nd the vertical mor
X
?
phisms form a category with identities vXy . The bimorphisms have horizontal a*
*nd vertical laws of
X
o ! o
? ?
o ! o ! o y y
? ? ?
composition y y y , o ! o under which they form a category wi*
*th identities
? ?
o ! o ! o y y
o ! o
o ! o ! o
? ? ?
X hX!X X f! Y y y y
?y ? ? ?
OE idOE yOE,vXy idf y vY. In the situation o ! o ! o , the result *
*of composing
? ? ?
Y !h Y X ! Y y y y
Y f
o ! o ! o
horizontally and then vertically is the same as the result of composing vertica*
*lly and then horizontally.
Furthermore, horizontal composition of vertical identities gives a vertical ide*
*ntity and vertical compo
sition of horizontal identities gives a horizontal identity. Finally, the hori*
*zontal and vertical identities
X hX!X X hX!X
v ?y ? ? ?
X idvX yvX , vXy idhX yvX coincide.]
X !h X X ! X
X hX
Example: Let C be a small category_then dbC is the double category whose ob*
*jects are those of
C , whose horizontal and vertical morphisms are those of C, and whose bimorphis*
*ms are the commutative
squares in C. All sources, targets, identities, and compositions come from C.
Let C be a category with pullbacks. Given an object O in C , an Ograph_is *
*an object
A and a pair of morphisms s; t : A ! O. OGR is the category whose objects a*
*re the
Ographs and whose morphisms (A; s; t) ! (A0; s0; t0) are the arrows f : A ! A0*
*such that
A xO?A0 ! *
*A0?
s = s0O f, t = t0O f. If A xO A0 is defined by the pullback square y *
*y t0 and
A !sO
0 t
if the structural morphisms are A xO A0! A0!s O, A xO A0! A ! O, then A xO A0is*
* an
Ograph. Therefore OGR is a monoidal category: Take A A0 to be A xO A0 and *
*let e
be (O; idO; idO). A monoid M in OGR is an internal category in C with objec*
*t element
O.
044
Let C be a category with pullbacks. Given an internal category M in C , th*
*e nerve_
nerM of M is the simplicial object in C definedabyener0M = O, ner1Mae= M, ne*
*rnM =
M xO . .x.OM (n factors). At the bottom, d0d : ner1M ! ner0M is t , while
1 s
higher up, in terms of the underlying projections, d0 = (ss1; : :;:ssn1), dn =*
* (ss2; : :;:ssn),
di = (ss1; : :;:c O (ssni; ssni+1); : :;:ssn) (0 < i < n), and at the bottom,*
* s0 : ner0M !
ner1M is e, while higher up, si= eiO oei, where oei inserts O at the n  i + 1*
* spot and ei is
idxO . .x.Oe xO . .x.Oidplaced accordingly (0 i n).
[Note: An internal functor M ! M 0induces a morphism nerM ! nerM 0of simp*
*licial
objects.]
Suppose that C is a small category. Consider nerC_then an element f of nern*
*Cis a diagram of
the form X0f0!X1 ! . .!.Xn1 fn1!Xn and
8
< X1 ! . .!.Xn (i = 0)f Of
dif = X0 ! . .!.Xi1 ii1!Xi+1! . .!.Xn (0 < i < n);
:
X0 ! . .!.Xn1 (i = n)
idXi
sif = X0 ! . .!.Xi! Xi! . .!.Xn. The abstract definition thus reduces to the*
*se formulas since
f corresponds to the ntuple (fn1; : :;:f0).
Let C be a category with pullbacks. Given an internal category M in C, a le*
*ft_M_object_
is an object T : Y ! O in C =O and a morphism : M xO Y ! Y such that
M xO M xO Y _______wcxOidM xOuY______eOxxOOYid
  
idxO   L
u u u
M xO Y _____________wY ____________ Y
M xO?Y ! Y?
and y yT commute, where M xO Y is defined by the pullback squ*
*are
M !t O
M xO?Y ! Y
y ?yT . Example: Take C = SET _then M is a small category and*
* the
M !s O
category of left M objects is equivalent to the functor category [M ; SET ].
[Note: A right_M_object_is an object S : X ! O in C =O and a morphism ae*
* :
X xO M ! X such that the analogous diagrams commute, where X xO M is defined
045
X xO?M ! M?
by the pullback square y yt . Example: Take C = SET _then M is a
X !S O
small category and the category of right M objects is equivalent to the functo*
*r category
[M OP ; SET ].]
Let C be a category with pullbacks. Given an internal category M in C a*
*nd a
left M object Y , the translation_category_tranY of Y is the category objec*
*t M Y =
(MY ; OY ; sY ; tY ; eY ; cY ) in C , where MY = M xO Y; OY = Y; sY is the*
* projection
M xO Y ! Y , tY is the action : M xO Y ! Y , and eY ; cY are derived from e : *
*O ! M,
c : M xO M ! M. Example: Take C = SET , let M be a small category, and suppose*
* that
G : M ! SET is a functor_then G determines a left M object YG and the transl*
*ation
category of YG can be identified with the Grothendieck construction on G.
Let G be a semigroup with unit, G the category having a single object * wit*
*h Mor(*; *) = G.
Suppose that Y is a left Gset, i.e., an object in LACT G or still, a left G *
*object. The translation
category of Y is (G x Y; Y; sY ; tY ; eY ; cY ), where sY (g; y) = y, tY (g; y)*
* = g . y, eY (y) = (e; y), cY ((g2; y2),
(g1; y1)) = (g2g1; y1). Specialize and let Y = G_then the objects of the transl*
*ation category of G are the
elements of G and Mor(g1; g2) {g : gg1 = g2}.
Let C be a category with pullbacks. Given an internal category M in C , *
*and a
right M object X and a left M object Y , the bar_construction_bar(X; M ; Y ) *
*on (X; Y )
is the simplicial object in C defined by barn(X; M ; Y ) = X xO nernM xO Y . N*
*ote that
ae appears only in dn and appears only in d0. The translation_category_tran(X*
*; Y ) of
(X; Y ) is the category object M X;Y = (MX;Y ; OX;Y ; sX;Y ; tX;Y ; eX;Y ; cX;Y*
* ) in C , where
MX;Y = X xO M xO Y; OX;Y = X xO Y; sX;Y = ae xO idY; tX;Y = idX xO ; eX;Y &
cX;Y being definable in terms of e & c. Therefore bar(X; M ; Y ) nerM X;Y . Ex*
*ample:
O can be viewed as a right M object via O xO M L!M s!O and as a left M object*
* via
M xO O R!M !tO, and M can be viewed as a right M object via M xO M c!M s!O and
as a left M object via M xO M c!M !tO, so bar(O; M ; O), bar(O; M ; M), bar(M;*
* M ; O),
bar(M; M ; M) are meaningful.
Let G be a group, G the groupoid having a single object * with Mor(*; *) = *
*G. View G as a
left Gset_then bar(*; G; G) is isomorphic to the nerve of grdG. In fact, the *
*objects of grdG are the
elements of G and the morphisms of grdG are the elements of G x G (s(g; h) = g,*
* t(g; h) = h, idg= (g; g),
(h; k)O(g; h) = (g; k)), thus nerngrdG = Gx. .x.G (n+1 factors) and di(g0; : :;*
*:gn) = (g0; : :;:bgi; : :;:gn),
si(g0; : :;:gn) = (g0; : :;:gi; gi; : :;:gn). On the other hand, bar(*; G; G) i*
*s the nerve of the translation
category of G. The functor tranG ! grdG which is the identity on objects and se*
*nds a morphism (g; h)
046
in tranG to the morphism (h; g . h) in grdG induces an isomorphism nertranG ! n*
*ergrdG of simplicial
sets. For (g0; : :;:gn) ! (gn; gn1gn; : :;:g0. .g.n) is the arrow nerntranG !*
* nerngrdG, its inverse
being (g0; : :;:gn) ! (gng1n1; gn1g1n2; : :;:g0). Both nertranG and nergrd*
*G are simplicial right G
sets, viz. (g0; : :;:gn) . g = (g0; : :;:gng) and (g0; : :;:gn) . g = (g0g; : :*
*;:gng), and the isomorphism
nertranG ! nergrdG is equivariant.
Let T = (T; m; ffl) be a triple in a category C _then a right_T_functor_in*
* a category V
is a functor F : C ! V plus a natural transformation ae : F OT ! F such that th*
*e diagrams
F O?T O T aeT!F O T F _____wFfflflflF O T
Fm y ?yae, flflflflflflcomaemute and a left T functorin a ca*
*tegory U is
u _____________
F O T !ae F F
a functor G : U ! C plus a natural transformation : T O G ! G such that the di*
*agrams
T O?T O G T! T O G Gf_____wfflGfllT O G
mG y ?y , ffllflflflflcommute. The bar constructionbar(F ; T*
*; G) on
u _______________
T O G ! G G
(F; G) is the simplicial object in [U ; V] defined by barn(F ; T; G) = F O T nO*
* G, where
d0 = aeT n1G, di = F T i1mT ni1G (0 < i < n), dn = F T n1, and si = F T if*
*flT niG.
In particular: bar1(F ; T; G) = F OT OG, bar0(F ; T; G) = F OG, and d0; d1 : F *
*OT OG ! F OG
are aeG; F , while s0 : F O G ! F O T O G is F fflG.
Example: If X is a T algebra in C with structural morphism : T X ! X, the*
*n X
determines a left T functor G : 111! C and one writes bar(F ; T; X) for the as*
*sociated bar
construction.
Take V = C, F = T, ae = m, and put o = fflTG (thus o : T O G ! T O T O G). *
*There is a commutative
diagram
T OGae______________________wG
aeo ffNlG 
aeo NNQ  
  _____T  
TO T OTGO Gfl w  
 AAADmG flflffl
T O G ____________________________wG
from which it follows that : T O G ! G is a coequalizer of (d0; d1) = (mG; T).*
* Consider the string
of arrows T O Tn O G d0!T O Tn1 O G ! . .!.T O T O G d0!T O G ! G fflG!T O G s*
*0!T O T O G ! . .!.
T O Tn1 O G s0!T O Tn O G. Viewing G as a constant simplicial object in [ OP;*
* [C ; V]], there are simplicial
morphisms G ! bar(T; T; G), bar(T; T; G) ! G, viz. sn0OfflG : G ! T OTn OG, Odn*
*0: T OTn OG ! G, and
the composition G ! bar(T; T; G) ! G is the identity. On the other hand, if hi:*
* T OTn OG ! T OTn+1 OG
047
is defined by hi= si0(fflTni+1G)di0(0 i n), then d0O h0 = id, dn+1 O hn = sn*
*0O fflG O O dn0, and
8
< hj1O di(i < j) aeh O s (i j)
diO hj = diO hi1(i = j > 0); siO hj = j+1 i :
: hjO si1(i > j)
hjO di1(i > j + 1)
[Note: Take instead U = C, G = T, = m_then with o = FTffl, ae : F O T ! F *
*is a coequalizer of
(d1; d0) = (Fm; aeT) and the preceding observations dualize.]
11
x1. COMPLETELY REGULAR HAUSDORFF SPACES
The reader is assumed to be familiar with the elements of general topology.*
* Even so,
I think it best to provide a summary of what will be needed in the sequel. Not *
*all terms
will be defined; most proofs will be omitted.
Let X be a locally compact Hausdorff space (LCH space).
PROPOSITION 1 A subspace of X is locally compact iff it is locally closed,*
* i.e., has
the form A \ U, where A is closed and U is open in X.
The class of nonempty LCH spaces is closed under the formation in TOP of f*
*inite products and
arbitrary coproducts.
[Note: An arbitrary product of nonempty LCH spaces is a LCH space iff all b*
*ut finitely many of the
factors are compact.]
In practice, various additional conditions are often imposed on a LCH spac*
*e X. The
connections among the most common of these can be summarized as follows:
NPmetrizable____________________wparacompact______wnorma*
*lu
N N 
compact metrizableae 
aeae 
aeo 
compact ______________________w'oecompact
' ' NP
') QNN
Lindel"of
EXAMPLE Let be the first uncountable ordinal and consider [0; ] (in the o*
*rder topology)_
then [0; ] is Hausdorff. And: (i) [0; ] is compact but not metrizable; (ii) [0;*
* [ is locally compact and
normal but not paracompact; (iii) [0; ] x [0; [ is locally compact but not norm*
*al.
Here are some important points to keep in mind.
(LCH 1) X is completely regular, i.e., X has enough real valued conti*
*nuous func
tions to separate points and closed sets in the sense that for every point x 2 *
*X and for every
closed subset A X not containing x, there exists a continuous function OE : X *
*! [0; 1]
such that OE(x) = 1, OEA = 0 .
(LCH 2) X is oecompact iff X possesses a sequence_of_exhaustion_, i.*
*e., an in
__
creasing sequence {Un} of relatively compact open sets Un X such that Un Un+1*
* and
S
X = Un.
n
12
`
(LCH 3) X is paracompact iff X admits a representation X = Xi, wher*
*e the
i
Xi are pairwise disjoint nonempty open oecompact subspaces of X.
(LCH 4) X is second countable iff X is oecompact and metrizable.
(a) If X is metrizable, then X is completely metrizable.
(b) If X is metrizable and connected, then X is second countable.
Let X be a topological space_then a collection S = {S} of subsets of X is s*
*aid to be:
point_finite_if each x 2 X belongs to at most finitely many S 2 S;
neighborhood_finite_if each x 2 X has a neighborhood meeting at most finite*
*ly many
S 2 S;
discrete_if each x 2 X has a neighborhood meeting at most8one S 2 S.
< point finite
A collection which is the union of a countable number of : neighborhood fin*
*ite
discrete
subcollections is said to be 8
< oepoint_finite_
: oeneighborhood_finite_oediscrete
: _______
A collection S = {S} of subsets of X is said to be closure_preserving_if fo*
*r every subcollection S0 S,
S __ ____S __ __
S0 = S0, S0the collection {S : S 2 S0}.
A collection which is the union of a countable number of closure preserving*
* subcollections is said to
be oeclosure_preserving_.
Every neighborhood finite collection of subsets of X is closure preserving *
*but the converse is certainly
false since any collection of subsets of a discrete space is closure preserving*
*. A point finite closure preserving
closed collection is neighborhood finite. However, this is not necessarily true*
* if "closed" is replaced by
"open" as can be seen by taking X = [0; 1], S = {]0; 1=n[: n 2 N}.
Let S = {S} be a collection of subsets of X. The order_of a point x 2 X wit*
*h respect
to S, written ord(x; S), is the cardinality of {S 2 S : x 2 S}. S is of finite_*
*order_if ord(S) =
sup ord(x; S) < !. The star_of a subset Y X with respect to S, written st(Y; S*
*), is the
x2X S
set {S 2 S : S \ Y 6= ;}. S is star_finite_if 8 S0 2 S : #{S 2 S : S \ S0 6= *
*;} < !.
Suppose that U = {Ui : i 2 I} is a covering of X_then a covering V = {Vj : *
*j 2 J}
of X is a refinement_(star_refinement_) of U if each Vj (st(Vj; V)) is containe*
*d in some Ui
and is a precise_refinement_of U if I = J and Vi Ui for every i. If U admits a *
*point finite
(open) or a neighborhood finite (open, closed) refinement, then U admits a prec*
*ise point
finite (open) or neighborhood finite (open, closed) refinement.
13
To illustrate the terminology, recall that if X is metrizable, then every o*
*pen covering
of X has an open refinement that is both neighborhood finite and oediscrete.
Let X be a completely regular Hausdorff space (CRH space).
(C) X is compact iff every open covering of X has a finite (neighborho*
*od finite,
point finite) subcovering.
(P) X is paracompact iff every open covering of X has a neighborhood f*
*inite
open (closed) refinement.
(M) X is metacompact iff every open covering of X has a point finite o*
*pen
refinement.
The following conditions are equivalent to paracompactness.
(P1) Every open covering of X has a closure preserving open refinement.
(P2) Every open covering of X has a oeclosure preserving open refinem*
*ent.
(P3) Every open covering of X has a closure preserving closed refineme*
*nt.
(P4) Every open covering of X has a closure preserving refinement.
PROPOSITION 2 A LCH space X is paracompact iff every open covering of X has
a star finite open refinement.
[Suppose that X is paracompact. Given an open covering U = {Ui} of X, choos*
*e a
__
relatively compact open refinement V = {Vj} of U such that each V jis contained*
* in some
Ui_then every neighborhood finite open refinement of V is necessarily star fini*
*te.]
A collection S = {S} of subsets of a CRH space X is said to be directed_if *
*for all S1, S2 2 S, there
exists S3 2 S such that S1[ S2 S3.
The following condition is equivalent to metacompactness.
(M)D Every directed open covering of X has a closure preserving closed*
* refinement.
Given an open covering U of X, denote by UF the collection whose elements a*
*re the unions of the finite
subcollections of U_then UF is directed and refines U if U itself is directed. *
*So the above characterization
of metacompactness can be recast:
(M)F For every open covering U of X, UF has a closure preserving close*
*d refinement.
*
* S
It is therefore clear that a LCH space X is metacompact iff X admits a repr*
*esentation X = Ki,
*
* i
where {Ki} is a closure preserving collection of compact subsets of X.
A CRH space X is said to be subparacompact_ if every open covering of X has*
* a
oediscrete closed refinement.
[Note: This definition is partially suggested by the fact that X is paraco*
*mpact iff
every open covering of X has a oediscrete open refinement.]
14
Suppose that X is subparacompact. Let U = {U} be an open covering of X_then*
* U
S
has a closed refinement A = An, where each An is discrete. Every A 2 An is co*
*ntained
n
in some UA 2 U. The collection
Vn = {UA  ([ An  A) : A 2 An} [ {U  [An : U 2 U}
is an open refinement of U and 8 x 2 X 9 nx : ord(x; Vnx) = 1.
FACT X is subparacompact iff every open covering of X has a oeclosure pres*
*erving closed refinement.
A CRH space X is said to be submetacompact_if for every open covering U of *
*X there
exists a sequence {Vn} of open refinements of U such that 8 x 2 X 9 nx : ord(x;*
* Vnx) < !.
FACT X is submetacompact iff every directed open covering of X has a oeclo*
*sure preserving closed
refinement.
These properties are connected by the implications:
metacompact _______wsubmetacompact
446
4 4 46
compact ______wparacompact' 4
' '') 44
subparacompact
Each is hereditary with respect to closed subspaces and, apart from compact*
*ness,
each is hereditary with respect to Foesubspaces (and all subspaces if this is *
*so of open
subspaces).
EXAMPLE (The_Thomas_Plank_)1Let L0 = {(x; 0) : 0 < x < 1} and for n 1, le*
*t Ln =
S
{(x; 1=n) : 0 x < 1}. Put X = Ln. Topologize X as follows: For n 1, each po*
*int of Ln except
0
for (0; 1=n) is isolated, basic neighborhoods of (0; 1=n) being subsets of Ln c*
*ontaining (0; 1=n) and having
finite complements, while for n = 0, basic neighborhoods of (x; 0) are sets of *
*the form {(x; 0)}[{(x; 1=m) :
m n} (n = 1; 2; : :):. X is a LCH space. Moreover, X is metacompact: Every ope*
*n covering of X has
an open refinement consisting of one basic neighborhood for each x 2 X and any *
*such refinement is point
finite since the order of each x 2 X with respect to it is at most three. But X*
* is not paracompact. In
fact, X is not even normal: A = {(0; 1=n) : n = 1; 2; : :}:and B = L0 are1disjo*
*int closed subsets of X
S
and every neighborhood of A contains all but countably many points of Ln, whi*
*le every neighborhood
1 1
S
of B contains uncountably many points of Ln. Finally, X is subparacompact. Th*
*is is because X is a
1
countable union of closed paracompact subspaces.
15
EXAMPLE (The_Burke_Plank_) Take X = [0; +[x[0; +[{(0; 0)}; + the cardinal*
* successor of
. For 0 < ff < +, put ae
Hff= [0; +[x{ff}
Vff= {ff} x [0; +[:
Topologize Xaasefollows: Isolate all pointsaexceptethose onatheevertical or hor*
*izontal axis, the basic neigh
borhoods of (0; ff)being the subsets of Hff containing (0; ff)and having f*
*inite complements.
(ff; 0) Vff (ff; 0)
X is a metacompact LCH space. But X is not subparacompact. To see this, first o*
*bserve that if S and
T are subsets of X such S \ Hffand T \ Vffare countable for every ff < +, then *
*X 6= S [ T. Let
U = {Hff: 0 < ff < +} [ {Vff: 0 < ff < +}. U is an open covering of X and the c*
*laim is: U does not
S
have a oediscrete closed refinement V = Vn. To get a contradiction suppose t*
*hat such a V does exist.
n
Let Sn and Tn be the elements of Vn whichaareecontained in {Hff:a0e< ff < +} an*
*d {Vff: 0 < ff < +},
S = [Sn Sn = [Sn
respectively_then Vn = Sn [ Tn. Write T = n[ ; where . Since*
* the Vn are
discrete, S \ Hffand T \ Vffare countable fonTnr every ff < +,Tnt=h[Tnus X 6= S*
* [ T = [V and so V does not
cover X.
[Note: Why does one work with + rather than ? Reason: In general, if the we*
*ight of X is ,
then X is subparacompact iff X is submetacompact.]
EXAMPLE (IsbellMrowka_Space_) Let D be an infinite set. Choose a maximal i*
*nfinite collection S
of almost disjoint countably infinite subsets of D, almost disjoint meaning tha*
*t 8 S1 6= S2 2 S, #(S1\S2) <
!. Observe that S is uncountable. Put (D) = S [ D. Topologize (D) as follows: I*
*solate the points of
D and take for the basic neighborhoods of a point S 2 S all sets of the form {S*
*} [ (S  F), F a finite
subset of S. (D) is a LCH space. In addition: S is closed and discrete, while D*
* is open and dense.
Specialize and let D = N_then X = (N ) is subparacompact, being a Moore space (*
*cf. p. 117), but is
not metacompact. In fact, since S is uncountable, the open covering {N } [ {{S}*
* [ S : S 2 S} cannot have
a point finite open refinement.
[Note: The IsbellMrowka space (N ) depends on S. Question: Up to homeomorp*
*hism how many
distinct (N ) are there? Answer: 22!.]
The coproduct of the Burke plank and the IsbellMrowka space provides an ex*
*ample of a submeta
compact X that is neither metacompact nor subparacompact.
EXAMPLE (The_van_Douwen_Line_) The object is to equip X = R with a first co*
*untable, separable
topology that is finer than the usual topology (hence Hausdorff) and under whic*
*h X = R is locally compact
but not submetacompact. Given x 2 R, choose a sequence {qn(x)} Q such that x *
* qn(x) < 1=n.
Next, let {Cff: ff < 2!} be an enumeration of the countable subsets Cffof R wit*
*h #(__Cff) = 2!. For
ff < 2!; N = 0; 1; 2; : :,:pick inductively a point
xffN2 __Cff (Q [ {xfiM: fi < ff orfi = ff andM < N}):
16
Put ae
S0 = {xff0: ff < 2!}
SN = {xffN: ff < 2! andCff S0} (N = 1; 2; : :):
1S
and write S in place of R  SN . Observe that Q [ S0 S and that the SN are p*
*airwise disjoint. Given
1
x = xffN2 R  S, choose a sequence {cm (x)} Cff( S0 S) such that x  cm (x)*
* < 1=m.aTopologizee
X = R as follows: Isolate the points of Q and take for the basic neighborhoods *
*of x 2 S  Q the
*
* x 2 R  S
sets ae
Kk(x) = {x} [ {qn(x) : n k} (k = 1; 2; : :)::
Kk(x) = {x} [ {cm (x) : m k} [ {qn(cm (x)) : m k; n m}
This prescription defines a first countable, separable topology on the line tha*
*t is finer than the usual
topology. And, since the Kk are compact, it is a locally compact topology. Ho*
*wever, it is not a sub
metacompact topology. Thus let UN = S [ SN _then UN is open and U = {UN } is an*
* open covering
of X. Consider any sequence {VM } of open refinements of U. For M = 1; 2; : :,:*
*and N = 1; 2; : :,:let
T S
WMN = S {V 2 VM : V \ SN 6= ;} and form W0 = S0 \ WMN = S0  (S0  WMN*
* ). Since
M;N M;N
#(S0) = 2! and since the S0  WMN are countable, W0 is nonempty. But any x0 i*
*n W0 necessarily
belongs to infinitely many distinct elements of VM (M = 1; 2; : :):. Conseque*
*ntly, the topology is not
submetacompact.
JONES' LEMMA If a Hausdorff space X contains a dense set D and a closed di*
*screte subspace
S with #(S) 2#(D), then X is not normal.
Application: The van Douwen line is not normal.
[In fact, each SN is closed and discrete with #(SN ) = 2!.]
Let X be a LCH space. Under what conditions is it true that X metacompact )*
* X
paracompact? For example, is it true that if X is normal and metacompact, then*
* X is
paracompact? This is an open question. There are no known counterexamples in ZF*
*C or
under any additional set theoretic assumptions. Two positive results have been *
*obtained.
(1) (Danielsy) A normal LCH space X is paracompact provided that it is*
* bound_
edly_metacompact_, i.e., every open covering of X has an open refinement of fin*
*ite order.
(2) (Gruenhagez) A normal LCH space X is paracompact provided that it *
*is
locally connected and submetacompact.
Suppose that X is normal and metacompact_then on general grounds all that o*
*ne can say is this.
Consider any open covering U of X: By metacompactness, U has a point finite ope*
*n refinement V which,
_________________________
yCanad. J. Math. 35 (1983), 807823; see also Topology Appl. 28 (1988), 113*
*125.
zTopology Proc. 4 (1979), 393405.
17
__
by normality, has a precise open refinement W with the property that W is a pre*
*cise closed refinement of
V.
FACT Let X be a CRH space. Suppose that X is submetacompact_then X is norma*
*l iff every
open covering of X has a precise closed refinement.
A Hausdorff space X is said to be perfect_if every closed subset of X is a *
*Gffi. The
IsbellMrowka space (N ) is perfect; however, it is not normal (cf. p. 112).
A Hausdorff space X is said to be perfectly_normal_if it is perfect and nor*
*mal. The
ordinal space [0; ], while normal, is not perfectly normal since the point {} i*
*s not a Gffi.
On the other hand, X metrizable ) X perfectly normal. Every perfectly normal L*
*CH
space X is first countable.
[Note: The assumption of perfect normality can be used to upgrade the stren*
*gth of a
covering property.
(1) (Arhangel'skiiy) Let X be a LCH space. If X is perfectly normal an*
*d meta
compact, then X is paracompact.
(2) (BennettLutzerz) Let X be a LCH space. If X is perfectly normal *
*and
submetacompact, then X is subparacompact.]
A CRH space X is said to be countably_paracompact_if every countable open c*
*overing
of X has a neighborhood finite open refinement. The ordinal space [0; [ is coun*
*tably para
compact (being countably compact) and normal, whereas the ordinal space [0; ]x[*
*0; [ is
countably paracompact (being compact x countably compact countably compact) but
not normal. On the other hand, X perfectly normal ) X countably paracompact.
To recapitulate:
paracompact
' ' '') u
metrizable' normalu countably paracompact
'')  4 46
4
perfectly normal
FACT Suppose that X is normal_then X is countably paracompact iff every co*
*untable open
covering of X has a oediscrete closed refinement.
So: In the presence of normality, X subparacompact ) X countably paracompac*
*t. This implication
is strict since the ordinal space [0; [ is normal and countably paracompact; ho*
*wever, it is not even
_________________________
ySoviet Math. Dokl. 13 (1972), 517520.
zGeneral Topology Appl. 2 (1972), 4954.
18
submetacompact (cf. p. 112). On the other hand: (i) The ordinal space [0; ] x *
*[0; [ is nonnormal and
countably paracompact but not subparacompact; (ii) The IsbellMrowka space (N )*
* is nonnormal and
subparacompact but not countably paracompact (cf. p. 112).
[Note: To verify that X = [0; ] x [0; [ is not subparacompact, let A = {(; *
*ff) : ff < } and
B = {(ff; ff) : ff < }_then A and B are disjoint closed subsets of X. Therefore*
* X = U [ V , where
U = X  A and V = X  B. Since the open covering {U; V } has no oediscrete clo*
*sed refinement, X is not
subparacompact.]
Is every normal LCH space countably paracompact? This question is a reinfor*
*cement
of the "Dowker problem". Dropping the supposition of local compactness, a Dowke*
*r_space_
is by definition a normal Hausdorff space which fails to be countably paracompa*
*ct or,
equivalently, whose product with [0; 1] is not normal. Do such spaces exist? Th*
*e answer is
"yes", the first such example within ZFC being a construction due to M.E. Rudin*
*y. Her
example is not locally compact and only by imposing assumptions beyond ZFC has *
*it been
possible to produce locally compact examples.
The ordinal space [0; ]x[0; [ is neither first countable nor separable. Can*
* one construct an example
of a nonnormal countably paracompact LCH space with both of these properties? T*
*he answer is "yes".
Let S and T be subsets of N. Write S T if #(S T) < !; write S < T if S T*
* and #(T S) = !.
ae + +
LEMMA (Hausdorff) There exist collections S = {Sff: ff < } of subsets o*
*f N with the
S = {Sff: ff < }
following properties:
(1) 8 ff : #(N  (S+ff[ Sff)) = !.
(2) 8 ff; 8 fi : fi < ff ) S+fi< S+ffand Sfi< Sff.
(3) 8 ff : #(S+ff\ Sff) < !.
(4) 8 ff; 8 n 2 N: #{fi : fi < ff & S+ff\ Sfi Fn} < ! (Fn = {1; : :;:*
*n}).
There is then no H N such that 8 ff : S+ff H and Sff N  H.
[We shall establish the existence of S+ and S by constructing their elemen*
*ts via induction on ff.
Start by setting S+0= ; and S0= ;. Given S+ffand Sff, decompose N  (S+ff[ S*
*ff) into three infinite
pairwise disjoint sets N+ff, Nff, and Nff. Put
ae + + +
Sff+1= Sff[ Nff () N  (S+ [ S ) N ):
Sff+1= Sff[ Nff ff+1 ff+1 ff
Then this definition handles the successor ordinals < . Suppose now that 0 < <*
* is a limit ordinal.
Choose a strictly increasing sequence {ffi} [0; [: ff1 = 0, supffi = . Fix ni *
*2 N such that S+ffi\
_________________________
yFund. Math. 73 (1971), 179186; see also Balogh, Proc. Amer. Math. Soc. 124*
* (1996), 25552560.
19
S S
Sffj Fniand write T+ for (S+ffi Fni). Note that 8 ff < : S+ff< T+ and 8 *
*i : #(T+ \ Sffi) < !.
ji i T S
If Ii= {ff : ffi ff < ffi+1& T+ Sff Fi} and if I = Ii, then each Ii is fin*
*ite and so I \ [0; ff[ is
i S
finite for every ff < . Assign to each nonzero ff 2 Ii the infinite set Sff *
*{Sffj: ffj < ff} and denote
by n(ff) its minimum element in N  Fi. Relative to this data, define S+ = T+ [*
* {n(ff) : ff 2 I (ff 6= 0)}.
Then it is not difficult to verify that
ae + + + 
8 ff < : Sff< S and 8 i : #(S \ Sffi) < !
8 n 2 N : #{ff : ff < & S+ \ Sff Fn} < !:
S  *
* +
As for S , observe that (N  S+ )  Sffjis infinite, thus there exists an in*
*finite set L (N  S )
ji
such that L \ Sffiis finite for every i. Defining S = N  (S+ [ L ), we have
ae  
8 ff < : Sff< S
S+ \ S = ;; #(N  (S+ [ S )) = !;
which completes the induction. There remains the assertion of nonseparation. To*
* deal with it, assume
that there exists an H N such that S+ff H and Sff\ H are both finite for eve*
*ry ff < . Choose an
n 2 N : W = {ff : Sff\ H Fn} is uncountable. Fix an ff 2 W with the property *
*that W \ [0; ff[ is
infinite. If S+ff H Fm , then {fi : fi < ff & S+ff\ Sfi Fmax(m;n)} contains *
*W \ [0; ff[. Contradiction.]
EXAMPLE (van_Douwen_Space_) Let
ae
X+ = {+1}x]0; [
X = {1}x]0; [
and put X = X+ [ X [ N. TopologizeaXeas follows: Isolate the points of N and t*
*ake for the basic
+
neighborhoods of a point (+1; ff) 2 X all sets of the form
(1; ff) 2 X
ae + +
K(+1; ff : fi; F) = {(+1; fl) : fi < fl ff} [ ((Sff Sfi)  F)
K(1; ff : fi; F) = {(1; fl) : fi < fl ff} [ ((Sff Sfi) *
* F);
where fi < ff and F N is finite. Since the K(1; ff : fi; F) are compact, X is *
*a LCH space. Obviously,
X is first countable and separable; in addition, X is countably paracompact, X *
* being a copy of ]0; [.
Still, X is not normal.
[Suppose that the disjoint closed sets X+ and X can be separated by disjoi*
*nt open sets U+ and
U . Given ff 2 ]0; [, select an ordinal f(ff) < ff and a finite subset F(ff) *
* N such that K(1; ff :
f(ff); F(ff)) U . Choose a < and a cofinal K [0; [ such that fK = (by "p*
*ressing down", i.e.,
Fodor's lemma). Put ae
H+ = (S+ [ (N \ U+ ))  S
H = (S [ (N \ U ))  S+ :
110
Then H+ \ H = ;. Let ff < be arbitrary. Using the cofinality of K and the rel*
*ation fK = , one finds
that Sff H . Contradiction.]
A CRH space X is said to be countably_compact_if every countable open cover*
*ing
of X has a finite subcovering or, equivalently, if every neighborhood finite co*
*llection of
nonempty subsets of X is finite. The ordinal space [0; [ is countably compact *
*but not
compact. The van Douwen space is not countably compact but is countably paracom*
*pact.
Associated with this ostensibly simple concept are some difficult unsolved *
*problems. Sample: Within
ZFC, does there exist a first countable, separable, countably compact LCH space*
* X that is not compact?
This is an open question. But under CH, e.g., such an X does exist (cf. p. 117*
*). Consider the asser
tion: Every perfectly normal, countably compact LCH space X is compact. While i*
*nnocent enough, this
statement is undecidable in ZFC (Ostaszewskiy, Weissz).
PROPOSITION 3 X is countably compact iff every point finite open covering *
*of X
has a finite subcovering.
[Suppose that X is countably compact. Let U be a point finite open covering*
* of X_
then, on general grounds, U admits an irreducible subcovering V. This minimal c*
*overing
must be finite: For otherwise there would exist an infinite subset S X such th*
*at each
x 2 X has a neighborhood containing exactly one point of S, an impossibility.
Suppose that X is not countably compact_then there exists a countably infin*
*ite
discrete closed subset D X, say D = {xn}. Choose a sequence {Un} of nonempty
open sets whose closures are pairwise disjoint such that 8 n : xn 2 Un. The co*
*llection
{X  D; U1; U2; : :}:is a point finite open covering of X which has no finite s*
*ubcovering.]
A CRH space X is said to be pseudocompact_if every countable open covering *
*of X
has a finite subcollection whose closures cover X or, equivalently, if every ne*
*ighborhood
finite collection of nonempty open subsets of X is finite. The IsbellMrowka sp*
*ace (N )
is pseudocompact but not countably compact (cf. p. 112).
PROPOSITION 4 X is pseudocompact iff every real valued continuous function*
* on
X is bounded.
[Suppose that X is not pseudocompact_then there exists a countably infinite*
* neigh
borhood finite collection {Un} of nonempty open subsets of X. Choose a point xn*
* 2 Un.
_________________________
yJ. London Math. Soc. 14 (1976), 505516.
zCanad. J. Math. 30 (1978), 243249.
111
Since X is completely regular, there exists a continuous function fn : X ! [0; *
*n] such that
P
fn(xn) = n, fnX  Un = 0. Put f = fn: f is continuous and unbounded.]
n
A CRH space X is said to be countably_metacompact_if every countable open c*
*overing
of X has a point finite open refinement. The ordinal space [0; [ is countably m*
*etacompact
but not metacompact (cf. p. 112). Every perfect X is countably metacompact.
The relative position of these conditions is shown by:
compact _______________wparacompact_______________wmetacompact
  
  
u u u
countably compact ______wcountably paracompact_____wcountably metacompact


u
pseudocompact
FACT X is countably metacompact iff for every countable open covering U of *
*X there exists a
sequence {Vn} of open refinements of U such that 8 x 2 X 9 nx : ord(x; Vnx) < !.
[The point here is to show that the stated condition forces X to be countab*
*ly metacompact. Enu
merate the elements of U : Un (n = 1; 2; : :):. Write Wn for the set of all x 2*
* Un such that 8 m n 9 V 2
S
Vm : x 2 V and V 6 Ui. Then W = {Wn} is a point finite open refinement of U *
*= {Un}. ]
i ff0 and if xff2 clR(C), then xff2 cl(C). Therefore clR(S)  cl(S) {xff: f*
*f ff0}.]
The fact that X is hereditarily separable is thus immediate. To establish *
*perfect normality, suppose
*
* T
that A X is closed_then it is a question of finding a sequence {Un} o such *
*that A = Un =
T *
* n
cl (Un). Since R is perfectly normal, there exists a sequence {On} of Ropen *
*sets such that clR(A) =
Tn T
On = clR(On). From the claim, clR(A)  A can be enumerated: {an}. Each an*
* 2 X  A, so
n n
9 Kn 2 o : an 2 Kn X  A, Kn clopen. Bearing in mind that o is finer than th*
*e usual topology on
R , we then have
A = T On \ T (X  Kn) = T cl(On) \ T (X  Kn):
n n n n
The final point is collectionwise normality. But as CH is in force, Jones' lemm*
*a implies that X , being
separable and normal, has no uncountable closed discrete subspaces.
[Note: X is not metacompact (cf. Proposition 10). However, X is countably*
* paracompact (being
perfectly normal).]
Retaining the assumption CH and working with
ae
X = N [ ({0} x [0; [)
Xff= N [ {(0; fi) : fi < ff};
one can employ the foregoing methods and construct an example of a first counta*
*ble, separable, countably
compact, noncompact LCH space (cf. p. 110). Recursive techniques can also be u*
*sed in conjunction with
set theoretic hypotheses other than CH to manufacture the same type of example.
A CRH space X is said to be a Moore_space_if it admits a development.
[Note: A development_for X is a sequence {Un} of open coverings of X such *
*that
8 x 2 X : {st(x; Un)} is a neighborhood basis at x.]
Every Moore space is first countable and perfect. Any first countable X th*
*at is
expressible as a countable union of closed discrete subspaces Xn is Moore, so, *
*e.g., the
IsbellMrowka space (N ) is Moore.
FACT Suppose that X is a Moore space_then X is subparacompact.
[Let O = {Oi : i 2 I} be an open covering of X_then the claim is that O has*
* a oediscrete closed
refinement. Fix a development {Un} for X. Equip I with a well ordering < and put
!
Ai;n= X  st(X  Oi; Un) [ S Oj Oi:
j 0, every x 2 X has a neighborhood U : oeiU < ffl for all but a f*
*inite number of i,
thus agrees locally with the maximum of finitely many of the oei and so is co*
*ntinuous.
P
Let oe = max {0; oei =2} and take for aei the normalization max {0; oei =2}*
*=oe.]
i
Suppose that H is a Hilbert space with orthonormal basis {ei: i 2 I}. Let X*
* be the unit sphere in
H and set oei(x) = 2(x 2 X)_then the oeisatisfy the above assumptions.
PROPOSITION 12 Every numerable open covering U = {Ui : i 2 I} of X has a
numerable open refinement that is both neighborhood finite and oediscrete.
125
[Let {i : i 2 I} be a partition of unity on X subordinate to U. Denote by F
theacollectioneof all nonempty finite subsets of I. Assign to each F 2 F the f*
*unctions
mF = min i (i 2 F )
MF = max i (i =2F )and put = maxF(mF  MF ), which is strictly positive. *
*Write
F in place of mF  MF  =2 , oeF in place of max {0; F } and set VF = {x : oeF *
*(x) > 0}_
__ T
then V F {x : mF (x) > MF (x)} Ui. The collection V = {VF : F 2 F} is a
i2F
neighborhood finite open refinement of U which is in fact oediscrete as may be*
* seen by
defining Vn = {VF : #(F ) = n}. In this connection, note that F 06= F 00& #(F*
* 0) =
#(F 00) ) {x : mF0(x) > MF0(x)} \ {x : mF00(x) > MF00(x)} = ;. The numerability*
* of V
P
follows upon considering the oeF =oe (oe = oeF ).]
F
Implicit in the proof of Proposition 12 is the fact that if U is a numerabl*
*e open covering of X, then
there exists a countable numerable open covering O = {On} of X such that 8 n; O*
*n is the disjoint union
of open sets each of which is contained in some member of U.
FACT (Domino_Principle_) Let U be a numerable open covering of X. Assume:
(D 1) Every open subset of a member of U is a member of U.
(D 2) The union of each disjoint collection of members of U is a membe*
*r of U.
(D 3) The union of each finite collection of members of U is a member *
*of U.
Conclusion: X is a member of U.
[Work with the On introduced above, noting that there is no loss of general*
*ity in assuming that
OnaeOn+1. Choose a precise open refinement P = {Pn} of O : 8 n, __Pn Pn+1. Pu*
*t Qn =
Pn (n = 1; 2)and write X = 1SQ = (1SQ ) [ (1SQ ) = X [ X .]
Pn  __Pn2(n 3) 1 n 1 2n1 1 2n 1 2
ae
Let X be a topological space_then by C(C(X)X; [0;w1])e shall understand t*
*he set of
ae
all continuous functions ff::XX!![R0;.1]Bear in mind that C(X) can consist of*
* constants
alone, even if X is regular Hausdorff.
A zero_set_in X is a set of the form Z(f) = {x : f(x) = 0}, where f 2 C(X).
The complement of a zero set is a cozero_set_. Since Z(f) = Z(min {1; f}), C*
*(X)aande
C(X; [0; 1]) determine the same collection of zero sets. All sets of the form *
* {x{:xf(x): 0}f(x) 0}
ae
(f 2 C(X)) are zero sets and all sets of the form {x{:xf(x):>f0}(x)( 0}.]
Application: Let U = {Ui: i 2 I} be an open covering of X_then U is numerab*
*le iff there exists a
numerable open covering O = {Oi: i 2 I} of crX such that 8 i : cr1(Oi) Ui.
EXAMPLE Let G be a topological group; let U be a neighborhood of the identi*
*ty in G_then the
open covering {xU : x 2 G} is numerable.
Suppose given a set X and a collection {Xi: i 2 I} of topological spaces Xi.
(FT) Let {fi : i 2 I} be a collection of functions fi : Xi ! X_then the
final_topology_on X determined by the fi is the largest topology for which each*
* fi is
continuous. The final topology is characterized by the property that if Y is a *
*topological
space and if f : X ! Y is a function, then f is continuous iff 8 i the compo*
*sition
f O fi: Xi! Y is continuous.
(IT) Let {fi : i 2 I} be a collection of functions fi : X ! Xi_then t*
*he
initial_topology_on X determined by the fi is the smallest topology for which e*
*ach fi is
continuous. The initial topology is characterized by the property that if Y is *
*a topological
space and if f : Y ! X is a function, then f is continuous iff 8 i the compos*
*ition
fiO f : Y ! Xi is continuous.
For example, in the category of topological spaces, coproducts carry the fi*
*nal topology
and products carry the initial topology. The discrete topology on a set X is t*
*he final
topology determined by the function ; ! X and the indiscrete topology on a set *
*X is
the initial topology determined by the function X ! *. If X is a topological sp*
*ace and if
f : X ! Y is a surjection, then the final topology on Y determined by f is the *
*quotient
topology, while if Y is a topological space and if f : X ! Y is an injection, t*
*hen the initial
topology on X determined by f is the induced topology.
EXAMPLE Let E be a vector space over R_then the finite_topology_on E is the*
* final topology
determined by the inclusions F ! E, where F is a finite dimensional linear subs*
*pace of E endowed with
128
its natural euclidean topology. E, in the finite topology, is a perfectly norm*
*al paracompact Hausdorff
space. Scalar multiplication R x E ! E is jointly continuous; vector addition E*
* x E ! E1is separately
*
* S
continuous but jointly continuous iff dimE !. For a concrete illustration, put*
* R1 = Rn , where
*
* 0
{0} = R0 R1 . ...The elements of R1 are therefore the real valued sequences h*
*aving a finite number
of nonzero values. Besides the finite topology, one can also give R1 the inher*
*ited product topology oP
or any of the topologies op(1 p 1) derived from the usual `p norm. It is clea*
*r that oP op0 op00
(1 p00< p0 1), each inclusion being proper. Moreover, o1 is strictly smaller t*
*han the finite topology.
To see this, let U = {x 2 R1 : 8 i; xi < 2i}_then U is a neighborhood of th*
*e origin in the finite
topology but U is not open in o1. These considerations exhibit uncountably many*
* distinct topologies on
R 1. Nevertheless, under each of them, R1 is contractible, so they all lead to*
* the same homotopy type.
[Note: The finite topology on R1 is not first countable, thus is not metri*
*zable.]
PROPOSITION 14 Suppose that X is Hausdorff_then X is completely regular iff*
* X
has the initial topology determined by the elements of C(X) (or, equivalently, *
*C(X; [0; 1]).
[Note: Therefore, if o0 and o00are two completely regular topologies on X, *
*then o0 = o00
iff, in obvious notation, C0(X) = C00(X).]
When constructing the initial topology, it is not necessary to work with fu*
*nctions whose domain is
all of X.
Suppose given a set X, a collection {Ui: i 2 I} of subsets Ui X, and a coll*
*ection {Xi: i 2 I} of
topological spaces Xi. Let {fi: i 2 I} be a collection of functions fi: Ui! Xi_*
*then the initial_topology_
on X determined by the fi is the smallest topology for which each Ui is open an*
*d each fi is continuous.
The initial topology is characterized by the property that if Y is a topologica*
*l space and if f : Y ! X is
a function, then f is continuous iff 8 i the composition f1(Ui) f!Uifi!Xiis co*
*ntinuous.
EXAMPLE Let X and Y be nonempty topologicalaspaces_thenethe join_X * Y is t*
*he quotient of
0; 0) ~ (x; y00; 0) *
*aeX * ; = X
X x Y x [0; 1]with respect to the relations (x; y . Conventionally *
* , so *
(x0; y;a1)e~ (x00; y; 1) *
* ; * Y = Y
is a functor TOP x TOP ! TOP . The projection p : X x Y x [0; 1] ! X *sYend*
*s X x Y x {0}
(x; y; t) ! [x; y; t]
(or X x Y x {1}) onto a closed subspace homeomorphic to X (or Y ).aConsiderenow*
* X * Y as merely
1([0;*
* 1[) ! X
a set. Let t : X * Y ! [0; 1] be the function [x; y; t] ! t; let x : t *
* be the functions
ae y : t1(]0;*
* 1]) ! Y
[x; y; t]_!txhen the coarse joinX * Y is X * Y equipped with the initial topo*
*logy determined by
[x; y; t] ! y _________ c
t, x, and y. The identity map X * Y ! X *c Y is continuous; it is a homeomorphi*
*sm if X and Y are
compact Hausdorff but not in general. The coarse join X *cY of Hausdorff X and *
*Y is Hausdorff, thus
so is X *Y . The join X *Y of path connected X and Y is path connected, thus so*
* is X *cY . Examples: (1)
129
The cone_X of X is the join of X and a single point; (2) The suspension_X of X *
*is the joinaofeX and a
pair of points. There are also coarse versions of both the cone and the suspens*
*ion, say cX . Complete
ae *
* cX
the picture by setting X *c; = X.
; *cY = Y
[Note: Analogous definitions can be made in the pointed category TOP *.]
FACT Let X and Y be topological spaces_then the identity map X * Y ! X *cY *
*is a homotopy
equivalence. 8
< [x; y; 0] (0*
* t 1=3)
[A homotopy inverse X *cY ! X * Y is given by [x; y; t] ! [x; y; 3t  1] *
*(1=3 t. 2=3)Since
:
[x; y; 1] (2*
*=3 t 1)
the homotopy type of X * Y depends only on the homotopy types of X and Y and si*
*nce the coarse join is
associative, it follows that the join is associative up to homotopy equivalence*
*.]
EXAMPLE (Star_Construction_) The cone X of a topological space X is contrac*
*tible and there
is an embedding X ! X. However, one drawback to the functor : TOP ! TOP is t*
*hat it does not
preserve embeddings or finite products. Another drawback is that while does pr*
*eserve HAUS , within
HAUS it need not preserve complete regularity (consider X, where X is the Tych*
*onoff plank). The star
construction eliminates these difficulties. Thus put ;? = ; and for X 6= ;, den*
*ote by X? the set of all
right continuous step functions f : [0; 1[! X. So, f 2 X? iff there is a partit*
*ion a0 = 0 < a1 < . .<.an <
1 = an+1 of [0; 1[ such that f is constant on [ai; ai+1[ (i = 0; 1; : :;:n). Th*
*ere is an injection i : X ! X?
that sends x 2 X to i(x) 2 X?, the constant step function with value x. Given a*
*; b : 0 a < b < 1, U an
open subset of X, and ffl > 0, let O(a; b; U; ffl) be the set of f 2 X? such th*
*at f is constant on [a; b[, U is a
neighborhood of f(a), and the Lebesgue measure of {t 2 [a; b[: f(t) 62 U} is < *
*ffl. Topologize X? by taking
the O(a; b; U; ffl) as a subbasis_then i : X ! X? is an embedding, which is clo*
*sed if X is Hausdorff. The
assignment X ! X? defines a functor TOP ! TOP that preserves embeddings and f*
*inite products. It
restricts to a functor HAUS ! HAUS that respects complete regularity.
Claim: Suppose that X is not empty_then X? is contractible and has a basis *
*of contractible open
sets. ae
[Fix f0 2 X? and define H : X? x [0; 1] ! X? by H(f; T)(t) = f0(t)(0 t <*
* T).]
f(t) (T t <*
* 1)
An expanding_sequence_of topological spaces is a system consisting of a seq*
*uence
of topological spaces Xn linked by embeddings fn;n+1: Xn ! Xn+1 . Denote by X1
the colimit in TOP associated with this data_then for every n there is an arr*
*ow fn;1 :
Xn ! X1 and the topology on X1 is the final topology determined by the fn;1 .*
* Each
S
fn;1 is an embedding and X1 = fn;1 (Xn ). One can therefore identify Xn w*
*ith
n
fn;1 (Xn ) and regard the fn;n+1 as inclusions.
130
[Note: If all the fn;n+1 are open (closed) embeddings, then the same holds *
*for all the
fn;1 .]
If all the Xn are T1, then X1 is T1. If all the Xn are Hausdorff, then X1 *
* need not
be Hausdorff but there are conditions that lead to this conclusion.
(A) If all the Xn are LCH spaces, then X1 is a Hausdorff space.
[Let x; y 2 X1 : x 6= y. Fix an index n0 such that x; y 2 Xn0. Choose open*
* relatively
__ __ __
compact subsets Un0, Vn0 Xn0 : x 2 Un0 & y 2 Vn0, with Un0 \ Vn0 = ;. Since Un*
*0 and
__
V n0 are compact disjoint subsets of Xn0+1, there exist open relatively compact*
* subsets
__ __
Un0+1, Vn0+1 Xn0+1 : Un0 Un0+1 & Vn0 Vn0+1, with U n0+1\ Vn0+1 = ;. Iterate
S S
the procedure to build disjoint neighborhoods U = Un and V = Vn of x an*
*d y
nn0 nn0
in X1 .]
(B) Suppose that all the Xn are Hausdorff. Assume: 8 n; Xn is a neighb*
*orhood
retract of Xn+1 _then X1 is Hausdorff.
(C) If all the Xn are normal (normal and countably paracompact, perfec*
*tly
normal, collectionwise normal, paracompact) Hausdorff spaces and if 8 n, Xn is *
*a closed
subspace of Xn+1 , then X1 is a normal (normal and countably paracompact, perf*
*ectly
normal, collectionwise normal, paracompact) Hausdorff space.
[The closure preserving closed covering {Xn } is absolute, so the generalit*
*ies on p. 54
can be applied.]
LEMMA Given an expanding sequence of T1 spaces, let OE : K ! X1 be a conti*
*nuous
function such that OE(K) is a compact subset of X1 _then there exists an index *
*n and a
continuous function OEn : K ! Xn such that OE = fn;1 O OEn.
EXAMPLE Working in the plane, fix a countable dense subset S = {sn} of {(x*
*; y) : x = 0}.
Put Xn = {(x; y) : x > 0} [ {s0; : :;:sn} and let fn;n+1: Xn ! Xn+1 be the incl*
*usion_then X1 is
Hausdorff but not regular.
EXAMPLE (Marciszewski_Space_) Topologize the set [0; 2] by isolating the po*
*ints in ]0; 2[, basic
neighborhoods of 0 or 2 being the usual ones. Call the resulting space X0. Give*
*n n > 0, topologize the
set ]0; 2[x[0; 1] by isolating the points of ]0; 2[x]0; 1] along with the point*
* (1; 0), basic neighborhoods of
(t; 0) (0 < t < 1 or 1 < t < 2) being the subsets of Ln that contain (t; 0) and*
* have a finite complement,
where Ln is the line segment joining (t; 0) and (t + 1  1=n; 1) (0 < t < 1) or*
* (t; 0) and (t  1 + 1=n; 1)
(1 < t < 2). Call the resulting space Xn. Form X0q X1q . .q.Xn and let Xn be th*
*e quotient obtained
by identifying points in ]0; 2[. Each Xn is Hausdorff and there is an embedding*
* fn;n+1: Xn ! Xn+1.
But X1 is not Hausdorff.
131
ae 0 1
FACT Suppose that X X . . .are expanding sequences of LCH spaces_then*
* X1 xY 1 =
Y 0 Y 1 . . .
colim(Xn x Y n).
Let X be a topological space_then a filtration_on X is a sequence X0; X1; :*
* :o:f
S
subspaces of X such that 8 n : Xn Xn+1 . Here, one does not require that Xn *
*= X.
n
A filtered_space_Xis a topological space X equipped with a filtration {Xn }. A *
*filtered_map_
f : X ! Y of filtered spaces is a continuous function f : X ! Y such that 8 n :*
* f(Xn )
Y n. Notation: f2 C (X ; Y). FILSP is the category whose objects are the filt*
*ered spaces
and whose morphisms are the filtered maps. FILSP is a symmetric monoidal cate*
*gory:
S
Take X Y to be X x Y supplied with the filtration n ! Xp x Y q, let e be *
*the one
p+q=n
point space filtered by specifying that the initial term is 6= ;, and make the *
*obvious choice
for >. There is a notion of homotopy in FILSP . Write I for I = [0; 1] endowed*
* with its
skeletal filtration, i.e., I0 = {0; 1}, In = [0; 1] (n 1)_then filtered maps f*
*; g: X ! Y
areasaideto be filter_homotopic_if there exists a filtered map H : X I ! Y s*
*uch that
H(x; 0) = f(x)
H(x; 1) = g(x)(x 2 X).
Geometric realization may be viewed as a functor ? : SISET ! FILSP via *
*consideration of
skeletons. To go the other way, equip n with its skeletal filtration and let n*
* be the associated filtered
space. Given a filtered space X , write sinX for the simplicial set defined by*
* sinX([n]) = sinnX =
C ( n; X)_then the assignment X ! sinX is a functor FILSP ! SISET and (?; si*
*n) is an adjoint
pair.
If C is a full subcategory of TOP (HAUS ) and if X is a topological spac*
*e (Hausdorff
topological space), then X is an object in the monocoreflective hull of C in TO*
*P (HAUS )
`
iff there exists a set {Xi} Ob C and an extremal epimorphism f : Xi! X (cf. *
*p. 021
i
ff.). Example: The monocoreflective hull in TOP of the full subcategory of TO*
*P whose
objects are the locally connected, connected spaces is the category of locally *
*connected
spaces.
[Note: The categorical opposite of "epireflective" is "monocoreflective".]
EXAMPLE (A_Spaces_) The monocoreflective hull in TOP of [0; 1]=[0; 1[ is *
*the category of A
spaces.
EXAMPLE (Sequential_Spaces_) A topological space X is said to be sequentia*
*l_provided that a
subset U of X is open iff every sequence converging to a point of U is eventual*
*ly in U. Every first
132
countable space is sequential. On the other hand, a compact Hausdorff space ne*
*ed not be sequential
(consider [0; ]). Example: The one point compactification of the IsbellMrowka *
*space (N ) is sequential
but there is no sequence in N converging to 1 2 __N. If SEQ is the full, isomor*
*phism closed subcategory
of TOP whose objects are the sequential spaces, then SEQ is closed under the *
*formation in TOP of
coproducts and quotients. Therefore SEQ is a monocoreflective subcategory of TO*
*P (cf. p. 021), hence
is complete and cocomplete. The coreflector sends X to its sequential_modificat*
*ion_sX. Topologically,
sX is X equipped with the final topology determined by the OE 2 C(N 1; X), wher*
*e N 1 is the one
point compactification of N (discrete topology). The monocoreflective hull in T*
*OP of N1 is SEQ , so
a topological space is sequential iff it is a quotient of a first countable spa*
*ce. SEQ is cartesian closed:
C(s(X x Y ); Z) C(X; ZY ). Here, s(X x Y ) is the product in SEQ (calculate th*
*e product in TOP and
apply s). As for the exponential object ZY , given any open subset P Z and any*
* continuous function
OE : N1 ! Y , put O(OE; P) = {g 2 C(Y; Z) : g(OE(N 1)) P} and call Cs(Y; Z) th*
*e result of topologizing
C(Y; Z) by letting the O(OE; P) be a subbasis_then ZY = sCs(Y; Z).
[Note: Every CW complex is sequential.]
A Hausdorff space X is said to be compactly_generated_provided that a subse*
*t U of
X is open iff U \ K is open in K for every compact subset K of X. Examples: (1)*
* Every
LCH space is compactly generated; (2) Every first countable Hausdorff space is *
*compactly
generated; (3) The product R , > !, is not compactly generated. A Hausdorff *
*space
is compactly generated iff it can be represented as the quotient of a LCH space*
*. Open
subspaces and closed subspaces of a compactly generated Hausdorff space are com*
*pactly
generated, although this is not the case for arbitrary subspaces (consider N [ *
*{p} fiN ,
where p 2 fiN N ). However, Arhangel'skiiy has shown that if X is a Hausdorff *
*space, then
*
* __
X and all its subspaces are compactly generated iff for every A X and each x 2*
* A there
exists a sequence {xn} A : limxn = x. The product X x Y of two compactly gener*
*ated
Hausdorff spaces may fail to be compactly generated (consider X = R  {1=2; 1=3*
*; : :}:
and Y = R =N ) but this will be true if one of the factors is a LCH space or if*
* both factors
are first countable.
EXAMPLE (Sequential_Spaces_) A Hausdorff sequential space is compactly gen*
*erated. In fact, a
Hausdorff space is sequential provided that a subset U of X is open iff U \K is*
* open in K for every second
countable compact subset K of X.
EXAMPLE Equip R1 with the finite topology and let H(R 1) be its homeomorp*
*hism group.
_________________________
yCzech. Math. J. 18 (1968), 392395.
133
Give H(R 1) the compact open topology_then H(R 1) is a perfectly normal paracom*
*pact Hausdorff
space. But H(R 1) is not compactly generated.
[The set of all linear homeomorphisms R1 ! R1 is a closed subspace of H(R 1*
*). Show that it is
not compactly generated. Incidentally, H(R 1) is contractible.]
For certain purposes of algebraic topology, it is desirable to single out a*
* full, isomor
phism closed subcategory of TOP , small enough to be "convenient" but large en*
*ough to
be stable for the "standard" constructions. A popular candidate is the category*
* CGH of
compactly generated Hausdorff spaces (Steenrody). Since CGH is closed under *
*the for
mation in HAUS of coproducts and quotients, CGH is a monocoreflective subca*
*tegory
of HAUS (cf. p. 021). As such, it is complete and cocomplete. The coreflecto*
*r sends
X to its compactly_generated_modification_kX. Topologically, kX is X equipped *
*with
the final topology determined by the inclusions K ! X, K running through the co*
*m
pact subsets of X. The identity map kX ! X is continuous and induces isomorphi*
*sms
of homotopy and singular homology and cohomology groups. If X and Y are compact*
*ly
generated, then their product in CGH is X xk Y k(X x Y ). Each of the func*
*tors
_ xk Y : CGH ! CGH has a right adjoint Z ! ZY , the exponential object ZY *
* being
kC(Y; Z), where C(Y; Z) carries the compact open topology. So one ofatheeadvant*
*ages of
0
CGH is that it is cartesian closed. Another advantage is that if X;YX; Ya0r*
*e in CGH and
ae 0
if fg::XY!!XY 0are quotient, then f xk g : X xk Y ! X0xk Y 0is quotient. But *
*there
are shortcomings as well. Item: The forgetful functor CGH ! TOP does not pr*
*eserve
colimits. For let A be a compactly generated subspace of X and consider the pu*
*shout
A? ! *?
square y y in CGH _then P = h(X=A), the maximal Hausdorff quotient of *
*the
X ! P
ordinary quotient computed in TOP . To appreciate the point, let X = [0; 1], A*
* = [0; 1[_
then [0; 1]=[0; 1[ is not Hausdorff and h([0; 1]=[0; 1[) is a singleton. Finall*
*y, it is clear that
CGH is the monocoreflective hull in HAUS of the category of compact Hausdor*
*ff spaces.
CGH *, the category of pointed compactly generated Hausdorff spaces, is a *
*closed category: Take
X Y to be the smash product X#kY (cf. p. 328) and let e be S0. Here, the inter*
*nal hom functor sends
(X; Y ) to the closed subspace of kC(X; Y ) consisting of the base point preser*
*ving continuous functions.
FACT Let X be a CRH space. Suppose that there exists a sequence {Un} of ope*
*n coverings of X
_________________________
yMichigan Math. J. 14 (1967), 133152.
134
T
such that 8 x 2 X : Kx st(x; Un) is compact and {st(x; Un)} is a neighborhoo*
*d basis at Kx (i.e., any
n
open U containing Kx contains some st(x; Un))_then X is compactly generated. Ex*
*ample: Every Moore
space is compactly generated.
[Note: Jiangy has shown that any CRH space X realizing this assumption is n*
*ecessarily submeta
compact.]
In practice, it can be troublesome to prove that a given space is Hausdorff*
* and
while this is something which is nice to know, there are situations when it is *
*irrele
vant. We shall therefore enlarge CGH to its counterpart in TOP , the categ*
*ory CG
of compactly_generated_spaces (Vogtz), by passing to the monocoreflective hull *
*in TOP of
the category of compact Hausdorff spaces. It is thus immediate that a topologic*
*al space
is compactly generated iff it can be represented as the quotient of a LCH space*
*. Con
sequently, if X is a topological space, then X is compactly generated provided *
*that a
subset U of X is open iff OE1(U) is open in K for every OE 2 C(K; X), K any co*
*mpact
Hausdorff space. What has been said above in the Hausdorff case is now applica*
*ble in
general, the main difference being that the forgetful functor CG ! TOP prese*
*rves co
limits. Also, like CGH , CG is cartesian closed: C(X xk Y; Z) C(X; ZY ). Of *
*course,
X xk Y k(X x Y ) and the exponential object ZY is defined as follows. Given an*
*y open
subset P Z and any continuous function OE : K ! Y , where K is a compact Hausd*
*orff
space, put O(OE; P ) = {g 2 C(Y; Z) : g(OE(K)) P } and call Ck(Y; Z) the resul*
*t of topolo
gizing C(Y; Z) by letting the O(OE; P ) be a subbasis_then ZY = kCk(Y; Z). Exam*
*ple: A
sequential space is compactly generated.
[Note: If X and Y are compactly generated and if f : X ! Y is a continuous *
*injection,
then f is an extremal monomorphism iff the arrow X ! kf(X) is a homeomorphism, *
*where
f(X) has the induced topology. Therefore an extremal monomorphism in CG need n*
*ot be
an embedding (= extremal monomorphism in TOP ). Extremal monomorphisms in CG
are regular. Call them CG__embeddings_.]
S
EXAMPLE Partition [1; 1] by writing [1; 1] = {1} [ {x; x} [ {1}. *
*Let X be the
0x<1
associated quotient space_then X is compactly generated (in fact, first countab*
*le). Moreover, X is
compact and T1 but not Hausdorff; X is also path connected.
ae
FACT Let X and Y be compactly generated_then the projections X xk Y ! X a*
*re open maps.
X xk Y ! Y
_________________________
yTopology Proc. 11 (1986), 309316.
zArch. Math. 22 (1971), 545555; see also Wyler, General Topology Appl. 3 (1*
*973), 225242.
135
Given any class K of compact spaces containing at least one nonempty space,*
* denote
by M the monocoreflective hull of K in TOP and let R : TOP ! M be the ass*
*ociated
coreflector. If X is a topological space, then a subset U of RX is open provid*
*ed that
OE1(U) is open in K for every OE 2 C(K; X), K any element of K. Write K for*
* the full,
isomorphism closed subcategory of TOP whose objects are those X which are sep*
*arated_
by K, i.e., such that X {(x; x) : x 2 X} is closed in R(X x X)_then K is
closed under the formation in TOP of products and embeddings. Therefore K*
* is an
epireflective subcategory of TOP (cf. p. 021). Examples: (1) Take for K the *
*class of all
finite indiscrete spaces_then an X in TOP is separated by K iff it is T0; (2*
*) Take for
K the class of all finite spaces_then an X in TOP is separated by K iff it i*
*s T1.
[Note: Recall that a topological space X is Hausdorff iff its diagonal is c*
*losed in X xX
(product topology).]
EXAMPLE (Sequential_Spaces_) Let X be a topological space_then every seque*
*nce in X has at
most one limit iff X is sequentially closed in X x X, i.e., iff X is separated*
* by K = {N 1}. When this
is so, X must be T1 and if X is first countable, then X must be Hausdorff.
[Note: Recall that a topological space X is Hausdorff iff every net in X ha*
*s at most one limit.]
If K is a compact space, then for any OE 2 C(K; X), OE(K) is a compact subs*
*et of X.
In general, OE(K) is neither closed nor Hausdorff.
(K1) A topological space X is said to be K1 provided that 8 OE 2 C(K; *
*X)
(K 2 K), OE(K) is a closed subspace of X.
(K2) A topological space X is said to be K2 provided that 8 OE 2 C(K; *
*X)
(K 2 K), OE(K) is a Hausdorff subspace of X.
A topological space X which is simultaneously K1 and K2 is necessarily sep*
*arated
by K.
Specialize the setup and take for K the class of compact Hausdorffaspacese(*
*McCordy),
so M = CG . Suppose that X is K1 (hence T1)_then X is K2. Proof: Let xy2 OE(*
*K)
ae ae 1
(OE 2 C(K; X)) : x 6= y, choose disjoint open sets UV K : OEOE(x)1U(y)anV*
*d consider
ae
OE(K)  OE(K  U)
OE(K)  OE(K  V ). Denote by CG the full subcategory of CG whose obje*
*cts are
separated by K. There are strict inclusions CGH CG CG . Example: Every
first countable X in CG is Hausdorff.
_________________________
yTrans. Amer. Math. Soc. 146 (1969), 273298; see also Hoffmann, Arch. Math.*
* 32 (1979), 487504.
136
LEMMA Let X be a separated compactly generated space_then X is K1.
[Let K, L 2 K; let OE 2 C(K; X), 2 C(L; X). Since OE x : K x L ! X xk X*
* is
continuous, (OEx )1(X ) is closed in KxL. Therefore 1 (OE(K)) = prL((OEx )1*
*(X ))
is closed in L.]
It follows from the lemma that every separated compactly generated space X*
* is T1.
More is true: Every compact subspace A of X is closed in X. Proof: For any OE 2*
* C(K; X)
(K 2 K), A \ OE(K) is a closed subspace of A, thus is compact, so A \ OE(K) is *
*a closed
subspace of OE(K), implying that OE1(A) = OE1(A \ OE(K)) is closed in K. Coro*
*llary: The
intersection of two compact subsets of X is compact.
Equalizers in CGH and CG are closed (e.g., retracts) but CG is bett*
*er behaved
than CGH when it comes to quotients. Indeed, if X is in CG and if E is an *
*equivalence
relation on X, then X=E is in CG iff E XxkX is closed. To see this, let p :*
* X ! X=E
be the projection. Because p xk p : X xk X ! X=E xk X=E is quotient, X=E is cl*
*osed
in X=E xk X=E iff (p xk p)1(X=E ) = E is closed in X xk X. Consequently, if A *
* X
is closed, then X=A is in CG .
[Note: Recall that if X is a topological space, then for any equivalence re*
*lation E on
X, X=E Hausdorff ) E X x X closed and E X x X closed plus p : X ! X=E open
) X=E Hausdorff.]
CG , like CG and CGH , is cartesian closed. For CG has finite prod*
*ucts and if
X is in CG and if Y is in CG , then kCk(X; Y ) is in CG .
[Note: Suppose that B is separated_then CG =B is cartesian closed (Booth
Browny).]
CG * and CG *are the pointed versions of CG and CG . Both are closed ca*
*tegories.
[Note: The pointed_exponential_object_ZY is hom(Y; Z).]
EXAMPLE Let X be a nonnormal LCH space. Fix nonempty disjoint closed subset*
*s A and B of X
that do not have disjoint neighborhoods_then X=A and X=B are compactly generate*
*d Hausdorff spaces
but neither X=A nor X=B is regular. Put E = A x A [ B x B [ X . The quotient X=*
*E is a separated
compactly generated space which is not Hausdorff. Moreover, X=E is not the cont*
*inuous image of any
compact Hausdorff space.
[Note: Take for X the Tychonoff plank. Let A = {(; n) : 0 n < !} and B = *
*{(ff; !) : 0
ff < }_then X=E is compact and all its compact subspaces are closed. By compari*
*son, the product
X=E x X=E, while compact, has compact subspaces that are not closed.]
_________________________
yGeneral Topology Appl. 8 (1978), 181195.
137
EXAMPLE (kSpaces_) The monocoreflective hull in TOP of the category of c*
*ompact spaces is
the category of kspaces. In other words, a topological space X is a kspace_pr*
*ovided that a subset U of
X is open iff U \ K is open in K for every compact subset K of X. Every compact*
*ly generated space is a
kspace. The converse is false: Let X be the subspace of [0; ] obtained by dele*
*ting all limit ordinals except
_then X is not discrete. Still, the only compact subsets of X are the finite se*
*ts, thus kX is discrete.
The one point compactification X1 of X is compact and contains X as an open sub*
*space. Therefore X1
is not compactly generated but is a kspace (being compact). The category of k*
*spaces is similar in many
respects to the category of compactly generated spaces. However, there is one m*
*ajor difference: It is not
cartesian closed (Cincuray).
[Note: If K is the class of compact spaces, then HAUS K and the inclusi*
*on is strict. Reason:
A topological space X is in K iff every compact subspace of X is Hausdorff.]
FACT Let X0 X1 . .b.e an expanding sequence of topological spaces. Assume*
*: 8 n; Xn is in
CG and is a closed subspace of Xn+1_then X1 is in CG .
[That X1 is in CG is automatic. Let K be a compact Hausdorff space; let O*
*E 2 C(K; X1 )_then,
from the lemma on p. 129, OE(K) Xn (9 n) ) OE(K) is closed in Xn ) OE(K) is c*
*losed in X1 .]
EXAMPLE (Weak_Products_) Let (X0; x0), (X1; x1); : :b:e a sequence of poi*
*nted spaces in
CG*. Put Xn = X0 xk . .x.kXn_then Xn is in CG* with base1point (x0; : :;*
*:xn). The
Q
pointed map Xn ! Xn+1 is a closed embedding. One writes (w) Xn in place of X*
*1 and calls it
1 1
Q
the weak_product_of the Xn. By the above, (w) Xn is in CG *(the base point *
*is the infinite string
1
made up of the xn).
[Note: The same construction can be carried out in TOP , the only differenc*
*e being that Xn is the
ordinary product of X0; : :;:Xn.]
Every Hausdorff topological group is completely regular. In particular, eve*
*ry Haus
dorff topological vector space is completely regular. Every Hausdorff locally *
*compact
topological group is paracompact.
[Note: Every topological group which satisfies the T0 separation axiom is n*
*ecessarily
a CRH space.]
EXAMPLE Take G = R ( > !)_then G is a Hausdorff topological group but G is *
*not compactly
generated. Consider kG: Inversion kG ! kG is continuous, as is multiplication k*
*G xk kG ! kG. But
kG is not a topological group, i.e., multiplication kG x kG ! kG is not continu*
*ous. In fact, kG, while
Hausdorff, is not regular.
_________________________
yTopology Appl. 41 (1991), 205212.
138
Let E be a normed linear space; let E* be its dual, i.e., the space of cont*
*inuous linear functionals on
E_then E* is also a normed linear space. The elements of E can be regarded as s*
*calar valued functions
on E*. The initial topology on E* determined by them is called the weak*_topolo*
*gy_. It is the topology
of pointwise convergence. In the weak* topology, E* is a Hausdorff topological*
* vector space, thus is
completely regular. If dimE !, then every nonempty weak* open set in E* is unb*
*ounded in norm. By
contrast, Alaoglu's theorem says that the closed unit ball in E* is compact in *
*the weak* topology (and
second countable if E is separable). However, the weak* topology is metrizable *
*iff dimE !.
[Note: Let E be a vector space over R_then Krusey has shown that E admits a*
* complete norm (so
that E is a Banach space) iff dimE < ! or (dimE)! = dimE. Therefore, the weak* *
*topology on the dual
of an infinite dimensional Banach space is not metrizable.]
The forgetful functor from the category of topological groups to the catego*
*ry of
topological spaces (pointed topological spaces) has a left adjoint X ! FgrX((X;*
* x0) !
Fgr(X; x0)), where FgrX (Fgr(X; x0)) is the free_topological_group_on X((X; x0)*
*). Alge
braically, FgrX (Fgr(X; x0)) is the free group on X (X  {x0}). Topologically,*
* FgrX
(Fgr(X; x0)) carries the finest topology compatible with the group structure fo*
*r which the
canonical injection X ! FgrX ((X; x0) ! Fgr(X; x0)) is continuous. There is a c*
*ommu
XA_______wFgrX
tative triangle AAC u and Fgr(X; x0) FgrX= ( the normal sub*
*group
Fgr(X; x0)
generated by the word x0). On the other hand, FgrX Fgr(X; x0) q Z (q the copro*
*duct
in the category of topological groups) and, of course, FgrX Fgr(X q *; *).
[Note: The arrow of adjunction X ! FgrX ((X; x0) ! Fgr(X; x0)) is an embedd*
*ing iff
X is completely regular and is a closed embedding iff X is completely regular +*
* Hausdorff
(Thomasz).]
LEMMA If X is a compact Hausdorff space, then Fgr(X) (Fgr(X; x0)) is a Hau*
*sdorff
topological group.
Application: If X is a CRH space, then Fgr(X) (Fgr(X; x0)) is a Hausdorff t*
*opological
group.
[Consider X ! Fgr(fiX) ((X; x0) ! Fgr(fiX; fix0)).]
_________________________
yMath. Zeit. 83 (1964), 314320.
zGeneral Topology Appl. 4 (1974), 5172; see also Quaestiones Math. 2 (1977)*
*, 355377.
139
EXAMPLE It is easy to construct nonnormal Hausdorff topological groups. Th*
*us, given a topo
logical space X, let FgrX be the free topological group on X_then, for X a CRH *
*space, the arrow
X ! FgrX is a closed embedding and FgrX is a Hausdorff topological group, so X *
*not normal ) FgrX
not normal.
FACT Given a topological space X, Fgr(X; x00) Fgr(X; x000) 8 x00; x0002 X.
[Let 0: (X; x00) ! Fgr(X; x00); 00: (X; x000) ! Fgr(X; x000) be the arrows *
*of adjunction and consider
the pointed continuous functions f0: (X; x00) ! Fgr(X; x000), f00: (X; x000) ! *
*Fgr(X; x00) defined by f0(x) =
00(x)00(x00)1, f00(x) = 0(x)0(x000)1.]
The forgetful functor from the category of abelian topological groups to th*
*e category
of topological spaces (pointed topological spaces) has a left adjoint X ! FAB X*
*((X; x0) !
FAB (X; x0)) and when given the quotient topology, FgrX=[FgrX; FgrX] FAB X (Fg*
*r(X; x0)=
[Fgr(X; x0); Fgr(X; x0)] FAB (X; x0)).
21
x2. CONTINUOUS FUNCTIONS
Apart from an important preliminary, namely a characterization of the expon*
*ential
objects in TOP , the emphasis in this x is on the properties possessed by C(X)*
*, where X
is a CRH space.
A topological space Y is said to be cartesian_if the functor _ x Y : TOP *
* ! TOP
has a right adjoint Z ! ZY . Example: A LCH space is cartesian.
PROPOSITION 1 A topological space Y is cartesian iff _ x Y preserves co*
*limits
(cf. p. 033) or, equivalently, iff _ x Y preserves coproducts and coequalizers.
[Note: The preservation of coproducts is automatic and the preservation of *
*coequal
izers reduces to whether _ x Y takes quotient maps to quotient maps.]
Notation: Given topological spaces X; Y; Z; : F (X x Y; Z) ! F (X; F (Y; Z*
*)) is the
bijection defined by the rule (f)(x)(y) = f(x; y).
Let o be a topology on C(Y; Z)_then o is said to be splitting_if 8 X, f 2 C*
*(X x
Y; Z) ) (f) 2 C(X; C(Y; Z)) and o is said to be cosplitting_if 8 X, g 2 C(X; C(*
*Y; Z)) )
1(g) 2 C(X x Y; Z).
LEMMA If o0 is a splitting topology on C(Y; Z) and o00is a cosplitting top*
*ology on
C(Y; Z), then o0 o00.
Application: C(Y; Z) admits at most one topology which is simultaneously sp*
*litting
and cosplitting, the exponential_topology_.
EXAMPLE 8 Y & 8 Z, the compact open topology on C(Y; Z) is splitting.
EXAMPLE If Y is locally compact, then 8 Z the exponential topology on C(Y;*
* Z) exists and is
the compact open topology.
[Note: A topological space Y is said to be locally_compact_if 8 open set P *
*and 8 y 2 P, there exists
a compact set K P with y 2 intK. Example: The one point compactification Q1 of*
* Q is compact but
not locally compact.]
FACT Let Y be a locally compact space_then for all X and Z, the operation *
*of composition
C(X; Y ) x C(Y; Z) ! C(X; Z) is continuous if the function spaces carry the com*
*pact open topology.
PROPOSITION 2 A topological space Y is cartesian iff the exponential topol*
*ogy on
C(Y; Z) exists for all Z.
22
EXAMPLE A locally compact space is cartesian.
FACT Suppose that Y is cartesian. Assume: 8 Z, the exponential topology o*
*n C(Y; Z) is the
compact open topology_then Y is locally compact.
Let Y be a topological space, oY its topology_then the open sets in the con*
*tinuous_
topology_on oY are those collections V oY such that (1) V 2 V, V 02 oY ) V 02 *
*V if
S
V V 0and (2) Vi2 oY (i 2 I), Vi2 V ) 9 i1; : :;:in : Vi1[ . .[.Vin 2 V.
i
LEMMA Let f 2 F (X; oY ), where X is a topological space and oY has the co*
*ntinuous
topology_then f is continuous if {(x; y) : y 2 f(x)} is open in X x Y .
Let T = {(P; y) : y 2 P} oY x Y _then a topology on oY is said to have pro*
*perty_Tif T is open
in oY x Y . Example: The discrete topology on oY has property T.
FACT The continuous topology on oY is the largest topology in the collecti*
*on of all topologies on
oY that are smaller than every topology on oY which has property T.
[If oY (T ) is oY in a topology having property T, then by the lemma, the i*
*dentity function oY (T ) ! oY
is continuous if oY has the continuous topology.]
Let Y be a topological space_then Y is said to be core_compact_if 8 open se*
*t P and
8 y 2 P , there exists an open set V P with y 2 V such that every open coverin*
*g of P
contains a finite covering of V . Example: A locally compact space is core comp*
*act.
There exists a core compact space with the property that every compact subs*
*et has an empty interior
(HofmanLawsony).
FACT Equip oY with the continuous topology_then Y is core compact iff 8 op*
*en set P and
8 y 2 P, there exists an open V oY such that P 2 V and y 2 int\ V.
EXAMPLE A topological space Y is core compact iff the continuous topology *
*on oY has property
T .
Let Y; Z be topological spaces_then theaIsbell_topology_oneC(Y; Z) is the i*
*nitial
topology on C(Y; Z) determined by the eQ : C(Y;fZ)!!foY1(Q)(Q 2 oZ ), where *
*oY has the
_________________________
yTrans. Amer. Math. Soc. 246 (1978), 285310 (cf. 304306).
23
continuous topology. Notation: isC(Y; Z). Examples: (1) isC(Y; [0; 1]=[0; 1*
*[) oY ; (2)
isC(*; Z) Z.
LEMMA The compact open topology on C(Y; Z) is smaller than the Isbell topo*
*logy.
EXAMPLE 8 Y & 8 Z, the Isbell topology on C(Y; Z) is splitting.
[Fix an f 2 C(X x Y; Z) and let g = (f)_then the claim is that g 2 C(X; isC*
*(Y; Z)). From
the definitions, this amounts to showing that 8 Q 2 oZ, eQ O g is continuous. W*
*rite f1(Q) as a union
of rectangles Ri = Uix Vi X x Y . Take an x 2 X and consider any V : eQ (g(x)*
*) 2 V. Since
S nS *
* nT
eQ (g(x)) = {y : (x; y) 2 Ri}, 9 ik (k = 1; : :;:n) : {y : (x; y) 2 Rik} 2*
* V, so 8 u 2 Uik,
i k=1 *
* k=1
eQ (g(u)) 2 V.]
FACT Let Y be a core compact space_then for all X and Z, the operation of*
* composition
C(X; Y ) x C(Y; Z) ! C(X; Z) is continuous if the function spaces carry the Isb*
*ell topology.
PROPOSITION 3 Let Y be a topological space_then Y is cartesian iff Y is*
* core
compact.
[Necessity: Let oi run through the topologies on oY which have property T a*
*nd put
`
Xi = (oY ; oi). Form the coproduct X = Xi and let f : X ! oY be the function *
*whose
i
restriction to each Xi is the identity, where oY carries the continuous topolog*
*y_then f
is a quotient map (cf. p. 22). Since Y is cartesian, it follows from Propo*
*sition 1 that
`
f x idY: X x Y ! oY x Y is also quotient. But X x Y Xix Y and, by hypothesis,
i
T is open in Xix Y 8 i. Therefore T must be open in oY x Y as well, i.e., the c*
*ontinuous
topology on oY has property T, thus Y is core compact (cf. p. 22).
Sufficiency: As has been noted above, the Isbell topology on C(Y; Z) is spl*
*itting, so to
prove that Y is cartesian it suffices to prove that the Isbell topology on C(Y;*
* Z) is cosplitting
when Y is core compact (cf. Proposition 2). Fix g 2 C(X; isC(Y; Z)) and put f =*
* 1(g).
Given a point (x; y) 2 X x Y , let Q be an open subset of Z such that f(x; y) 2*
* Q. Choose
an open P Y : y 2 P & f({x} x P ) Q. Because Y is core compact, there exists *
*an
open V oY : P 2 V and y 2 int\ V. But eQ (g(x)) P ) eQ (g(x)) 2 V and, from t*
*he
continuity of eQ O g, 9 a neighborhood O of x : eP (g(O)) V, hence f(O x int\ *
*V) Q.]
Remark: Suppose that Y is core compact_then 8 Z, "the" exponential object Z*
*Y is
isC(Y; Z), the exponential topology on C(Y; Z) being the Isbell topology.
[Note: The Isbell topology and the compact open topology on C(Y; Z) are one*
* and
the same if Y is locally compact.]
24
FACT Let f; g 2 C(Y; Z). Assume: f; g are homotopic_then f; g belong to th*
*e same path com
ponent of isC(Y; Z).
FACT Let f; g 2 C(Y; Z). Assume: f; g belong to the same path component of*
* isC(Y; Z)_then
f; g are homotopic if Y is core compact.
What follows is a review of the elementary properties possessed by C(X; Y )*
* when
equipped with the compact open topology (omitted proofs can be found in Engelki*
*ngy).
Notation: Given Hausdorff spaces X and Y , let coC(X; Y ) stand for C(X; Y *
*) in the
compact open topology.
[Note: The point open topology on C(X; Y ) is smaller than the compact open*
* topol
ogy. Therefore coC(X; Y ) is necessarily Hausdorff. Of course, if X is discrete*
*, then "point
open" = "compact open".]
PROPOSITION 4 Suppose that Y is regular_then coC(X; Y ) is regular.
PROPOSITION 5 Suppose that Y is completely regular_then coC(X; Y ) is com
pletely regular.
EXAMPLE It is false that Y normal ) coC(X; Y ) normal. Thus take X = {0; *
*1} (discrete
topology)_then coC({0; 1}; Y ) Y x Y and there exists a normal Hausdorff space*
* Y whose square is not
normal (e.g., the Sorgenfrey line (cf. p. 511)).
O'Mearaz has shown that if X is a second countable metrizable space and Y i*
*s a metrizable space,
then coC(X; Y ) is perfectly normal and hereditarily paracompact.
EXAMPLE The loop space Y of a pointed metrizable space (Y; y0) is paracomp*
*act.
A Hausdorff space X is said to be countable_at_infinity_if there is a seque*
*nce {Kn} of
compact subsets of X such that if K is any compact subset of X, then K Kn for *
*some
n. Example: A LCH space is countable at infinity iff it is oecompact.
[Note: X countable at infinity ) X oecompact. Example: P is not oecompact*
*, hence
is not countable at infinity.]
FACT Suppose that X is countable at infinity. Assume: X is first countable*
*_then X is locally
compact.
_________________________
yGeneral Topology, Heldermann Verlag (1989).
zProc. Amer. Math. Soc. 29 (1971), 183189.
25
EXAMPLE Q is oecompact but Q is not countable at infinity.
EXAMPLE Fix a point x 2 fiN N _then X = N[{x}, viewed as a subspace of fi*
*N , is countable
at infinity but it is not first countable.
[Note: The compact subsets of X are finite. However X is not compactly gene*
*rated.]
EXAMPLE Let E be an infinite dimensional Banach space_then E* in the weak**
* topology is
countable at infinity.
PROPOSITION 6 Suppose that X is countable at infinity_then for every metri*
*zable
Y , coC(X; Y ) is metrizable.
PROPOSITION 7 Suppose that X is countable at infinity and compactly genera*
*ted_
then for every completely metrizable Y , coC(X; Y ) is completely metrizable.
Notation: Given a topological space X, write H(X) for its set of homeomorph*
*isms_
then H(X) is a group under composition.
Let us assume that X is a LCH space. Endow H(X) with the compact open topol*
*ogy.
Question: Is H(X) thus topologized a topological group? In general, the answer *
*is "no"
(cf. infra) but there are situationsainewhich the answer is "yes".
[Note: The composition H(X)(xfH(X);!gH(X)) ! giOsfcontinuous, so the prob*
*lem is
whether the inversion f ! f1aisecontinuous.]
Remark: The evaluation H(X)(xfX; x) !if(x)s continuous.
Given subsets A and B of X, put = {f 2 H(X) : f(A) B}_then by
definition, the collection {} (K compact and U open) is a subbasis for th*
*e compact
open topology on H(X).
PROPOSITION 8 If X is a compact Hausdorff space, then H(X) is a topological
group in the compact open topology.
[For f 2 , f1 2 .]
FACT If X is a compact metric space, then H(X) is completely metrizable.
LEMMA Let X be a locally connected LCH space_then the collection {},
where L is compact & connected with intL 6= ; and V is open, constitute a subba*
*sis for
the compact open topology on H(X).
26
PROPOSITION 9 If X is a locally connected LCH space, then H(X) is a topolo*
*gical
group in the compact open topology.
[Fix an f 2 H(X) and choose per the lemma: f1 2 . Deter*
*mine
__ __
relatively compact open O & P : f1 (L) O O P P V () f((X  O) \
__
P ) (X  L) \ f(V )). Let x be any point such that f(x) 2 intL_then <{x}; int*
*L> \
__
<(X  O) \ P ; (X  L) \ f(V )> is a neighborhood of f in H(X), call it Hf. Cl*
*aim:
__
g 2 Hf ) g1 2 . To check this, note that g((X  O) \ P) (X  L) \ f(V *
*) )
__ __
L [ (X  f(V )) g(O) [ g(X  P). But g(O), g(X  P) are nonempty disjoint open*
* sets,
__
so L is contained in either g(O) or g(X  P) (L being connected). Since the con*
*tainment
__ __
L g(X  P) is impossible (g(x) 2 intL and x 62 X  P), it follows that L g(O)*
* or
still, g1 (L) O V , i.e., g1 2 . Therefore inversion is a continuous*
* function.]
Application: The homeomorphism group of a topological manifold is a topolo*
*gical
group in the compact open topology.
EXAMPLE Let X = {0; 2n(n 2 Z)}_then in the induced topology from R, X is a*
* LCH space
but H(X) in the compact open topology is not a topological group.
Suppose that X is a LCH space, X1 its one point compactification_then H(X)*
* can
be identified with the subgroup of H(X1 ) consisting of those homeomorphisms X1*
* ! X1
which leave 1 fixed. In the compact open topology, H(X1 ) is a topological gro*
*up (cf.
Proposition 8). Therefore H(X) is a topological group in the induced topology. *
*As such,
H(X) is a closed subgroup of H(X1 ).
[Note: This topology on H(X) is the complemented compact open topology. It *
*has
for a subbasis all sets of the form , where K is compact and U is open, a*
*s well as
all sets of the form , where V is open and L is compact.]
Anaisotopy_ofea topological space X is a collection {ht : 0 t 1} of homeo*
*morphisms of X such
that h : X x [0; 1] !iXs continuous.
h(x; t) = ht(x)
[Note: When X is a LCH space, isotopies correspond to paths in H(X) (compac*
*t open topology).]
EXAMPLE A homeomorphism h : Rn ! Rn is said to be stable_if 9 homeomorphis*
*ms h1; : :;:hk :
R n! Rn such that h = h1O. .O.hk, where each hihas the property that for some n*
*onempty open Ui Rn,
hiUi= idUi. Every stable homeomorphism of Rn is isotopic to the identity.
[Take k = 1 and consider a homeomorphism h : Rn ! Rn forawhichehU = idU. D*
*efine an isotopy
h(x + 2tu)  2tu (*
*0 t 1=2)
{ht: 0 t 1} of Rn as follows. Fix u 2 U and put ht(x) = __1__ *
* &
h1(x) = x.] 2  2th1=2((2  2t)*
*x)(1=2 t < 1)
27
FACT Equip H(R n) with the compact open topology and write HST(R n) for th*
*e subspace of
H(R n) consisting of the stable homeomorphisms_then HST(R n) is an open subgrou*
*p of H(R n).
[Note: Therefore HST(R n) is also a closed subgroup of H(R n) (since H(R n)*
* is a topological group
in the compact open topology).]
Application: The path component of idRnin H(R n) is HST(R n).
[In view of the example, there is a path from every element of HST(R n) to *
*idRn. On the other hand,
if o : [0; 1] ! H(R n) is a path with o(1) = idRnbut o(0) 62 HST(R n), then o1*
*(HST(R n)) would be a
nontrivial clopen subset of [0; 1].]
[Note: It can be shown that H(R n) is locally path connected (indeed, local*
*ly contractible (cf. p.
617)).]
An isotopy {ht: 0 t 1} is said to be invertible_if the collection {h1t: *
*0 t 1} is an isotopy.
LEMMA An isotopy {ht : 0 t 1} is invertible iff the function H : X x [0;*
* 1] ! X x [0; 1]
defined by the rule (x; t) ! (ht(x); t) is a homeomorphism.
[Note: H is necessarily onetoone, onto, and continuous.]
FACT Let X be a LCH space_then every isotopy {ht: 0 t 1} of X is inverti*
*ble.
[Show first that 8 x 2 X, h1t(x) is a continuous function of t.]
FACT Let X be a LCH space_then every isotopy {ht: 0 t 1} of X extends to*
* an isotopy of
X1 .
[Define __ht: X1 ! X1 by __htX = ht & __ht(1) = 1. To verify that __his *
*continuous, extend H
to X1 x [0; 1] via the prescription __H(1; t) = (__ht(1); t), so __h= ss1 O __H*
*, where ss1 is the projection of
X1 x [0; 1] onto X1 . Establish the continuity of __Hby utilizing the continuit*
*y of H1 (the substance of
the previous result).]
EXAMPLE Every isotopy {ht: 0 t 1} of Rn extends to an isotopy of Sn.
Let X be a CRH space, (Y; d) a metric space. Given f 2 C(X; Y ) and OE 2 C(*
*X; R>0 ),
put NOE(f) = {g : d(f(x); g(x)) < OE(x) 8 x}.
Observations: (1) If OE1; OE2 2 C(X; R>0 ), then NOE(f) NOE1(f) \ NOE2(f)*
*, where
OE(x) = min{OE1(x); OE2(x)}; (2) If g 2 NOE(f), then N (g) NOE(f), where (x) *
*= OE(x) 
d(f(x); g(x)).
Therefore the collection {NOE(f)} is a basic system of neighborhoods at f. *
*Accordingly,
varying f leads to a topology on C(X; Y ), the majorant_topology_.
28
[Note: Each OE 2 C(X; R>0 ) determines a metric dOEon C(X; Y ), viz. dOE(*
*f; g) =
min {1; sup d(f(x);_g(x))_}, and their totality defines the majorant topology o*
*n C(X; Y ),
x2X OE(x)
which is thus completely regular. However, in general, the majorant topology on*
* C(X; Y )
need not be normal (Wegenkittly).]
Here is a proof that C(X; Y ) (majorant topology) is completely regular. Fi*
*x a closed subset A
C(X; Y ) and an f 2 C(X; Y )  A. Choose OE 2 C(X; R>0) : NOE(f) C(X; Y )  A.*
* Define a function
: C(X; Y ) ! [0; 1] by (g) = sup d(f(x);_g(x))_if g 2 NOE(f) and let it be 1 o*
*therwise_then is
x2X OE(x)
continuous and (f) = 0, A = 1.]
*
* _____
[Note: The verification of the continuity of hinges on the observation tha*
*t g 2 NOE(f)) d(f(x);
_____ d(f(x); g(x))
g(x)) OE(x) 8 x, hence 8 g 2 NOE(f) NOE(f), sup __________= 1.]
x2X OE(x)
Example: Suppose that the sequence {fk} converges to f in C(R n; Rn) (major*
*ant
topology)_then 9 a compact K R n and an index k0 such that fk(x) = f(x) 8 k > *
*k0
& 8 x 2 R n K.
EXAMPLE Suppose that f : R n! R nis a homeomorphism_then f has a neighborh*
*ood of
surjective maps in C(R n; Rn) (majorant topology).
EXAMPLE Equip H(R n) with the majorant topology_then the path component of*
* idRn in
H(R n) consists of those homeomorphisms that are the identity outside some comp*
*act set.
FACT The majorant topology on C(R n; Rn) is not first countable.
LEMMA The compact open topology on C(X; Y ) is smaller than the majorant t*
*opol
ogy.
[Fix a compact K X, an open V Y , and a continuous f : X ! Y such that
f(K) V . Choose ffl > 0 such that 8 y 2 f(K); d(y; y0) < ffl ) y02 V . Let OE *
*2 C(X; R>0 )
be the constant function x ! ffl_then 8 g 2 NOE(f), g(K) V .]
Remark: The uniform_topology_on C(X; Y ) is the topology induced by the me*
*tric
d(f; g) = min{1; supd(f(x); g(x))}. The proof of the lemma shows that the compa*
*ct open
x2X
topology on C(X; Y ) is smaller than the uniform topology (which in turn is sma*
*ller than
the majorant topology).
_________________________
yAnn. Global Anal. Geom. 7 (1989), 171178; see also van Douwen, Topology Ap*
*pl. 39 (1991), 332.
29
FACT The compact open topology on C(X; Y ) equals the uniform topology if *
*X is compact.
FACT The uniform topology on C(X; Y ) equals the majorant topology if X is*
* pseudocompact.
Let M(Y ) be the set of all metrics on Y which are compatible with the topo*
*logy of
Y _then the limitation_topology_on C(X; Y ) has for a neighborhood basis at f t*
*he Nm (f)
(m 2 M(Y )), where Nm (f) = {g : sup m(f(x); g(x)) < 1}.
x2X
[Note: If m1; m2 2 M(Y ), then Nm1+m2 (f) Nm1 (f) \ Nm2 (f) and if g 2 Nm *
*(f),
then N(2_ (g) Nm (f), where m(f(x); g(x)) 1  ffl 8 x.]
ffl)m
The limitation topology is defined by the metrics (f; g) ! min{1; supm(f(x)*
*; g(x))} (m 2 M(Y )),
x2X
thus the uniform topology on C(X; Y ) is smaller than the limitation topology.
LEMMA Suppose that X is paracompact_then the limitation topology on C(X; Y*
* )
is smaller than the majorant topology.
[Fix m 2 M(Y ) and let f 2 C(X; Y ). By compatibility, 8 x 2 X, 9 ffl(x) >*
* 0 :
d(f(x); y) < ffl(x) ) m(f(x); y) < 1_4. Put Ox = {x0 : d(f(x); f(x0)) < ffl(x)_*
*2}_then {Ox}
is an open covering of X. Let {Ux} be a precise neighborhood finite open refine*
*ment and
P ffl(x)
choose a subordinated partition of unity {x}. Definition: OE = ____x. Consid*
*er now
x 2
any x0 2 X and assume that d(f(x0); y) < OE(x0). Let x1; : :;:xn be an enumera*
*tion
of those x whose support contains x0 and fix i between 1 and n : ffl(xj)_2 ffl(*
*xi)_2(j =
1; : :;:n) to get OE(x0) ffl(xi)_2. But x0 2 Uxi Oxi. Therefore d(f(xi); f(x0)*
*) < ffl(xi)_2()
m(f(xi); f(x0)) < 1_4) ) d(f(xi); y) < ffl(xi) ) m(f(xi); y) < 1_4) m(f(x0); y)*
* < 1_2. And
this shows that NOE(f) Nm (f).]
[Note: In general, the limitation topology is strictly smaller than the maj*
*orant topol
ogy. To see this, observe that C(R ; R) is a topological group under addition i*
*n the majorant
topology. On the other hand, there is a countable basis at a given f 2 C(R ; R)*
* (limitation
topology) iff f is bounded, thus C(R ; R) is not a topological group under addi*
*tion in the
limitation topology.]
FACT Take X = Y _then in the limitation topology, H(X) is a topological gr*
*oup.
REFINEMENT PRINCIPLE Let (Y; d) be a metric space_then for any open cover
ing V = {V } of Y , 9 m 2 M(Y ) such that the collection {Vy} is a refinement o*
*f V, where
Vy = {y0: m(y; y0) < 1}.
210
[A proof can be found in Dugundjiy.]
LEMMA Let (Y; d) be a metric space_then for any ffi 2 C(Y; R>0 ), 9 m 2 M(*
*Y ) :
d(y; y0) < ffi(y) whenever m(y; y0) < 1.
[Choose an open covering V = {V } of Y such that the diameter of a given V*
* is
1_2infffi(V ). Using the refinement principle, fix an m 2 M(Y ) such that the *
*collection
{Vy} refines V. If (y; y0) is a pair with m(y; y0) < 1, then Vy V for some V*
* , hence
y; y02 V ) d(y; y0) 1_2ffi(y) < ffi(y).]
PROPOSITION 10 Take X = Y _then the limitation topology on H(X) is equal to
the majorant topology.
[Fix f 2 H(X) and OE 2 C(X; R>0 ). Thanks to the lemma, 9 m 2 M(X) : d(x; x*
*0) <
OE O f1 (x) whenever m(x; x0) < 1. If g 2 H(X) and supm(f(x); g(x)) < 1, th*
*en
x2X
d(f(x); g(x)) < OE O f1 (f(x)) = OE(x) 8 x, i.e., NOE(f) \ H(X) is open in H(X*
*) (limi
tation topology).]
Application: The homeomorphism group of a metric space is a topological gro*
*up in
the majorant topology.
EXAMPLE Let X be a second countable topological manifold of euclidean dime*
*nsion n_then
in the majorant topology, H(X) is a topological group. Moreover, Cernavskiiz ha*
*s shown that H(X) is
locally contractible.
[Note: X is metrizable (cf. x1, Proposition 11), so 9 d : (X; d) is a metri*
*c space.]
Notation: 8 f 2 C(X; Y ), grf X x Y is its graph.
Given an open subset O X x Y , let O = {f : grf O}_then the collection {O }
is a basis for a topology on C(X; Y ), the graph_topology_.
[Note: In this connection, observe that O \ P = O\P .]
LEMMA The majorant topology on C(X; Y ) is smaller than the graph topology.
[The function (x; y) ! OE(x)  d(f(x); y) from X x Y to R is continuous, t*
*hus O =
{(x; y) : d(f(x); y) < OE(x)} is an open subset of X x Y . But O = NOE(f).]
_________________________
yTopology, Allyn and Bacon (1966), 196; see also BessagaPelczynski, Selecte*
*d Topics in Infinite
Dimensional Topology, PWN (1975), 63.
zMath. Sbornik 8 (1969), 287333.
211
Rappel: A function f : X ! R is lower_semicontinuous_(upper_semicontinuous_*
*) if for
each real number c, {x : f(x) > c} ({x : f(x) < c}) is open. Example: The chara*
*cteristic
function of a subset S of X is lower semicontinuous (upper semicontinuous) iff *
*S is open
(closed).
HAHN'S EINSCHIEBUNGSATZ Suppose that X is paracompact. Let g : X ! R be
lower semicontinuous and G : X ! R upper semicontinuous. Assume: G(x) < g(x) 8 *
*x 2
X_then 9 a continuous function f : X ! R such that G(x) < f(x) < g(x) 8 x 2 X.
*
* S
[Put Ur = {x : G(x) < r}\{x : g(x) > r} (r rational). Each Ur is open and X*
* = Ur.
P *
* r
Let {r} be a partition of unity subordinate to {Ur} and take f = rr.]
r
The following result characterizes the class of X satisfying the conditions*
* of Hahn's einschiebungsatz.
FACT Let X be a CRH space_then X is normal and countably paracompact iff f*
*or every lower
semicontinuous g : X ! R and upper semicontinuous G : X ! R such that G(x) < g(*
*x) 8 x 2 X,
9 f 2 C(X; R) : G(x) < f(x) < g(x) 8 x 2 X.
[Necessity: With r running through the rationals, there exists a neighborho*
*od finite open covering
{Or} of X : Or {x : G(x) < r < g(x)} 8 r and a neighborhood finite open coveri*
*nga{Pr}eof X :
__P 1 *
* (x 62 Or)
r Or 8 r. Fix a continuous function fr : X ! [1; r] such that fr(x) = r *
* (x 2 __P. Put
*
* r)
f(x) = suprfr(x)_then f has the required properties.
Sufficiency: There are two parts.
X is normal. ThusaleteA; B be disjoint closed subsets of X. With G the*
* characteristic function
of A, let g be defined by g(x) = 1(x 2 B): g is lower semicontinuous, G is up*
*per semicontinuous,
g(x) = 2(x 62 B)
and G(x) < g(x) 8 x 2 X.aeChoose f 2 C(X; R) per the assumptionaandelet U = {x *
*: f(x) > 1},
V = {x : f(x) < 1}_then U are disjoint open subsets of X and A U , hence X*
* is normal.
V B V
X is countably paracompact. Thus consider any decreasing sequence {An}*
* of closed sets such
T 1
that An = ;. Put g(x) = _____(x 2 An  An+1; n = 0; 1; : :):(A0 = X): g is lo*
*wer semicontinuous.
n n + 1
Take f 2 C(X; R) : 0 < f(x) < g(x) and let Un = {x : f(x) < __1__n}+_1then {Un}*
* is a decreasing sequence
T
of open sets with An Un for every n and Un = ;. Since X is normal, this guar*
*antees that X is also
n
countably paracompact (via CP (cf. p. 113)).]
LEMMA Assume that X is paracompact and suppose given a neighborhood finite
closed covering {Aj : j 2 J} of X and 8 j, a positive real number aj_then 9 a c*
*ontinuous
function OE : X ! R >0such that OE(x) < aj if x 2 Aj.
[The function from X to R defined by the rule x ! min {aj : x 2 Aj} is low*
*er
semicontinuous and strictly positive.]
212
PROPOSITION 11 The majorant topology on C(X; Y ) is independent of the cho*
*ice
of d provided that X is paracompact.
[It suffices to show that the graph topology on C(X; Y ) is smaller than th*
*e majorant
topology (cf. p. 210). So fix an f 2 O and consider any x0 2 X. Choose a neigh*
*borhood
U0 of x0 and a positive real number a0 such that x 2 U0 & d(f(x0); y) < 2a0 ) (*
*x; y) 2 O.
Choose further a neighborhood V0 of x0 such that V0 U0 & d(f(x0); f(x)) < a0 8*
* x 2
V0_then {(x; y) : x 2 V0 & d(f(x); y) < a0} O. From this, it follows that one *
*can find
a neighborhood finite closed covering {Aj : j 2 J} of X and a set {aj : j 2 J} *
*of positive
real numbers for which {(x; y) : x 2 Aj & d(f(x); y) < aj} O. In view of the l*
*emma, 9
a continuous function OE : X ! R >0with OE(x) < aj whenever x 2 Aj, hence NOE(f*
*) O ,
i.e., every point of O is an interior point in the majorant topology.]
To reiterate: If X is paracompact, then the majorant topology on C(X; Y ) e*
*quals the
graph topology.
[Note: The assumption of paracompactness can be relaxed (see below).]
Let X be a CRH space, (Y; d) a metric space. Given f 2 C(X; Y ) and a lowe*
*r semicontinuous
oe : X ! R>0, put Noe(f) = {g : d(f(x); g(x)) < oe(x) 8 x}.
Observations: (1) If oe1; oe2 : X ! R>0 are lower semicontinuous, then Noe(*
*f) Noe1(f) \ Noe2(f),
where oe(x) = min{oe1(x); oe2(x)}; (2) If g 2 Noe(f), then No(g) Noe(f), where*
* o(x) = oe(x)d(f(x); g(x)).
[Note: The minimum of two lower semicontinuous functions is lower semiconti*
*nuous, so oe is lower
semicontinuous. On the other hand, the sum of two lower semicontinuous function*
*s is lower semicontinuous.
But x ! d(f(x); g(x)) is continuous, thus x ! d(f(x); g(x)) is lower semiconti*
*nuous, so o is lower
semicontinuous.]
Therefore the collection {Noe(f)} is a basic system of neighborhoods at f. *
*Accordingly, varying f
leads to a topology on C(X; Y ), the semimajorant_topology_.
LEMMA The semimajorant topology on C(X; Y ) is smaller than the graph topo*
*logy.
[Let O = {(x; y) : d(f(x); y) < oe(x)}_then O is open in C(X; Y ). Proof: *
* Fix (x0; y0) 2 O,
put ffl = 1_3(oe(x0)  d(f(x0); y0)), and note that the subset of O consisting *
*of those (x; y) such that
oe(x) > oe(x0)  ffl, d(f(x); f(x0)) < ffl, and d(y; y0) < ffl is open. And: No*
*e(f) = O .]
LEMMA The graph topology on C(X; Y ) is smaller than the semimajorant topo*
*logy.
[Fix an f 2 O . Define a strictly positive function oe : X ! R by letting o*
*e(x0) be the supremum of
those a0 2]0; 1] for which x0 has a neighborhood U0 such that x 2 U0 & d(f(x0);*
* y) < a0 ) (x; y) 2 O.
Since Noe(f) O , the point is to prove that oe is lower semicontinuous, i.e., *
*that 8 c 2 R, {x : c < oe(x)} is
open. This is trivial if c 0 or c 1, so take c 2]0; 1[ and fix x0 : c < oe(x0*
*). Put ffl = (oe(x0)  c)=3_then
213
c + 2ffl < oe(x0), thus 9 a neighborhood U0 of x0 such that x 2 U0 & d(f(x0); y*
*) < c + 2ffl ) (x; y) 2 O.
Supposing further that x 2 U0 ) d(f(x0); f(x)) < ffl, one has x 2 U0 & d(f(x); *
*y) < c + ffl ) (x; y) 2 O )
c < c + ffl oe(x).]
FACT The semimajorant topology on C(X; Y ) equals the graph topology.
A CRH space X is said to be a CB_space_if for every strictly positive lower*
* semicontinuous oe : X ! R
there exists a strictly positive continuous OE : X ! R such that 0 < OE(x) oe(*
*x) 8 x 2 X.
Example: If X is normal and countably paracompact, then X is a CB space (cf*
*. p. 211).
Examples (Macky): (1) Every countably compact space is a CB space; (2) Ever*
*y CB space is count
ably paracompact.
EXAMPLE The IsbellMrowka space (N ) is a pseudocompact LCH space which is*
* not countably
paracompact (cf. p. 112), hence is not a CB space.
FACT The majorant topology on C(X; Y ) equals the graph topology 8 pair (Y*
*; d) iff X is a CB
space.
[Necessity: Fix a strictly positive lower semicontinuous oe : X ! R. Specia*
*lized to the case Y = R,
the assumption is that the majorant topology on C(X) equals the semimajorant to*
*pology, so working with
Noe(0), 9 OE : NOE(0) Noe(0) ) (1  ffl)OE 2 NOE(0) Noe(0) (0 < ffl < 1) ) 0 *
*< OE(x) oe(x) 8 x 2 X, thus
X is a CB space.
Sufficiency: Since NOE(f) Noe(f), the semimajorant topology on C(X; Y ) is*
* smaller than the majo
rant topology.]
If (Y; d) is a complete metric space, then coC(X; Y ) need not be Baire. E*
*xamples:
(1) coC([0; [; R) is not Baire; (2) coC(Q ; R) is not Baire.
[Note: Recall, however, that if X is countable at infinity and compactly ge*
*nerated,
then coC(X; Y ) is completely metrizable (cf. Proposition 7), hence is Baire.]
PROPOSITION 12 Assume: (Y; d) is a complete metric space_then C(X; Y ) (ma
jorant topology) is Baire.
[Let {On} be a sequence of dense open subsets of C(X; Y ). Let U be a nonem*
*pty open
subset of C(X; Y ). Since U \ O1 is nonempty and open and since C(X; Y ) is com*
*pletely
regular (cf. p. 28), 9 f1 2 U \ O1 & OE1 2 C(X; R>0 ) : {g : d(f1(x); g(x)) O*
*E1(x) 8 x}
U \O1, where OE1 < 1. Next, 9 f2 2 NOE1(f1)\O2 & OE2 2 C(X; R>0 ) : {g : d(f2(x*
*); g(x))
_________________________
yProc. Amer. Math. Soc. 16 (1965), 467472.
214
OE2(x) 8 x} NOE1(f1) \ O2, where OE2 < OE1=2. Proceeding, 9 fn+1 2 NOEn(fn) \*
* On+1
& OEn+1 2 C(X; R>0 ) : {g : d(fn+1(x); g(x)) OEn+1(x) 8 x} NOEn(fn) \ On+1, w*
*here
OEn+1 < OEn=2. So, 8 x, d(fn+1(x); fn(x)) __1__2n1, thus {fn(x)} is a Cauchy*
* sequence
in Y . Definition: f(x) = limfn(x). Because the convergence is uniform, f 2 C*
*(X; Y ).
T
Moreover, d(fn(x); f(x)) OEn(x) 8 n & 8 x, which implies that f 2 U \ ( On).]
n
FACT Assume: (Y; d) is a complete metric space_then C(X; Y ) (limitation t*
*opology) is Baire.
Convention: Maintaining the assumption that X is a CRH space, C(X) hencefor*
*th
carries the compact open topology.
Let K be a compact subset of X. Put pK (f) = supf(f 2 C(X))_then pK : C(X*
*) !
K
R is a seminorm on C(X), i.e., pK (f) 0, pK (f +g) pK (f)+pK (g), pK (cf) = *
*cpK (f).
[Note: More is true, viz. pK is multiplicative in the sense that pK (fg) p*
*K (f)pK (g).]
Remark: The initial topology on C(X) determined by the pK as K runs through*
* the
compact subsets of X is the compact open topology.
[Note: In the compact open topology, C(X) is a Hausdorff locally convex top*
*ological
vector space.]
Observation: If K X is compact and if f 2 C(K), then 9 F 2 BC(X) : FK = f*
*. Proof: Apply
the Tietze extension theorem to K regarded as a compact subset of fiX.
A CRH space X is said to be a kR_space_provided that a real valued functio*
*n f :
X ! R is continuous whenever its restriction to each compact subset of X is con*
*tinuous.
Example: A compactly generated X is a kR space (but not conversely (cf. infra)*
*).
EXAMPLE Let X be a kR space. Assume: X is countable at infinity_then X i*
*s compactly
generated.
[Fix a "defining" sequence {Kn} of compact subsets of X with Kn Kn+1 8 n. *
*Claim: A subset
A of X is closed if A \ Kn is closed in Kn for each n. For if not, then A has *
*an accumulation point
a0 : a0 62 A, which can be taken in K1 (adjust the notation). Choose a continuo*
*us function f1 : K1 ! R
such that f1(A \ K1) = {0} and f1(a0) = 1. Extend f1 to a continuous function f*
*2 : K2 ! R such that
f2(A \ K2) = {0}. Repeat the process to get a function f : X ! R such that f(x)*
* = fn(x) (x 2 Kn).
Since X is a kR space, f is continuous. This, however, is a contradiction: f(A*
*) = {0}, f(a0) = 1.]
FACT A kR space X is compactly generated iff kX is completely regular.
[If X is a kR space, then C(X) = C(kX). So, the supposition that kX is com*
*pletely regular forces
X = kX (cf. x1, Proposition 14).]
215
[Note: Recall that in general, X completely regular 6) kX completely regula*
*r (cf. p. 136).]
PROPOSITION 13 C(X) is complete as a topological vector space iff X is a k*
*R 
space.
[Necessity: Suppose that f : X ! R is a real valued function such that f*
*K is
continuous 8 compact K X. Let fK 2 C(X) be an extension of fK_then {fK } is a
Cauchy net in C(X), thus is convergent, say limfK = F . But f = F .
Sufficiency: Let {fi} be a Cauchy net in C(X)_then 8 compact K X, the net
{fiK} is Cauchy in C(K), hence has a limit, call it fK . If K1 K2, then fK2 *
*K1 = fK1 ,
so the prescription f(x) = fK (x) (x 2 K) defines a function f : X ! R . Since*
* X is a
kR space, f is continuous. And: limfi= f.]
EXAMPLE Let be a cardinal > !_then N is a kR space but N is not compac*
*tly generated.
[Note: N !is homeomorphic to P, thus is compactly generated.]
FACT Suppose that the closed bounded subsets of C(X) are complete_then X i*
*s a kR space.
PROPOSITION 14 C(X) is metrizable iff X is countable at infinity (cf. Prop*
*osition
6).
[Let d be a compatible metric on C(X). Put Un = {f : d(f; 0) < 1=n}. Choo*
*se a
compact Kn X and a positive ffln : f(Kn) ] ffln; ffln[) f 2 Un_then for any c*
*ompact
subset K of X, 9 n : K Kn. Therefore X is countable at infinity.]
PROPOSITION 15 C(X) is completely metrizable iff X is countable at infinit*
*y and
compactly generated (cf. Proposition 7).
[If C(X) is completely metrizable, then C(X) is complete as a topological v*
*ector space,
so X is a kR space (cf. Proposition 13), thus X, being countable at infinity, *
*is compactly
generated (cf. p. 214).]
A CRH space X is said to be topologically_complete_if X is a Gffiin fiX or *
*still, if X is a Gffiin any
Hausdorff space containing it as a dense subspace. Example: P is topologically *
*complete but Q is not.
Examples: (1) Every completely metrizable space is topologically complete a*
*nd every topologically
complete metrizable space is completely metrizable; (2) Every LCH space is topo*
*logically complete.
[Note: A topologically complete space is necessarily compactly generated an*
*d Baire (Engelkingy).]
_________________________
yGeneral Topology, Heldermann Verlag (1989), 197198.
216
Remark: It can be shown that Proposition 15 goes through if the hypothesis *
*"completely metrizable"
is weakened to "topologically complete" (McCoyNtantuy).
EXAMPLE Let X be a LCH space. Assume: X is paracompact_then C(X) is Baire.
`
[Using LCH3 (cf. p. 12), write X = Xi, where the Xi are pairwise disjoin*
*t nonempty open oe
i *
* Q
compact subspaces of X. Each Xiis countable at infinity and there is a homeomor*
*phism C(X) C(Xi).
*
* i
But the C(Xi) are completely metrizable (cf. Proposition 15), hence are topolog*
*ically complete, and it is
a fact that a product of topologically complete spaces is Baire (Oxtobyz).]
[Note: The paracompactness assumption on X cannot be dropped. Example: Take*
* X = [0; [_then
S
C(X) is not Baire. Proof: Since X is pseudocompact, On = {f : n < f(x) < n + *
*1} is a dense open
T x
subset of C(X) and On = ;.]
n
FACT Suppose that X is first countable and C(X) is Baire_then X is locally*
* compact.
STONEWEIERSTRASS THEOREM Let X be a compact Hausdorff space. Suppose
that A is a subalgebra of C(X) which contains the constants and separates the p*
*oints of
X_then A is uniformly dense in C(X).
EXAMPLE Let 0 < a < b < 1_then every f 2 C([a; b]) can be uniformly approx*
*imated by
Pd
polynomials nkxk, nk integral.
1
[It is enough to show that f = 1_2can be so approximated. Given an odd pri*
*me p, put OEp(x) =
1_(1  xp  (1  x)p) : OE is a polynomial with integral coefficients, no cons*
*tant term, and pOE ! 1
p p fi fi *
* p
uniformly on [a; b] as p ! 1. Now write p = 2q + 1, note that fifi1_2fq_pifi< *
*1_p, and consider qOEp.]
PROPOSITION 16 Suppose that X is a compact Hausdorff space_then C(X) is
separable iff X is metrizable.
[Necessity: If {fn} is a uniformly dense sequence in C(X), then the {x : f*
*n(x) > 1_2}
constitute a basis for the topology on X, therefore X is second countable, henc*
*e metrizable.
Sufficiency: Let d be a compatible metric on X. Choose a countable basis {U*
*n} for
its topology and put fn(x) = d(x; X  Un) (x 2 X)_then the fn separate the poin*
*ts of
X, thus the subalgebra of C(X) generated by 1 and the fn is uniformly dense in *
*C(X), so
_________________________
ySLN 1315 (1988), 75.
zFund. Math. 49 (1961), 157166.
217
the same is true of the rational subalgebra of C(X) generated by 1 and the fn. *
*But the
latter is a countable set.]
EXAMPLE Assume that X is not compact and consider BC(X), viewed as a Banac*
*h space in
the supremum norm: kfk = supf_then BC(X) can be identified with C(fiX) (f ! f*
*if : kfk = kfifk).
X
Since fiX is not metrizable, it follows that BC(X) is not separable.
[Note: To see that fiX is not metrizable, fix a point x0 2 fiX  X and, arg*
*uing by contradiction,
choose a sequence {xn} X of distinct xn having x0 for their limit. Put A = {x2*
*n}, B = {x2n+1}_then
A and B are disjoint closed subsets of X, so, by Urysohn, 9 OE 2 BC(X) such tha*
*t 0 OE 1 with OE = 1
on A and OE = 0 on B. Therefore 1 = OE(x2n) ! fiOE(x0) & 0 = OE(x2n+1) ! fiOE(x*
*0), an absurdity.]
PROPOSITION 17 C(X) is separable iff X admits a smaller separable metrizab*
*le
topology.
[Necessity: Fix a countable dense set {fn} in C(X)_then {fn} separates the *
*points of
X and the initial topology on X determined by the fn is a separable metrizable *
*topology.
Reason: The arrow X ! R !defined by the rule x ! {fn(x)} is an embedding.
Sufficiency: Let X0 stand for X equipped with a smaller separable metrizabl*
*e topology.
Embed X0 in [0; 1]!. Fix a countable dense set {OEn} in C([0; 1]!) (cf. Propo*
*sition 16)
and put fn = OEnX0_then the sequence {fn} is dense in C(X0), thus C(X0) is sep*
*arable.
Indeed, given a compact subset K0 of X0 and f0 2 C(X0), 9 OE0 2 C([0; 1]!) : OE*
*0K0 =
f0K0 & 8 ffl > 0, 9 OEn : pK0 (OEn  OE0) < ffl ) pK0 (fn  f0) < ffl. Finally*
*, the separability
of C(X0) forces the separability of C(X). This is because a compact subset K of*
* X is a
compact subset of X0 and the two topologies induce the same topology on K.]
Example: Take X = R (discrete topology)_then C(X) is separable.
S
EXAMPLE If X = Kn, where each Kn is compact and metrizable, then C(X) is*
* separable.
n
[There is no loss of generality in supposing that Kn Kn+1 8 n. Choose a co*
*untable dense subset
{fn;m} in C(Kn) (cf. Proposition 16) and let Fn;m be a continuous extension of *
*fn;m to X_then the
initial topology on X determined by the Fn;m is a separable metrizable topology*
* which is smaller than
the given topology on X, so C(X) is separable (cf. Proposition 17).]
FACT Let X be a LCH space_then C(X) is separable and metrizable iff X is s*
*eparable and
metrizable.
FACT Let X be a LCH space_then C(X) is separable and completely metrizable*
* iff X is separable
and completely metrizable.
218
PROPOSITION 18 C(X) is first countable iff X is countable at infinity.
PROPOSITION 19 C(X) is second countable iff X is countable at infinity and*
* all
the compact subsets of X are metrizable.
[Necessity: C(X) second countable ) C(X) first countable ) X countable at i*
*nfinity
(cf. Proposition 18). In addition, C(X) second countable ) C(X) separable. S*
*o, by
Proposition 17, X admits a smaller separable metrizable topology which, however*
*, induces
the same topology on each compact subset of X.
Sufficiency: The hypotheses on X guarantee that C(X) is separable (via the *
*example
above) and metrizable (cf. Proposition 14).]
EXAMPLE Let E be an infinite dimensional locally convex topological vector*
* space. Assume:
E is second countable and completely metrizable_then the AndersonKadec theorem*
* says that E is
homeomorphic to R! (for a proof, see BessagaPelczynskiy). Consequently, if X i*
*s countable at infinity
and compactly generated and if all the compact subsets of X are metrizable, the*
*n C(X) is homeomorphic
to R!.
FACT Suppose that X is second countable_then C(X) is Lindel"of.
Up until this point, the playoff between X and C(X) has been primarily "top*
*ological",
little use having been made of the fact that C(X) is also a locally convex topo*
*logical
vector space. It is thus only natural to ask: Can one characterize those X for *
*which C(X)
has a certain additional property (e.g., barrelled or bornological)? While this*
* theme has
generated an extensive literature, I shall present just two results, namely Pro*
*positions 20
and 21, these being due independently to Nachbinz and Shirotak.
FACT C(X) is reflexive iff X is discrete.
[Assuming that C(X) is reflexive, its bounded weakly closed subsets are wea*
*kly compact. Therefore
the compact subsets of X are finite which means that C(X) is a dense subspace o*
*f RX (product topology).
But the reflexiveness of C(X) also implies that its closed bounded subsets are *
*complete, hence X is a kR 
space (cf. p. 215). Thus C(X) is complete (cf. Proposition 13), so C(X) = RX a*
*nd X is discrete.]
A subset A of X is said to be bounding_if every f 2 C(X) is bounded on A. E*
*xample:
X is pseudocompact iff X is bounding.
_________________________
ySelected Topics in Infinite Dimensional Topology, PWN (1975), 189.
zProc. Nat. Acad. Sci. U.S.A. 40 (1954), 471474.
kProc. Japan Acad. Sci. 30 (1954), 294298.
219
Given a subset W of C(X), let K(W ) be the subset of X consisting of those *
*x with the
property that for every neighborhood Ox of x there exists an f 2 C(X): f(X  Ox*
*) = {0}
& f 62 W .
BOUNDING LEMMA If W is a barrel in C(X), then K(W ) is bounding.
[Suppose that K(W ) is not bounding and fix an infinite discrete collection*
* O = {O}
of open subsets of X such that O \ K(W ) 6= ; 8 O 2 O. Choose an element O1 2 *
*O.
Since O1 \ K(W ) 6= ;, 9 f1 2 C(X) : f1(X  O1) = {0} & f1 62 W . On the other *
*hand,
W , being a barrel, is closed, so 9 a compact K1 X and a positive ffl1 : {g : *
*pK1 (f1  g) <
ffl1} \ W = ;. Choose next an element O2 2 O : O2 \ K1 = ; and continue. The up*
*shot
is that there exist sequences {On}, {fn}, {Kn}, {ffln} with the following prope*
*rties: (1)
nS
On+1 \ ( Ki) = ;; (2) fn(X  On) = {0} & fn 62 W ; (3) {g : pKn (fn  g) < ff*
*ln} \ W = ;.
i=1
Take c1 = 1 and determine cn+1 : 0 < cn+1 < __1__n,+s1ubject to the requirement*
* that
Pn 1 P1 1
cn+1pKn+1( __fi) < ffln+1 8 n. Put f = __fi_then by (2) and the discreten*
*ess of
i=1ci i=1ci
{On}, f is continuous, and (1)(3) combine to imply that cn+1f 62 W 8 n, thus W*
* does
not absorb the function f, a contradiction.]
LEMMA OF DETERMINATION If W is a barrel in C(X) and if f is an element of
C(X) such that f(x) = 0 8 x 2 U, where U is an open set containing K(W ), then *
*f 2 W .
[Suppose false. Choose a compact K X and a positive ffl : {g : pK (f g) <*
* ffl}\W =
;, and for each x 2 K  U, choose a neighborhood Ox of x : g(X  Ox) = {0} ) g *
*2 W .
Fix fx 2 C(X; [0; 1]) : fx(x) = 1 & fxX  Ox = 0, and let Ux = {y : fx(y) > 1=*
*2}.
The Ux comprise an open covering of K  U, thus one can extract a finite subcov*
*ering
Pn
Ux1; : :;:Uxn. Put xi= __________fxi___________max{1=2;(fi = 1; : :;:n)_ then *
* xiK U =
x1 + . .+.fxn} i*
*=1
1. Since xi(X  Oxi) = {0}, cxif 2 W (c 2 R ), therefore F = x1f + . .+.xnf =
_1_(n f + . .+.n f) 2 W . But by its very construction, F K = fK ) F 62 W .]
n x1 xn
PROPOSITION 20 C(X) is barrelled iff every bounding subset of X is relativ*
*ely
compact.
[Necessity: Rephrased, the assertion is that for any closed noncompact subs*
*et S of
X, 9 f 2 C(X) : f is unbounded on S. Thus let BS = {f : sup f 1}_then BS is
S
balanced and convex. Since BS is also closed and since the requirement that the*
*re be some
f 2 C(X) which is unbounded on S amounts to the failure of BS to be absorbing, *
*it need
only be shown that BS does not contain a neighborhood of 0. Assuming the oppos*
*ite,
choose a compact K and a positive ffl : {f : pK (f) < ffl} BS. Claim: S K. *
*Proof:
220
If x 2 S  K, 9 f 2 C(X) : f(K) = {0} & f(x) = 2, an impossibility. Therefore *
*S is
compact (being closed), contrary to hypothesis.
Sufficiency: Fix a barrel W in C(X)_then the contention is that W contains *
*a neigh
borhood of 0. Owing to the bounding lemma, K(W ) is compact (inspect the defini*
*tions to
see that K(W ) is closed). Accordingly, it suffices to produce a positive ffl :*
* {f : pK(W) (f) <
ffl} W . To this end, consider BC(X) viewed as a Banach space in the supremum *
*norm.
Because BC(X) is barrelled and W \ BC(X) is a barrel in BC(X), 9 ffl > 0 : kOEk
2ffl ) OE 2 W (OE 2 BC(X)). Assuming that pK(W) (f) < ffl, fix an open set U co*
*ntaining
K(W ) such that f(x) < ffl 8 x 2 U. Let F (x) = max {ffl; f(x)} + min{ffl; *
*f(x)}_then
2F (x) = 0 (x 2 U), thus the lemma of determination implies that 2F 2 W . But 8*
* x 2 X,
2(f(x)  F (x)) < 2ffl ) k2(f  F )k 2ffl ) 2(f  F ) 2 W , so 1_2(2F ) + 1_*
*2(2(f  F )) 2 W ,
i.e., f 2 W .]
Example: C([0; [) is not barrelled.
EXAMPLE If X is a paracompact LCH space, then C(X) is Baire (cf. p. 216).*
* Since Baire )
barrelled, it follows from Proposition 20 that the bounding subsets of X are re*
*latively compact.
Notation: Every f 2 C(X) can be regarded as an element of C(X; R1 ), hence *
*admits
a unique continuous extension f1 : fiX ! R 1.
T
[Note: Put AEfX = {x 2 fiX : f1 (x) 2 R }_then the intersection AEfX *
*is AEX.]
f2C(X)
FACT The elements of fiX  AEX are those x with the property that there ex*
*ists a Gffiin fiX
containing x which does not meet X.
Let W be a balanced, convex subset of C(X)_then W is said to contain_a_ball*
*_if
9 r > 0 : {f : supf r} W .
X
Example: Every balanced, convex bornivore W in C(X) contains a ball.
[Given f; g 2 C(X) with f g, let [f; g] = {OE : f OE g}. Since 8 compact*
* K X,
pK (OE) max {pK (f); pK (g)}, [f; g] is bounded, thus is absorbable by W . In*
* particular:
9 r > 0 such that [r1; r1] W .]
FACT Suppose that W contains a ball. Let K be a compact subset of X. Assum*
*e: f(K) = {0} )
f 2 W_then 9 ffl > 0 : {f : pK (f) < ffl} W.
Let W be a balanced, convex subset of C(X)_then a compact subset K of fiX i*
*s said
to be a hold_of W if f 2 W whenever f1 (K) = {0}. Example: fiX is a hold of W .
221
LEMMA Suppose that W contains a ball_then a compact subset K of fiX is a h*
*old
of W provided that f 2 W whenever f1 vanishes on some open subset O of fiX con*
*taining
K.
Application: Under the assumption that W contains a ball, if K and L are ho*
*lds of
W , then so is K \ L.
[Consider any f : f1 (O) = {0}, where O is some open subset of fiX containi*
*ng K \L.
Choose disjoint open subsets U; V of fiX : K U, LO V and let U0; V 0be open *
*subsets
__0 __0 __0
of fiX : K U0 U U, L  O V 0 V V . Fix OE 2 C(X; [0; 1]) : fiOE(U ) = {1*
*},
__0 _________*
*___
fiOE(V ) = {0}. Note that 2fOE vanishes on (O [ V 0) \ X. But O [ V 0 (O [ V 0)*
* \ X)
(2fOE)1 (O [ V 0) = {0}. On the other hand, L O [ V 0, thus by the lemma, 2fOE*
* 2 W .
Similarly, 2f(1  OE) 2 W . Therefore f = 1_2(2fOE) + 1_2(2f(1  OE)) 2 W .]
Let W be a balanced, convex subset of C(X)_then the support_of W , written *
*sptW ,
is the intersection of all the holds of W .
LEMMA Suppose that W contains a ball_then sptW is a hold of W .
[Since fiX is a compact Hausdorff space, for any open O fiX containing spt*
*W , 9
Tn
holds K1; : :;:Kn of W such that Ki O.]
i=1
PROPOSITION 21 C(X) is bornological iff X is R compact.
[Necessity: Assuming that X is not R compact, fix a point x0 2 AEX  X_the*
*n the
assignment f ! f1 (x0) defines a nontrivial homomorphism bx0: C(X) ! R , which *
*is
necessarily discontinuous (cf. p. 224). So, to conclude that C(X) is not borno*
*logical, it
suffices to show that bx0takes bounded sets to bounded sets. If this were untru*
*e, then there
would be a bounded subset B C(X) and a sequence {fn} B such that bx0(fn) ! 1.
T
The intersection {x 2 fiX : (fn)1 (x) > (fn)1 (x0)  1} is a Gffiin fiX conta*
*ining x0,
n
thus it must meet X (cf. p. 220), say at x00 hence fn(x00) ! 1. But then, a*
*s B is
bounded, __fn___f! 0 in C(X), which is nonsense.
n(x00)
Sufficiency: It is a question of proving that every balanced, convex borniv*
*ore W in
C(X) contains a neighborhood of 0. Because W contains a ball, the lemma implies*
* that
sptW is a hold of W , thus the key is to establish the containment sptW X sinc*
*e this
will allow one to say that 9 ffl > 0 : {f : psptW (f) < ffl} W (cf. p. 220)*
*. So take a
point x0 2 fiX  X and choose closed subsets A1 A2 . . .of fiX : 8 n, x0 2 in*
*tAn
T
& ( An) \ X = ; (possible, X being R compact (cf. p. 220)). Claim: At lea*
*st one
n
of the fiX  intAn is a hold of W () x0 62 sptW ) sptW X). If not, then 8 n,
222
9 fn : (fn)1 (fiX  intAn) = 0 & fn 62 W . The sequence {X  An} is an increas*
*ing
sequence of open subsets of X whose union is X. Therefore f = supnnfn is in C*
*(X). Fix
d > 0: [f; f] dW _then nfn 2 dW 8 n ) fn 2 W 8 n d, a contradiction.]
LEMMA A subset A of X is bounding iff its closure in fiX is contained in A*
*EX.
FACT If C(X) is bornological, then C(X) is barrelled.
[Note: Recall that in general, bornological 6) barrelled and barrelled 6) b*
*ornological.]
Remark: There are completely regular Hausdorff spaces X whose bounding subs*
*ets are relatively
compact but that are not Rcompact (GillmanHenrikseny). For such X, C(X) is th*
*erefore barrelled but
not bornological.
Given a closed subset A of X, let IA = {f : fA = 0}_then IA is a closed id*
*eal in
C(X). Examples: (1) I; = C(X); (2) IX = {0}.
SUBLEMMA Suppose that X is compact. Let I C(X) be an ideal. Assume:
8 x 2 X, 9 fx 2 I : fx(x) 6= 0_then I = C(X).
[8 x 2 X, 9 a neighborhood Ux of x : fxUx 6= 0. Choose points x1; : :;:xn*
* : X =
nS Pn 1
Uxi and let f = f2xi: f 2 I ) 1 = f . __2 I ) I = C(X).]
i=1 i=1 f
LEMMA Suppose that X is compact. Let I C(X) be an ideal and put A =
T
Z(f). Assume: A U Z(OE), where U is open and OE 2 C(X)_then OE 2 I.
f2I
[The restriction IX  U is an ideal in C(X  U) (Tietze), hence by the sub*
*lemma,
equals C(X  U). Choose an f 2 I : fX  U = 1 to get OE = fOE 2 I.]
PROPOSITION 22 Suppose that X is compact. Let I C(X) be an ideal_then
__ T
I = IA , where A = Z(f).
f2I __
[Since I IA , it need only be shown that IA I. So let f be a nonzero elem*
*ent of
IA and fix ffl > 0. Choose OE 2 C(X; [0; 1]) : {x : f(x) ffl=2} Z(OE) & {x*
* : f(x)
3ffl=4} Z(1  OE). Because A {x : f(x) < ffl=4} Z(fOE), the lemma gives f*
*OE 2 I.
__
And: kf  fOEk = supf  fOE < ffl ) f 2 I.]
X
PROPOSITION 23 The closed subsets of X are in a onetoone correspondence *
*with
the closed ideals of C(X) via A ! IA .
_________________________
yTrans. Amer. Math. Soc. 77 (1954), 340362 (cf. 360362).
223
[Due to the complete regularity of X, the map A ! IA is injective. To see*
* that
it is surjective, it suffices to prove that for any closed ideal I in C(X) : I *
*= IA , where
T
A = Z(f). Obviously, I IA . On the other hand, 8 compact K X, the restrict*
*ion
f2I ____
IK is an ideal in C(K) (cf. p. 214), thus IK = IA\K (cf. Proposition 22), a*
*nd from
__
this it follows that IA I= I.]
Application: The points of X are in a onetoone correspondence with the c*
*losed
maximal ideals of C(X) via x ! I{x}.
By comparison, recall that the points of fiX are in a onetoone correspond*
*ence with the maximal
ideals of C(X).
[Note: Assign to each x 2 fiX the subset mx of C(X) consisting of those f s*
*uch that x 2 clfiX(Z(f))_
then mx is a maximal ideal and all such have this form. For the details, see Wa*
*lkery.]
A character_of C(X) is a nonzero multiplicative linear functional on C(X), *
*i.e., a
homomorphism C(X) ! R of algebras.
LEMMA If O : R ! R is a nonzero ring homomorphism, then O = idR.
[In fact, O is order preserving and the identity on Q .]
Application: Every ring homomorphism C(X) ! R is R linear, thus is a chara*
*cter.
LEMMA If O : C(X) ! R is a character of C(X), then 8 f, O(f) = O(f).
[For O(f)2 = O(f)2 = O(f2) = O(f2) = O(f)2 and O(f) is 0.]
By way of a corollary, if O : C(X) ! R is a character of C(X) and if O(f) =*
* 0, then O(min{1; f}) = 0.
Proof: 2O(min{1; f}) = O(1) + O(f)  O(1  f) = 1  O(1  f) = 1  1 = 0.
FACT Write AEf for the unique extension of f 2 C(X) to C(AEX)_then C(X) "i*
*s" C(AEX) and the
characters of C(X) are parameterized by the points of AEX : f ! AEf(x) (x 2 AEX*
*).
[If X is Rcompact and if O : C(X) ! R is a character, then in the terminol*
*ogy of p. 196 & p.
197, FO = {Z(f) : O(f) = 0} is a zero set ultrafilter on1X. Claim: FO has the *
*countable intersection
P min{1; fn} 1T
property. Thus let {Z(fn)} FO be a sequence and put f = __________n_then *
*Z(fn) = Z(f).
n 1 2 1
P min{1; fi}
To prove that O(f) = 0, write f = _________i+ gn, where 0 gn 2n, apply O *
*to get O(f) =
i=1 2
_________________________
yThe StoneCech Compactification, Springer Verlag (1974), 18.
224
O(gn) 2n, and let n ! 1. It therefore follows that \FO is nonempty, say x 2 \*
*FO (cf. p. 197). And:
O(f  O(f)) = 0 ) x 2 Z(f  O(f)) ) O(f) = f(x).]
Notation: "C(X) is the set of continuous characters of C(X).
From the above, there is a onetoone correspondence X ! "C(X), viz. x ! Ox*
*, where
Ox(f) = f(x).
If X is not Rcompact, then the elements of AEX  X correspond to the disco*
*ntinuous characters of
C(X).
Topologize "C(X) by giving it the initial topology determined by the functi*
*ons O !
O(f) (f 2 C(X))_then the correspondence X ! "C(X) is a homeomorphism (cf. x1,
Proposition 14).
ae ae
PROPOSITION 24 Let XY be CRH spaces. Assume: C(X)C(Ya)re isomorphic as
ae
topological algebras_then XY are homeomorphic.
Xfl Yfl
[Schematically, fl fl and ! is a homeomorphism.]
"C(X) ! C"(Y )
ae ae *
* ae
FACT Let X be CRH spaces. Assume: C(X) are isomorphic as algebras_then*
* AEXare
Y C(Y ) *
* AEY
homeomorphic.
31
x3. COFIBRATIONS
The machinery assembled here is the indispensable technical prerequisite fo*
*r the study
of homotopy theory in TOP or TOP *.
Let X and Y be topological spaces. Let A ! X be a closed embedding and let
f : A ! Y be a continuous function_then the adjunction_space_X tf Y correspon*
*ding
A? f! Y?
to the 2source X A f!Y is defined by the pushout square y y , f*
* being
X ! X tf Y
the attaching_map_. Agreeing to identifyaAewith its image in X,atheerestrictio*
*n of the
projection p : X q Y ! X tf Y to XY A is a homeomorphism of XY A onto an
ae ae
open p(X  A)
closed subset of X tf Y and the images p(Y ) partition X tf Y .
[Note: The adjunction space X tf Y is unique only up to isomorphism. For ex*
*ample,
if OE : X ! X is a homeomorphism such that OEA = idA, then there arises anothe*
*r pushout
square equivalent to the original one.]
(AD 1) If A is not empty and if X and Y are connected (path connected)*
*, then
X tf Y is connected (path connected).
(AD 2) If X and Y are T 1, then X tf Y is T 1but if X and Y are Hau*
*sdorff,
then X tf Y need not be Hausdorff.
(AD 3) If X and Y are Hausdorff and if A is compact, then X tf Y is Ha*
*usdorff.
(AD 4) If X and Y are Hausdorff and if A is a neighborhood retract of *
*X such
__
that each x 2 X  A has a neighborhood U with A \ U = ;, then X tf Y is Hausdor*
*ff.
(AD 5) If X and Y are normal (normal and countably paracompact, perfe*
*ctly
normal, collectionwise normal, paracompact) Hausdorff spaces, then X tf Y is a *
*normal
(normal and countably paracompact, perfectly normal, collectionwise normal, par*
*acom
pact) Hausdorff space.
(AD 6) If X and Y are in CG ( CG ), then X tf Y is in CG ( CG ).
EXAMPLE Working with the IsbellMrowka space (N ) = S [ N, consider the pu*
*shout square
S f! fiS
?y ?y
. Due to the maximality of S, every open covering of (N )*
*tffiS has a finite
(N ) ! (N ) tf fiS
subcovering. Still, (N ) tf fiS is not Hausdorff.
32
ae
TOP ! TOP
The cylinder_functor_I is the functor I : , where X x [0; 1*
*] carries
Xa!eX x [0; 1]
X ! IX
the product topology. There are embeddings it : (0 t 1) and a proj*
*ection
ae x ! (x; t) ae
IX ! X TOP ! TOP
p : . The path_space_functor_P is the functor P : *
*, where
(x; t) ! x X ! C([0; 1];aX)e
X !*
* P X
C([0; 1]; X) carries the compact open topology. There is an embedding j : *
* ,
ae x !*
* j(x)
P X ! X
with j(x)(t) = x, and projections pt : (0 t 1), with pt(oe) = oe(*
*t).
oe ! pt(oe)
(I;aPe) is an adjoint pair: C(IX; Y ) C(X; P Y ). Accordingly, two continuous *
*functions
f : X ! Y
determine the same morphism in HTOP , i.e. are homotopic (f ' g*
*), iff
g : X ! Y ae
H O i0 = f
9 H 2 C(IX; Y ) such that or, equivalently, iff 9 G 2 C(X; P Y ) s*
*uch that
ae H O i1 = g
p0 O G = f
.
p1 O G = g
Let A and X be topological spaces_then a continuous function i : A ! X is s*
*aid
to be a cofibration_if it has the followingaproperty:eGiven any topological spa*
*ce Y and
F : X ! Y
any pair (F; h) of continuous functions such that F O i = h O i0,*
* there is a
h : IA ! Y
continuous function H : IX ! Y such that F = H O i0 and H O Ii = h. Thus H is a*
* filler
for the diagram
A __________________wiX
 
 
 
 
i0 Y i0 :
 557 
 5 
 5 
u u
IA _________________wIiIX
[Note: One can also formulate the definition in terms of the path space fun*
*ctor, viz.
A ________wP Y
 iij p
i i  0:]
u i u
X ________wY
A continuous function i : A ! X is a cofibration iff the commutative diagram
33
A? i! X
iy0 ?yi0 is a weak pushout square. Homeomorphisms are cofibrations. *
*Maps
IA !IiIX
with an empty domain are cofibrations. The composite of two cofibrations is a c*
*ofibration.
EXAMPLE Let p : X ! B be a surjective continuous function. Consider Cp = *
*IX q B=~,
where (x0; 0) ~ (x00; 0) & (x; 1) ~ p(x) (no topology). Let t : Cp ! [0; 1] be*
* the function [x; t] ! t;
let x : t1(]0; 1[) ! X be the function [x; t] ! x; let p : t1(]0; 1]) ! B be *
*the function [x; t] ! p(x).
Definition: The coordinate_topology_on Cp is the initial topologyadeterminedeby*
* t; x; p. There is a closed
embedding j : B ! Cp which is a cofibration. For suppose that F : Cp ! Yare c*
*ontinuous functions
h : IB ! Y
such that F O j = h O i0_then the formulas H(j(b); T) = h(b; T),
8 T
< F[x; t + __2] (t 1=2; T 2  2t)
H([x; t]; T) = h(p(x); 2t + T (2)t 1=2; T 2  2t)
:
F[x; t + tT] (t 1=2)
specify a continuous function H : ICp ! Y such that F = H O i0 and H O Ij = h.
[Note: Cp also carries another (finer) topology (cf. p. 322). When X = B &*
* p = idX, Cp is cX,
and when B = * & p(X) = *, Cp is cX, i.e., the coordinate topology is the coars*
*e topology (cf. p. 127
ff.).]
LEMMA Suppose that i : A ! X is a cofibration_then i is an embedding.
A? i! X?
[Form the pushout square iy0 yF corresponding to the 2source IA *
*i0A
IA !h Y
!i X. The definitions imply that there is a continuous function G : Y ! IX suc*
*h that
ae ae
G O F = i0 H O i0 = F
G O h = Ii and a continuous function H : IX ! Y such that H O Ii = h. Becau*
*se
H O G = idY, G is an embedding. On the other hand, h O i1 : A ! Y is an embeddi*
*ng,
hence G O h O i1 : A ! i(A) x {1} is a homeomorphism.]
For a subspace A of X, the cofibration condition is local in the sense that*
* if there exists a numerable
covering U = {U} of X such that 8 U 2 U, the inclusion A \ U ! U is a cofibrati*
*on, then the inclusion
A ! X is a cofibration (cf. p. 45).
When A is a subspace of X and the inclusion A ! X is a cofibration, the com*
*mutative
i0A? ! IA?
diagram y y is a pushout square and there is a retraction r : *
*IX !
i0X ! i0X [ IA
34
ae
i0X [ IA. If ae : i0X [ IA ! IX is the inclusion and if uv::XX!!IXIXare defi*
*ned by
ae
u = i1
v = ae O r O,i1then A is the equalizer of (u; v). Therefore the inclusion A *
*! X is a
closed cofibration provided that X is Hausdorff or in CG .
PROPOSITION 1 Let A be a subspace of X_then the inclusion A ! X is a cofi
bration iff i0X [ IA is a retract of IX.
Why should the inclusion A ! X be a cofibration if i0X [IA is a retract of *
*IX? Here
is the problem. Suppose that OE : i0X [ IA ! Y is a function such that OEi0X &*
* OEIA are
continuous. Is OE continuous? That the answer is "yes" is a consequence of a *
*generality
(which is obvious if A is closed).
LEMMA If i0X [ IA is a retractaofeIX, then a subsetaOeof i0X [ IA is open *
*in
i0X [ IA iff its intersection with i0XIAis open in i0XIA.
[Let r be the retraction in question and assume that O has the stated prope*
*rty. Put
XO = {x : (x; 0) 2 O}. Write Un for the union of all open U X : A \ U x [0; 1=*
*n[ O.
1S 1S __ 1S
Note that A \ XO = A \ Un and X  Un A . Claim: XO Un. Turn it
1 1 1 1
S __
around and take an x 2 X  Un_then for any t 2 ]0; 1], r(A x {t}) = A x {t}, *
*so
1 1
S
r(x; t) 2 (A  Un) x [0; 1] = (A  XO ) x [0; 1] (X  XO ) x [0; 1] ) (x; 0)*
* = r(x; 0) 2
1
(X  XO ) x [0; 1] ) x 2 X  XO , from which the claim. Thus O = O0[ O00, where
1S
O0= O \ (Ax]0; 1]) and O00= (i0X [ IA) \ (XO \ Un x [0; 1=n[) are open in i0X *
*[ IA.]
1
EXAMPLE Not every closed embedding is a cofibration: Take X = {0} [ {1=n :*
* n 1} and let
A = {0}. Not every cofibration is a closed embedding: Take X = [0; 1]=[0; 1[= {*
*[0]; [1]} and let A = {[0]}.
ae
EXAMPLE Given nonempty topological spaces X , form their coarse join X **
*c Y _then the
ae Y
closed embeddings X ! X *cY are cofibrations.
Y
[It suffices to exhibit a retraction r : I(X*cY ) ! i0(X*cY )[IY . To this *
*end, consider r([x; y; 1]; T) =
([x; y; 1]; T), (
([x; y; _2t__]; 0)(0 t 2__T_)
r([x; y; t]; T) = 2  TT + 2t  22  2T :]
([x; y; 1]; ________t)(_____2 t 1)
35
FACT Let X0 X1 . .b.e an expanding sequence of topological spaces. Assum*
*e: 8 n, the
inclusion Xn ! Xn+1 is a cofibration_then 8 n, the inclusion Xn ! X1 is a cofi*
*bration.
[Fix retractions rk : IXk+1 ! i0Xk+1 [ IXk. Noting that IX1 = colimIXn, wor*
*k with the rk to
exhibit i0X1 [ IXn as a retract of IX1 .]
LEMMA Let X and Y be topologicalaspaces;elet A X and B Y be subspaces.
Suppose that the inclusions AB!!XY are cofibrations_then the inclusion AxB ! *
*X xY
is a cofibration.
[Consider the inclusions figuring in the factorization A x B ! X x B ! X x *
*Y .]
Given t : 0 t 1, the inclusion {t} ! [0; 1] is a closed cofibration and t*
*herefore, for
any topological space X, the embedding it : X ! IX is a closed cofibration. Ana*
*logously,
the inclusion {0; 1} ! [0; 1] is a closed cofibration and it too can be multipl*
*ied.
Z? g! Y?
?j
PROPOSITION 2 Let fy y be a pushout square and assume that f is a
X ! P
cofibration_then j is a cofibration.
[The cylinder functor preserves pushouts.]
Application: Let A ! X be a closed cofibration and let f : A ! Y be a conti*
*nuous
function_then the embedding Y ! X tf Y is a closed cofibration.
The inclusion S n1 ! D n is a closed cofibration. Proof: Define a retrac*
*tion r :
ID n ! i0D n[ IS n1 by letting r(x; t) be the point where the line joining (0;*
* 2) 2 R nx R
and (x; t) meets i0D n [ IS n1. Consequently, if f : Sn1 ! A is a continuous *
*function,
then the embedding A ! D ntf A is a closed cofibration. Examples: (1) The embed*
*ding
D n ! Sn of D n as the northern or southern hemisphere of Sn is a closed cofibr*
*ation;
(2) The embedding Sn1 ! Sn of Sn1 as the equator of Sn is a closed cofibratio*
*n, so
8 m n, the embedding Sm ! Sn is a closed cofibration.
FACT Let f : Sn1 ! A be a continuous function. Suppose that A is path co*
*nnected_then
D ntf A is path connected and the homomorphism ssq(A) ! ssq(D ntf A) is an isom*
*orphism if q < n  1
and an epimorphism if q = n  1.
VAN KAMPEN THEOREM Suppose that the inclusion A ! X is a closed cofibratio*
*n. Let
36
f
A ! Y
? ?
f : A ! Y be a continuous function_then the commutative diagram y y*
* is a
X ! (X t*
*f Y )
pushout square in GRD.
[Note: If in addition, A, X and Y are path connected, then for every x 2 A,*
* the commutative
ss1(A; x)f*!ss1(Y; f(x))
? ?
diagram y y is a pushout square in GR.]
ss1(X; x)! ss1(X tf Y; f(x))
Let A be a subspace of X, i : A ! X the inclusion.
(DR) A is said to be a deformation_retract_of X if there is a continuo*
*us function
r : X ! A such that r O i = idA and i O r ' idX.
(SDR) A is said to be a strong_deformation_retract_of X if there is a *
*continuous
function r : X ! A such that r O i = idA and i O r ' idXrelA.
If i0X [ IA is a retract of IX, then i0X [ IA is a strong deformation retra*
*ct of IX.
Proof: Fix a retraction r : IX ! i0X [ IA, say r(x; t) = (p(x; t), q(x; t)), an*
*d consider the
homotopy H : I2X ! IX defined by H((x; t); T ) = (p(x; tT ); (1  T )t + T q(x;*
* t)).
PROPOSITION 3 Let A be a closed subspace of X and let f : A ! Y be a conti*
*nuous
function. Suppose that A is a strong deformation retract of X_then the image of*
* Y in
X tf Y is a strong deformation retract of X tf Y .
EXAMPLE The house with two rooms is a strong deformation *
*retract of
[0; 1]3.
LEMMA Suppose that the inclusion A ! X is a cofibration_then the inclusion
i0X [ IA [ i1X ! IX is a cofibration.
[Fix a homeomorphism : I[0; 1] ! I[0; 1] that sends I{0}[i0[0; 1][I{1} to *
*i0[0; 1]_
then the homeomorphism idX x : I2X ! I2X sends i0IX [ I(i0X [ IA [ i1X) to
i0IX [ I2A. Since the inclusion IA ! IX is a cofibration, i0IX [ I2A is a retra*
*ct of I2X
and Proposition 1 is applicable.]
[Note: A similar but simpler argument proves that the inclusion i0X [ IA ! *
*IX is a
cofibration.]
37
PROPOSITION 4 If A is a deformation retract of X and if i : A ! X is a cof*
*ibration,
then A is a strong deformation retract of X.
[Choose a homotopy H : IX ! X such that H O i0 = idX and H O i1 = i O r, wh*
*ere
r : X ! A is a retraction. Define a function h : I(i0X [ IA [ i1X) ! X by
8
< h((x; 0); T ) = x (x 2 X)
: h((a;ht);(T()x=;H(a;1(1);TT)t)) (a=2HA)(r(x);:1  T ) *
*(x 2 X)
Observing that i0X [ IA [ i1X can be written as the union of i0X [ A x [0; 1=2]*
* and
Ax[1=2; 1][i1X, the lemma used in the proof of Proposition 1 implies that h is *
*continuous.
But the restriction of H to i0X [ IA [ i1X is h O i0, so there exists a continu*
*ous function
G : IX ! X which extends h O i1. Obviously, G O i0 = idX, G O i1 = i O r, and 8*
* a 2 A,
8 t 2 [0; 1] : G(a; t) = a. Therefore A is a strong deformation retract of X.]
PROPOSITION 5 If i : A ! X is both a homotopy equivalence and a cofibratio*
*n,
then A is a strong deformation retract of X.
[To say that i : A ! X is a homotopy equivalence means that there exists a *
*continuous
function r : X ! A such that r O i ' idA and i O r ' idX. However, due to the c*
*ofibration
assumption, the homotopy class of r contains an honest retraction, thus A is a *
*deformation
retract of X or still, a strong deformation retract of X (cf. Proposition 4).]
EXAMPLE (The_Comb_) Consider the subspace X of R2 consisting of the union*
* ([0; 1] x {0}) [
({0} x [0; 1]) and the line segments joining (1=n; 0) and (1=n; 1) (n = 1; 2; :*
* :):_then X is contractible.
Moreover, {0} x [0; 1] is a deformation retract of X. But it is not a strong de*
*formation retract. Therefore
the inclusion {0} x [0; 1] ! X, while a homotopy equivalence, is not a cofibrat*
*ion.
Let A be a subspace of X_then a Strom_structure_on (X; A) consists of a con*
*tinuous
function OE : X ! [0; 1] such that A OE1(0) and a homotopy : IX ! X of idX r*
*elA
such that (x; t) 2 A whenever t > OE(x).
[Note: If the pair (X; A) admits a Strom structure (OE; ) and if A is close*
*d in X, then
A = OE1(0). Proof: OE(x) = 0 ) x = (x; 0) = lim(x; 1=n) 2 A.]
If the pair (X; A) admits a Strom structure (OE0; 0) for which OE0 < 1 thro*
*ughout X,
then A is a strong deformation retract of X. Conversely, if A is a strong defor*
*mation retract
of X and if the pair (X; A) admits a Strom structure (OE; ), then the pair (X; *
*A) admits
a Strom structure (OE0; 0) for which OE0 < 1 throughout X. Proof: Choose a homo*
*topy
H : IX ! X of idX relA such that H O i1(X) A and put OE0(x) = min {OE(x); 1=2},
0(x; t) = H((x; t); min{2t; 1}).
38
COFIBRATION CHARACTERIZATION THEOREM The inclusion A ! X is a
cofibration iff the pair (X; A) admits a Strom structure (OE; ).
[Necessity: Fix a retraction r : IX ! i0X [ IA and let X p IX q![0; 1] be*
* the
projections. Consider OE(x) = sup t  qr(x; t), (x; t) = pr(x; t).
0t1
Sufficiency: Given a Strom structure (OE; ) on (X; A), define a retraction *
*r : IX !
i0X [ IA by ae
r(x; t) = ((x;(t);(0)x;(tt)OE(x)); t :OE(x))] (t OE(x))
One application of this criterion is the fact that if the inclusion A ! X i*
*s a cofibration,
__
then the inclusion A ! X is a closed cofibration. For let (OE; ) be a Strom str*
*ucture on
__ __ *
* __
(X; A)_then (OE; ), where (x; t) = (x; min{t; OE(x)}), is a Strom structure *
*on (X; A).
Another application is that if the inclusion A ! X is a closed cofibration, the*
*n the inclusion
kA ! kX is a closed cofibration. Indeed, a Strom structure on (X; A) is also a*
* Strom
structure on (kX; kA).
EXAMPLE Let A [0; 1]n be a compact neighborhood retract of Rn_then the in*
*clusion A !
[0; 1]n is a cofibration.
EXAMPLE Take X = [0; 1] ( > !) and let A = {0 }, 0 the "origin" in X_then*
* A is a strong
deformation retract of X but the inclusion A ! X is not a cofibration (A is not*
* a zero set in X).
FACT Let A be a nonempty closed subspace of X. Suppose that the inclusion *
*A ! X is a co
fibration_then 8 q, the projection (X; A) ! (X=A; *A) induces an isomorphism Hq*
*(X; A) ! Hq(X=A; *A),
*A the image of A in X=A.
[With U running over the neighborhoods of A in X, show that Hq(X; A) limHq*
*(X; U) and then
use excision.]
LEMMA Let X and Y be Hausdorff topological spaces. Let A be a closed subsp*
*ace of X and let
f : A ! Y be a continuous function. Assume: The inclusion A ! X is a cofibratio*
*n_then X tf Y is
Hausdorff.
As we shall now see, the deeper results in cofibration theory are best appr*
*oached by
implementation of the cofibration characterization theorem.
PROPOSITION 6 Let K be a compact Hausdorff space. Suppose that the inclusi*
*on
A ! X is a cofibration_then the inclusion C(K; A) ! C(K; X) is a cofibration (c*
*ompact
open topology).
39
[Let (OE; ) be a Strom structure on (X; A). Define OEK : C(K; X) ! [0; 1] b*
*y OEK (f) =
sup OE O f and K : IC(K; X) ! C(K; X) by K (f; t)(k) = (f(k); t)_then (OEK ; K *
*) is
K
a Strom structure on (C(K; X); C(K; A)).]
EXAMPLE If A is a subspace of X, then the inclusion PA ! PX is a cofibrati*
*on provided that
the inclusion A ! X is a cofibration.
EXAMPLE Take A = {0; 1}, X = [0; 1]_then the inclusion A ! X is a cofibrat*
*ion but the
inclusion C(N ; A) ! C(N ; X) is not a cofibration (compact open topology).
[The Hilbert cube is an AR but the Cantor set is not an ANR.]
ae
PROPOSITION 7 Let AB X Y, with A closed, and assume that the correspondi*
*ng
inclusions are cofibrations_then the inclusion A x Y [ X x B ! X x Y is a cofib*
*ration.
[Let (OE; ) and ( ; ) be Strom structures on (X; A) and (Y; B). Define ! : *
*X x Y !
[0; 1] by !(x; y) = min{OE(x); (y)} and define : I(X x Y ) ! X x Y by
((x; y); t) = ((x; min{t; (y)}); (y; min{t; OE(x)})):
Since A is closed in X, OE(x) < 1 ) (x; OE(x)) 2 A, so (!; ) is a Strom structu*
*re on
(X x Y; A x Y [ X x B).]
[Note: If in addition, A (B) is a strong deformation retract of X (Y ), the*
*n AxY [XxB
is a strong deformation retract of X x Y . Reason: OE < 1 ( < 1) throughout X *
*(Y ) )
! < 1 throughout X x Y .]
EXAMPLE If the inclusion A ! X is a cofibration, then the inclusion A x X *
*[ X x A ! X x X
need not be a cofibration. To see this, let X = [0; 1]=[0; 1[= {[0]; [1]}, A = *
*{[0]} and, to get a contradiction,
assume that the pair (X x X; A x X [ X x A) admits a Strom structure (OE; ). Ob*
*viously, OE1([0; 1[ )
_____A[x_X___X=xXAx X (since __A= X), so there exists a retraction r : X x X ! *
*A x X [ X x A. But
________ _________ ________ ____
([1]; [1]) 2 {([0]; [1])}) r([1]; [1]) 2 {r([0]; [1])}= {([0]; [1])}= {[0]}x {[*
*1]} ) r([1]; [1]) = ([0]; [1]) and
________
([1]; [1]) 2 {([1]; [0])}) . .).r([1]; [1]) = ([1]; [0]).
LEMMA Let A be a subspace of X and assume that the inclusion A ! X is a
cofibration. Suppose that K; L : IX ! Y are continuous functions that agree on *
*i0X [
IA_then K ' L reli0X [ IA.
[The inclusion i0X[IA[i1X ! IX is a cofibration (cf. the lemma preceding th*
*e proof
of Proposition 4). With this in mind, define a continuous functionaFe: IX ! Y b*
*y F (x; t) =
K(x; 0) and a continuous function h : I(i0X [ IA [ i1X) ! Y by h((x;h0);(T()x*
*=;K(x;1T)); T ) = L(x; T )
310
& h((a; t); T ) = K(a; T ) = L(a; T ). Since the restriction of F to i0X [ IA*
* [ i1X is
equal to h O i0, there exists a continuous function H : I2X ! Y such that F = *
*H O i0
and HI(i0X [ IA [ i1X) = h. Let : [0; 1] x [0; 1] ! [0; 1] x [0; 1] be the i*
*nvolution
(t; T ) ! (T; t)_then H O(idX x) : I2X ! Y is a homotopy between K and L reli0X*
* [IA.]
ae PROPOSITION 8 Let A and B be closed subspaces of X. Suppose that the inclu*
*sions
A ! X
B ! X , A \ B ! X are cofibrations_then the inclusion A [ B ! X is a cofibra*
*tion.
[In IX, write (x; t) ~ (x; 0) (x 2 A \ B), call Xe the quotient IX= ~, and *
*let
p : IX ! Xe be the projection. Choose continuous functions OE; : X ! [0; 1] s*
*uch that
A = OE1(0), B = 1 (0). Define : X ! eX by (x) = x; ___OE(x)___OE(x)i+f (x)*
*x 62 A \ B,
ae
(x) = [x; 0] if x 2 A \ B_then is continuous and (x)(=x[x;)0]=on[Ax;.1]ConoB*
*nsider now
ae
a pair (F; h) of continuous functions Fh::XI!(YA [ B) !fYor which F (A [ B) *
*= h O i0.
ae ae
Fix homotopies HAH: IX ! Y such that HA IA = hIA & F = HA O i0 = HB O *
*i0
B : IX ! Y HB IB = hIB
and, using the lemma, fix a homotopy H : I2X ! Y between HA and HB reli0X [
I(A \ B). With as in the proof above, the composite H O (idX x ) factors thro*
*ugh
I2X pxid!IXe, thus there is a continuous function eH : IXe ! Y that renders t*
*he diagram
I2X? idXx!I2X
pxidy ?yH commutative. An extension of (F; h) is then given by the co*
*mposite
IXe ! Y
eH
He O ( x id) : IX ! IXe ! Y .]
FACT Let A and B be closed subspaces of a metrizable space X. Suppose tha*
*t the inclusions
A \ B ! A, A \ B ! B, B ! X, A  B ! X  B are cofibrations_then the inclusion *
*A ! X is a
cofibration.
Let A be a subspace of X. Suppose given a continuous function : X ! [0; 1*
*] such
that A 1 (0) and a homotopy : I 1 ([0; 1]) ! X of the inclusion 1 ([0; 1*
*]) !
X relA such that (x; t) 2 A whenever t > (x)_then the inclusion A ! X is a cof*
*ibra
tion. Proof: Define a Strom structure (OE; ) on (X; A) by OE(x) = min{2 (x); 1},
8
< (x; t) (2 (x) 1)
(x; t) = : (x; t(2  2 (x))) (1 2 (x) 2) :
x ( (x) 1)
311
LEMMA Let A be a subspace of X and assume that the inclusion A ! X is a
cofibration. Suppose that U is a subspace of X with the property that there ex*
*ists a
__
continuous function ss : X ! [0; 1] for which A \ U ss1 (]0; 1]) U_then the *
*inclusion
A \ U ! U is a cofibration.
[Fix a Strom structure (OE; ) on (X; A). Set ss0(x) = 0inft1ss((x; t)) (x 2*
* X). Define
a continuous function : U ! [0; 1] by (x) = OE(x)=ss0(x). This makes sense *
*since
OE(x) = 0 ) ss0(x) > 0 (x 2 U). Next, (x) 1 ) ss0(x) > 0 ) ss((x; t)) > 0 )
(x; t) 2 U (8 t). One can therefore let : I 1 ([0; 1]) ! U be the restriction*
* of and
apply the foregoing remark to the pair (U; A \ U).]
Let A; U be subspaces of a topological space X_then U is said to be a halo_*
*of A
in X if there exists a continuous function ss : X ! [0; 1] (the haloing_functio*
*n_) such
that A ss1 (1) and ss1 (]0; 1]) U. For example, if X is normal (but not nec*
*essarily
Hausdorff), then every neighborhood of a closed subspace A of X is a halo of A *
*in X but
in a nonnormal X, a closed subspace A of X may have neighborhoods that are not *
*halos.
__
(HA 1) If U is a halo of A in X, then U is a halo of A in X.
(HA 2) If U is a halo of A in X, then there exists a closed subspace B*
* of X : A
B X, such that B is a halo of A in X and U is a halo of B in X.
[A haloing function for ss1 ([1=2; 1]) is max {2ss(x)  1; 0}.]
Observation: If the inclusion A ! X is a cofibration and if U is a halo of *
*A in X,
then the inclusion A ! U is a cofibration.
[This is a special case of the lemma.]
PROPOSITION 9 If j : B ! A and i : A ! X are continuous functions such tha*
*t i
and i O j are cofibrations, then j is a cofibration.
[Take i and j to be inclusions. Using the cofibration characterization theo*
*rem, fix a
halo U of A in X and a retraction r : U ! A. Since U is also a halo of B in X,*
* the
B? g! P*
*?Y
inclusion B ! U is a cofibration. Consider a commutative diagram jy y*
* p0.
A !F Y
B? g! P?Y
To construct a filler for this, pass to its counterpart y y p0 over U, *
*which thus
U !FOrY
admits a filler G : U ! P Y . The restriction GA : A ! P Y will then do the tr*
*ick.]
EXAMPLE (Telescope_Construction_) Let X0 X1 . .b.e an expanding sequenc*
*e of topo
312
logical spaces. Assume: 8 n, the inclusion Xn ! Xn+1 is a closed cofibration_th*
*en18 n, the inclusion
`
Xn ! X1 is a closed cofibration (cf. p. 35). Write telX1 for the quotient *
*Xn x [n; n + 1]=~. Here,
0
~ means that the pair (x; n+1) 2 Xn x{n+1} is identified with the pair (x; n+1)*
* 2 Xn+1x{n+1}. One
calls telX1 the telescope_of X1 . It can be viewed as a closed subspace of X1 *
*x [0; 1[. The inclusion
Sn
telnX1 Xk x [k; k + 1] ! X1 x [0; 1[ is a closed cofibration (cf. Proposit*
*ion 8), so the same is
k=0
true of the inclusion telnX1 ! teln+1X1 (cf. Proposition 9) and telX1 = coli*
*mtelnX1 . Denote by
p1 the composite telX1 ! X1 x [0; 1[! X1 .
Claim: p1 is a homotopy equivalence.
[It suffices to establish that telX1 is a strong deformation retract of X1*
* x [0; 1[. One approach is
to piece together strong deformation retractions Xn+1 x [0; n + 1] ! Xn+1 x {n *
*+ 1} [ Xn x [0; n + 1].]
ae 0 1
Let X X . . .be expanding sequences of topological spaces. Assume: 8 *
*n, the inclusions
ae Y 0 Y 1 . . .
Xn ! Xn+1 are closed cofibrations. Suppose given a sequence of continuous fu*
*nctions OEn : Xn ! Y n
Y n! Y n+1
Xn ! Xn+1
? ?
such that 8 n, the diagram OEyn y OEn+1commutes. Associated with the OE*
*n is a continuous
Y n ! Y n+1
function OE1 : X1 ! Y 1 and a continuous function telOE : telX1 ! telY 1, the l*
*atter being defined by
ae n n
telOE(x; n + t) = (OE (x); n + 2t) 2 Y x [n; n + 1] (0: t 1=*
*2)
(OEn(x); n + 1) 2 Y n+1x {n + 1} (1=2 t 1)
telX1 ! X1
? ?
There is then a commutative diagram teylOE yOE1. The horizontal arrows *
*are homotopy
telY 1 ! Y 1
equivalences. Moreover, telOE is a homotopy equivalence if this is the case of *
*the OEn, thus, under these
circumstances, OE1 : X1 ! Y 1 itself is a homotopy equivalence.
[Note: One can also make the deduction from first principles (cf. Propositi*
*on 15).]
PROPOSITION 10 Let A be a closed subspace of a topological space X. Suppose
that A admits a halo U with A = ss1 (1) for which there exists a homotopy : I*
*U ! X
of the inclusion U ! X relA such that O i1(U) A_then the inclusion A ! X is a
closed cofibration.
[Define a retraction r : IX ! i0X [ IA as follows: (i) r(x; t) = (x; 0) (ss*
*(x) = 0); (ii)
r(x; t) = ((x; 2ss(x)t); 0) (0 < ss(x) 1=2); (iii) r(x; t) = ((x; t=2(1  ss(x*
*))); 0) (1=2
ss(x) < 1 & 0 t 2(1  ss(x))) and r(x; t) = ((x; 1); t  2(1  ss(x))) (1=2 *
*ss(x) < 1
& 2(1  ss(x)) t 1); (iv) r(x; t) = (x; t) (ss(x) = 1).]
313
EXAMPLE If A is a subcomplex of a CW complex X, then the inclusion A ! X i*
*s a closed
cofibration.
A topological space X is said to be locally_contractible_provided that for *
*any x 2
X and any neighborhood U of x there exists a neighborhood V U of x such that
the inclusion V ! U is inessential. If X is locally contractible, then X is l*
*ocally path
connected. Example: 8 X, X? is locally contractible (cf. p. 128).
[Note: The empty set is locally contractible but not contractible.]
A topological space X is said to be numerably_contractible_if it has a nume*
*rable covering {U} for
which each inclusion U ! X is inessential. Example: Every locally contractible *
*paracompact Hausdorff
space is numerably contractible.
[Note: The product of two numerably contractible spaces is numerably contra*
*ctible.]
FACT Numerable contractibility is a homotopy type invariant. Proof: If X i*
*s dominated in ho
motopy by Y and if Y is numerably contractible, then X is numerably contractibl*
*e.
Examples: (1) Every topological space having the homotopy type of a CW comp*
*lex is numerably
contractible; (2) If the Xn of the telescope construction are numerably contrac*
*tible, then X1 is numerably
contractible (consider telX1 ).
A topological space X is said to be uniformly_locally_contractible_provided*
* that there
exists a neighborhood U of the diagonal X X x X and a homotopy H : IU ! X
between p1U and p2U relX , where p1 and p2 are the projections onto the first*
* and
second factors. Examples: (1) R n, D n, and Sn1 are uniformly locally contract*
*ible; (2)
The long ray L+ is not uniformly locally contractible.
EXAMPLE (Stratifiable_Spaces_) Suppose that X is stratifiable and in NES(*
*stratifiable)_then
X is uniformly locally contractible. Thus put A = X x i0X [ (IX ) [ X x i1X,aae*
*closed subspace of the
stratifiable space I(X x X). Define a continuous function OE : A ! X by (x; y*
*; 0) !&x(x; x; t) ! x_
(x; y*
*; 1) ! y
then OE extends to a continuous function : O ! X, where O is a neighborhood of*
* A in I(X x X). Fix a
neighborhood U of X in X x X : IU O and consider H = IU.
[Note: Every CW complex is stratifiable (cf. p. 630) and in NES(stratifiab*
*le) (cf. p. 643). Every
metrizable topological manifold is stratifiable (cf. p. 629 ff.: metrizable ) *
*stratifiable) and, being an
ANR (cf. p. 628), is in NES(stratifiable) (cf. p. 644: stratifiable ) perfect*
*ly normal + paracompact).]
FACT Let K be a compact Hausdorff space. Suppose that X is uniformly local*
*ly contractible_
then C(K; X) is uniformly locally contractible (compact open topology).
314
LEMMA A uniformly locally contractible topological space X is locally cont*
*ractible.
[Take a point x0 2 X and let U0 be a neighborhood of x0_then I{(x0; x0)}
H1 (U0). Since H1 (U0) is open in IU, hence open in I(X x X), there exists a*
* neigh
borhood V0 U0 of x0 : I(V0 x V0) H1 (U0). To see that the inclusion V0 ! U0*
* is
inessential, define H0 : IV0 ! U0 by H0(x; t) = H((x; x0); t).]
[Note: The homotopy H0 keeps x0 fixed throughout the entire deformation. In*
* addi
tion, the argument shows that an open subspace of a uniformly locally contracti*
*ble space
is uniformly locally contractible.]
EXAMPLE (A_Spaces_) Every A space is locally contractible. In fact, if X*
* is a nonempty A
space, then 8 x 2 X, Ux is contractible, thus X has a basis of contractible ope*
*n sets, so X is locally
contractible.aButeanaAespace need not be uniformly locally contractible. Consid*
*er, e.g., X = {a; b; c; d},
where c a, c b.
d a d b
FACT Let X be a perfectly normal paracompact Hausdorff space. Suppose tha*
*t X admits a
covering by open sets U, each of which is uniformly locally contractible_then X*
* is uniformly locally
contractible.
[Use the domino principle.]
When is X uniformly locally contractible? A sufficient condition is that th*
*e inclusion
X ! XxX be a cofibration. Proof: Fix a Strom structure (OE; ) on the pair (XxX;*
* X ),
put U = OE1([0; 1[ ) and define H : IU ! X by
ae
H((x; y); t) = p1(((x;py); 2t)) (0 t 1=2) :
2(((x; y); 2(12t))=2 t 1)
FACT Suppose that X is a perfectly normal Hausdorff space with a perfectly*
* normal square_then
X is uniformly locally contractible iff the diagonal embedding X ! X x X is a c*
*ofibration.
[Use Proposition 10, noting that X is a zero set.]
Application: If X is a CW complex or a metrizable topological manifold, the*
*n the diagonal embedding
X ! X x X is a cofibration.
FACT Let A be a closed subspace of a metrizable space X such that the incl*
*usion A ! X is a
cofibration. Suppose that A and X  A are uniformly locally contractible_then X*
* is uniformly locally
contractible.
[Show that the inclusion X ! X xX is a cofibration by applying the result o*
*n p. 310 to the triple
(X x X; X ; A x A).]
315
PROPOSITION 11 Suppose that A X admits a halo U such that the inclusion
U ! U x U is a cofibration. Assume that the inclusion A ! X is a cofibration_th*
*en
the inclusion A ! A x A is a cofibration.
A? A! A x?A
[Consider the commutative diagram y y . The vertical arrows*
* are
U ! U x U
U
cofibrations, as is U . That A is a cofibration is therefore implied by Proposi*
*tion 9.]
PROPOSITION 12 Let X be a Hausdorff space and suppose that the inclusion X*
* !
X x X is a cofibration. Let f : X ! [0; 1] be a continuous function such that A*
* = f1 (0)
is a retract of f1 ([0; 1[ )_then the inclusion A ! X is a closed cofibration.
[Write r for the retraction f1 ([0; 1[ ) ! A, fix a Strom structure (OE; )*
* on the pair
(X x X; X ), and let H : IU ! X be as above. Define OEf : X ! [0; 1] by OEf(x)*
* =
max {f(x); OE(x; r(x))} (f(x) < 1) & OEf(x) = 1 (f(x) = 1)_then OE1f(0) = A. *
*Put
Hf(x; t) = H((x; r(x)); t) to obtain a homotopy Hf : IOE1f([0; 1[ ) ! X of the*
* inclusion
OE1f([0; 1[ ) ! X relA such that Hf O i1(OE1f([0; 1[ )) A. Finish by citing *
*Proposition 10.]
Application: Let X be a Hausdorff space and suppose that the inclusion X ! *
*X xX
is a cofibration. Let e 2 C(X; X) be idempotent: e O e = e_then the inclusion e*
*(X) ! X
is a closed cofibration.
[Define f : X ! [0; 1] by f(x) = OE(x; e(x)).]
So, if X is a Hausdorff space and if the inclusion X ! X xX is a cofibratio*
*n, then for
any retract A of X, the inclusion A ! X is a closed cofibration. In particular:*
* 8 x0 2 X,
the inclusion {x0} ! X is a closed cofibration, which, as seen above, is a cond*
*ition realized
by every CW complex or metrizable topological manifold.
[Note: Let X be the Cantor set_then 8 x0 2 X, the inclusion {x0} ! X is clo*
*sed
but not a cofibration.]
FACT Let X be in CG and suppose that the inclusion X ! X xk X is a cofi*
*bration_then
for any retract A of X, the inclusion A ! X is a closed cofibration.
[Rework Proposition 12, noting that for any continuous function f : X ! X, *
*the function X !
X xk X defined by x ! (x; f(x)) is continuous.]
ae
LEMMA Suppose that the inclusions A ! X are closed cofibrations and th*
*at X is a closed
ae A0! X0
subspace of X0 with A = X \ A0. Let f : A ! Y be continuous functions. Assum*
*e that the dia
f0: A0! Y 0
316
f
X  A ! Y
? ? ?
gram y y y commutes and that the vertical arrows are cofibration*
*s_then the induced
X0  A0 !f0Y 0
map X tf Y ! X0tf0Y 0is a cofibration and (X tf Y ) \ Y 0= Y .
X tf Y g! PZ
? ?
[Consider a commutative diagram y yp0. To construct a fille*
*r H0 for this,
X0tf0Y 0 !F0 Z
Y ! X tf Y g! PZ
? ? ?
work first with y y yp0 to get an arrow G : Y 0! PZ. Nex*
*t, look at
Y 0 ! X0tf0Y 0 !F0 Z
ae f0 G
A0!Y 0!PZ . Since equality obtains on A = X \ A0, 9 G02 C(X [ A0; PZ) *
*: G0A0= G O f0.
X!X tf Y g!PZ
But the inclusion X [ A0 ! X00is a cofibration (cf. Proposition 8), so the com*
*mutative diagram
X [ A0____________________wGPZ
 p0
u uadmits a filler H : X0 ! PZ which agrees with G *
*O f0 on A0
X0 _______wX0tf0Y 0______wF0Z
and therefore determines H0: X0tf0Y 0! PZ.]
FACTaeLet A ! X be a closed cofibration andaletef : A ! Y be a continuous f*
*unction. Suppose
that X are in CG and that the inclusions X ! X xk X are cofibrations_*
*then the inclusion
Y Y ! Y xk Y
Z ! Z xk Z is a cofibration, Z the adjunctionaspaceeX tf Y .
[There are closed cofibrations A xk A ! X xk A [ A xk.XPrecompose these a*
*rrows with the
Y xk Y ! Z xk Y [ Y xk Z
diagonal embeddings, form the commutative diagram
X u______________A___________________wfY
   ;
u u u
X xk X u____X_xk A [ A xk X ______wZ xk Y [ Y xk Z
and apply the lemma.]
[Note: Proposition 7 remains in force if the product in TOP is replaced by*
* the product in CG .
Take U = X in Proposition 11 to see that the inclusion A ! A xk A is a cofibrat*
*ion.]
Application: Let X and Y be CW complexes. Let A be a subcomplex of X and le*
*t f : A ! Y be a
continuous function_then the inclusion Z ! Z xk Z is a cofibration, Z the adjun*
*ction space X tf Y .
317
ae
[The inclusions X ! X x X are cofibrations (cf. p. 314), thus the same*
* is true of the inclu
ae Y ! Y x Y
sions X ! X xk X (cf. p. 38). Z itself need not be a CW complex but, in vi*
*ew of the skeletal
Y ! Y xk Y
approximation theorem, Z at least has the homotopy type of a CW complex.]
FACTaeLet A ! X be a closed cofibration and let f : A ! Y be a continuous f*
*unction. Suppose
that X are uniformly locally contractible perfectly normal Hausdorff spaces *
*with perfectly normal
Y
squares_then X tf Y is uniformly locally contractible provided that its square *
*is perfectly normal.
[Note: A priori, X tf Y is a perfectly normal Hausdorff space (cf. AD 5).]
A pointed space (X; x0) is said to be wellpointed_if the inclusion {x0} ! X*
* is a
__
cofibration. X is the full subgroupoid of X whose objects are the x0 2 X such*
* that
(X; x0) is wellpointed. Example: Let X be a CW complex or a metrizable topolo*
*gical
manifold_then 8 x0 2 X, (X; x0) is wellpointed (cf. p. 315).
[Note: Take X = [0; ], x0 = _then (X; x0) is not wellpointed.]
The full subcategory of HTOP *whose objects are the wellpointed spaces is *
*not isomorphism closed,
i.e., if (X; x0) (Y; y0) in HTOP *, then it can happen that the inclusion {x0*
*} ! X is a cofibration but
the inclusion {y0} ! Y is not a cofibration (cf. p. 38).
EXAMPLE Let X be a topological manifold_then 8 x0 2 X, (X; x0) is wellpoin*
*ted.
FACT Let K be a compact Hausdorff space. Suppose that (X; x0) is wellpoint*
*ed_then 8 k0 2 K,
C(K; k0; X; x0) is wellpointed (compact open topology).
[Note: The base point in C(K; k0; X; x0) is the constant map K ! x0.]
ae __
Given topological spaces XY , the base point functor X x Y ! SET sen*
*ds
anaobjecte(x0; y0) to the set [X; x0; Y; y0]. To describe its behavior on morp*
*hisms, let
x0; x1 2 X
aey0; y1 2 Y and suppose thatabothe(X; x0) and (X; x1) are wellpointed. Let oe *
*2 P X :
oe(0) = x0 o(0) = y0
oe(1) = x1& let o 2 P Y : o(1) = y1_then the pair (oe; o) determines a bij*
*ectionae
[oe; o]# : [X; x0; Y; y0] ! [X; x1; Y; y1] that depends only on the path class*
*es of oeoin
ae
X
Y . Here is the procedure. Fix a homotopy H : IX ! X such that H O i0 = i*
*dX,
H(x1; t) = oe(1  t), and put e = H O i1. Take anafe 2 C(X; x0; Y; y0) and def*
*ine a
continuous function F : i0X [ I{x1} ! X x Y by (x;(0)x! (e(x); f(e(x)))_then*
* the
1; t) ! (oe(t); o(t))
318
i0X [?I{x1} F! X x?Y
diagram y y p commutes, where G(x; t) = H(x; 1  t). To co*
*n
IX !G X
struct a filler Hf : IX ! X x Y , let q : X x Y ! Y be the projection, cho*
*ose
a retraction r : IX ! i0X [ I{x1} and set Hf(x; t) = (G(x; t); qF (r(x; t))). *
* Write
f# = q O Hf O i1 2 C(X; x1; Y; y1). Definition: [oe; o]# [f] = [f# ]. The f*
*undamen
tal group ss1(Y; y0) thus operates to the left on [X; x0; Y; y0] : ([o]; [f]) !*
* [oe0; o]# [f],
oe0 the constant path in X at x0. If f, g 2 C(X; x0; Y; y0), then f ' g in TO*
*P iff
9 [o] 2 ss1(Y; y0) : [oe0; o]# [f] = [g]. Therefore the forgetful function [X; *
*x0; Y; y0] ! [X; Y ]
passes to the quotient to define an injection ss1(Y; y0)\[X; x0; Y; y0] ! [X; Y*
* ] which, when
Y is path connected, is a bijection. The forgetful function [X; x0; Y; y0] ! [X*
*; Y ] is oneto
one iff the action of ss1(Y;ay0)eon [X; x0; Y; y0] is trivial. Changing Y to Z *
*by a homotopy
equivalence in TOP : Yy! Z leads to an arrow [X; x0; Y; y0] ! [X; x0; Z; *
*z0]. It is a
0 ! z0
bijection.
FACT Suppose that X and Y are path connected. Let f 2 C(X; Y ) and assume *
*that 8 x 2 X, f* :
ss1(X; x) ! ss1(Y; f(x)) is surjective_then 8 x 2 X, f* : ssn(X; x) ! ssn(Y; f(*
*x)) is injective (surjective)
iff f* : [Sn; X] ! [Sn; Y ] is injective (surjective).
LEMMA Suppose that the inclusion i : A ! X is a cofibration. Let f 2 C(X; *
*X) :
f O i = i & f ' idX_then 9 g 2 C(X; X) : g O i = i & g O f ' idXrelA.
[Let H : IX ! X be a homotopy with H O i0 = f and H O i1 = idX; let G : IX *
*! X
beaaehomotopy with G O i0 = idX and G O Ii = H O Ii. Define F : IX ! X by F (x;*
* t) =
G(f(x); 1  2t) (0 t 1=2)
H(x; 2t  1) (1=2 t 1) and put
ae
k((a; t); T ) = G(a;H1(a2t(1;1T))2(1 (0(tt)1=2)(11=T2)) t*
* 1)
to get a homotopy k : I2A ! X with F OIi = kOi0. Choose a homotopy K : I2X ! X *
*such
that F = K O i0 and K O I2i = k. Write K(t;T): X ! X for the function x ! K((x;*
* t); T ).
Obviously, K(0;0)' K(0;1)' K(1;1)' K(1;0), all homotopies being relA. Set g = G*
* O i1_
then g O f = F O i0 = K(0;0)is homotopic relA to K(1;0)= F O i1 = idX.]
ae
PROPOSITION 13 Suppose that ij::AA!!XYare cofibrations. Let OE 2 C(X; Y *
*) :
OE O i = j. Assume that OE is a homotopy equivalence_then OE is a homotopy equi*
*valence in
A\TOP .
319
[Since j is a cofibration, there exists a homotopy inverse : Y ! X for O*
*E with
O j = i, thus, from the lemma, 9 0 2 C(X; X) : 0O i = i & 0O O OE ' idX *
*reli(A).
This says that OE0= 0O is a homotopy left inverse for OE under A. Repeat the*
* argument
with OE replaced by OE0 to conclude that OE0 has a homotopy left inverse OE00un*
*der A, hence
that OE0is a homotopy equivalence in A\TOP or still, that OE is a homotopy eq*
*uivalence in
A\TOP .]
ae
Application: Suppose that (X;(x0)Y;ayre wellpointed. Let f 2 C(X; x0; Y; *
*y0)_then
0)
f is a homotopy equivalence in TOP iff f is a homotopy equivalence in TOP *.
FACT Suppose that (X; x0) is wellpointed. Let f 2 C(X; Y ) be inessential_*
*then f is homotopic
in TOP * to the function x ! f(x0).
A? i! X? ae
LEMMA Suppose given a commutative diagram OyE y in which ijare
B !j Y
ae
cofibrations and OE are homotopy equivalences. Fix a homotopy inverse OE0 for*
* OE and a
homotopy hA : IA ! A between OE0O OE and idA_then there exists a homotopy inver*
*se 0
for with i O OE0=ae0O j and a homotopy HX : IX ! X between 0O and idX such*
* that
HX (i(a); t) = i(hAi(a;(2t))(0a)t(11=2)=2. t 1)
[Fix some 0 with i O OE0 = 0O j (possible, j being a cofibration). Put h *
*= i O hA :
h O i0 = i O hA O i0 = i O OE0O OE = 0 O j O OE = 0 O O i ) 9 H : IX ! X su*
*ch
that 0 O = H O i0 and H O Ii = h. Put f = H O i1 : f O i = i O hA O i1 = i*
* &
f ' H O i0 = 0O ' idX ) 9 g 2 C(X; X) : g O i = i & g O f ' idX reli(A). Let
G : IX ! X beaaehomotopy between g O f and idX reli(A). Define HX : IX ! X by
HX (x; t) = g(H(x;G2t))((0x;t2t1=2)(1)1=2: tHX1)is a homotopy between g O 0*
*O and idX
and HX O Ii = i O h0A, where h0A(a; t) = hA (a; min{2t; 1}) is a homotopy betwe*
*en OE0O OE and
idA. Make the substitution 0! g O 0 to complete the proof.]
A? i! X?
PROPOSITION 14 Suppose given a commutative diagram OyE y in which
B !j Y
ae ae
i OE
j are cofibrations and are homotopy equivalences_then (OE; ) is a homoto*
*py equiv
alence in TOP (!).
320
[The lemma implies that (OE0; 0) is a homotopy left inverse for (OE; ) in*
* TOP (!).]
ae
EXAMPLE Let f : X ! Y be objects in TOP (!). Write [f; f0] for the set *
*of homotopy
f0: X0! Y 0 ae
classes of maps in TOP (!) from f to f0. Question: Is it true that if f ' g (*
*in TOP ), then [f; f0] =
f0' g0
[g; g0]? The answer is "no". Let f = g be the constant map S1! (1; 0); let f0: *
*S1! D2 be the inclusion
and let g0: S1! D2 be the constant map at (1; 0)_then [f; f0] 6= [g; g0].
X0? ! X1? ! . . .
PROPOSITION 15 Let y y be a commutative ladder con
Y 0 ! Y 1 ! . . .
nectingatwoeexpanding sequences of topological spaces. Assume: 8 n, the inclu*
*sions
Xn ! Xn+1 n n n
Y n! Y n+1 are cofibrations and the vertical arrows OE : X ! Y are hom*
*otopy
equivalences_then the induced map OE1 : X1 ! Y 1 is a homotopy equivalence.
[Using the lemma, inductively construct a homotopy left inverse for OE1 .]
FACT Let X0 X1 . .b.e an expanding sequence of topological spaces. Assum*
*e: 8 n, the
inclusion Xn ! Xn+1 is a cofibration and that Xn is a strong deformation retrac*
*t of Xn+1_then X0 is
a strong deformation retract of X1 .
[Bearing in mind Proposition 5, recall first that the inclusion X0 ! X1 is *
*a cofibration (cf. p. 35).
X0 ! X0 ! . . .
? ?
Consider the commutative ladder y y to see that the inclusion*
* X0 ! X1 is also
X0 ! X1 ! . . .
a homotopy equivalence.]
FACT Let X0 X1 . .b.e an expanding sequence of topological spaces. Assum*
*e: 8 n, the
inclusion Xn ! Xn+1 is a cofibration and inessential_then X1 is contractible.
EXAMPLE Take Xn = Sn_then X1 = S1 is contractible.
Let f : X ! Y be a continuous function_then the mapping_cylinder_Mf of f is
X? f! Y?
defined by the pushout square iy0 y : Special case: The mapping cylind*
*er of
IX ! Mf
X ! * is X, the cone_of X (in particular, S n1 = D n, so ; = *). There is a
closed embedding j : Y ! Mf, a homotopy H : IX ! Mf, and a unique continuous
function r : Mf ! Y such that r O j = idY and r O H = f O p (p : IX ! X). One *
*has
321
j O r ' idMfrelj(Y ). The composition H O i1 is a closed embedding i : X ! Mf *
*and
f = r O i.
Suppose that X is a subspace of Y and that f : X ! Y is the inclusion_then *
*there is a continuous
bijection Mf ! i0Y [IX. In general, this bijection is not a homeomorphism (cons*
*ider X =]0; 1], Y = [0; 1])
but will be if X is closed or f is a cofibration.
LEMMA j is a closed cofibration and j(Y ) is a strong deformation retract *
*of Mf.
LEMMA i is a closed cofibration.
[Define F : X q X ! Y q X by F = f q idX and form the pushout square
X q?X F! Y q X
i0y i1 ?y _then IX tF (Y q X) can be identified with Mf, i b*
*e
IX ! IX tF (Y q X)
coming the composite of the closed cofibrations X ! Y q X ! IX tF (Y q X).]
It is a corollary that the embedding i of X into its cone X is a closed cof*
*ibration.
EXAMPLE The mapping_telescope_is the functor tel: FIL(TOP ) ! FILSP defin*
*ed on an object
`
(X ; f) by tel(X ; f) = IXn=~, where (xn; 1) ~ (fn(xn); 0), and on a morphism*
* OE : (X ; f) ! (Y ; g) by
n
` `
telOE([xn; t]) = [OEn(xn); t]. Let teln(X ; f) be the image of IXk i0*
*Xn, so teln(X ; f) is obtained
kn1
from Xn via iterated application of the mapping cylinder construction. The emb*
*edding teln(X ; f) !
teln+1(X ; f) is a closed cofibration and tel(X ; f) = colimteln(X ; f). There*
* is a homotopy equivalence
teln(X ; f) ! Xn, viz. the assignment [xk; t] ! (fn1 O . .O.fk)(xk) (0 k n *
* 1), [xn; 0] ! xn and the
teln(X?; f)!teln+1(X?; f)
diagram y y commutes. Consequently, if all the fn are cof*
*ibrations, then it
Xn ! Xn+1
follows from Proposition 15 that the induced map tel(X ; f) ! colimXn is a homo*
*topy equivalence.
[Note: Up to homeomorphism, the telescope construction is an instance of th*
*e above procedure.]
PROPOSITION 16 Every morphism in TOP can be written as the composite of a
closed cofibration and a homotopy equivalence.
PROPOSITION 17 Let f : X ! Y be a continuous function_then f is a homotopy
equivalence iff i(X) is a strong deformation retract of Mf.
[Note that f is a homotopy equivalence iff i is a homotopy equivalence and *
*quote
Proposition 5.]
322
Let f : X ! Y be a continuous function_then the mapping_cone_Cf of f is def*
*ined
X? f! Y?
by the pushout square yi y . Special case: The mapping cone of X ! * i*
*s X,
X ! Cf
the suspension_of X (in particular, S n1= Sn, so ; = S0). There is a closed co*
*fibration
j : Y ! Cf and an arrow Cf ! X. By construction, j O f is inessential and for*
* any
g : Y ! Z with g O f inessential, there exists a OE : Cf ! Z such that g = OE O*
* j.
[Note: The mapping_cone_sequence_associated with f is given by X f!Y ! Cf*
* !
X ! Y ! Cf ! 2X ! . . .. Taking into account the suspension isomorphism
Heq(X) eHq+1(X), there is an exact sequence
. .!.eHq(X) ! eHq(Y ) ! eHq(Cf) ! eHq1(X) ! eHq1(Y ) ! . .:.]
The mapping cylinder and the mapping cone can be viewed as functors TOP (!)*
* ! TOP . With
this interpretation, i, j, and r are natural transformations.
[Note: Owing to AD4, these functors restrict to functors HAUS (!) ! HAUS . *
*Consequently, if X
and Y are in CGH , then for any continuous function f : X ! Y , both Mf and Cf *
*remain in CGH . On
the other hand, stability relative to CG or CG is automatic.]
ae
FACT Suppose that f : X ! Yare homotopic_then in HTOP 2, (Mf; i(X)) (M*
*g; i(X)),
g : X ! Y
and in HTOP , Cf Cg.
FACT Let f 2 C(X; Y ). Suppose that OE : X0 ! X ( : Y ! Y 0) is a homotop*
*y equivalence_
then the arrow (MfOOE; i(X0)) ! (Mf; i(X)) ((Mf; i(X)) ! (M Of; i(X))) is a hom*
*otopy equivalence (in
TOP 2) and the arrow CfOOE! Cf (Cf ! C Of) is a homotopy equivalence (in TOP ).
EXAMPLE The suspension X of X is the union of two closed subspaces  X and*
* +X, each
homeomorphic to the cone X of X, with  X \+X = X (identify the section i1=2X w*
*ith X). Therefore
X ! +X
? ?
X is numerably contractible. The commutative diagram y y is a pusho*
*ut square and
 X ! X
ae
the inclusions X ! X are closed cofibrations.
+X ! X
FACT Let f : X ! Y be a continuous function. Suppose that Y is numerably c*
*ontractible_then
Cf is numerably contractible.
323
[The image of X x [0; 1[ in Cf is contractible. On the other hand, the imag*
*e of Xx]0; 1] q Y in Cf
has the same homotopy type as Y , hence is numerably contractible (cf. p. 313)*
*.]
[Note: Y and Mf have the same homotopy type, so Y numerably contractible ) *
*Mf numerably
contractible (cf. p. 313).]
Let X f Z g!Y be a 2source_then the double_mapping_cylinder_Mf;gof f; g is*
* de
Z q?Z fqg!X q?Y
fined by the pushout square i0 y i1 y . The homotopy type of Mf;g*
*de
IZ ! Mf;g
pends only on the homotopy classesaofef and g and Mf;g is homeomorphic to Mg;f.
There are closed cofibrations ij::XY!!Mf;gMand an arrow Mf;g! Z. The diagram
f;g
Z? g! Y? Z g! Y?
fy ?yj is homotopy commutative and if the diagram f?y ?yj is *
*ho
X !i Mf;g X ! W
ae
motopy commutative, then there exists a OE : Mf;g! W such that j==OEOOEiO.jEx*
*ample:
The double mapping cylinder of X X x Y ! Y is X * Y , the join_of X and Y .
[Note: The mapping cylinder and the mapping cone are instances of the doubl*
*e map
ping cylinder (homeomorphic models arise from the parameter reversal t ! 1  t)*
*. Con
ae Z? ! Mg?
sideration of ZZxx[0;[1=2]1=2;l1]eads to a pushout square y y .]
Mf ! Mf;g
EXAMPLE (The_Mapping_Telescope_) tel(X ; f) can be identified with the do*
*uble mapping cylin
` ` `
der of the 2source X2n Xn ! X2n+1. Here, the left hand arrow is def*
*ined by x2n ! x2n
n0 n0 n0
& x2n+1! f2n+1(x2n+1) and the right hand arrow is defined by x2n+1! x2n+1 & x2n*
* ! f2n(x2n).
Z? g! Y?
?
Every 2source X f Z g!Y determines a pushout square fy yj and th*
*ere
X ! P
ae
is an arrow OE : Mf;g! P characterized by the conditions j==OEOOEiO&jIZ ! Mf;*
*gOE!P =
8
< O f O p
: j Okg O p.
324
PROPOSITION 18 If f is a cofibration, then OE : Mf;g! P is a homotopy equi*
*valence
in Y \TOP .
[The arrow Mf ! IX admits a left inverse IX ! Mf.]
Application: Suppose that f : X ! Y is a cofibration_then the projection C*
*f !
Y=f(X) is a homotopy equivalence.
[Note: If in addition X is contractible, then the embedding Y ! Cf is a hom*
*otopy
equivalence. Therefore in this case the projection Y ! Y=f(X) is a homotopy equ*
*ivalence.]
EXAMPLE Let A be a nonempty finite subset of Sn(n 1)_then Sn=A has the ho*
*motopy type
of the wedge of Sn with (#(A)  1) circles.
[The inclusion A ! Snis a cofibration (cf. Proposition 8).]
( f
Consider the 2sources XX AA!!Y , where the arrow A ! X is a closed cof*
*ibration.
g Y
Assume that f ' g_then Proposition 18 implies that X tf Y and X tg Y have the s*
*ame
homotopy type relY . Corollary: If f0 : A ! Y 0is a continuous function and if *
*OE : Y ! Y 0
is a homotopy equivalence such that OE O f ' f0, then there is a homotopy equiv*
*alence
: X tf Y ! X tf0Y 0with  Y = OE.
FACT Suppose that A ! X is a closed cofibration. Let f : A ! Y be a homoto*
*py equivalence_
then the arrow X ! X tf Y is a homotopy equivalence.
Denote by ; idTOP the comma category corresponding to the diagonal funct*
*or : TOP !
TOP x TOP and the identity functor idon TOP x TOP . So, an object in ; idT*
*OP is a 2source
X f Z g*
*! Y
? ? *
* ?
X f Z g!Y and a morphism of 2sources is a commutative diagram y y *
* y . The
X0 f0 Z0 *
*!g0Y 0
double mapping cylinder is a functor ; idTOP ! TOP . It has a right adjoint T*
*OP ! ; idTOP , viz.
the functor that sends X to the 2source X p0PX p1!X.
X f Z g! Y
? ? ?
FACT Let y y y be a commutative diagram in which the verti*
*cal arrows
X0 f0 Z0 !g0Y 0
are homotopy equivalences_then the arrow Mf;g! Mf0;g0is a homotopy equivalence.
325
ae ae
Application: Suppose that A ! X are closed cofibrations. Let f : A ! *
*Y be continuous
A0! X0 f0: A0! *
*Y 0
X  A f! Y
? ? ?
functions. Assume that the diagram y y y commutes and that the v*
*ertical arrows
X0  A0 !f0Y 0
are homotopy equivalences_then the induced map X tf Y ! X0tf0Y 0is a homotopy e*
*quivalence.
ae ae
EXAMPLE Suppose that X = A [ B, where A are closed and the inclusions *
*A \ B ! A
B *
*A \ B ! B
are cofibrations. Assume: A and B are contractible_then the arrow (A \ B) ! X *
*is a homotopy
equivalence.
SEGALSTASHEFF CONSTRUCTION Let X be a topological space. Fix a covering U*
* =
{Ui : i 2 I} of X. Equip I with a well ordering < and put I[n] = {[i] (i0; : :*
*;:in) : i0 < . .<.in}.
Every strictly increasing ff 2 Mor([m]; [n]) defines a map I[n] ! I[m]. Set U[i*
*]= Ui0\ . .\.Uinand form
`
U([n]) = U[i], a coproduct in TOP . Give U([n]) x n the product topology and*
* call BU the quotient
` I[n]
U([n]) x n=~, the equivalence relation being generated by writing ((x; [i]); *
*fft) ~ ((x; ff[i]); t). Let
n `
BU(n)be the image of U([m]) x m in BU, so BU = colimBU(n). The commutative *
*diagram
mn `
U[i]x _n ! BU(n1)
I[n]
 
` u u
U[i]x n ! BU(n)
I[n]
is a pushout square in TOP and the vertical arrows are closed cofibrations. Th*
*ere is a projection pU :
BU ! X induced by the arrows U[i]x n ! U[i], i.e., ((x; [i]); t) ! x. Moreover*
*, pU is a homotopy
equivalence provided that U is numerable. Indeed, any partition of unity {i: i *
*2 I} on X subordinate
to U determines a continuous function sU : X ! BU (since 8 x; #{i 2 I : x 2 spt*
*i} < !). Obviously,
pU O sU = idXand sU O pU can be connected to the identity on BU via a linear ho*
*motopy.
ae
FACT Let X be topological spaces and let f : X ! Y be a continuous funct*
*ion. Suppose that
ae Y ae
U = {Ui: i 2 I}are numerable coverings of X such that 8 i : f(U ) V . Assu*
*me: 8 [i], the induced
V = {Vi: i 2 I} Y i i
map f[i]: U[i]! V[i]is a homotopy equivalence_then f is a homotopy equivalence.
BU F! BV
? ?
[There is an arrow F : BU ! BV and a commutative diagram pUy ypV .*
* Due to the
X !f Y
numerability of U and V, pU and pV are homotopy equivalences. Claim: 8 n, the *
*restriction F(n) :
326
BU(n)! BV(n)is a homotopy equivalence. This is clear if n = 0. For n > 0, consi*
*der the commutative
diagram
` U n ` n (n1)
[i]x  U[i]x _ ! BU
I[n] I[n]
` u ` u u
V[i]x n  V[i]x _n ! BV(n1)
I[n] I[n]
By induction, F(n1)is a homotopy equivalence, thus F(n) is too. Proposition 1*
*5 then implies that
F : BU ! BV is a homotopy equivalence, so the same is true of f.]
Let u; v : X ! Y be a pair of continuous functions_then the mapping_torus_T*
*u;vof
X q X u!!Y
u; v is defined by the pushout square i0?yi1 v ?y . There is a closed co*
*fibration
IX ! Tu;v
j : Y ! Tu;v. From the definitions, j O u ' j O v and for any g : Y ! Z with g *
*O u ' g O v,
there exists a OE : Tu;v! Z such that g = OE O j.
[Note: If u = v = idX, then Tu;vis the product X x S1.]
EXAMPLE (The_Scorpion_) Let ss : Sn! Dn be the restriction of the canonic*
*al map Rn+1 !
R n; let p : Dn ! Dn=Sn1 = Sn be the projection. Put f = p O ss_then f : Sn ! *
*Sn is inessential.
The scorpion_Sn+1 is the quotient of ISn with respect to the relations (x; 0) ~*
* (f(x); 1), i.e., Sn+1 is the
mapping torus of x ! f(x) & x ! x (x 2 Sn). One may also describe Sn+1 as the q*
*uotient Dn+1=~,
where x ~ p(2x) (x 2 (1=2)D n). Fix a point x0 2 (1=2)Sn1, let L0 be the line *
*segment from x0 to
p(2x0), and let C0 be the circle L0=~ _then the inclusion C0 ! Sn+1 is a homoto*
*py equivalence, thus
Sn+1 is a homotopy circle. The dunce_hat_Dn+1 is the quotient Sn+1=C0. It is co*
*ntractible.
The formalities in TOP *run parallel to those in TOP , thus a detailed ac*
*count of the
pointed theory is unnecessary. Of course, there is an important differenceabetw*
*eeneTOP
and TOP *: TOP * has a zero object but TOP does not. Consequently, if (X;*
*(x0)Y;ayre
*
* 0)
in TOP *, then [X; x0; Y; y0] is a pointed set with distinguished element [0],*
* the pointed
homotopy class of the zero morphism, i.e., of the constant map X ! y0. Function*
*s f 2 [0]
are said to be nullhomotopic_: f ' 0.
[Note: The forgetful functor TOP * ! TOP has a left adjoint TOP ! TOP *
* * that
sends the space X to the pointed space X+ = X q *.]
The computation of pushouts in TOP * is expedited by noting that a pushout*
* in
TOP of a 2source in TOP * is a pushout in TOP *. Examples: (1) The pusho*
*ut
327
*? ! (Y;?y0)
square y y defines the wedge_ X _ Y ; (2) The pushout s*
*quare
(X; x0) ! X _ Y
X _?Y ! *
y ?y defines the smash_product_X#Y .
X x Y ! X#Y
[Note: Base points are suppressed if there is no need to display them.]
ae
The wedge is the coproduct in TOP *. If both of the inclusions {x0} ! X a*
*re cofibrations and if
{y0} ! Y
at least one is closed, then the embedding X _ Y ! X x Y is a cofibration (cf. *
*Proposition 7) and X _ Y
is wellpointed (cf. Proposition 9).
ae
FACT Suppose that (X; x0)are in TOP *_then 8 n > 1, there is a split sho*
*rt exact sequence
(Y; y0)
0 ! ssn+1(X x Y; X _ Y ) ! ssn(X _ Y ) ! ssn(X x Y ) ! 0:
Griffithsy proved that if (X; x0) is a path connected pointed Hausdorff spa*
*ce which is both first
countable and locally simply connected at x0, then for any path connected point*
*ed Hausdorff space (Y; y0),
the arrow ss1(X; x0) * ss1(Y; y0) ! ss1((X; x0) _ (Y; y0)) is an isomorphism.
[Note: X is locally_simply_connected_at x0 provided that for any neighborho*
*od U of x0 there exists
a neighborhood V U of x0 such that the induced homomorphism ss1(V; x0) ! ss1(U*
*; x0) is trivial.]
Edazhas constructed an example of a path connected CRH space X which is loc*
*ally simply connected
at x0 with the property that ss1(X; x0) = 1 but ss1((X; x0) _ (X; x0)) 6= 1. Mo*
*ral: The hypothesis of first
countability cannot be dropped.
EXAMPLE (The_Hawaiian_Earring_) Let X be the subspace of R 2consisting of*
* the union of
the circles Xn, where Xn has center (1=n; 0) and radius 1=n (n 1). Take x0 = (*
*0; 0)_then X is first
countable at x0, X is not locally simply connected at x0, the inclusion {x0} ! *
*X is not a cofibration,
and the arrow ss1(X; x0) * ss1(X; x0) ! ss1((X; x0) _ (X; x0)) is injective but*
* not surjective. Denote now
by X0 the result of assigning to X the final topology determined by the inclusi*
*ons Xn ! X. X0 is a CW
complex. Take x0 = (0; 0)_then X0 is not first countable at x0, X0 is locally s*
*imply connected at x0, the
_________________________
yQuart. J. Math. 5 (1954), 175190.
zProc. Amer. Math. Soc. 109 (1990), 237241; see also MorganMorrison, Proc.*
* London Math. Soc.
53 (1986), 562576.
328
inclusion {x0} ! X0 is a cofibration, and the arrow ss1(X0; x0) * ss1(X0; x0) !*
* ss1((X0; x0) _ (X0; x0)) is
an isomorphism (Van Kampen).
FACT Given a wellpointed space (X; x0),asupposeethataXe= A [ B, where x0 2*
* A \ BaandeA \ B
is contractible. Assume: The inclusions A \ B ! A& A ! X are cofibrations. *
*Take a0 = x0_
A \ B ! B B ! X *
* b0 = x0
then the arrow A _ B ! X is a pointed homotopy equivalence.
The smash product # is a functor TOP * x TOP * ! TOP *. It respects homoto*
*pies, thus the
pointed homotopyatypeeof X#Y depends only on the pointed homotopy types of X an*
*d Y . If both of the
inclusions {x0} ! X are cofibrations and if at least one is closed, then X#Y *
*is wellpointed.
{y0} ! Y
[Note: Suppose that Y is a pointed LCH space_then it is clear that the func*
*tor _#Y : TOP *!
TOP *has a right adjoint Z ! ZY which passes to HTOP *: [X#Y; Z] [X; ZY ]; Z*
*Y the set of pointed
continuous functions from Y to Z equipped with the compact open topology. One c*
*an say more: In fact,
Cagliariy has shown that for any pointed Y , the functor _#Y has a right adjoin*
*t in TOP *iff the functor
_ xY has a right adjoint in TOP , i.e., iff Y is core compact (cf. p. 22).]
(#1) X#Y is homeomorphic to Y #X.
(#2) (X#Y )#Z is homeomorphic to X#(Y #Z) if both X and Z are LCH spac*
*es or if two of
X; Y; Z are compact Hausdorff.
[Note: The smash product need not be associative (consider (Q #Q )#Z and Q#*
*(Q #Z)).]
(#3) (X _ Y )#Z is homeomorphic to (X#Z) _ (Y #Z).
(#4) (X * Y ) is homeomorphic to X#Y if X and Y are compact Hausdorff.
[Note: The suspension can be viewed as a functor TOP ! TOP *. This is beca*
*use the suspension
is the result of collapsing to a point the embedded image of a space in its con*
*e. Example: Sm1 * Sn1=
Sm+n1 ) Sm#Sn = Sm+n.]
All the homeomorphisms figuring in the foregoing are natural and preserve t*
*he base points.
LEMMA The smash product of two pointed Hausdorff spaces is Hausdorff.
X _ Y ! *
? ?
The pushout square y y defines the smash_product_X#kY in CG*
* , CG , or
X xk Y ! X#kY
CGH . It is associative and distributes over the wedge.
[Note: With #k as the multiplication and S0 as the unit, CG *, CG *, and *
*CGH *are closed
categories.]
_________________________
yProc. Amer. Math. Soc. 124 (1996), 12651269.
329
The pointed_cylinder_functor_I : TOP * ! TOP * is the functor that sends *
*(X; x0)
to the quotient X x [0; 1]={x0} x [0; 1], i.e., I(X; x0) = IX=I{x0}. Variant: *
* Let I+ =
[0; 1] q *_then I(X; x0) is the smash product X#I+ . The pointed_path_space_fu*
*nctor_
P : TOP * ! TOP * is the functor that sends (X; x0) to C([0; 1]; X) (compact*
* open
topology), the base point for the latter being the constant path [0; 1] ! x0. *
*As in the
unpointed situation, (I; P ) is an adjoint pair.
Using I and P , one can define the notion of pointed cofibration. Since all*
* maps and
homotopies must respect the base points, an arrow A ! X in TOP * may be a poin*
*ted
cofibration without being a cofibration. For example, 8 x0 2 X, the arrow ({x0}*
*; x0) !
(X; x0) is a pointed cofibration but in general the inclusion {x0} ! X is not a*
* cofibration.
On the other hand, an arrow A ! X in TOP *which is a cofibration, when conside*
*red as an
arrow in TOP , is necessarily a pointed cofibration. Pointed cofibrations are *
*embeddings.
If x0 2 A X and if {x0} is closed in X, then the inclusion A ! X is a pointed *
*cofibration
iff i0X [ IA=I{x0} is a retract of I(X; x0). Observe that for this it is not ne*
*cessary that
A itself be closed.
Let (X; A; x0) be a pointed pair_then a Strom_structure_on (X; A; x0) consi*
*sts of
a continuous function OE : X ! [0; 1] such that A OE1(0), a continuous functi*
*on :
X ! [0; 1] such that {x0} = 1 (0), and a homotopy : IX ! X of idX relA such *
*that
(x; t) 2 A whenever min{t; (x)} > OE(x).
[Note: is therefore a pointed homotopy.]
POINTED COFIBRATION CHARACTERIZATION THEOREM Let x0 2 A X
and suppose that {x0} is a zero set in X_then the inclusion A ! X is a pointed *
*cofibration
iff the pointed pair (X; A; x0) admits a Strom structure.
[Necessity: Fix 2 C(X; [0; 1]) : {x0} = 1 (0) and let X p IX q![0; 1] *
*be the
projections. Put Y = {(x; t) 2 i0X [ IA : t (x)}. Define a continuous func*
*tion
f : i0X [ IA ! Y by f(x; t) = (x; min{t; (x)}) and let F : IX ! Y be some cont*
*inuous
extension of f. Consider OE(x) = sup  min{t; (x)}  qF (x; t), (x; t) = pF *
*(x; t).
0t1
Sufficiency: Given a Strom structure (OE; ; ) on (X; A; x0), define a retr*
*action r :
I(X; x0) ! i0X [ IA=I{x0} by
ae
r(x; t) = ((x;(t);(0)x; t); t  O(tE(x)(xOE(x)))=((x))t:(x)]>*
* OE(x))
ae
LEMMA Let (X; A; x0) be a pointed pair. Suppose that the inclusions {x0}*
*{!xA
*
* 0} ! X
are closed cofibrations and that the inclusion A ! X is a pointed cofibration_t*
*hen the
pair (X; x0) has a Strom structure (f; F ) for which F (IA) A.
330
[Fix a Strom structure (fX ; FX ) on (X; x0). Choose a Strom structure (OE*
*; ; ) on
(X; A; x0) such that OE = fX . Fix a Strom structure (fA ; FA ) on (A; x0).*
* Extend
__
the pointed homotopy i O FA : IA ! A i!X to a pointed homotopy F : IX ! X with
__
F O i0 = idX. Put
__ ae(1  OE(x)= (x))fA ((x; 1)) + OE(x)(OE(x) < (x))
f(x) = (x) (OE(x) = (x)):
__ __ __1 *
*__
Then f 2 C(X; [0; 1]); fA = fA , and f (0) = {x0}. Consider f(x) = min {1; *
*f(x) +
__
fX (F (x; 1))}, ae__ __ __
F (x; t) = F(x;Ft=f(x))_ (t_< f(x))_:]
X (F (x; 1); t (f(x))t f(x))
ae PROPOSITION 19 Let (X; A; x0) be a pointed pair. Suppose that the inclusi*
*ons
{x0} ! A
{x0} ! X are closed cofibrations_then the inclusion A ! X is a cofibration *
*iff it is a
pointed cofibration.
[To establish_the_nontrivial assertion, take (f; F ) as in the lemma and ch*
*oose a Strom
__ __ __
structure (OE; ; _)_on (X;_A; x0) with_OE = f. Define a Strom structure (O*
*E; ) on
__
(X; A) by OE(x) = OE(x)  (x) + sup ( (x; t)),
0t1
__ __ __
(x; t) = F ( (x; t); min{t; OE(x)= (x)}) (x 6= x0)
and (x0; t) = x0.]
So, under conditions commonly occurring in practice, the pointed and unpoin*
*ted
notions of cofibration are equivalent.
Let X f Z g!Y be a pointed 2source_then there is an embedding M*;*! Mf;gand
the quotient Mf;g=M*;*is the pointed double mapping cylinder of f; g. Here, M*;*
**is the
double mapping cylinder of the 2source * * ! *, which, being * x [0; 1], is *
*contractible.
Thus if X, Y , and Z are wellpointed, then Mf;g=M*;*is wellpointed and the proj*
*ection
Mf;g! Mf;g=M*;*is a homotopy equivalence (cf. p. 324).
[Note: The pointed mapping torus of a pair u; v : X ! Y of pointed contin*
*uous
functions is Tu;v=T*;*, where T*;*is * x S1, which is not contractible.]
Iz0  z0q z0 ! x0q y0
? ? ?
The commutative diagram y y y leads to an induced ma*
*p of pushouts
IZ i0iZ q Z ! X q Y
1 fqg
331
ae
Iz0 ! Mf;gwhich we claim is a cofibration. Thus, since X are wellpointed, the*
* arrow x0qy0 ! X qY
Y
is a cofibration. On the other hand, the pushout of the 2source Iz0 z0q z0 !*
* Z q Z can be identified
with i0Z [ Iz0[ i1Z (even though z0 is not assumed to be closed) and the inclus*
*ion i0Z [ Iz0[ i1Z ! IZ
is a cofibration (cf. p. 36). The claim is then seen to be a consequence of th*
*e proof of Proposition 4 in x12
(which depends only on the fact that cofibrations are pushout stable (cf. Propo*
*sition 2)). Consideration of
Iz0 ! *
? ?
the pushout square y y now implies that Mf;g=M*;*is wellpointed*
*. Finally, one can
Mf;g ! Mf;g=M*;*
view Mf;gitself as a wellpointed space (take [z0; 1=2] as the base point). The *
*projection Mf;g! Mf;g=M*;*
is therefore a homotopy equivalence between wellpointed spaces, hence is actual*
*ly a pointed homotopy
equivalence (cf. p. 319).
In particular: There are pointed versions X and X of the cone and suspensio*
*n of
a pointed space X. Each is a quotient of its unpointed counterpart (and has th*
*e same
homotopy type if X is wellpointed). X is a cogroup object in HTOP *. In terms*
* of the
smash product, X = X#[0; 1] (0 the base point of [0; 1]) and X = X#S 1((1; 0) t*
*he
base point of S1). Example: (X _ Y ) = X _ Y and (X _ Y ) = X _ Y . The
mapping_space_functor_ : TOP *! TOP * is the functor that sends (X; x0) to th*
*e sub
space of C([0; 1]; X) consisting of those oe such that oe(0) = x0 and the loop_*
*space_functor_
: TOP * ! TOP * is the functor that sends (X; x0) to the subspace of C([0; 1*
*]; X)
consisting of those oe such that oe(0) = x0 = oe(1), the base point in either c*
*ase being the
constant path [0; 1] ! x0. X is a group object in HTOP *. (; ) and (; ) are a*
*djoint
pairs. Both drop to HTOP *: [X; Y ] [X; Y ] and [X; Y ] [X; Y ].
[Note: If X is wellpointed, then so are X and X.]
X *
*! X
? *
* ?
The mapping space X is contractible and there is a pullback square y *
* yp1 in TOP ,
{x0} *
*! X
hence in TOP *.
EXAMPLE (The_Moore_Loop_Space_) Given a pointed space (X; x0), let M X be *
*the set of all
pairs (oe; roe) : oe 2 C([0; roe]; X) (0 roe< 1) and oe(0) = x0 = oe(roe). Att*
*ach to each (oe; roe) 2 M X
the function __oe(t) = oe(min{t; roe}) on R 0 _then the assignment (oe; roe) ! *
*(__oe; roe) injects M X into
C(R 0 ; X) x R0 . Equip M X with the induced topology from the product (compact*
* openatopologyeon
C(R 0 ; X)). Define an associative multiplication on M X by writing (o+oe)(t) =*
* oe(t) (0 t roe) ,
*
* o(t  roe)(roe t ro+oe)
where ro+oe= ro + roe, the unit thus being (0; 0) (0 ! x0). Since "+" is contin*
*uous, M X is a monoid in
332
TOP , the Moore_loop_space_of X, and M is a functor TOP *! MON TOP. The inclu*
*sion X ! M X
is an embedding (but it is not a pointed map).
Claim: X is a deformation retract of M X.
[Consider the homotopy H : IM X ! M X defined as follows. The domain of H((*
*oe; roe); t) is the
interval [0; (1  t)roe+ t] and there
i Tr j
H((oe; roe); t)(T) = oe _____oe___(1  t)r
oe+ t
if roe> 0, otherwise H((0; 0); t)(T) = x0.]
One can also introduce M X, the Moore_mapping_space_of X. Like X, M X is co*
*ntractible and
evaluation at the free end defines a Hurewicz fibration M X ! X whose fiber ove*
*r the base point is
M X.
Let f : X ! Y be a pointed continuous function, Cf its pointed mapping cone.
LEMMA If f is a pointed cofibration, then the projection Cf ! Y=f(X) is a *
*pointed
homotopy equivalence.
In general, there is a pointed cofibration j : Y ! Cf and an arrow Cf ! X. *
*Iterate
CfA____wCj
AC
to get a pointed cofibration j0 : Cf ! Cj_then the triangle u commutes*
* and
X
by the lemma, the vertical arrow is a pointed homotopy equivalence. Iterate aga*
*in to get
CjA___wCj0
AC
a pointed cofibration j00: Cj ! Cj0_then the triangle ucommutes and by*
* the
Y
lemma, theaverticalearrow is a pointed homotopy equivalence. Example: Given po*
*inted
X __
spaces , let X# Y be the pointed mapping cone of the inclusion f : X _ Y ! *
*X x Y _
Y
then in HTOP *, Cj (X _ Y ) and Cj0 (X x Y ).
Let f : X ! Y be a pointed continuous function_then the pointed_mapping_co*
*ne_
sequence_associated with f is given by X f!Y ! Cf ! X ! Y ! Cf ! 2X ! . ...
Example: When f = 0, this sequence becomes X 0!Y ! Y _ X ! X ! Y !
Y _ 2X ! 2X ! . ...
X? f! Y?
[Note: If the diagram y y commutes in HTOP *and if the vertical *
*arrows
X0 !f0Y 0
333
are pointed homotopy equivalences, then the pointed mapping cone sequences of f*
* and
f0 are connected by a commutative ladder in HTOP *, all of whose vertical arr*
*ows are
pointed homotopy equivalences.]
REPLICATION THEOREM Let f : X ! Y be a pointed continuous function_then
for any pointed space Z, there is an exact sequence
. .!.[Y; Z] ! [X; Z] ! [Cf; Z] ! [Y; Z] ! [X; Z]
in SET *.
[Note: A sequence of pointed sets and pointed functions (X; x0) OE!(Y; y0) *
*! (Z; z0) is
said to be exact_in SET * if the range of OE is equal to the kernel of .]
EXAMPLE Let f : X ! Y be a pointed continuous function, Z a pointedaspace.*
*eGiven pointed
continuous functions ff : X ! Z, OE : Cf ! Z, write (ff . OE)[x; t] = ff(x; 2*
*t) (0 t 1=2)(x 2 X)
OE(x; 2*
*t (1)1=2 t 1)
& (ff . OE)(y) = OE(y) (y 2 Y )_then this prescription defines a left action of*
* [X; Z] on [Cf; Z] and the
orbits are the fibers of the arrow [Cf; Z] ! [Y; Z].
FACT Given a pointed continuous function f : X ! Y and a pointed space Z, *
*put fZ = f#idZ_
then there is a commutative ladder
X#Z? ! Y #Z? ! CfZ? ! (X#Z)? ! (Y?#Z) ! . . .
iyd iyd y y y
X#Z ! Y #Z ! Cf#Z ! X#Z ! Y #Z ! . . .
in HTOP *, all of whose vertical arrows are pointed homotopy equivalences.(
OE : *
*Cf#Z ! CfZ
[Show that there are mutually inverse pointed homotopy equivalences *
* for which
: C*
*fZ ! Cf#Z
the triangles
NPCf#Z NP Cf#Zu
Y #Z N OE Y #Z N 
u 
CfZ CfZ
commute.]
_
Given a pointed space (X; x0), let X be the mapping cylinder of the inclusi*
*on {x0} !
_ _ _
X and denote by x_0the image of x0 under the embedding i : {x0} ! X _then (X ; *
*x0)
_ *
* _
is wellpointed and {x_0} is closed in X (cf. p. 321). The embedding j : X *
*! X is a
*
* _
closed cofibration (cf. p. 321). It is not a pointed map but the retraction r *
*: X ! X is
334
both a pointed map and a homotopy equivalence. We shall term (X; x0) nondegener*
*ate_if
_
r : X ! X is a pointed homotopy equivalence.
_
[Note: Consider X _ [0; 1], where x0 = 0_then X is homeomorphic to X _ [0; *
*1] with
x_0$ 1.]
ae ae
FACT Suppose that (X; x0)are nondegenerate. Assume: X are numerably c*
*ontractible_
(Y; y0) Y
then X _ Y and X#Y are numerablyacontractible.e ae
[To discuss X#Y , take (X; x0)wellpointed with {x0} X closed. The ma*
*pping cone of
(Y; y0) {y0} Y
the inclusion X _ Y ! X x Y is numerably contractible (cf. p. 322) and has th*
*e homotopy type of
X x Y=X _ Y = X#Y , which is therefore numerably contractible.]
ae
FACT Suppose that (X; x0)are nondegenerate. Let f 2 C(X; x0; Y; y0)_the*
*n the pointed
(Y; y0)
mapping cone Cf is numerably contractible provided that Y is numerably contract*
*ible.
X _ [0; 1]f_id!Y _ [0; 1]
? ?
[Consider the commutative diagram y y . By hypothesis,*
* the vertical
X !f Y
arrows are pointed homotopy equivalences, so Cf_idand Cf have the same pointed *
*homotopy type. Look
at the unpointed mapping cone of f _ id.]
Application: The pointed suspension of any nondegenerate space is numerably*
* contractible.
A pointed space (X; x0) is said to satisfy Puppe's_condition_provided that *
*there exists
a halo U of {x0} in X and a homotopy : IU ! X of the inclusion U ! X rel{x0} s*
*uch
that O i1(U) = {x0}. Every wellpointed space satisfies Puppe's condition.
LEMMA Let (X; A; x0) be a pointed pair. Suppose that there exists a point*
*ed
homotopy H : IX ! X of idX such that H Oi1(A) = {x0} and H Oit(A) A (0 t 1)_
then the projection X ! X=A is a pointed homotopy equivalence.
PROPOSITION 20 Let (X; x0) be a pointed space_then (X; x0) is nondegenerate
iff it satisfies Puppe's condition.
_
[Necessity: Let ae : X ! X be a pointed homotopy inverse for r. Fix a ho*
*motopy
H : IX ! X of idX rel{x0} such that H O i1 = r O ae. Put U = ae1({x0}x]0; 1])_*
*then
_ _
U is a halo of {x0} in X with haloing function ss the composite X ae!X! X =X = *
*[0; 1].
Consider = HIU.
335
Sufficiency: One can assume that U is closed (cf. p. 311). Set
ae _
0(x; t) = (x; 2t) (2 X X) _ (0 t 1=2) (x 2 U):
2t  1 (2 [0; 1] X)(1=2 t 1)
_ _
Define a pointed homotopy H : IX ! X by
ae
(H O itX)(x) = x0(x; tss(x)(x)62(U)x 2 U)
and ae
(H O it[0; 1])(T ) = T1  (1  T )(2  2(0t)t(11=2)=2: t 1)
_ _
The lemma implies that r : X ! X=[0; 1] = X is a pointed homotopy equivalence.]
EXAMPLE Take X = [0; 1] ( > !) and let x0 = 0 , the "origin" in X_then (X;*
* x0) is not
wellpointed (cf. p. 38) but is nondegenerate.
FACT A pointed space (X; x0) is nondegenerate iff it has the same pointed *
*homotopy type as
_ _
(X ; x0).
Application: Nondegeneracy is a pointed homotopy type invariant.
[Note: Compare this with the remark on p. 317.]
ae
FACT Suppose that (X; x0)are nondegenerate. Let f 2 C(X; x0; Y; y0)_then*
* f is a homotopy
(Y; y0)
equivalence in TOP iff f is a homotopy equivalence in TOP *.
EXAMPLE (The_Moore_Loop_Space_) Suppose that the pointed space X is nondeg*
*enerate_then
X and M X are nondegenerate. Since the retraction of M X onto X is not only a h*
*omotopy equiva
lence in TOP but a pointed map as well, it follows that X and M X have the sam*
*e pointed homotopy
type.
PROPOSITION 21 Let (X; x0) be a pointed space_then (X; x0) is wellpointed *
*and
{x0} is closed in X iff (X; x0) is nondegenerate and {x0} is a zero set in X.
[This is a consequence of Propositions 10 and 20.]
As noted above, nondegeneracy is a pointed homotopy type invariant. It is *
*also a
relatively stable property: X nondegenerate ) X; X; X; X nondegenerate and X; Y
nondegenerate ) X x Y; X _ Y; X#Y nondegenerate.
336
ae _ _
_ _ {x0} ! X
To illustrate, consider X#Y . In HTOP *, X#Y X#Y, and since _ _ ar*
*e closed cofi
_ _ {y0} ! Y
brations, X#Y is wellpointed (cf. p. 328), hence a fortiori, nondegenerate. Th*
*us the same is true of
X#Y .
Given pointed spaces (X1; x1); : :;:(Xn; xn), write X1 . .X.nfor the subspa*
*ce
({x1} x X2 x . .x.Xn) [ . .[.(X1 x . .x.Xn1 x {xn})
of X1 x . .x.Xn and let X1# . .#.Xn be the quotient X1 x . .x.Xn=X1 . .X.n.
PROPOSITION 22 Let X; Y; Z be nondegenerate_then (X#Y )#Z and X#(Y #Z)
have the same pointed homotopy type.
[There is a pointed 2source (X#Y )#Z X#Y #Z ! X#(Y #Z) arising from
the identity. Both arrows are continuous bijections and it will be enough to s*
*how that
they are pointed homotopy equivalences. For this purpose, consider instead the*
* pointed
_ _ _ _ _ _ _ _ _
2source (X #Y )#Z X #Y #Z ! X #(Y #Z ) and, to be specific,aworkeon_the_left*
*,acalle
ing the arrow OE. Define pointed continuous functions uv::XY!_X!bY_y (uX)(*
*x)(=vxY&)(y) = y
(u[0; 1])(t) = max {0; 2t  1} _ *
*_ _
(v[0; 1])(t) = max {0; 2t_t1}hen uxvxidZ induces a pointed functionae : (X #*
*Y )#Z !
_ _ _ A _ _
X #Y #Z . To check that is continuous, introduce closed subspaces B of X#Y*
* : Points
_ _
of A are represented by pairs (x; y), where xae1=2 (y 2 Y)aorey 1=2 (x 2 X), a*
*nd points
of B are represented by pairs (x; y), where xy22XY or xy 1=2 (y12=Y2)(xo2rX*
*)x 1=2
ae
_ _ _ _ _ _ *
* AZ
& y 1=2. Since the projection (X #Y ) x Z ! (X #Y )#Z is closed, the images *
* B
ae _ *
* Z
_ _ _ _ _ _
of ABxxZZ_in (X #Y )#Z are closed and their union fills out (X #Y )#Z . The *
*continu
ity of is a consequence of the continuity of AZ and BZ (BZ is homeomor*
*phic
_ _ _ _ _ _ _ _ _
to B xaZ=Bex {z0} and B x Z is closed in both (X #Y ) x Z and X x Y x Z). To s*
*ee
that OEare mutually inverse pointed homotopy equivalences, define pointed hom*
*otopies
ae _ _ ae ae ae oe
H : IX_ ! X_by (H O itX)(x) = x& (H O it[0; 1])(T=)max 0; 2T__t_. H a*
*nd
G : IY ! Y (G O itY )(y) = y (G O it[0;_1])(T_)_ 2  t
G combine with idZto define a pointed homotopy on X xY xZ which (i) induces a p*
*ointed
_ _ _
homotopy on X#Y #Z between the identity and O OE and (ii) induces a pointed h*
*omotopy
_ _ _
on (X #Y )#Z between the identity and OE O .]
Application: If X and Y are nondegenerate, then in HTOP *, (X#Y ) X#Y
X#Y .
337
[Note: Nondegeneracy is not actually necessary for the truth of this conclu*
*sion (cf. p.
333).]
Within the class of nondegenerate spaces, associativity of the smash produc*
*t is natural, i.e., if f :
X ! X0, g : Y ! Y 0, h : Z ! Z0are pointed continuous functions, then the diagr*
*am
(X#Y )#Z ! X#(Y #Z)
?y ??
(f#g)#h yf#(g#h)
(X0#Y 0)#Z0 ! X0#(Y 0#Z0)
commutes in HTOP *.
[Note: The horizontal arrows are the pointed homotopy equivalences figuring*
* in the proof of Propo
sition 22.]
__ PROPOSITION 23 Suppose that X and Y are nondegenerate_then the projection
X# Y ! X#Y is a pointed homotopy equivalence.
____ _ _
X# Y ! X #Y
[Consider the commutative diagram ?y ?y . The upper horizontal*
* ar
__
X# Y ! X#Y
row and the two vertical arrows are pointed homotopy equivalences, thus so is t*
*he lower
horizontal arrow.]
ae
Given pointed spaces XY, the pointed mapping cone sequence associated wit*
*h the
__
inclusion f : X _Y ! X xY reads: X _Y !fX xY ! X# Y ! (X _Y ) ! (X xY ) !
. ...
__
LEMMA The arrow F : X# Y ! (X _ Y ) is nullhomotopic.
__
[There is a pointed injection X# Y ! (X x Y ). Itaisecontinuous (but not ne*
*cessarily
an embedding). Write (X _ Y ) = X _ Y to realize F : FF[x;[y0;xt] = [x; t] 2&X
ae__ 0; y; t] = [y;*
* t] 2 Y
X = X={[x; t] : x 2 X; t 1=2}
F [x; y; 1] = *, the base point. Put _then th*
*e arrows
ae __ __Y = Y ={[y; t] : y 2 Y; t 1=2}
X ! X
are pointed homotopy equivalences, hence the same holds for their*
* wedge:
Y ! __Y ae
__ [x; t] (t 1=2)
X _ Y ! X _ __Y . The assignment [x; y; t] ! defines a poin*
*ted
__ [y; t]_(t 1=2) __
continuous function (X xY )_!_ X ___Y . The composite X# Y ! (X xY ) ! X ___Y
__
is equal to the composite X# Y F!X _ Y ! X _ __Y . But the first composite *
*is
338
nullhomotopic. Therefore the second composite is nullhomotopic and this implie*
*s that
F ' 0.]
PUPPE FORMULA Suppose that X and Y are nondegenerate_then in HTOP *,
(X x Y ) X _ Y _ (X#Y ). __
[The third term_of the pointed mapping cone sequence of_0_: X# Y ! (X _ Y )
is (X _ Y ) _ (X# Y ), so from the lemma, CF (X _ Y ) _ (X# Y ). Using now
__ j0
X# Y A______wCj
the notation of p. 332, there is a commutative triangle FAC u in *
*which the
(X _ Y )
vertical arrow_is a pointed homotopy equivalence, thus Cj0 CF or still, (X x Y*
* )
(X _ Y ) _ (X# Y ) X _ Y _ (X#Y ) (cf. Proposition 23).]
Thanks to Proposition 22, this result can be iterated. Let X1; : :;:Xn be n*
*ondegener
W
ate_then (X1x. .x.Xn) has the same pointed homotopy type as ( # Xi), where N
N i2N W
runs over the nonempty subsets of {1; : :;:n}. Example: (S k1x . .x.Skn) SN *
*; SN
P N
a sphere of dimension 1 + ki.
i2N
ae
EXAMPLE (Whitehead_Products_) Let X be nondegenerate_then for any point*
*ed space E,
Y
there is a short exact sequence of groups
0 ! [(X#Y ); E] ! [(X x Y ); E] ! [(X _ Y ); E] ! 0:
Here, composition is written additively even though the groupsainvolvedemay not*
* be abelian. This data
generates a pairing [X; E] x [Y; E] ! [(X#Y ); E]. Take ff 2 [X; E]and use t*
*he embeddings
ae fi 2 [Y; E]
[X; E] ! [(X xY ); E] to form the commutator ff+fi fffi in [(X xY ); E]. Be*
*cause it lies in the
[Y; E]
kernel of the homomorphism [(X x Y ); E] ! [(X _ Y ); E], by exactness there ex*
*ists a unique element
[ff; fi] 2 [(X#Y ); E] with image ff + fi  ff  fi. [ff; fi] is called the Whi*
*tehead_product_of ff; fi. [ff; fi] and
[fi; ff] are connected by the relation [ff; fi] + [fi; ff] O > = 0, where > : X*
*#Y ! Y #X is the interchange.
Of course, [ff; 0] = [0; fi] = 0. In general, [ff; fi] = 0 if E is an H space (*
*since then [(X xY ); E] is abelian),
hence, always [ff; fi] = 0 (look at the arrow E ! E). There are left actions
ae ae
[X; E] x [(X#Y ); E] ! [(X#Y ); E]: (ff; ) ! ff . = ff +(abffuse of n*
*otation).
[Y; E] x [(X#Y ); E] ! [(X#Y ); E] (fi; ) ! fi . = fi +  fi
ae 0 0 *
* ae
One has [ff + ff ; fi] = ff . [ff.; fi]T+h[ff;efi]se relations simplify if th*
*e cogroup objects X are
[ff; fi + fi0] = [ff; fi] + fi . [ff;afi0]e ae *
* Y
0 X0
commutative (as would be the case, e.g., when X = X for nondegenerate )*
*. Indeed, under this
Y = Y 0 Y 0
339
ae 0 0
assumption, [(X#Y ); E] is abelian. Therefore the ff . [ff ; fi]mu[ffs;tfi]v*
*anish ("being commuta
ae fi . [ff; fi0]  [ff; fi0]
0; fi] = [ff; fi] + [ff0; fi]
tors"), implying that [ff + ff . The Whitehead product also satis*
*fies a form of the
[ff; fi + fi0] = [ff; fi] + [ff; fi0]
Jacobi identity. Precisely: Suppose given nondegenerate X; Y; Z whose associate*
*d cogroup objects X,
Y , Z are commutative_then
[[ff; fi]; fl] + [[fi; fl]; ff] O oe + [[fl; ff]; fi] O *
*o = 0
ae
in the group [(X#Y #Z); E], where oe : X#Y #Z ! Y #Z#X(cf. Proposition 22). T*
*he verification
o : X#Y #Z ! Z#X#Y
is a matter of manipulating commutator identities.]
*
* L
A graded_Lie_algebra_over a commutative ring R with unit is a graded Rmodu*
*le L = Ln together
*
* n0
with bilinear pairings [ ; ] : Ln x Lm ! Ln+m such that [x; y] = (1)xy+1[y*
*; x] and
(1)xz[[x; y]; z] + (1)yx[[y; z]; x] + (1)zy[[z; *
*x]; y] = 0:
L
L is said to be connected_if L0 = 0. Example: Let A = An be a graded Ralgeb*
*ra. For x 2 An,
n0
y 2 Am , put [x; y] = xy  (1)xyyx_then with this definition of the bracke*
*t, A is a graded Lie algebra
over R.
[Note: As usual, an absolute value sign stands for the degree of a homogeno*
*us element in a graded
Rmodule.]
ae
EXAMPLE Let X be a path connected topological space. Given ff 2 ssn(X), *
*the Whitehead
fi 2 ssm (X)
product [ff; fi] 2 ssn+m1 (X). One has [ff; fi] = (1)nm+n+m [fi; ff]. Moreove*
*r, if fl 2 ssr(X), then
(1)nr+m [[ff; fi]; fl] + (1)mn+r [[fi; fl]; ff] + (1)rm+n [[f*
*l; ff]; fi] = 0:
L
Assume now that X is simply connected. Consider the graded Zmodule ss*(X) = *
* ssn(X). Since
n0
ssn+1(X) = ssn(X), the Whitehead product determines a bilinear pairing [ ; ] : *
*ssn(X) x ssm (X) !
ssn+m (X) with respect to which ss*(X) acquires the structure of a connected gr*
*aded Lie algebra over
Z.
FACT Suppose that X is simply connected_then the Hurewicz homomorphism ss**
*(X) ! H*(X)
is a morphism of graded Lie algebras, i.e., preserves the brackets.
[Note: Recall that H*(X) is a graded Zalgebra (Pontryagin product), hence *
*can be regarded as a
graded Lie algebra over Z.]
A pair (X; A) is said to be nconnected_(n 1) if each path component of X *
*meets A
and ssq(X; A; x0) = 0 (1 q n) for all x0 2 A or, equivalently, if every map (*
*D q; Sq1) !
340
(X; A) is homotopic relSq1 to a map D q ! A (0 q n). If A is path connected,
then 8 x00; x0002 A, ssn(X; A; x00) ssn(X; A; x000) (n 1). Examples: (1) (D*
* n+1; Sn) is
nconnected; (2) (B n+1; Bn+1  {0}) is nconnected.
[Note: Take A = {x0}_then ssq(X; {x0}; x0) = ssq(X; x0), so X is nconnecte*
*d_(n 1)
provided that X is path connected and ssq(X) = 0 (1 q n). Example: S n+1 is *
*n
connected.]
EXAMPLE If X is nconnected and Y is mconnected, then X *Y is ((n+1)+(m+1*
*))connected.
[Note: If X is path connected and Y is nonempty but arbitrary, then X * Y i*
*s 1connected.]
ae
EXAMPLE Suppose that X are nondegenerate and X is nconnected and Y is m*
*connected_
Y
then X#Y is (n + m + 1)connected.
FACT Let f : Sn! A be a continuous function. Put X = Dn+1tfA_then (X; A) i*
*s nconnected.
EXAMPLE The pair (Sn x Sm; Sn_ Sm) is n + m  1 connected.
ae
HOMOTOPY EXCISION THEOREM Suppose that X1X are subspaces of X with
ae 2ae
X = intX1 [ intX2. Assume: (X1;(X1X\ X2) is nconnected _then the arrow
2; X2 \ X1)mconnected
ssq(X1; X1\X2) ! ssq(X1[X2; X2) induced by the inclusion (X1; X1\X2) ! (X1[X2; *
*X2)
is bijective for 1 q < n + m and surjective for q = n + m.
[This is dealt with at the end of the x.]
LEMMA Let X be a strong deformation retract of Y and let A X be a strong
deformation retract of B Y _then 8 n 1, ssn(X; A) ssn(Y; B).
[Use the exact sequence for a pair and the five lemma.]
ae
PROPOSITION 24 Let ABbe closed subspaces of X with X = A[B. Put C = A\
ae ae ae
B. Assume: The inclusions CC!!AB are cofibrations and (A;(C)B;iC)s ncon*
*nectedmconnected_
then the arrow ssq(A; C) ! ssq(X; B) is bijective for 1 q < n + m and surjecti*
*ve for
q = n + m. ae_ ae __ __
__ X 1 = i0A [ IC __ __ intX1 X  i1B
[Set X = i0A[IC[i1B; __X : X 1\X 2 = IC and __ __ )
2 = IC [ i1Bae __ intX2aeX_ i0A
__ __ __ ssq(A; C) ssq(X 1; IC) (X 1; IC)
X = intX 1 [ intX 2. From the lemma, ss __ ) __ is
ae q(B; C) ssq(X 2; IC) (X 2; IC)
nconnected *
* __ __ __
mconnected , thus theahomotopyeexcision theorem is applicable to the triple *
*(X ; X1; X2).
Because the inclusions CC!!AB are cofibrations, i0A [ IC is a strong deformat*
*ion retract
341
*
* __
of IA and IC [ i1B is a strong deformation retract of IB (cf. p. 36). Therefor*
*e X is a
__ __
strong deformation retract of IA [ IB = IX, so ssq(X ; X2) ssq(IX; IB) ssq(X;*
* B).]
LEMMA Let f : (X; A) ! (Y; B) be a homotopy equivalence in TOP 2_then 8 x*
*0 2
A and any q 1, the induced map f* : ssq(X; A; x0) ! ssq(Y; B; f(x0)) is biject*
*ive.
PROPOSITION 25 Let A be a nonempty closed subspace of X. Assume: The inclu
sion A ! X is a cofibration and A is nconnected, (X; A) is mconnected_then th*
*e arrow
ssq(X; A) ! ssq(X=A; *) is bijective for 1 q n + m and surjective for q = n +*
* m + 1.
[Denote by Ci theaunpointedemapping cone of the inclusion i : A ! X. There*
* are
closed cofibrations AX!!CiC and Ci= A[X, with A\X = A. Since the pair (A; A)
i
is (n + 1)connected, it follows from Proposition 24 that the arrow ssq(X; A) !*
* ssq(Ci; A)
is bijective for 1 q n+m and surjective for q = n+m+1. But A is contractible,*
* hence
the projection (Ci; A) ! (Ci=A; *) is a homotopy equivalence in TOP 2(cf. Prop*
*osition
14). Taking into account the lemma, it remains only to observe that X=A can be *
*identified
with Ci=A.]
FREUDENTHAL SUSPENSION THEOREM Suppose that X is nondegenerate and
nconnnected_then the suspension homomorphism ssq(X) ! ssq+1(X) is bijective for
0 q 2n and surjective for q = 2n + 1.
[Take X wellpointed with a closed base point and, for the moment, work with*
* its
unpointed suspension X. Using the notation of p. 322, write X =  X [ + X_
then 8 q; ssq(X) ssq( X \ + X) ssq+1( X;  X \ + X). On the other hand,
Proposition 25 implies that the arrow ssq+1( X;  X \ + X) ! ssq+1(X) is a bij*
*ection
for 1 q + 1 2n + 1 and a surjection for q + 1 = 2n + 2. Moreover, X is wellpo*
*inted,
therefore its pointed and unpointed suspensions have the same homotopy type.]
[Note: This result is true if X is merely path connected, i.e., n = 0 is a*
*dmissible
(inspect the proof of Proposition 25).]
Application: Suppose that n 1_then (i) ssq(S n) = 0 (0 q < n); (ii) ssq(S*
* n)
ssq+1(S n+1) (0 q 2n  2); (iii) ssn(S n) Z.
[As regards the last point, note that in the sequence ss1(S 1) ! ss2(S 2) !*
* ss3(S 3) ! . .,.
the first homomorphism is an epimorphism, the others are isomorphisms, and ss1(*
*S 1) Z,
ss2(S 2) Z (a piece of the exact sequence associated with the Hopf map S3 ! S*
* 2is
ss2(S 3) ! ss2(S 2) ! ss1(S 1) ! ss1(S 3)).]
342
The infinite cyclic group ssn(S n) is generated by [n], n the identity Sn !*
* Sn. Form
the Whitehead product [n; n] 2 ss2n1(S n)_then the kernel of the suspension ho*
*momor
phism ss2n1(S n) ! ss2n(S n+1) is generated by [n; n] (Whiteheady).
The proof of the homotopy excision theorem is elementary but complicated. T*
*his is the downside.
The upside is that the highpowered approaches are cluttered with unnecessary as*
*sumptions, hence do not
go as far.
ae
OPEN HOMOTOPY EXCISION THEOREM Suppose that X1 are open subspaces of X
ae ae X2
with X = X1 [ X2. Assume: (X1; X1\ X2)is nconnected_then the arrow ssq(X1;*
* X1 \ X2) !
(X2; X2\ X1) mconnected
ssq(X1[ X2; X2) induced by the inclusion (X1; X1\ X2) ! (X1[ X2; X2) is bijecti*
*ve for 1 q < n + m
and surjective for q = n + m.
[Note: Goodwilliez has extended the open homotopy excision theorem to "(N +*
* 1)ads".]
Admit the open homotopy excision theorem.
ae
CW HOMOTOPY EXCISION THEOREM Suppose that K1 are subcomplexes of a CW
ae K2ae
complex K with K = K1[K2. Assume: (K1; K1\ K2)is nconnected_then the arrow *
*ssq(K1; K1\
(K2; K2\ K1) mconnected
K2) ! ssq(K1[ K2; K2) induced by the inclusion (K1; K1\ K2) ! (K1[ K2; K2) is b*
*ijective for 1 q <
n + m and surjective foraqe= n + m. ae
[Fix a neighborhood U of K1 \ K2 in K1 such that K1 \ K2 is a strong de*
*formation retract
ae ae V ae K2 ae ae
0 = K1[ V U = O \ K1 O *
* K0 = P[
of U and put K1 . Write , where are open in K_then *
* 1
V K02=aK2[eU V = P \ K2 P ae *
* K02=aO[e
(K  K2), hence K01are open in K and K = K0 [ K0. Since K1 & V are closed *
*in K01, the
(K  K1) K02ae 1 2 K2a&eU *
* K02 ae
0 *
* K1
homotopy deforming V into K1\ K2 can be extended to all of K1 in the obviou*
*s way, so is
U ae K02 ae*
* K2
0 U
a strong deformation retract of K1. On the other hand, K01\ K02= U [ V and *
* is closed in U [ V ,
K02 V
thus the union of the deformingahomotopieseisacontinuouseand K1 \ K2 is a stron*
*g deformation retract
0; K0\ K0) nconnected
of K01\ K02. Therefore (K1 1 2 is and the open homotopy exci*
*sion theorem is
(K02; K02\ K01) mconnected
applicable to the triple (K; K01; K02). Consider the commutative triangle
_________________________
yElements of Homotopy Theory, Springer Verlag (1978), 549.
zMemoirs Amer. Math. Soc. 431 (1990), 1317.
343
ssq(K1; K1\fK2)l_____wssq(K01; K01\ K02)
flflffl AADA :]
ssq(K1[ K2; K2)
The CW homotopy excision theorem implies the homotopy excision theorem. Fo*
*r choose a CW
resolution L ! X1 \ X2. There exist: (1) A CW complex K1 L and a CW resolution*
* f1 : K1 ! X1
K1 ! X1
such that the square x? x? commutes; (2) A CW complex K2 L and a CW*
* resolution f2 :
L ! X1\ X2
K2  ! X2 ae ae
K2 ! X2 such that the square x? x? commutes. Note that (K1; L)is *
* nconnected.
(K2; L) *
* mconnected
L  ! X2\ X1
L ! K2
Define a CW complex K by the pushout square ?y ?y : K = K1[ K2 & L = K*
*1\ K2_then
aeK1 ! aKe
there is an arrow f : K ! X determined by f1, viz. fK1 = f1.
f2 fK2 = f2
LEMMA f is a weak homotopy equivalence.ae_ ae
[Set __K= i0K1 [ IL [ i1K2 : U1 = K_ i1K2_then U1 are open in __Kand _*
*_K= U1 [ U2.
__ U2 = K  i0K1 U2 __
Leta_p:eK_! K beatheerestriction_of the projection p : IK ! K and denote by f t*
*he composite f O _p:
f(U1) X1 fU1 __
__f(U and __ & fU1 \ U2 are weak homotopy equivalences. But by assu*
*mption X =
2) X2 fU2 *
* ae
intX1 [ intX2. Therefore __fis a weak homotopy equivalence (cf. p. 452). The i*
*nclusions K1 ! K
__ *
* K2 ! K
are closed cofibrations (cf. p. 313), hence K is a strong deformation retract *
*of IK. Consequently, _pis a
homotopy equivalence, so f is a weak homotopy equivalence.]
The CW homotopy excision theorem is applicable to the triple (K; K1; K2). *
*Examination of the
commutative square
ssq(K1; K1\ K2)! ssq(K1[ K2; K2)
?y ?y
ssq(X1; X1\ X2)! ssq(X1[ X2; X2)
thus justifies the claim. Accordingly, it is the open homotopy excision theorem*
* which is the heart of the
matter.
344
Given a pdimensional cube C in Rq (q 1; 0 p q), denote by skdC its ddi*
*mensional skeleton,
i.e., the set of its ddimensional faces. Put _C= [skp1C_then the inclusion _C*
*! C is a closed cofibration.
Analytically, C is specified by a point (c1; : :;:cq) 2 Rq, a positive number f*
*fi, and a subset P of {1; : :;:q}
of cardinality p : C is the set of x 2 Rq such8that ci xi ci+ffi (i 2 P) & xi= *
*ci(i 62 P). Here, if P = ;,
< Kd(C) = {x 2 C : xi< ci+ ffi_for a*
*tdleastindicesi 2 P}
then C = {(c1; : :;:cq)}. For 1 d q, let 2 *
* .
: Ld(C) = {x 2 C : xi> ci+ ffi_for a*
*tdleastindicesi 2 P}
ae 2
When d > p, it is understood that Kd(C) = ;.
Ld(C) = ;
COMPRESSION LEMMA Fix a pdimensional cube C in Rq (q 1; 1 p q), a posi*
*tive
integer d p, and a pair (X; A). Suppose that f : C ! X is a continuous functio*
*n such that 8 D 2 skp1C,
f1(A) \ D Kd(D) (Ld(D))_then there exists a continuous function g : C ! X wit*
*h f ' g rel_Cand
g1(A) Kd(C) (Ld(C)).
[Take p = q, C = [0; 1]q, and put x0 = (1=4; : :;:1=4). Given an x 2 [0; 1]*
*q, let `(x0; x) be the ray that
starts at x0 and passes through x. Denote by P(x) the intersection of `(x0; x) *
*with the frontier of [0; 1=2]q,
Q(x) the intersection of `(x0; x) with the frontier of [0; 1]q. Let OE : [0; 1]*
*q ! [0; 1]q be the continuous
function that sends the line segment joining P(x) and Q(x) to the point Q(x) an*
*d maps the line segment
joining x0 and P(x) linearly onto the line segment joining x0 and Q(x). Note th*
*at OE ' id[0;1]qrel[fr0; 1]q.
Now set g = f O OE. Assume: x 2 g1(A). Case 1: xi < 1=2 (8 i) ) x 2 Kq([0; 1*
*]q) Kd([0; 1]q).
Case 2: xi 1=2 (9 i) ) OE(x) 2 fr[0; 1]q ) OE(x) 2 D (9 D 2 skq1[0; 1]q) ) OE*
*(x) 2 Kd(D) ) 1=2 >
OE(x)i = 1=4 + t(xi 1=4) for at least d indices i ) 1=2 > OE(x)i xi (t 1) fo*
*r at least d indices
i ) x 2 Kd([0; 1]q).]
[Note: The parenthetical assertion is analogous.]
Notation: Put Iq = [0; 1]q, _Iq= fr[0; 1]q, Iq10= Iq1 x {0} (q > 1) & I0*
*0= {0} (q = 1),
Jq1 = _Iq1xI [Iq1x{1} (q > 1) & J0 = {1} (q = 1), so _Iq= Iq10[Jq1 and _Iq*
*10= Iq10\Jq1_
then for any pointed pair (X; A; x0), ssq(X; A; x0) = [Iq; _Iq; Jq1; X; A; x0].
[Note: A continuous function f : (Iq; _Iq; Jq1) ! (X; A; x0) represents 0 *
*in ssq(X; A; x0) iff there
exists a continuous function g : Iq ! A such that f ' g rel_Iq.]
There are two steps in the proof of the open homotopy excision theorem: (1)*
* Surjectivity in the range
1 q n + m; (2) Injectivity in the range 1 q < n + m. The argument in either *
*situation is founded
on the same iterative principle.
Starting with surjectivity, let ff 2 ssq(X1 [ X2; X2; x0), x0 2 X1 \ X2 the*
* ambient base point.
Represent ff by an f : (Iq; _Iq; Jq1) ! (X1 [ X2; X2; x0). It will be shown b*
*elow that 9 F 2 ff :
pro(F1 (X  X1)) \ pro(F1 (X  X2)) = ;, pro: Iq ! Iq1 the projection. Grant*
*ed this, choose a
continuous function OE : Iq1 ! [0; 1] which is 1 on pro(F1 (X X1)) and 0 on *
*_Iq1[pro(F1 (X X2)).
345
Define : Iq ! Iq by (x1; : :;:xq) = (x1; : :;:xq1; t + (1  t)xq), where t = *
*OE(x1; : :;:xq1), and
put g = F O _then g : (Iq; _Iq; Jq1) ! (X1; X1 \ X2; x0) is a continuous funct*
*ion whose class fi 2
ssq(X1; X1\ X2; x0) is sent to ff under the inclusion.
There remains the task of producing F. Since {f1(X1); f1(X2)} is an open *
*covering of Iq, one
can subdivide Iq into a collection C of qdimensional cubes C such that either *
*f(C) X1 or f(C) X2.
Enumerateatheeelements in skdC (C 2 C; d = 0; 1; : :;:q) : D = {D}. In D, disti*
*nguish two subcollections
{Dk : k = 1; : :;:r} : f(Dk)buX2t f(Dk) 6 X1, arranging the indexing so that *
*dimD dimD .
{Dl : l = 1; : :;:s} : f(Dl) X1 f(Dl) 6 X2 *
* j j+1
() There exist continuous functions 0 = f, k : Iq ! X (k = 1; : :;:r)*
*asuchethat 8 k : k '
0 (as a map of triples), 1k(X2 X1\ X2) \ Dj Kn+1(Dj) (j k), and 8 D 2 D : *
* 0(D) X1 )
ae *
* 0(D) X2
k(D) X1 or (D) X \ X ) (D) X \ X . This is seen via induction on k, *
* = f being
k(D) X2 0 1 2 k 1 2 0
the initial step. Assume that k1 has been constructed.
Claim: 9 a homotopy hk : IDk ! X2rel_Dksuch that hkO i0 = k1Dk and (hkO i*
*1)1(X2 X1\
X2) Kn+1(Dk).
[Case 1: dim Dk = 0. Here, Kn+1(Dk) = ; and the point k1(Dk) 2 X2 can be *
*joined by a
path in X2 to some point of X1 \ X2. Case 2: 0 < dimDk < n + 1. Here, Kn+1(Dk*
*) = ; and the
induction hypothesis forces the containment k1(D_k) X1\ X2, hence k1Dk repr*
*esents an element
of ssdk(X2; X1\ X2) = 0 (dk = dimDk). Case 3: dimDk n + 1. Apply the compressi*
*on lemma.]
k1S
Extend hk to a homotopy Hk : Iq x I ! X of k1rel[ {D : f(D) X1} [ Dj s*
*uch that
r j=1
S
Hk(IDj) X2. Complete the induction by taking k = Hk O i1.
j=k+1
() There exist continuous functions 0 = r, l: Iq ! X (l = 1; :a:;:s)e*
*such that 8 l : l'
0rel[{D : f(D) X2}, 1l(X1X1\X2)\Dj Lm+1 (Dj) (j l), and 8 D 2 D : 0(D) *
*X1 )
ae 0(D) *
*X2
l(D) X1 or (D) X \X ) (D) X \X . As above, this is seen via induction *
*on l, =
l(D) X2 0 1 2 l 1 2 *
* 0 r
being the initial step. Observe that [{D : f(D) X2} _Iq Jq1.
Definition: F = s () F 2 ff). If pro(F1 (X  X1))a\epro(F1 (X  X2)) were*
* nonempty, then
there would exist an x 2 Iq1 and a cube D Iq1 : x 2 Kn(D), an impossibilit*
*y since q 1 < n+m.
x 2 Lm (D)
Turning to injectivity, let f; g : (Iq; _Iq; Jq1) ! (X1; X1 \ X2; x0) be c*
*ontinuous functions such
that u O f ' u O g as maps of triples, u : (X1; X1 \ X2;ax0)e! (X1 [ X2; X2; x0*
*) the inclusion. Fix a
homotopy h : (Iq; _Iq; Jq1) x I ! (X1[ X2; X2; x0) : h O i0 = u.OUfsing the *
*techniques employed in
h O i1 = u O g
the proof of surjectivity, one can replace h by another homotopy H such that pr*
*ox idI(H1(X  X1)) \
prox idI(H1(X  X2)) = ;. It is this extra dimension that accounts for the res*
*triction q < n + m.
Choose a continuous function OE : Iq1 x I ! [0; 1] which is 1 on prox idI(H1(*
*X  X1)) and 0 on
(_Iq1x I) [ (Iq1 x _I) [ prox idI(H1(X  X2)). Define : Iq x I ! Iq x I by *
*(x1; : :;:xq; xq+1) =
346
(x1; : :;:xq1; t+(1t)xq; xq+1), where t = OE(x1; : :;:xq1; xq+1)_then the co*
*mposite HO is a homotopy
between f and g : H O (_Iqx I) X1\ X2 & H O (Jq1 x I) = {x0}.
Given a pair (X; A), let ss0(X; A) be the quotient ss0(X)=~, where ~ means *
*that the path components
of X which meet A are identified. With this agreement, ss0(X; A) is a pointed s*
*et. If f : (X; A) ! (Y; B)
is a map of pairs, then f* : ss0(X; A) ! ss0(Y; B) is a morphism of pointed set*
*s and the sequence * !
ss0(X; A) ! ss0(Y; B) is exact in SET *iff (f*)1im(ss0(B) ! ss0(Y )) = im(ss0(*
*A) ! ss0(X)).
LEMMA Let f : (X; A) ! (Y; B) be a continuous function. Fix q 0_then 8 x*
*0 2 A, f* :
ssq(X; A; x0) ! ssq(Y; B; f(x0)) is injective and f* : ssq+1(X; A; x0) ! ssq+1(*
*Y; B; f(x0)) is surjective iff in
(X;xA) f! (Y;xB)
any diagram O?E ? , where f O OE ' on Jq by h : (Jq; _Iq0) x*
* I ! (Y; B), there
(Jq; _Iq0)!(Iq+1; Iq0)
exists a : (Iq+1; Iq0) ! (X; A) such that (Jq; _Iq0) = OE and an H : (Iq+1; I*
*q0) x I ! (Y; B) such that
H(Jq; _Iq0) x I = h and f O ' on Iq+1 by H.
[Note: When q = 0, replace injectivity by the statement "* ! ss0(X; A) ! ss*
*0(Y; B)" is exact.
Observe that f O OE = on Jq is permissible (h = constant homotopy) and implie*
*s by specialization the
direct assertion. In addition, if & H exist in this case, then & H exist in g*
*eneral. Thus the point is
to show that the direct assertion entails the existence of & H under the assum*
*ption that f O OE = on
Jq.]
ae ae ae ae
FACT Suppose that X1 & Y1 are open subspaces of X with X = X1[ X2*
* . Let f :
X2 Y2 ae Y Y = Y1[ Y2
1(Y1)
X ! Y be a continuous function such that X1 = f . Fix n 1. Assume: Th*
*e sequence
X2 = f1(Y2)
* ! ss0(Xi; X1\ X2) ! ss0(Yi; Y1\ Y2) is exact (i = 1; 2) and that f* : ssq(Xi;*
* X1\ X2) ! ssq(Yi; Y1\ Y2)
is bijective for 1 q < n and surjective for q = n (i = 1; 2)_then the sequence*
* * ! ss0(X; Xi) ! ss0(Y; Yi)
is exact (i = 1; 2) and f* : ssq(X; Xi) ! ssq(Y; Yi) is bijective for 1 q < n *
*and surjective for q = n
(i = 1; 2).
[Fix i0 2 {1; 2}, 0 q < n, and maps OE : (Jq; _Iq0) ! (X; Xi0), : (Iq+1;*
* Iq0) ! (Y; Yi0) satisfying
f O OE = on Jq. In view of the lemma, it suffices to exhibit an extension : *
*(Iq+1; Iq0) ! (X; Xi0)
of OE and a homotopy H : (Iq+1; Iq0) x I ! (Y; Yi0) such that H(Jq; _Iq0) x I *
*is the constant homotopy
at f O OE and f O ' on Iq+1 by H. Subdivide Iq+1 into a collection Caofe(q +*
* 1)dimensional cubes
1(X *
* X1) [ 1(Y  Y1)
C : 8 C 2 C, 9 iC 2 {1; 2} : OE(C \ Jq) XiC and (C) YiC (possible, OE
OE1(X *
* X2) [ 1(Y  Y2)
being disjoint and closed). Regard Iq+1 as Iq x I_then C restricts to a subdivi*
*sion of Iq and induces a
partition of I into subintervals Ik = [ak1; ak] : 0 = a0 < a1 < . .<.ar = 1. B*
*reak the subdivision of
Iq into its skeletal constituents D. Construct on D x Ik & H on I(D x Ik) via *
*downward induction
on k and for fixed k, via upward induction on dimD. Arrange matters so that: (1*
*) (D x Ik) Yi )
347
(D x Ik) Xi & H(I(D x Ik)) Yi; (2) (D x {ak1}) Y1 \ Y2 ) (D x {ak1}) X1 *
*\ X2
& H(I(D x {ak1})) Y1 \ Y2. The first condition plus the second when k = 1 yi*
*eld (Iq0) Xi0
& H(Iq0x I) Yi0. At each stage, the induction hypothesis secures on _Dx Ik [ *
*D x {ak} & H on
I(D_x Ik [ D x {ak}). Case 1: If either (D x {ak1}) is not contained in Y1 \*
* Y2 or (D x Ik) is
contained in Y1 \ Y2, use the fact that _Dx Ik [ D x {ak} is a strong deformati*
*on retract of D x Ik to
specify on D x Ik & H on I(D x Ik). Case 2: If (D x {ak1}) is contained in Y*
*1\ Y2 and (D x Ik)
is contained in just one of the Yi, realize : (D_x Ik [ D x {ak}; _Dx {ak1}) *
*! (Xi; X1 \ X2) &
H : (D_x Ik [ D x {ak}; _Dx {ak1}) x I ! (Yi; Y1 \ Y2): Apply the lemma to pro*
*duce the required
extension of to D x Ik & H to I(D x Ik). Here, of course, the assumption on f *
*comes in.]
41
x4. FIBRATIONS
The technology developed below, like that in the preceding x, underlies the*
* foundations
of homotopy theory in TOP or TOP *.
Let B be a topological space. An object in TOP =B is a topological space X*
* together
with a continuous function p : X ! B called the projection_. For O B, put XO =*
* p1(O),
which is therefore an object in TOP =O (with projection pO = pXO ). The notat*
*ion XO
is also used. In particular: Xb = p1(b) is the fiber_over b 2 B. A morphism in*
* TOP =B
is a continuous function f : X ! Y over B, i.e., an f 2 C(X; Y ) such that the *
*triangle
X _________wf'')Y
p [[^qcommutes. Notation: f 2 CB (X; Y ); fO = fXO (O B). The base *
*space
B
B is an object in TOP =B, where p = idB. An element s 2 CB (B; X) is called a *
*section_
of X, written s 2 secB(X).ae
[Note: The product of pq::XY!!BB in TOP =B is the fiber product: X xB Y*
* . If
B0 is a topological space and if 02 C(B0; B), then 0 determines a functor TOP *
*=B !
TOP =B0 that sends X to X0 = B0xB X. Obviously, (X xB Y )0= X0xB0 Y 0.]
EXAMPLE Let X be in TOP =B_then the assignment O ! secO(XO ), O open in B,*
* defines a
sheaf of sets on B, the sheaf_of_sections_X of X.
[Note: Recall that for any sheaf of sets F on B, there exists an X in TOP =*
*B with p : X ! B a
local homeomorphism such that F is isomorphic to X . In fact, the category of s*
*heaves of sets on B is
equivalent to the full subcategory of TOP =B whose objects are those X for whic*
*h p : X ! B is a local
homeomorphism.]
FACT Let X be in TOP =B_then the projection p : X ! B is a local homeomorp*
*hism iff both it
and the diagonal embedding X ! X xB X are open maps.
FACT Let X be in TOP =B. Assume: X & B are path connected Hausdorff space*
*s and the
projection p : X ! B is a local homeomorphism_then p is a homeomorphism iff p i*
*s proper and p* :
ss1(X) ! ss1(B) is surjective.
There is a functor TOP ! TOP =B that sends a topological space T to B x*
* T
(product topology) with projection B x T ! B. An X in TOP =B is said to be tri*
*vial_if
there exists a T in TOP such that X is homeomorphic over B to B x T , locally*
*_trivial_if
there exists an open covering {O} of B such that 8 O, XO is trivial over O.
42
[Note: Spelled out, local triviality means that 8 O there exists a topologi*
*cal space TO
and a homeomorphism XO ! O x TO over O. If the TO can be chosen independent of *
*O,
so 8 O, TO = T , then X is said to be locally_trivial_with_fiber_T. When B is c*
*onnected,
this can always be arranged.]
FACT Let X be in TOP =IB. Suppose that X(B x[0; 1=2]) and X(B x[1=2; 1])*
* are trivial_then
X is trivial.
EXAMPLE Let X be in TOP =[0; 1]n (n 1). Suppose that X is locally trivial*
*_then X is trivial.
A fiber_homotopy_is a homotopy over B : f 'Bg (f; g 2 CB (X; Y )). Isomorph*
*isms in
*
* ae
the associated homotopy category are the fiber homotopy equivalences and any tw*
*o XY
in TOP =B for which there exists a fiber homotopy equivalence X ! Y have the s*
*ame fiber
homotopy type. The fiber homotopy type of X xB Y depends only on the fiber homo*
*topy
types of X and Y . The objects in TOP =B that have the fiber homotopy type of *
*B itself
are said to be fiberwise_contractible_. Example: The path space P B with projec*
*tion p0 is
in TOP =B and is fiberwise contractible (consider the fiber homotopy H : IP B *
*! P B
defined by H(oe; t)(T ) = oe(tT )).
[Note: A fiber homotopy with domain IB is called a vertical_homotopy_.]
LEMMA Let X be in TOP =B. Assume: X is fiberwise contractible_then for any
02 C(B0; B), X0 is fiberwise contractible.
Let f : X ! Y be a continuous function. View its mapping cylinder Mf as an *
*object in TOP =Y
with projection r : Mf ! Y _then j 2 secY(Mf) and Mf is fiberwise contractible.
Let X; Y be in TOP =B_then a fiber preserving function f : X ! Y is said*
* to be
fiberwise_constant_if f = t O p for some section t : B ! Y . Elements of CB (X;*
* Y ) that are
fiber homotopic to a fiberwise constant function are fiberwise_inessential_.
Suppose that B is not in CG _then the identity map kB ! B is continuous and*
* constant on fibers
but not fiberwise constant.
LEMMA Let X be in TOP =B_then X is fiberwise contractible iff idX is fibe*
*rwise
inessential.
EXAMPLE Take X = ([0; 1] x {0; 1}) [ ({0} x [0; 1]), B = [0; 1], and let p*
* be the vertical
projection_then X is contractible but not fiberwise contractible.
43
EXAMPLE Let X be a subspace of B x Rn and suppose that there exists an s 2*
* secB(X), say
b ! (b; s(b)), such that 8 b 2 B, 8 x 2 Xb, {(b; (1  t)s(b) + tx) : 0 t 1} *
*Xb_then X is fiberwise
contractible.
FACT Let Xabeein TOP =B; let f; g 2 CB (X; X). Suppose that {O; P} is a nu*
*merable covering
of B for which fO are fiberwise inessential_then g O f is fiberwise inessenti*
*al.
gP ae ae ae
[Fix fiber homotopies K : IXO ! XO between fO & k O pO, where k 2 secO(X*
*O ). Through
L : IXP ! XP ae gP & l O pP l 2 secP(X*
*P )
reparametrization, it can be assumed that K O itare independent of t when 0 *
*t 1=4, 3=4
ae ae L O it
t 1. Choose 2 C(B; [0; 1]) : spt O & + = 1. Let be the triangle *
*in R 2with
spt P
vertexes (0; 0), (1; 0), (0; 1). Note that the transformation (; j) ! (; (1  *
*)j) takes I[0; 1]  I{1}
homeomorphically onto  {(1; 0)}. The continuous fiber preserving function :*
* I2XO\P ! XO\P
defined by (x; (; j)) = L(K(x; j); ) is independent of j when = 1, thus it ind*
*uces a continuous fiber
preserving function : XO\P x ! XO\P . On XO\P x fr, one has (x; (t;81  t)*
*) = L(k(p(x)); t),
< L(k(b); *
*(b))(b 2 O \ P)
(x; (0; t)) = g(K(x; t)), (x; (t; 0)) = L(f(x); t). Write s(b) = g(k(b)) *
* (b 2 O  P)_then
:
l(b) *
* (b 2 P  O)
s 2 secB(X) and g O f is fiber homotopic to s O p via
8
< (x; t((b); (b)))(b 2 O \ P)
H(x; t) = g(K(x; t)) (b 2 O  P)(x 2 Xb):]
:
L(f(x); t) (b 2 P  O)
Consequently, if f1; : :;:fn 2 CB (X; X) and if O1; : :;:On is a numerable *
*covering of B such that 8 i,
fOi is fiberwise inessential, then f1O . .O.fn is fiberwise inessential. Exampl*
*e: XOi fiberwise contractible
(i = 1; : :;:n) ) X fiberwise contractible (cf. p. 426).
Let X be in TOP =B_then X is said to have the section_extension_property_(*
*SEP)
provided that for each A B, every section sA of XA which admits an extension s*
*O to a
halo O of A in B can be extended to a section s of X : sA = sA .
[Note: If X has the SEP, then secB(X) is nonempty (take A = ; = O).]
Let X be in TOP =B and suppose that X has the SEP. Let s be a section of X*
*OE1(]0; 1]), where
OE 2 C(B; [0; 1])_then 8 ffl, 0 < ffl < 1, sOE1([ffl; 1]) can be extended to *
*a section sfflof X but it is false in
general that s can be so extended.
EXAMPLE Suppose that B is a CW complex of combinatorial dimension n + 1 a*
*nd T is
nconnected_then B x T has the SEP.
44
aePROPOSITION 1 Let X; Y be in TOP =B and suppose that Y has the SEP. Assum*
*e:
9 fg22CBC(X; Y ): g O f ' idX_then X has the SEP.
B (Y; X) B
[Fix a fiber homotopy H : IX ! X between idX and g O f. Given A B, let sA *
*be
a section of XA which admits an extension sO to a halo O of A in B. Choose a c*
*losed
halo P of A in B : A P O and O a halo of P in B (cf. HA 2, p. 311). Since*
* Y
has the SEP, there exists a sectionateof Y : tP = f O sO P . With ss a haloin*
*g function of
1 (0))
P , define s : B ! X by s(b) = gHO(t(b)s (b 2 ss to get a sectio*
*n s of
O (b); 1 (ss(b))b 2 P )
X : sA = sA .]
Application: Fiberwise contractible spaces have the SEP.
LEMMA Let X be in TOP =B and suppose that X has the SEP. Let O be a cozero
set in B_then XO has the SEP.
[There is no loss of generality in assuming that A = f1 (]0; 1]), where f *
*2 C(O; [0; 1]).
Accordingly, given a section sA of XA , it will be enough to construct a secti*
*on s of
XO which agrees with sA on f1 (1). Fix OE 2 C(B; [0; 1]) : O = OE1(]0; 1])*
*. Claim:
There exist sections s2; s3; : :o:f X such that sn+1(b) = sn(b) (OE(b) > 1_n)an*
*d sn(b) =
sA (b) (f(b) > 1  1_n& OE(b) > __1__n)+.1 Granted the claim, we are done. Put*
* F (b) =
ae
f(b)OE(b) (b 2 O)
0 (b 2 B  O) : F 2 C(B; [0; 1]). Since X has the SEP and sA is de*
*fined
on F 1(]0; 1]), a halo of F 1([1=6; 1]) in B, there exists a section of X th*
*at agrees
with sA on f1 (]1=2; 1]) \ OE1(]1=3; 1]). Call it s2, thus setting the stage*
* for induction.
Choose continuous functions n, n : [0; 1] ! [0; 1] subject to __1__n<+n3(x) < n*
*(x) 1_n
with n(x) __1__n +(2x 1  __1__n)+a1nd n(x) __1__n +(1x 1  1_n)(n = 2; 3; *
*: :):. Let
An = {b 2 O : OE(b) > n(f(b))}, On = {b 2 O : OE(b) > n(f(b))}_then On is a hal*
*o of
An in B, a haloing function being 1 on {b 2 O : n(f(b)) OE(b)},
__OE(b)__n(f(b))_on {b 2 O : (f(b)) OE(b) (f(b))};
n(f(b))  n(f(b)) n n
and 0 on {b 2 O :8OE(b) n(f(b))} [ B  O. To pass from n to n + 1, note that *
*the
> __1__)
prescription b ! > n + 1defines a section of XOn . Its restrict*
*ion to An
: sA (b) (f(b) > 1  1_)
n
can therefore be extended to a section sn+1 of X with the required properties.]
SECTION EXTENSION THEOREM Let X be in TOP =B. Suppose that O = {Oi:
i 2 I} is a numerable covering of B such that 8 i, XOi has the SEP_then X has t*
*he SEP.
45
[Given A B, let sA be a section of XA which admits an extension sO to a h*
*alo
O of A in B. Fix a haloing function ss for O and let {ssi : i 2 I} be a partit*
*ion of
P
unity on B subordinate to O. Put S = (1  ss)ssi + ss (S I). Consider the*
* set
i2S
S of all pairs (S; s) : s is a section of X1S(]0; 1]) & sA = sA : S is nonem*
*pty (take
S = ;, s = sO ss1 (]0; 1])). Order S by stipulating that (S0; s0) (S00; s00)*
* iff S0 S00and
s0(b) = s00(b) when S0(b) = S00(b) > 0. One can check that every chain in S has*
* an upper
bound, so by Zorn, S has a maximal element (S0; s0). Since I = 1, to finish it *
*need only
be shown that S0 = I. Suppose not. Select an i0 2 I S0, set 0 = S0 & ss0 = (1*
*ss)ssi0,
and define a continuous function OE0 : ss10(]0; 1]) ! [0; 1] by OE0(b) = min{1*
*; 0(b)=ss0(b)}.
Owing to the lemma, Xss10(]0; 1]) has the SEP (ss10(]0; 1]) Oi0). On the ot*
*her hand,
OE10(]0; 1]) is a halo of OE10(1) in ss10(]0; 1]) and s0OE10(1) admits an *
*extension to OE10(]0; 1]),
viz. s0OE10(]0; 1]). Therefore s0OE10(1)acanebe extended to a section si0of*
* Xss10(]0; 1]).
Let T = S0[{i0} and write t(b) = s0(b)s (ss0(b) 0(b))(T (b) > 0)_then (T; t)*
* 2 S
i0(b)(ss0(b) 0(b))
and (S0; s0) < (T; t), contradicting the maximality of (S0; s0).]
FACT Let A be a subspace of X. Suppose that there exists a numerable cover*
*ing U = {Ui: i 2 I}
of X such that 8 i, the inclusion A \ Ui! Uiis a cofibration_then the inclusion*
* A ! X is a cofibration.
[Let {i : i 2 I} be a partition of unity on X subordinate to U. The lemma o*
*n p. 311 implies
that 8 i, the inclusion A \ 1i(]0; 1]) ! 1i(]0; 1]) is a cofibration. Therefo*
*re one canaassumeethat U is
numerable and open. Fix a topological space Y and a pair (F; h) of continuous f*
*unctions F : X ! Y
*
* h : IA ! Y
such that FA = h O i0. Define a sheaf of sets F on X by assigning to each open*
* set U the set of all
continuous functions H : IU ! Y such that FU = H O i0 and HI(A \ U) = hI(A \*
* U). Choose a
topological space E and a local homeomorphism p : E ! X for which F(U) = secU(E*
*U ) at each U. Show
that 8 i, EUi has the SEP. The section extension theorem then says that 9 H 2 F*
*(X).]
Let X be in TOP =B. Let E be in TOP ; let OE 2 C(E; B)_then a continuous *
*function
: E ! X is a lifting_of OE provided that p O = OE. Example: Every s 2 secB(X)*
* is a
lifting of idB.
FACT Suppose that X is fiberwise contractible. Let OE 2 C(E; B)_then for a*
*ny halo U of any A
in E and all 2 C(U; X) : p O = OEU, there exists a lifting of OE : A = *
*A.
[Note: The condition is also characteristic. First take E = B, A = ; = U, a*
*nd OE = idBto see that
9 s 2 secB(X). Next let E = IX, A = i0X [ i1X,aUe= X x [0; 1=2[ [ X x ]1=2; 1],*
* and define OE : IX ! B
by OE(x; t) = p(x), : U ! X by (x; t) = x (t < 1=2). Since U is a hal*
*o of A in IX, every
s O p(x)(t > 1=2)
lifting of OE with A = A is a fiber homotopy between idXand s O p, i.e., X *
*is fiberwise contractible.]
46
(HLP) Let Y be a topological space_then the projection p : X ! B is sa*
*id to
have the homotopy_lifting_property_with_respect_to_Y(HLPaw.r.t.eY ) if given co*
*ntinuous
functions Fh::YI!YX! Bsuch that pOF = hOi0, there is a continuous function H *
*: IY ! X
such that F = H O i0 and p O H = h.
ae
If p : X ! B has the HLP w.r.t. Y and if f 2 C(Y; B)are homotopic, then *
*f has a lifting
g 2 C(Y; B)
F 2 C(Y; X) iff g has a lifting G 2 C(Y; X).
EXAMPLE Take X = [0; 1] q *, B = [0; 1] and define p : X ! B by p(t) = t, *
*p(*) = 0.
Fix a nonempty Y and let f be the constant map Y ! 0_then the constant map Y ! *
** is a lifting
F 2 C(Y; X) of f. Put h(y; t) = t, so h : IY ! B. Obviously, p O F = h O i0 but*
* there does not exist
H 2 C(IY; X) : F = H O i0 and p O H = h.
ae Let X be in TOP =B. Given a topological space Y and continuous functio*
*ns
F : Y ! X
h : IY ! B such that p O F = h O i0, let W be the subspace of Y x P X consis*
*ting
of the pairs (y; oe) : F (y) = oe(0) & h(y; t) = p(oe(t)) (0 t 1). View W as *
*an object in
TOP =Y with projection (y; oe) ! y.
Y? F! X?
LEMMA The commutative diagram iy0 ypadmits a filler H : IY ! X iff
IY !h B
secY(W ) 6= ;.
PROPOSITION 2 Suppose that p : X ! B has the HLP w.r.t. Y _then 8 pair
(F; h), W has the SEP.
[Fix A Y and let V be a halo of A in Y for which there exists a homotopy H*
*V :
IV ! X such that F V = HV O i0 and p O HV = hIV . To construct a homotopy
H : IY ! X such that F = H O i0 and p O H = h, with HIA =_HV IA, take V closed
(cf. HA 2, p. 311) and using a haloing functionass,eput_h(y; t) = h(y; min{1; *
*ss(y) + t}), so
__ __ H V(y; 0) = F (y) __
h : IY ! B. Define H V : i0Y [ IV ! X by __H and define F : Y *
*! X
__ __ __ __ V(y; t) = HV (y; t) *
*__
by F (y) = H V(y; ss(y)). Since p O F = h O i0, there is a continuous function *
*H : IY ! X
__ __ __ __
such that F = H O i0 and p O H = h. The rule
ae__
H(y; t) = H_V(y;Ht)(y; t(0(tssss(y))(y))ss(y) t 1)
then specifies a homotopy H : IY ! X having the properties in question.]
47
Let Y be a class of topological spaces_then p : X ! B is said to be a Y_fib*
*ration_if
8 Y 2 Y, p : X ! B has the HLP w.r.t. Y .
(H) Take for Y the class of topological spaces_then a Y fibration p : *
*X ! B is
called a Hurewicz_fibration_.
(S) Take for Y the class of CW complexes_then a Y fibration p : X ! B *
*is
called a Serre_fibration_.
Every Hurewicz fibration is a Serre fibration. The converse is false (cf. p*
*. 48).
Observation: Let Y 2 Y and suppose that p : X ! B is a Y fibration_then a*
*ny
inessential f 2 C(Y; B) admits a lifting F 2 C(Y; X).
[Note: It is thus a corollary that if B 2 Y is contractible, then secB(X) i*
*s nonempty.]
Other possibilities suggest themselves. For example, one could consider p :*
* X ! B, where both X
and B are in CG , and work with the class Y of compactly generated spaces. This*
* leads to the notion of
CG_fibration_. Any CG fibration is a Serre fibration. In general, if p : X ! B*
* is a Hurewicz fibration,
then kp : kX ! kB is a CG fibration. Another variant would be to consider point*
*ed spaces and pointed
homotopies. Via the artifice of adding a disjoint base point (cf. p. 326), one*
* sees that every pointed
Hurewicz fibration is a Hurewicz fibration. In the opposite direction, an f 2 C*
*B (X; Y ) is said to be a
fiberwise_Hurewicz_fibration_if it has the fiber homotopy lifting property with*
* respect to all E in TOP =B.
Of course, if f is a Hurewicz fibration, then f is a fiberwise Hurewicz fibrati*
*on. On the other hand, for
any X in TOP =B, the projection p : X ! B is always a fiberwise Hurewicz fibrat*
*ion.
FACT Suppose that p : X ! B is a Hurewicz fibration. Let E be a topologica*
*l space with the
homotopy type of a compactly generated space_then a OE 2 C(E; B) has a lifting *
*E ! X iff kOE 2
C(kE; kB) has a lifting kE ! kX.
[The identity map kE ! E is a homotopy equivalence.]
EXAMPLE For any topological space T, the projection B xT ! B is a Hurewicz*
* fibration. Take,
e.g., T = Dn, let X0 B x Sn1, and put X = B x Dn X0_then the restriction to *
*X of the projection
B x Dn ! B is a Hurewicz fibration.
EXAMPLE (Covering_Spaces_) A continuous function p : X ! B is said to be a*
* covering_projection_
if each b 2 B has a neighborhood O such that XO is trivial with discrete fiber.*
* Every covering projection
is a Hurewicz fibration.
[Note: A sheaf of sets F on B is locally_constant_provided that each b 2 B *
*has a basis B of neigh
borhoods such that whenever U; V 2 B with U V , the restriction map F(V ) ! F(*
*U) is a bijection.
If p : X ! B is a covering projection, then its sheaf of sections X is locally *
*constant. Moreover, every
locally constant sheaf of sets F on B can be so realized.]
48
EXAMPLE Let X be the triangle in R 2with vertexes (0; 0), (1; 0), (0; 1)_t*
*hen the vertical
projection p : X ! [0; 1] is a Hurewicz fibration but X is not locally trivial.
[Note: Ferryy has constructed an example of a Hurewicz fibration p : X ! [0*
*; 1] whose fibers are
connected nmanifolds but such that X is not locally trivial.]
LEMMA Let X be in TOP =B_then p : X ! B is a Serre fibration iff it has t*
*he
HLP w.r.t. the [0; 1]n (n 0).
1S
EXAMPLE Take X = {(x; x) : 0 x 1} [ ([0; 1] x {1=n}); B = [0; 1], and *
*let p be the
1
vertical projection_then p is a Serre fibration but not a Hurewicz fibration.
[Note: p1(0) and p1(1) do not have the same homotopy type.]
EXAMPLE Let B be a topological space which is not compactly generated_then*
* B is not
compactly generated and the identity map kB ! B is a Serre fibration but not a *
*Hurewicz fibration.
[For any compact Hausdorff space K, the arrow C(K; kB) ! C(K; B) is a bijec*
*tion.]
EXAMPLE Let B = [0; 1]!, the Hilbert cube. Put X = B x B  B and let p be *
*the vertical
projection, q the horizontal projection_then p : X ! B is a Serre fibration. Mo*
*reover, B is an AR as are
the Xb (each being homeomorphic to B x [0; 1[) but p : X ! B is not a Hurewicz *
*fibration.
[If so, then there would exist an s 2 secB(X). Consider q O s: It is a cont*
*inuous function B ! B
without a fixed point, contradicting Brouwer.]
Ungarz has shown that if X and B are compact ANRs of finite topological dim*
*ension, then a Serre
fibration p : X ! B is necessarily a Hurewicz fibration.
The projection p : X ! B is a Hurewicz fibration iff the commutative diagr*
*am
P?X p0!X
Ppy ?yp is a weak pullback square. Homeomorphisms are Hurewicz fibrat*
*ions.
P B !p B
0
Maps with an empty domain are Hurewicz fibrations. The composite of two Hurewi*
*cz
fibrations is a Hurewicz fibration.
ae
PROPOSITION 3 Let p1p: X1 ! B1 be Hurewicz fibrations_then p1 x p2 : X1 x
2 : X2 ! B2
X2 ! B1 x B2 is a Hurewicz fibration.
_________________________
yTrans. Amer. Math. Soc. 327 (1991), 201219; see also Husch, Proc. Amer. Ma*
*th. Soc. 61 (1976),
155156.
zPacific J. Math. 30 (1969), 549553.
49
X0? ! X?
PROPOSITION 4 Let py0 yp be a pullback square. Suppose that p is a
B0 ! B
Hurewicz fibration_then p0 is a Hurewicz fibration.
Application: Let p : X ! B be a Hurewicz fibration_then 8 O B, pO : XO ! O
is a Hurewicz fibration.
PROPOSITION 5 Let p : X ! B be a Hurewicz fibration_then for any LCH space
Y , the postcomposition arrow p* : C(Y; X) ! C(Y; B) is a Hurewicz fibration (c*
*ompact
open topology).
[Convert
E ____wC(Y; X) E x Y _____wX
 ""] to OOP :]
u" u u O u
IE ___wC(Y; B) I(E x Y ) ___wB
Application: Let p : X ! B be a Hurewicz fibration_then P p : P X ! P B is*
* a
Hurewicz fibration.
PROPOSITION 6 Let i : A ! X be a closed cofibration, where X is a LCH spac*
*e_
then for any topological space Y , the precomposition arrow i* : C(X; Y ) ! C(A*
*; Y ) is a
Hurewicz fibration (compact open topology).
[Convert
E ____wC(X; Y ) E x X ________wYu
 ""] to 556  :]
u" u u 5 5 
IE ___wC(A; Y ) I(E x X)u___I(E x A)
Application: Let X be a topological space_then pt : P X ! X (0 t 1) is a
Hurewicz fibration.
EXAMPLE Let i : A ! X be a closed cofibration, where X is a LCH space. Fix*
* a0 2 A and put
x0 = i(a0)_then for any pointed topological space (Y; y0), the precomposition a*
*rrow i* : C(X; x0; Y; y0) !
C(A; a0; Y; y0) is a Hurewicz fibration (compact open topology).
C(X; x0; Y; y0)!C(X; Y )
? ?
[The commutative diagram y y is a pullback square.]
C(A; a0; Y; y0)!C(A; Y )
410
ae
FACT Let X be a topological space_then : PX ! X x X is a Hurewicz fib*
*ration. More
oe ! (oe(0); oe(1))
over, X is locally path connected iff is open.
[Note: Fix x0 2 X_then the fiber of over (x0; x0) is X, the loop space of *
*(X; x0).]
STACKING LEMMA Given a topological space Y , let {Pi : i 2 I} be a numerab*
*le
covering of IY _then there exists a numerable covering {Yj : j 2 J} of Y and p*
*ositive
real numbers fflj (j 2 J) such that 8 t0, t002 [0; 1] with t0 t00& t00 t0 < f*
*flj, 9 i 2 I :
Yj x [t0; t00] Pi.
[Let {aei : i 2 I} be a partition of unity on IY subordinate to {Pi : i 2 *
*I}. Put
1S
J = Ir. Take j 2 J, say j = (i1; : :;:ir) 2 Ir, define ssj 2 C(Y; [0; 1]) by
1
Yr ae k  1 k + 1oe
ssj(y) = min aeik(y; t) : t 2 _____; _____
k=1 r + 1 r + 1
rTae k  1 k + *
*1 oe
and set Yj = ss1j(]0; 1]), fflj = 1=2r. Since Yj y : {y} x _____; ____*
*_ Pik ,
k=1 r + 1 r + 1
the fflj will work. Moreover, due to the compactness of [0; 1], for each y 2 Y *
*there is: (1)
An index j 2 Ir such that {y} x k__1_r;+k1+_1_r +a1e1ik(]0; 1]) (k = 1; : :;*
*:r) and (2) A
neighborhood V of y such that IV meets but a finite number of the ae1i(]0; 1])*
*. Therefore
1S
{Yj : j 2 J} = {Yj : j 2 Ir} is a oeneighborhood finite cozero set covering *
*of Y , hence is
1
numerable.]
LOCALGLOBAL PRINCIPLE Let X be in TOP =B. Suppose that O = {Oi: i 2 I}
is a numerable covering of B such that 8 i, pOi : XOi ! Oi is a Hurewicz fibrat*
*ion_then
p : X ! B is a Hurewicz fibration. ae
[Fix a topological space Y and a pair (F; h) of continuous functions Fh:*
*:YI!YX! B
such that p O F = h O i0. To establish the existence of an H : IY ! X such *
*that
F = H O i0 and p O H = h is equivalent to proving that secY(W ) 6= ; (cf. p. 4*
*6). For
this, we shall use the section extension theorem and show that W has the SEP, w*
*hich
suffices. Set Pi = h1(Oi) : {Pi : i 2 I} is a numerable covering of IY and the*
* stacking
lemma is applicable. Given j, put Wj = W Yj, choose tk : 0 = t0 < t1 < . .<.tn*
* = 1,
tk  tk1 < fflj, and select i accordingly: h(Yj x [tk1; tk]) Oi. The claim *
*is that Wj
has the SEP. So let A Yj, let V be a halo of A in Yj, and let HV : IV ! X b*
*e a
homotopy such that F V = HV O i0 and p O HV = hIV . With ss a haloing funct*
*ion of
V , put Ak = ss1 ([tk; 1]) (k = 1; : :;:n) : Ak is a halo of Ak+1 in Yj and V *
* is a halo
411
of A1 in Yj. Owing to Proposition 2, there exist homotopies Hk : Yj x [tk1; t*
*k] ! X
having the following properties: p O Hk = hYj x [tk1; tk], Hk(y; tk1) = Hk1*
*(y; tk1)
(k > 1), H1(y; 0) = F (y), HkAk x [tk1; tk] = HV Ak x [tk1; tk]. The Hk thu*
*s combine
to determine a homotopy H : IYj ! X such that F Yj = H O i0, p O H = hIYj, and
HIA = HV IA.]
Application: Suppose that B is a paracompact Hausdorff space. Let X be in T*
*OP =B.
Assume: X is locally trivial_then p : X ! B is a Hurewicz fibration.
EXAMPLE Let B = L+, the long ray. Put X = {(x; y) 2 L+ x L+ : x < y} and l*
*et p be the
vertical projection_then X is locally trivial but p : X ! B is not a Hurewicz f*
*ibration.
FACT Let X be in TOP =B. Suppose that O = {Oi: i 2 I} is an open covering *
*of B such that
8 i, pOi : XOi ! Oiis a Hurewicz fibration_then the projection p : X ! B is a Y*
* fibration, where Y is
the class of paracompact Hausdorff spaces.ae
[Given Y 2 Y and continuous functions F : Y ! Xsuch that pOF = hOi0, cons*
*ider the pullback
h : IY ! B
IY xB X ! X
square ?y ?yp, observing that IY 2 Y.]
IY !h B
[Note: It follows that p : X ! B is a Serre fibration.]
Let f : X ! Y be a continuous function_then the mapping_track_Wf of f is de*
*fined
Wf? ! P?Y
by the pullback square y yp0. Special case: 8 y0 2 Y , the mapping *
*track
X !f Y
of the inclusion {y0} ! Y is the mapping space Y of (Y; y0). There is a proj*
*ection
p : Wf ! X, a homotopy G : Wf ! P Y , and a unique continuous function s : X ! *
*Wf
such that p O s = idX and G O s = j O f (j : Y ! P Y ). One has s O p 'XidWf.*
* The
composition p1 O G is a projection q : Wf ! Y and f = q O s.
[Note: The mapping track is a functor TOP (!) ! TOP .]
LEMMA p is a Hurewicz fibration and Wf is fiberwise contractible over X.
LEMMA q is a Hurewicz fibration.
412
E? ! Wf? ae
[To construct a filler for iy0 y q , write (e) = (xe; oe) : xeo2 *
*X &
e *
*2 P Y
IE !h Y
__
f(xe) = oe(0), and define H : IE ! Wf by H(e; t) = (xe; h(e; t)), where
__ aeoe(2T (2  t)1)(T 1  t=2)
h(e; t)(T ) = h(e; 2T + t  2)(T 1  t=2) :]
PROPOSITION 7 Every morphism in TOP can be written as the composite of a
homotopy equivalence and a Hurewicz fibration.
*
*ae
FACT Let f : X ! Y be a continuous function_then f can be factored as f = *
* O k, where
ae ae ae *
* O l
is a Hurewicz fibration, k is a closed cofibration, and k is a homotop*
*y equivalence.
l
[Per Proposition 7, write f = q O s, form S = Is(X) [ Wfx]0; 1] IWf, and l*
*et ! : IWf ! [0; 1]
be the projection. The restriction to S ofatheeHurewicz fibration IWf ! Wf is a*
* Hurewicz fibration, call
it p. Proof: Given continuous functions F : Y ! S such that p O F = h O i0, *
*consider H : IY ! S,
h : IY ! Wf
where H(y; t) = (h(y; t); t + (1  t)!(F(y))). Next, if k : X ! S is defined by*
* k(x) = (s(x); 0), then k(X)
is both a strong deformation retract of S and a zero set in S (being (!S)1(0)*
*). Therefore k is a closed
cofibration (cf. x3, Proposition 10). And: f = q O p O k. To derive the other f*
*actorization, write f = r O i
(cf. x3, Proposition 16) and decompose r as above.]
Let X be in TOP =B. Define : P X ! Wp by oe ! (oe(0); p O oe).
PROPOSITION 8 The projection p : X ! B is a Hurewicz fibration iff has a *
*right
inverse .
[Note: is called a lifting_function_.]
FACT Let p : X ! B be a Hurewicz fibration. Suppose that A is a subspace *
*of X for which
there exists a fiber preserving retraction r : X ! A_then the restriction of p *
*to A is a Hurewicz fibration
A ! B.
EXAMPLE Let X be a nonempty compact subspace of R n. Realize X in R n+1by*
* writing
S S
X = {(t; tx) : 0 t 1}, so 2X is {(s; st; stx) : 0 s 1 & 0 t 1}, a sub*
*space of Rn+2.
x ae2 x
Claim: The projection p : X ! [0; 1] is a Hurewicz fibration. To see this, c*
*onsider [0; 1] x X =
S (s; st; stx) ! s
{(s; t; tx) : 0 s 1 & 0 t 1} with projection (s; t; tx) ! s and define a *
*fiber preserving retraction
x
413
ae
r : [0; 1] x X ! 2X by r(s; t; tx) = (s; s; sx)(t .s)The fibers of p over the*
* points in ]0; 1] can be
(s; t; tx)(t s)
identified with X, while p1(0) = *.
[Note: If X is the Cantor set, then X is not an ANR.]
XA_________wflWp
Let X be in TOP =B_then there is a morphism pAC q . Here, in a cha*
*nge
B
of notation, fl sends x to (x; j(p(x))), j : B ! P B the embedding.
PROPOSITION 9 Suppose that p : X ! B is a Hurewicz fibration_then fl : X !
Wp is a fiber homotopy equivalence.
[Choose a lifting function : Wp ! P X. Define a fiber homotopy H : IX ! X *
*by
H(x; t) = (fl(x))(t) and a fiber homotopy G : IWp ! Wp by G((x; o); t) = ((x; o*
*)(t),
ot) (ot(T ) = o(t + T  tT ))_then it is clear that the assignment (x; o) ! (x;*
* o)(1) is a
fiber homotopy inverse for fl.]
Application: The fibers of a Hurewicz fibration over a path connected base *
*have the
same homotopy type.
[Note: This need not be true if "Hurewicz" is replaced by "Serre" (cf. p. 4*
*8). It can
also fail if "path connected" is weakened to "connected". Indeed, for a connect*
*ed B whose
path components are singletons, every p : X ! B is a Hurewicz fibration.]
X4____*
*_____wflWp
A Hurewicz fibration p : X ! B is said to be regular_if the morphism p46 *
* hhkqhas a left
inverse in TOP =B. B
FACT The Hurewicz fibration p : X ! B is regular iff there exists a liftin*
*g function 0 : Wp ! PX
with the property that 0(x; o) 2 j(X) whenever o 2 j(B).
[Given a left inverse for fl, consider the lifting function 0 : Wp ! PX de*
*fined by 0(x; o)(t) =
(x; ot), where ot(T) = o(tT).]
*
* Y F! X
*
* ? ?
FACT The Hurewicz fibration p:X !B is regular iff every commutative diagram*
*i0y yp
*
* IY !h B
admits a filler H : IY ! X such that H is stationary with h, i.e., hI{y0} cons*
*tant ) HI{y0} constant.
[Note: The localglobal principle is valid in the regular situation (work w*
*ith a suitable subspace of
W to factor in the stationary condition).]
414
A sufficient condition for the regularity of the Hurewicz fibration p : X !*
* B is that j(B) be a
zeroasetein PB. Thus let OE 2 C(PB; [0; 1]) : j(B) = OE1(0). Define 2 C(PB;*
* PB) by (o)(t) =
o(t=OE(o))t < OE(o)). Take any lifting function and put (x; o)(t) = (x; (o)*
*)(OE(o)t) to get
o(1) (OE(o) t 1) 0
a lifting function 0 : Wp ! PX with the property that 0(x; o) 2 j(X) whenever o*
* 2 j(B). Example:
j(B) is a zero set in PB if B is a zero set in B x B, e.g., if the inclusion B *
*! B x B is a closed
cofibration, a condition satisfied by a CW complex or a metrizable topological *
*manifold (cf. p. 314).
EXAMPLE Let B = [0; 1]=[0; 1[_then the Hurewicz fibration p0 : PB ! B is n*
*ot regular.
FACT Suppose that p : X ! B is a regular Hurewicz fibration_then 8 x0 2 X,*
* p : (X; x0) !
(B; b0) is a pointed Hurewicz fibration (b0 = p(x0)).
Let X be in TOP =B_then the projection p : X ! B is said to have the slicin*
*g_structure_property_
if there exists an open covering O = {Oi : i 2 I} of B and continuous functions*
* si : Oix XOi ! XOi
(i 2 I) such that si(p(x); x) = x and p O si(b; x) = b. Note that p is necessar*
*ily open. Example: X locally
trivial ) p : X ! B has the slicing structure property (but not conversely).
Observation: Suppose that p : X ! B has the slicing structure property_then*
* 8 i; pOi : XOi ! Oi
is a regular Hurewicz fibration.
[Consider the lifting function idefined by i(x; o)(t) = si(o(t); x).]
So, if p : X ! B has the slicing structure property, then p : X ! B must be*
* a Serre fibration and is
even a regular Hurewicz fibration provided that B is a paracompact Hausdorff sp*
*ace.
FACT Let X be in TOP =B, where B is uniformly locally contractible. Assume*
*: The projection
p : X ! B is a regular Hurewicz fibration_then p has the slicing structure prop*
*erty.
Application: Suppose that B is a uniformly locally contractible paracompact*
* Hausdorff space. Let
X be in TOP =B_then the projection p : X ! B is a regular Hurewicz fibration if*
*f p has the slicing
structure property.
[Note: It therefore follows that if B is a CW complex or a metrizable topol*
*ogical manifold, then the
Hurewicz fibrations with base B are precisely the p : X ! B which have the slic*
*ing structure property.]
FACT Let p : X ! B be a Serre fibration, where X and B are CW complexes_th*
*en p is a CG
fibration.
[An open subset of a CW complex is homeomorphic to a retract of a CW comple*
*x (cf. p. 512).]
[Note: If X x B is compactly generated, then p is a Hurewicz fibration.]
Cofibrations are embeddings (cf. p. 33). By analogy, one might expect that*
* surjective
Hurewicz fibrations are quotient maps. However, this is not true in general. *
*Example:
415
Take X = Q (discrete topology), B = Q (usual topology), p = idQ_then p : X ! B *
*is a
surjective Hurewicz fibration but not a quotient map.
PROPOSITION 10 Let p : X ! B be a Hurewicz fibration. Assume: p is surject*
*ive
and B is locally path connected_then p is a quotient map.
P?X ! Wp?
[Consider the commutative diagram p1y y q . Since and p1 have rig*
*ht
X !p B
inverses, they are quotient, so p is quotient iff q is quotient. Take a nonemp*
*ty subset
O B : WO is open in Wp. Fix b 2 O, x 2 Xb, and choose a neighborhood Ob of
b : ({x} x P Ob) \ Wp WO . The path component O0 of Ob containing b is open. G*
*iven
b0 2 O0, 9 o 2 P Ob connecting b and b0. But (x; o) 2 WO ) b0 = q(x; o) 2 O ) O*
*0 O.
Therefore O is open in B, hence q is quotient.]
Application: Every connected locally path connected nonempty space B is the*
* quo
tient of a contractible space.
[Fix b0 2 B and consider the mapping space B of (B; b0) with projection o !*
* o(1).]
Let p : X ! B be a Hurewicz fibration_then for any path component A of X, p*
*(A)
is a path component of B and A ! p(A) is a Hurewicz fibration. Therefore p(X) *
*is a
union of path components of B. So, if B is path connected and X is nonempty, th*
*en p is
surjective.
FACT Let p : X ! B be a Hurewicz fibration. Assume: B is path connected an*
*d Xb is path
connected for some b 2 B_then X is path connected.
[Note: The fibers of a Hurewicz fibration p : X ! B need not be path connec*
*ted but if X is path
connected, then any two path components of a given fiber have the same homotopy*
* type.]
FACT Suppose that B is path connected_then B is locally path connected iff*
* every Hurewicz
fibration p : X ! B is open.
PROPOSITION 11 Let p : X ! B be a Hurewicz fibration. Suppose that the
inclusion O ! B is a closed cofibration_then the inclusion XO ! X is a closed c*
*ofibration.
[Fix a Strom structure (OE; ) on (B; O). Let H : IX ! X be a filler for th*
*e com
X? idX!X?
mutative diagram iy0 yp , where h = O Ip. Define a Strom structure ( ; *
*) on
IX !h B
(X; XO ) by = OE O p, (x; t) = H(x; min{t; (x)}).]
416
Application: Let p : X ! B be a Hurewicz fibration. Let A be a subspace of *
*X and
suppose that the inclusion A ! X is a closed cofibration. View A as an object i*
*n TOP =B
with projection pA = pA_then the inclusion WpA ! Wp is a closed cofibration.
EXAMPLE Let (X; x0) be a pointed space. Assume: The inclusion {x0} ! X i*
*s a closed
cofibration_then Proposition 11 implies that the inclusion j : X ! X is a close*
*d cofibration. Call
the continuous function X ! X that sends [oe; t] to oet, where oet(T) = oe(tT)*
*. The arrow i :
*
* i
ae *
* X ! X
X ! X is a closed cofibration and Oi = j. Consider the commutative diagra*
*m k ?y .
oe ! [oe; 1]
*
* X !j X
Because X and X are contractible, it follows from x3, Proposition 14 that the a*
*rrow (idX ; ) is a
homotopy equivalence in TOP (!).
LEMMA Let OE 2 C(Y; [0; 1]) : A = OE1(0) is a strong deformation retract *
*of
Y . Suppose that p : X ! B is a Hurewicz fibration_then every commutative dia
A? g! X?
gram yi yp has a filler F : Y ! X.
Y !f B
[Fix a retraction r : Y ! A and a homotopyae: IY ! Y between i O r and idYr*
*elA.
Define a homotopy h : IY ! Y by h(y; t) = (y;(t=OE(y))y(t;<1OE(y)))(t. OE(y*
*))Since p is a
Hurewicz fibration, there exists a homotopy H : IY ! X such that g O r = H O i0*
* and
p O H = f O h. Take for F : Y ! X the continuous function y ! H(y; OE(y)).]
[Note: The hypotheses on A are realized when the inclusion i : A ! Y is b*
*oth a
homotopy equivalence and a closed cofibration (cf. x3, Proposition 5).]
FACT Let i : A ! Y be a continuous function with a closed image_then i is *
*both a homotopy
A ! X
? ?
equivalence and a closed cofibration iff every commutative diagram yi yp,*
* where p is a Hurewicz
Y ! B
fibration, has a filler Y ! X.
[First take X = PB, p = p0 to see that i is a closed cofibration. Next, ide*
*ntify A with i(A) and
*
* j
A idA!A *
* A ! PY
*
* ? ?
produce a retraction r : Y ! A from a filler for ?yi ?y. Finally, consider*
* yi y ,
Y ! * *
* Y !aeY x Y
where ae(y) = (y; r(y)) ( as on p. 410).]
417
FACT Let p : X ! B be a continuous function_then p is a Hurewicz fibration*
* iff every commuta
A ! X
tive diagram ?yi ?yp, where i is both a homotopy equivalence and a closed *
*cofibration, has a filler
Y ! B
Y ! X.
X0  X1  . . .
FACT Let ?y ?y be a commutative ladder of topological spac*
*es. Assume:
Y0  Y1 ae . . .
8 n, the horizontal arrows Xn Xn+1 are Hurewicz fibrations and the vertical*
* arrows OEn : Xn ! Yn
Yn Yn+1
are homotopy equivalences_then the induced map OE : limXn ! limYn is a homotopy*
* equivalence.
[The mapping cylinder is a functor TOP (!) ! TOP , so there is an arrow ssn*
* : MOEn+1! MOEn.
X0 ________widX0
hj
Use x3, Proposition 17 to construct a commutative trianglei h hr0 . The *
*lemma then provides
u h
MOE0
X1 id!X1
? ?
a filler r1 : MOE1! X1 for yi y , hence, by induction, a filler rn+1 *
*: MOEn+1! Xn+1
MOE1 !r X0
id 0Oss0
Xn+1 ! Xn+1
? ? j r
for yi y . Give the composite Yn ! MOEn!nXn a name, say n, and t*
*ake limits to get
MOEn+1 r! Xn
nOssn
a left homotopy inverse for OE.]
PROPOSITION 12 Let A be a closed subspace of Y and assume that the inclusi*
*on
A ! Y is a cofibration. Suppose that p : X ! B is a Hurewicz fibration_then e*
*very
i0Y?[ IA F! X?
commutative diagram y y p has a filler H : IY ! X.
IY !h B
[Quote the lemma: i0Y [ IA is a strong deformation retract of IY (cf. p. 3*
*6) and
i0Y [ IA is a zero set in IY .]
Application: Let p : X ! B be a Hurewicz fibration, where B is a LCH space.
Suppose that the inclusion O ! B is a closed cofibration_then the arrow of rest*
*riction
secB(X) ! secO(XO ) is a Hurewicz fibration.
418
EXAMPLE (Vertical_Homotopies_) Let p : X ! B be a Hurewicz fibration. Sup*
*pose that s0,
s002 secB(X) are homotopic_then s0, s00are vertically homotopic.
ae [Take any homotopy H : IB ! X between s0 and s00. Define G : IB ! X by G(b*
*; t) =
H(b; 2t) (0 t 1=2). Since p O G(b; t) = p O G(b; 1  t), it follows*
* that p O G is homotopic
s00O p O H(b; 2 (2t)1=2 t 1)
relB x {0; 1} to the projection B x [0; 1] ! B.]
LEMMA Let A be a closed subspace of Y and assume that the inclusion A ! Y *
*is
a cofibration. Suppose that p : X ! B is a Hurewicz fibration. Let F : i0Y [ IA*
* ! X
be a continuous function such that 8 a 2 A : p O F (a; t) = p O F (a; 0) (0 t *
* 1)_then
there exists a continuous function H : IY ! X which extends F such that 8 y 2 *
*Y :
p O H(y; t) = p O H(y; 0) (0 t 1).
[Choose OE 2 C(Y; [0; 1]) : A = OE1(0) and fix a retraction r : IY ! i0Y [*
* IA. Put
f = p O F O r. Define G 2 C(IY; P B) as follows: G(y; t)(T ) = (i) f(y; (tOE(*
*y)  T (2 
OE(y)))=OE(y)) (0 T tOE(y)=2 & OE(y) 6= 0); (ii) f(y; t) (0 T tOE(y)=2 & OE*
*(y) = 0);
(iii) f(y; tOE(y)  T ) (tOE(y)=2 T tOE(y)); (iv) f(y; 0) (tOE(y) T 1). Ta*
*ke a lifting
function : Wp ! P X and set H(y; t) = (F O r(y; t), G(y; t))(tOE(y)).]
LIFTING PRINCIPLE Let p : X ! B be a Hurewicz fibration. Let A be a subspa*
*ce
of X and suppose that the inclusion A ! X is a closed cofibration. View A as an
object in TOP =B with projection pA = pA and assume that pA : A ! B is a Hure*
*wicz
fibration. Let A : WpA ! P A be a lifting function_then there exists a lifting *
*function
X : Wp ! P X such that X WpA = A .
[The inclusion WpA ! Wp is a closed cofibration (cf. p. 416). Therefore th*
*e inclusion
i0Wp [ IWpA ! IWp is a closed cofibration (cf. p. 36 or x3, Proposition 7). Fi*
*x a lifting
function : Wp ! P X. Define a continuous function F : i0IWp [ I(i0Wp [ IWpA) !*
* X
by F ((x; o); t; T ) = (i) (x; o)(t) (T = 0 & (x; o) 2 Wp); (ii) x (t = 0 & (x;*
* o) 2 Wp);
(iii) A (a; o)(t) (0 t T & (a; o) 2aWpA);e(iv) (A (a; o)(T ); o * T )(t  T )*
* (T t 1
& (a; o) 2 WpA). Here, o * T (t) = o(to+(T1)(t)(1t T1).T )Apply the lemma *
*to get a
continuous function H : I2Wp ! X which extends F such that 8 ((x; o); t) 2 IWp *
*: p O
H((x; o); t; T ) = pOH((x; o); t; 0). Put X (x; o)(t) = H((x; o); t; 1)_then X *
*: Wp ! P X
is a lifting function that restricts to A .]
ae PROPOSITION 13 Let X be in TOP =B. Suppose thataXe = A1 [ A2, where
A1 A1
aeA2 are closed and the inclusions A0 = A1 \ A2 ! A2 are cofibrations. Ass*
*ume:
p1 = pA1 : A1 ! B
p2 = pA2 : A2 ! B & p0 = pA0 : A0 ! B are Hurewicz fibrations_then p : X ! B *
*is
419
a Hurewicz fibration.
[Choose a liftingafunctione0 : Wp0 ! P A0.aeUse the lifting principle to se*
*cure
lifting functions 1 : Wp1 ! P A1 such that 1Wp0 = 0 . Define a lifting *
*function
2 :aWp2e! P A2 2Wp0 = 0
: Wp ! P X by (x; o) = 1(x; o) ((x; o) 2 Wp1)and cite Proposition 8.]
2(x; o)((x; o) 2 Wp2)
ae
FACT (MayerVietoris_Condition_) Suppose that B = B1 [ B2, where B1 are*
* closed and the
ae ae B2
inclusions B0 = B1 \ B2 ! B1 are cofibrations. Let X1 ! B1 be Hurewicz fibr*
*ations. Assume:
ae B2 X2 ! B2
X1B0 have the same fiber homotopy type_then there exists a Hurewicz fibratio*
*n X ! B such that
aeX2B0
X1 & XB1 have the same fiber homotopy type.
X2 & XB2
X0 p0!B0 q0Y0
? ? ?
FACT Let y y y be a commutative diagram in which the verti*
*cal arrows
X !p B q Y
ae
are inclusions and closed cofibrations. Assume that the projections p0 are Hu*
*rewicz fibrations_then
p
the induced map X0xB0 Y0 ! X xB Y is a closed cofibration.
[The inclusion p1(B0) ! X is a closed cofibration (cf. Proposition 11). Si*
*nce X0 is contained in
p1(B0) and since the inclusion X0 ! X is a closed cofibration, the inclusion X*
*0 ! p1(B0) is a closed
cofibration (cf. x3, Proposition 9). Proposition 13 then implies that the arrow*
* i0p1(B0) [ IX0 ! B0 is
a Hurewicz fibration. Consequently (cf. Proposition 12), the commutative diagram
i0p1(B0) [ IX0id!i0p1(B0) [ IX0
?y ?y
Ip1(B0) ! B0
has a filler r : Ip1(B0) ! i0p1(B0) [ IX0. Therefore the inclusion X0 xB0 Y0 *
*! p1(B0) xB Y0 is a
closed cofibration. On the other hand, the projection X xB Y ! Y is a Hurewicz *
*fibration (cf. Proposition
4) and the inclusion Y0 ! Y is a closed cofibration, so the inclusion p1(B0) x*
*B Y0 ! X xB Y is a closed
cofibration (cf. Proposition 11).]
Application: Consider the 2sink X p!B q Y , where p : X ! B is a Hurewicz *
*fibration. Assume:
The inclusions X ! X x X, B ! B x B, Y ! Y x Y are closed cofibrations_then the*
* diagonal
embedding X xB Y ! (X xB Y ) x (X xB Y ) is a closed cofibration.
420
p q
Let X ! B Y be a 2sink_then the fiber_join_X *B Y is the double mapping *
*cylin
der of the 2source X X xB Y !jY . The fiber homotopy type of X *B Y depends *
*only
on the fiber homotopy types of X and Y . There is a projection X *B Y ! B and*
* the
fiber over b is Xb * Yb. Examples: (1) The fiber join of X p!B B x {0} is B *
*X, the
fiber_cone_of X; (2) The fiber join of X p!B B x {0; 1} is B X, the fiber_sus*
*pension_
of X; (3) The fiber join of B x T1 ! B B x T2 is B x (T1 * T2); (4) The fiber*
* join of
{b0} ! B p X is the mapping cone Cb0 of the inclusion Xb0! X.
Let X be in TOP =B_then B X can be identified with the mapping cylinder Mp *
*and B X can be
identified with the double mapping cylinder Mp;p.
ae
LEMMA Let f 2 CB (X; Y ). Suppose that pq::XY!!BB are Hurewicz fibration*
*s_
then the projection ss : Mfa!eB is a Hurewicz fibration.
[Fix lifting functions X : Wp ! P X . Define a lifting function : Wss!*
* P Mf as
Y : Wq ! P Y
follows: Given ((x; t); o) 2 IX xB P B, put
8
< (X (x; o)(T ); (t  1=2)(1 + T ) + (1  T()=2)1=2 t 1)
((x; t); o)(T ) = : (X (x; o)(T ); t  T=2) (0 t 1=2 & T 2t)
Y (f(X (x; o)(2t)); o2t)(T  2t) (0 t 1=2 & T 2t*
*);
where o2t(T ) = o(min {2t + T; 1}), and given (y; o) 2 Y xB P B, put (y; o) = Y*
* (y; o).]
ae
PROPOSITION 14 Suppose that pq::XY!!BB are Hurewicz fibrations_then the
projection X *B Y ! B is a Hurewicz fibration.
X xB?Y ! Mj?
[Consider the pushout square y y (cf. p. 323). *
*Here,
M ! X *B Y
ae
the arrows X xB Y ! MjM ! X *B Y are closed cofibrations and the project*
*ions
ae
X xB Y ! B, MjM ! B are Hurewicz fibrations. That the projection X *B Y ! B is
a Hurewicz fibration is therefore a consequence of Proposition 13.]
ae
Application: Let p : X ! B be a Hurewicz fibration_then the projections B*
* X ! B
*
*B X ! B
are Hurewicz fibrations.
Let X p!B q Y be a 2sink, where p is a Hurewicz fibration. There is a com*
*mutative diagram
421
p q
X ! B  Y
?
k k yfland fl is a homotopy equivalence, thus the induced map X *
*xB Y ! X xB Wq
X !p B  Wq
X  X xB Y !*
* Y
is a homotopy equivalence (cf. p. 425). Consideration of k ?y *
* k then leads
X  X xB Wq !*
* Y
to a homotopy equivalence X *B Y ! X *B Wq (cf. p. 324). Example: 8 b0 2 B, X *
**B B and Cb0
have the same homotopy type.
*
* X xB Y j!Y
*
* ? ?
Assume in addition that q is a closed cofibration and define P by the pusho*
*ut square y y
*
* X ! P
_then Proposition 11 implies that is a closed cofibration. Therefore the arro*
*w X *B Y ! P of x3,
Proposition 18 is a homotopy equivalence. Example: 8 b0 2 B such that the inclu*
*sion {b0} ! B is a
closed cofibration, B *B B and B=B have the same homotopy type.
ae
PROPOSITION 15 Suppose that pq::XY!!BB are Hurewicz fibrations. Let OE 2
CB (X; Y ). Assume that OE is a homotopy equivalence_then OE is a homotopy equi*
*valence
in TOP =B.
[This is the analog of x3, Proposition 13. It is a special case of Proposit*
*ion 16 below.]
Application: Let p : X ! B be a homotopy equivalence_then Wp is fiberwise c*
*on
tractible.
[Write p = q O fl : p and fl are homotopy equivalences, thus so is q.]
[Note: Similar reasoning leads to another proof of Proposition 9.]
EXAMPLE Let p : X ! B be a Hurewicz fibration. View PX as an object in TOP*
* =Wp with
projection : PX ! Wp_then PX is fiberwise contractible.
FACT Let p : X ! B be a continuous function_then p is both a homotopy equi*
*valence and a
A ! X
? ?
Hurewicz fibration iff every commutative diagram yi yp, where i is a clos*
*ed cofibration, has a
Y ! B
filler Y ! X.
[To discuss the necessity, use Proposition 12, noting that X is fiberwise c*
*ontractible, hence 9 s 2
secB(X) : s O p 'BidX.]
422
X0 ! X
? ?
Application: Let py0 yp be a pullback square. Suppose that p is a Hu*
*rewicz fibration and
B0 ! B
a homotopy equivalence_then p0is a Hurewicz fibration and a homotopy equivalenc*
*e.
FACT Let i : A ! Y be a continuous function_then i is a closed cofibration*
* iff every commutative
A ! X
? ?
diagram yi yp, where p is both a homotopy equivalence and a Hurewicz fibr*
*ation, has a filler
Y ! B
Y ! X.
A ! PX
? ?
[To establish the sufficiency, first consider yi y p0 to see that i i*
*s a cofibration. Taking i
Y ! X
to be an inclusion, put X = IA [ Y x]0; 1]_then the restriction to X of the Hur*
*ewicz fibration IY ! Y
is a Hurewicz fibration (cf. p. 412), call it p. Since p is also a homotopy eq*
*uivalence, the commutative
A ! X
diagram ?yi ?yphas a filler f : Y ! X (a ! (a; 0) (a 2 A)), therefore*
* A is a zero set in Y ,
Y =======================Y
thus is closed.]
FACT Let X p!B q Y be a 2sink, where p : X ! B is a Hurewicz fibration. D*
*enote by W* the
mapping track of the projection X *B Y ! B_then X *B Wq and W* have the same fi*
*ber homotopy type.
LEMMA Suppose that 2 CB (X; E) is a fiberwise Hurewicz fibration. Let f 2
C(X; X) : O f = & f 'BidX_then 9 g 2 C(X; X) : O g = & f O g 'EidX.
[Let H : IX ! X be a fiber homotopy with H O i0 = f and H O i1 = idX ; let
G : IX ! X be aafiberehomotopy with GOi0 = idX and OG = OH. Define F : IX ! X
by F (x; t) = fHO(G(x;x1;22t)(0t(t11=2)1)=2antd 1)put
ae
k((x; t); T ) = OOG(x;H1(x2t(1;1T))2(1 (0(tt)1=2)(11=T2)) *
*t 1)
to get a fiber homotopy k : I2X ! E with O F = k O i0. Choose a fiber homotopy
K : I2X ! X such that F = K O i0 and O K = k. Write K(t;T): X ! X for the func*
*tion
x ! K((x; t); T ). Obviously, K(0;0)' K(0;1)' K(1;1)' K(1;0), all fiber homotop*
*ies being
over E. Set g = G O i1_then f O g = F O i0 = K(0;0)'EK(1;0)= F O i1 = idX.]
[Note: Take B = *, E = B, = p, so p : X ! B is a Hurewicz fibration_then t*
*he
lemma asserts that 8 f 2 CB (X; X), with f ' idX, 9 g 2 CB (X; X) : f O g 'BidX*
*.]
423
ae
PROPOSITION 16 Suppose that j22CBC(X; E) are fiberwise Hurewicz fibratio*
*ns.
B (Y; E)
Let OE 2 C(X; Y ) : j O OE = . Assume that OE is a homotopy equivalence in TOP *
* =B_then
OE is a homotopy equivalence in TOP =E.
[Since is a fiberwise Hurewicz fibration, there exists a fiber homotopy in*
*verse :
Y ! X for OE with O = j, thus, from the lemma, 9 0 2 C(Y; Y ) : j O 0 = j*
* &
OE O O 0'EidY. This says that OE0 = O 0 is a homotopy right inverse for *
*OE over E.
Repeat the argument with OE replaced by OE0 to conclude that OE0 has a homotopy*
* right
inverse OE00over E, hence that OE0 is a homotopy equivalence in TOP =E or stil*
*l, that OE is
a homotopy equivalence in TOP =E.]
[Note: To recover Proposition 15, take B = *, E = B, = p, and j = q.]
X? p! B?
PROPOSITION 17 Suppose given a commutative diagram OyE y in which
Y !q A
ae ae
p OE
q are Hurewicz fibrations and are homotopy equivalences_then (OE; ) is *
*a homo
topy equivalence in TOP (!).
[This is the analog of x3, Proposition 14.]
Let X f!Z g Y be a 2sink_then the double_mapping_track_Wf;gof f; g is defi*
*ned by
Wf;g? ! P?Z
the pullback square y py0p1 . The homotopy type of Wf;gdepends only*
* on
X x Y !fxgZ x Z
the homotopy classes of f and g and Wf;gis homeomorphic to Wg;f. There are Hure*
*wicz
q
ae Wf;g? ! Y?
fibrations pq::Wf;g!WX . The diagram py yg is homotopy commutative *
*and
f;g! Y X ! Z
f
W? j! Y?
if the diagram y yg is homotopy commutative, then there exists a OE : *
*W ! Wf;g
X !f Z
ae
such that j==pqOOOEOE.
424
Wf;g? ! Y?
[Note: The commutative diagram y yg is a pullback square (f = q *
*O s).]
Wf !q Z
FACT Let X f!Z g Y be a 2sink_then the assignment (x; y; o) ! o(1=2) defi*
*nes a Hurewicz
fibrationaWf;g!eZ.
+ = {(x; o) : f(x) = o(0); o 2 C([0; 1=2]; Z)} aeW+ ! Z *
* aeWg
[Let Wf . The projections f ,
Wg= {(y; o) : g(y) = o(1); o 2 C([1=2; 1]; Z)} (x; o) ! o(1=*
*2) (y; o)
Wf;g ! Wg
! Z are Hurewicz fibrations and the commutative diagram ?y ?y is *
*a pullback square.]
! o(1=2)
W+f ! Z
P? j! Y?
Every 2sink X f!Z g Y determines a pullback square y yg and there *
*is an
X !f Z
ae
arrow OE : P ! Wf;gcharacterized by the conditions j==pqOOOEOE& P OE!Wf;g! P *
*Z =
8
< j O f O
: j Okg O j.
PROPOSITION 18 If f is a Hurewicz fibration, then OE : P ! Wf;gis a homoto*
*py
equivalence in TOP =Y .
[Use Proposition 9 and the fact that the pullback of a fiber homotopy equiv*
*alence is
a fiber homotopy equivalence.]
ae0
Application: Let p : X ! B be a Hurewicz fibration. Suppose that 10 2 C(B*
*0; B)
ae 2
0
are homotopic_then X1X0have the same homotopy type over B0.
2
For example, under the assumption that p : X ! B is a Hurewicz fibration, i*
*f 0 :
B0! B is homotopic to the constant map B0! b0, then X0 is fiber homotopy equiva*
*lent
to B0x Xb0.
FACT Suppose that p : X ! B is a Hurewicz fibration. Let 0 : B0 ! B be a *
*homotopy
equivalence_then the arrow X0! X is a homotopy equivalence.
Denote by id; TOP the comma category corresponding to the identity functo*
*r idon TOP x TOP
and the diagonal functor : TOP ! TOP x TOP . So, an object in id; TOP is a*
* 2sink X f!Z g Y
425
f g
X? ! Z?  Y?
and a morphism of 2sinks is a commutative diagram y y y . The d*
*ouble mapping
X0 !f0 Z0 g0 Y 0
track is a functor id; TOP ! TOP . It has a left adjoint TOP ! id; TOP , *
*viz. the functor that
sends X to the 2sink X i0!IX i1X.
X? f! Z? g Y?
FACT Let y y y be a commutative diagram in which the vertic*
*al arrows are
X0 !f0Z0 g0Y 0
homotopy equivalences_then the arrow Wf;g! Wf0;g0is a homotopy equivalence.
ae ae
Application: Suppose that p : X ! B are Hurewicz fibrations. Let g : *
*Y ! B be con
p0: X0! B0 g0: *
*Y 0! B0
X? p! B? g Y?
tinuous functions. Assume that the diagram y y y commutes and th*
*at the vertical
X0 !p0 B0 g0 Y 0
arrows are homotopy equivalences_then the induced map XxB Y ! X0xB0Y 0is a homo*
*topy equivalence.
X? p! B? ae
EXAMPLE Suppose given a commutative diagram yOE y in which p are *
*Hurewicz
q
ae Y !q A
fibrations and OEare homotopy equivalences_then 8 b 2 B, the induced map Xb !*
* Y (b)is a homotopy
equivalence.
[Note: Let f : X ! Y be a homotopy equivalence, fix x0 2 X and put y0 = f(x*
*0), form the com
X? p1!X?  {x0}?
mutative diagram y y y , and conclude that the arrow X ! Y is *
*a homotopy
Y !p Y  {y0}
equivalence.] 1
Given a 2sink X p!B q Y , let X ___BY beatheedouble mapping cylinder of *
*the 2source X Wp;q!
Y . It is an object in TOP =B with projection x ! p(x), ((x; y; o); t) ! o(t).
y ! q(y)
FACT There isaaehomotopy equivalence X ___BY OE!Wp *B Wq.
[Define OE by OE(x) = fl(x)& OE((x; y; o); t) = ((x; ot); (y; _ot); t), w*
*here ot(T) = o(tT) and _ot(T) =
OE(y) = fl(y)
o(tT + 1  T).]
[Note: More is true if p : X ! B is a Hurewicz fibration: X ___BY and X **
*B Y have the same
homotopy type. Indeed, Wp *B Wq has the same fiber homotopy type as X *B Wq whi*
*ch in turn has the
same homotopy type as X *B Y (cf. p. 420 ff.).]
426
Application: 8 b0 2 B, B and B *B B have the same homotopy type.
[Note: The suspension is taken in TOP , not TOP *.]
Given f 2 CB (X; Y ), let W be the subspace of X x P Y consisting of the*
* pairs
(x; o) : f(x) = o(0) and p(x) = q(o(t)) (0 t 1)_then W is in TOP =Y with pro*
*jec
tion (x; o) ! o(1) and is fiberwise contractible if f is a fiber homotopy equiv*
*alence (cf.
Proposition 16).
[Note: W is an object in TOP =B with projection (x; o) ! p(x). Viewed as a*
*n object
in TOP =Y , its projection (x; o) ! o(1) is therefore a morphism in TOP =B an*
*d as such,
is a fiberwise Hurewicz fibration.]
LEMMA f admits a right fiber homotopy inverse iff secY(W ) 6= ;.
PROPOSITION 19 Let f 2 CB (X; Y ). Suppose that there exists a numerable c*
*ov
ering O = {Oi : i 2 I} of B such that 8 i, fOi : XOi ! YOi is a fiber homotopy
equivalence_then f is a fiber homotopy equivalence.
[It need only be shown that secY(W ) 6= ;. For then, by the lemma, f has a *
*right fiber
homotopy inverse g and, repeating the argument, g has a right fiber homotopy in*
*verse h,
which means that g is a fiber homotopy equivalence, thus so is f. This said, w*
*ork with
fOi 2 COi(XOi; YOi) and, as above, form WOi XOi x P YOi. Obviously, W YOi = W*
*Oi.
The assumption that fOi is a fiber homotopy equivalence implies that WOi is fib*
*erwise
contractible, hence has the SEP. But {YOi : i 2 I} is a numerable covering of Y*
* . Therefore,
on the basis of the section extension theorem, W has the SEP. In particular: se*
*cY(W ) 6= ;.]
Application: Let X be in TOP =B. Suppose that there exists a numerable cov*
*ering
O = {Oi : i 2 I} of B such that 8 i, XOi is fiberwise contractible_then X is fi*
*berwise
contractible.
ae
PROPOSITION 20 Let pq::XY!!BBbe Hurewicz fibrations, where B is numerably
contractible. Suppose that f 2 CB (X; Y ) has the property that fb : Xb ! Yb is*
* a homotopy
equivalence at one point b in each path component of B_then f : X ! Y is a fib*
*er
homotopy equivalence.
[Fix a numerable covering O = {Oi: i 2 I} of B for which the inclusions Oi!*
* B are
inessential, say homotopic to Oi ! bi, where fbi: Xbi! Ybiis a homotopy equival*
*ence_
then 8 i, fOi : XOi ! YOi is a fiber homotopy equivalence (cf. p. 424), so Pro*
*position
19 is applicable.]
427
EXAMPLE Take B = {0} [ {1=n : n = 1; 2; : :}:, T = B [ {n : n = 1; 2; : :}*
*:, and put X =
B x T. Observe that B is not numerably contractible. Let k = 1; 2; : :;:1, l = *
*0; 1; 2; : :,:and define
f 2 CB (X; X) as follows: (i) f(1=k; l) = (1=k; l) (l < k), (1=k; 1=k) (l = k 6*
*= 1), (1=k; l  1) (l > k); (ii)
f(1=k; 1=l) = (1=k; 1=l) (0 < l < k), (1=k; 1=(l +1)) (l k)_then f is bijectiv*
*e and 8 b 2 B, fb : Xb ! Xb
is a homeomorphism (Xb = {b} x T). Nevertheless, f is not a fiber homotopy equi*
*valence. For if it were,
then f would have to be a homeomorphism, an impossibility (f1 is not continuou*
*s at (0; 0)).
ae
EXAMPLE (Delooping_Homotopy_Equivalences_) Suppose that X are path conne*
*cted and nu
Y
merably contractible. Let f : X ! Y be a continuous function. Fix x0 2 X and pu*
*t y0 = f(x0)_then
f : X ! Y is a homotopy equivalence iff f : X ! Y is a homotopy equivalence. In*
* fact, the necessity
is true without any restriction on X or Y (cf. p. 425). Turning to the suff*
*iciency, write f = q O s,
where q : Wf ! Y . Since s is a homotopy equivalence, one need only deal with q*
*. Form the pullback
X xY?Y !Y? ae
square y y p1. The map X ! X xY Y is a morphism in TOP =X whi*
*ch, when
X !fY oe ! (oe(1); f O oe)
restricted to the fibers over x0, is f, thus is a fiber homotopy equivalence (c*
*f. Proposition 20). In
WfA__*
*_______wPY
particular: X xY Y is contractible. Consider now the commutative triangle qA*
*C p1 . The
*
* Y
fiber of p1 over y0 is contractible; on the other hand, the fiber of q over y0 *
*is homeomorphic to X xY Y
(parameter reversal). The arrow Wf ! PY is therefore a homotopy equivalence (cf*
*. Proposition 20). But
p1 is a homotopy equivalence, hence so is q.
EXAMPLE (H_Groups_) In any H group (= cogroup object in HTOP *), the ope*
*rations of
left and right translation are homotopy equivalences (so all path components ha*
*ve the same homo
topy type). Conversely, let (X; x0) be a nondegenerate homotopy associative H s*
*pace with the property
that the operations of left and right translation are homotopy equivalences. A*
*ssume: X is numerably
contractible_thenaXeadmits a homotopy inverse, thus is an H group. To see this,*
* consider the shearing
map sh: X x X ! X x X . Agreeing to view X x X as an object in TOP =X via t*
*he first pro
(x; y) ! (x; xy)
jection, Proposition 20 implies that sh is a homotopy equivalence over X. There*
*fore sh is a homotopy
equivalence or still, sh is a pointed homotopy equivalence, (X x X; (x0; x0)) b*
*eing nondegenerate (cf. p.
335). Consequently, X is an H group.
[Note: If (X; x0) is a homotopy associative H space and if ss0(X) is a grou*
*p, then the operations of
left and right translation are homotopy equivalences.]
Example: Let K be a compact ANR. Denote by HE(K) the subspace of C(K; K) (c*
*ompact open
topology) consisting of the homotopy equivalences_then HE(K) is open in C(K; K)*
*, hence is an ANR (cf.
428
x6, Proposition 6). In particular: (HE(K); idK) is wellpointed (cf. p. 614) an*
*d numerably contractible
(cf. p. 313). Because HE(K) is a topological semigroup with unit under composi*
*tion and ss0(HE(K))
is a group, it follows that HE(K) is an H group.
EXAMPLE (Small_Skeletons_) In algebraic topology, it is often necessary t*
*o determine whether
a given category has a small skeleton. For instance, if B is a connected, local*
*ly path connected, locally
simply connected space, then the full subcategory of TOP =B whose objects are t*
*he covering projections
X ! B has a small skeleton. Here is a less apparent example. Fix a nonempty t*
*opological space F.
Given a numerably contractible topological space B, let FIBB;F be the category *
*whose objects are the
Hurewicz fibrations X ! B such that 8 b 2 B, Xb has the homotopy type of F, and*
* whose morphisms
X ! Y are the fiber homotopy classes [f] : X ! Y . The functor FIBB;F ! FIBB0;F*
*determined by a
homotopy equivalence 0: B0! B induces a bijection Ob____FIBB;F! Ob____FIBB0;F, *
*hence FIBB;F has a
small skeleton iff this is the case of FIBB0;F. ae
Claim: Consider a 2source B1OE1B0OE2!B2, where B0, B1 are numerably cont*
*ractible. Suppose
ae B2
that FIBB0;F, FIBB1;F have small skeletons_then FIBMOE ;OE;Fhas a small skele*
*ton.
FIBB2;F 1 2 *
* ae
[Observing that the double mapping cylinder MOE1;OE2is numerably contractib*
*le, write OE1 = r1O i1,
ae ae *
* OE2 = r2O i2
where r1are homotopy equivalences and i1are closed cofibrations (cf. x3, Pr*
*oposition 16). There is
r2 i2
Mi1? i1 B0 i2!Mi2?
a commutative diagram ry1 k yr2 and the arrow Mi1;i2! MOE1;OE2i*
*s a homotopy
B1 OE B0 ! B2
1 OE2 ae
equivalence (cf. p. 324). Thus one can assume that OE1are closed cofibration*
*s. But then if B is defined
OE OE2
B0? 2! B2?
by the pushout square OEy1 y , the arrow MOE1;OE2! B is a homotopy equiva*
*lence (cf. x3, Propo
B1 ! Bae *
* ae
sition 18). So, with B0 = B1\ B2; B1 B , take an X in FIBB;F and put X0 = X*
*B0, X1 = XB1
B2 B *
* X2 = XB2
X1? 1 X0? 2! X2? ae
to get a commutative diagram y y y in which 1 are closed co*
*fibrations (cf.
B1 OE B0 ! B2 2
1 OE2ae ae
Proposition 11). In the skeletons of FIBB0;F, FIBB1;F, choose objects Y0, Y*
*1 and fiber homotopy
ae ae FIBB2;F Y*
*2ae
equivalences f0 : Y0 ! X0, f1 : Y1 ! X1: p1O f1 = q1(obvious notation). Let*
* g1 : X1 ! Y1be
f2 : Y2 ! X2 p2O f2 = q2 *
* g2 : X2 ! Y2
429
ae ae ae *
* ae
a fiber homotopy inverse for f1. Set F1 = g1O 1O f0: f1O F1 ' 1O f0. Wr*
*ite F1 = 1O l1,
ae f2 F2 = g2O 2O f0 f2O F2a'e 2O f0 *
* F2 = 2O l2
where 1 are Hurewicz fibrations and homotopy equivalences and l1 are close*
*d cofibrations (cf.
2 ae __ __ ae__ l2 ae
p. 412), say l1 : Y0 ! Y1&_1 : Y1!_Y1 . Here: Y1_is an object in TOP =B1*
* with projec
ae l2a:eY0 ! Y2&_2 : Y2! Y2 Y2 TOP =B2
tion q1O 1 and f1O 1 : Y1!_X1 is a fiber homotopy equivalence (cf. Proposit*
*ion 15). Change
ae q2O 2 f2O 2 : Y2!aX2e ae ae
f1O 1 by a homotopy over B1 into a map G1 such that G1O l1 = 1O f0. F*
*orm the pushout
f2O 2 l __ B2 G2 G2O l2 = 2O f0
Y0? 2! Y2?
square ly1 y _then Y is in TOP =B and there is a fiber homotopy equivale*
*nce f : Y ! X,
__Y
1 ! Y
i.e., this process picks up all the isomorphism classes in FIBB;F.]
Example: Let B be a CW complex_then B is numerably contractible (cf. p. 31*
*3) and FIBB;F
has a small skeleton. In fact, B = colimB(n), so by induction, FIBB(n);Fhas a s*
*mall skeleton 8 n. On
the other hand, B and telB have the same homotopy type (cf. p. 312) and telB i*
*s a double mapping
cylinder calculated on the B(n)(cf. p. 323).
FACT Let X be in TOP =B. Suppose that U = {Ui : i 2 I} is a numerable cove*
*ring of X such
T
that for every nonempty finite subset F I, the restriction of p to Ui is a *
*Hurewicz fibration_then
i2F
p : X ! B is a Hurewicz fibration.
[Equip I with a well ordering < and use the SegalStasheff construction to *
*produce a lifting function
: Wp ! PX. Compare this result with Proposition 13 when I = {1; 2}.]
The property of being a Hurewicz fibration is not a fiber homotopy type inv*
*ariant, i.e.,
if X and Y have the same fiber homotopy type and if p : X ! B is a Hurewicz fib*
*ration,
then q : Y ! B need not be a Hurewicz fibration. Example: Take X = [0; 1] x *
*[0; 1],
Y = ([0; 1] x {0}) [ ({0} x [0; 1]), B = [0; 1], and let p; q be the vertical p*
*rojections_then
X and Y are fiberwise contractible and p : X ! B is a Hurewicz fibration but q *
*: Y ! B is
not a Hurewicz fibration. This difficulty can be circumvented by introducing st*
*ill another
notion of "fibration".
Let X be in TOP =B. Let Y be in TOP _then the projection p : Xa!eB is sai*
*d to
have the HLP w.r.t. Y up_to_homotopy_if given continuous functions Fh::YI!YX!*
* Bsuch
that p O F = h O i0, there is a continuous function H : IY ! X such that F 'BH *
*O i0 and
p O H = h.
[Note: To interpret the condition F 'BH O i0, view Y as an object in TOP *
*=B with
projection p O F .]
430
LEMMA The projectionape: X ! B has the HLP w.r.t. Y up to homotopy iff giv*
*en
continuous functions Fh::YI!YX! Bsuch that p O F = h O it (0 t 1=2), there *
*is a
continuous function H : IY ! X such that F = H O i0 and p O H = h.
Let X be in TOP =B_then p : X ! B is said to be a Dold_fibration_if it has*
* the
HLP w.r.t. Y up to homotopy for every Y in TOP . Obviously, Hurewicz ) Dold, b*
*ut
Dold 6) Serre and Serre 6) Dold. The pullback of a Dold fibration is a Dold fib*
*ration and
the localglobal principle remains valid.
PROPOSITION 21 LetaX;eY be in TOP =B and suppose that q : Y ! B is a Dold
fibration. Assume: 9 fg22CBC(X; Y ): g O f ' idX_then p : X ! B is a Dold fibr*
*ation.
B (Y; X) B ae
[Fix a topological space E and continuous functions ::EI!EX! B such that*
* p O =
O i0. Since q O f = p, 9 G : IE ! Y with f O 'BG O i0 and q O G = . Put
= g O G : 'Bg O f O 'B O i0 & p O = p O g O G = q O G = .]
The property of being a Dold fibration is therefore a fiber homotopy type i*
*nvariant.
Example: Take X = ([0; 1] x {0}) [ ({0} x [0; 1]), B = [0; 1], and let p be th*
*e vertical
projection_then p : X ! B is a Dold fibration but not a Hurewicz fibration (nor*
* is p an
open map (cf. p. 415)).
EXAMPLE Define f : [1; 1] ! [1; 1] by f(x) = 2x  1. Put X = I[1; 1]=*
*~, where (x; 0) ~
(f(x); 1), and let p : X ! S1be the projection_then p is an open map and a Dold*
* fibration but not a
Hurewicz fibration.
FACT Suppose that B is numerably contractible, so B admits a numerable cov*
*ering {O} for which
each inclusion O ! B is inessential. Let X be in TOP =B_then the projection p :*
* X ! B is a Dold
fibration iff 8 O there exists a topological space TO and a fiber homotopy equi*
*valence XO ! O x TO over
O.
The homotopy theory of Hurewicz fibrations carries over to Dold fibrations.*
* The
proofs are only slightly more complicated. Specifically: Propositions 15, 17, 1*
*8, and 20 are
true if "Hurewicz" is replaced by "Dold".
PROPOSITION 22 Let X be in TOP =B_then X is fiberwise contractible iff p :
X ! B is a Dold fibration and a homotopy equivalence.
[The necessity is a consequence of Proposition 21 and the sufficiency is a *
*consequence
of Proposition 15.]
431
PROPOSITION 23 Let X be in TOP =B_then p : X ! B is a Dold fibration iff
fl : X ! Wp is a fiber homotopy equivalence.
[Bearing in mind that q : Wp ! B is a Hurewicz fibration, the reasoning is *
*the same
as that used in the proof of Proposition 22.]
Application: The fibers of a Dold fibration over a path connected base have*
* the same
homotopy type.
[Note: Take X = ([0; 1] x {0; 1}) [ ({0} x [0; 1]), B = [0; 1], and let p b*
*e the vertical
projection_then p : X ! B is not a Dold fibration.]
ae
EXAMPLE Let p : X ! Bbe Hurewicz fibrations_then the projection X ___B*
*Y ! B is a
q : Y ! B __
Dold fibration, hence X *B Y and X _BY have the same fiber homotopy type.
EXAMPLE Let X be a topological space. Fix a numerable covering U = {Ui: i *
*2 I} of X_then,
in the notation of p. 325, the projection pU : BU ! X is a Dold fibration (for*
* BU, as an object in
TOP =X, is fiberwise contractible).
Notation: Given b0 2 B, put B0 = B  {b0} and for X; Y in TOP =B, write X0;*
* Y0 in place of
Xb0; Yb0.
FACT (Expansion_Principle_) Let X be in TOP =B. Suppose that pB0 : XB0 ! *
*B0 is a Dold
fibration and b0 has a halo O B contractible to b0, with O  {b0} numerably co*
*ntractible. Assume:
r : XO ! X0 is a homotopy equivalence which 8 b 2 O induces a homotopy equivale*
*nce rb : Xb ! X0_
thenathereeexists a Y in TOP =B and an embeddingaXe! Y over B such that q : Y !*
* B is a Dold fibration
and X is a strong deformation retract of Y .
X0 Y0
X00 ! X0
? ?
[The commutative diagram y y (X00= O x X0) is a pullback square. Si*
*nce O ! b0 is a
O ! b0
homotopy equivalence, X00! X0 is a homotopy equivalence (cf. p. 424), thus the*
* arrow r0: XO ! X00
defined by x ! (p(x); r(x)) is a homotopy equivalence. Let Y be the double map*
*ping cylinder of the
0
2source X XO r!X00: Y is in TOP =B and there is an embedding X ! Y over B. I*
*t is a closed
cofibration. YO is the mapping cylinder of r0, so XO is a strong deformation r*
*etract of YO (cf. x3,
Proposition 17). Therefore X is a strong deformation retract of Y (cf. x3, Prop*
*osition 3). Similar remarks
apply to X0 and Y0. Finally, to see that q is a Dold fibration, note that {O; B*
*0} is a numerable covering
of B. Accordingly, taking into account the localglobal principle, it is enough*
* to verify that qO : YO ! O
and qB0 : YB0 ! B0 are Dold fibrations. Consider, e.g., the latter. The hypothe*
*ses on r, in conjunction
432
with Proposition 20, imply that the embedding XB0 ! YB0 is a fiber homotopy equ*
*ivalence. But pB0 is
a Dold fibration, hence the same holds for qB0.]
Let f : X ! Y be a pointed continuous function_then the mapping_fiber_Ef of*
* f is
Ef? ! Wf?
defined by the pullback square y yq , i.e., Ef is the double mapping*
* track of
{x0} !f Y
the 2sink X f!Y {y0}. Example: The mapping fiber E0 of 0 : X ! Y is X x Y .
EXAMPLE Let f : X ! Y be a pointed continuous function. Assume: f is a Hur*
*ewicz fibration.
Denote by Cy0 the mapping cone of the inclusion Xy0! X_then the mapping fiber o*
*f Cy0! Y has the
same homotopy type as Xy0* Y (cf. p. 420 ff.).
FACT Let X p!B q Y be a 2sink. Denote by W ___the mapping track of the *
*projection X ___BY !
B_then Wp *B Wq and W ___have the same fiber homotopy type.
Application: The mapping fiber of the projection X ___BY ! B has the same*
* homotopy type as
Ep * Eq.
Let f : X ! Y be a pointed continuous function_then Wf and Ef are pointed s*
*paces,
the base point in either case being (x0; j(y0)). The pointed homotopy type of W*
*f or Ef
depends only on the pointed homotopy class of f. The projection q : Wf ! Y is a*
* pointed
Hurewicz fibration and the restriction ss of the projection p : Wf ! X to Ef is*
* a pointed
Hurewicz fibration with ss1 (x0) = Y . By construction, f O ss is nullhomotopi*
*c and for
any g : Z ! X with f O g nullhomotopic, there is a OE : Z ! Ef such that g = ss*
* O OE.
When is a pointed continuous function which is a Hurewicz fibration actuall*
*y a pointed Hurewicz
fibration? Regularity, suitably localized, is what is relevant. Thus let p : X *
*! B be a Hurewicz fibration
taking x0 to b0. Assume: 9 a lifting function such that (x0; j(b0)) = j(x0)_t*
*hen p is a pointed
Hurewicz fibration.
[Note: For this, it is sufficient that {b0} be a zero set in B, any Hurewic*
*z fibration p : X ! B
automatically becoming a pointed Hurewicz fibration 8 x0 2 Xb0(argue as on p. 4*
*14). The condition is
satisfied if the inclusion {b0} ! B is a closed cofibration.]
ae
LEMMA Let X; Y; Z be pointed spaces; let f : X ! Zbe pointed continuous *
*functions_then
ae g : Y ! Z
the projections Wf;g! X & Wf;g! X x Y are pointed Hurewicz fibrations, the b*
*ase point of Wf;g
Wf;g! Y
being the triple (x0; y0; j(z0)).
433
[To deal with p : Wf;g! X, define a lifting function : Wp ! PWf;gby ((x; y*
*; o); oe)(t) =
(oe(t); y; ot), where ae
fiO oe(tj 2T)(0 T t=2)
ot(T) = o 2T__t :
2  t (t=2 T 1)
Obviously, ((x0; y0; j(z0)); j(x0)) = j(x0; y0; j(z0)), so p : Wf;g! X is a poi*
*nted Hurewicz fibration.]
X0? ! X?
PROPOSITION 24 Consider the pullback square y yp, where p is a Hur*
*ewicz
B0 !0 B
ae ae
fibration. Suppose that XB & B0 are wellpointed, that the inclusions {x0}{*
*!bX &
*
* 0} ! B
{b00} ! B0 are closed, and that p(x0) = b0 = 0(b00). Put x00= (b00; x0)_then th*
*e inclusion
{x00} ! X0 is a closed cofibration.
[The arrow Xb0 ! X is a closed cofibration (cf. Proposition 11). Therefor*
*e the
composite X0b00! X0 ! X is a closed cofibration. On the other hand, the compos*
*ite
{x00} ! X0b00! X0 ! X is a closed cofibration. Therefore the inclusion {x00} ! *
*X0b00is a
closed cofibration (cf. x3, Proposition 9). But the arrow X0b00! X0 is a closed*
* cofibration
(cf. Proposition 11), thus the inclusion {x00} ! X0 is a closed cofibration.]
ae
Application: Let f : X ! Y be a pointed continuous function. Assume: XY*
* are
wellpointed with closed base points_then Wf and Ef are wellpointed with closed *
*base
points.
[P Y is wellpointed with a closed base point (cf. x3, Proposition 6).]
FACT Let f : X ! Y be a pointed continuous function. Suppose that OE : X0!*
* X ( : Y ! Y 0)
is a pointed homotopy equivalence_then the arrow EfOOE! Ef (Ef ! E Of) is a poi*
*nted homotopy
equivalence.
Application: Let X be wellpointed with {x0} X closed_then the mapping fibe*
*r of the diagonal
embedding X ! X x X has the same pointed homotopy type as X. ae
[The embedding j : X ! PX is a pointed homotopy equivalence and : PX ! X*
* x X is a
oe ! (*
*oe(0); oe(1))
pointed Hurewicz fibration.]
ae ae
EXAMPLE Let X be wellpointed with {x0} X closed.
Y {y0} Y
(1) The mapping fiber of the inclusion X _ Y ! X x Y has the same poin*
*ted homotopy type
as X * Y .
434
(2) The mapping fiber of the projection X _ Y ! Y has the same pointed*
* homotopy type as
X x Y={x0} x Y .
[In both situations, replace by as on p. 416.]
ae
FACT Let f : X ! Ybe pointed continuous functions_then there is a homoto*
*py equivalence
g : Y ! Z
EgOf! W, where W is the double mapping track of the 2sink X f!Y ssEg.
EgOf ! Eg ! *
? ? ?
[Consider the diagram y y y .]
X ! Y ! Z
Let f : X ! Y be a pointed continuous function, Ef its mapping fiber.
LEMMA If f is a pointed Hurewicz fibration, then the embedding Xy0 ! Ef is*
* a
pointed homotopy equivalence.
In general, there is a pointed Hurewicz fibration ss : Ef ! X and an embedd*
*ing
Y ! Ef. Iterate to get a pointed Hurewicz fibration ss0 : Ess! Ef_then the tr*
*ian
Ess____wEfu
gle  [[] commutes and by the lemma, the vertical arrow is a pointed homot*
*opy
Y
equivalence. Iterate again to get a pointed Hurewicz fibration ss00: Ess0! Ess_*
*then the
uEss0___wEss
triangle [[] commutes and by the lemma, the vertical arrow is a pointed h*
*omotopy
X ae
equivalence. Example: Given pointed spaces XY, let X[Y be the mapping fiber o*
*f the
inclusion f : X _ Y ! X x Y _then in HTOP *, Ess (X x Y ) and Ess0 (X _ Y ).
ae ae
LEMMA Let X be wellpointed with {x0} X closed. Denote by S the subspa*
*ce of X * Y
Yae {y0} Y
consisting of the [x; y0;_t]then X * Y=S = (X#Y ) and the projection X * Y ! *
*X * Y=S is a pointed
[x0; y; t]
homotopy equivalence.
[Note: The base point of X * Y is [x0; y0; 1=2] and is the pointed suspens*
*ion.]
ae ae
Application: Let X be wellpointed with {x0} X closed_then X[Y has the*
* same pointed
Y {y0} Y
homotopy type as (X#Y ).
EXAMPLE Suppose that X and Y are nondegenerate_then the Puppe formula says*
* that in
HTOP *, (X x Y ) X _ Y _ (X#Y ), and by the above, (X#Y ) X[Y .
435
*
* __
EXAMPLE (The_Flat_Product_) In contrast to the smash product # (or its mo*
*dification #),
the flat product [ does not possess the properties that one might expect to hol*
*d by analogy. Specifically,
for nondegenerate spaces, it is generally false that in HTOP *: (1) (X[Y )[Z *
* X[(Y [Z); (2) (X x
Y )[Z (X[Z) x (Y [Z); (3) (X[Y ) X[Y . Counterexamples: (1) Take X = Y = P1 *
*(C ), Z =
P 1(H ); (2) Take X = Y = Z = P 1(C ); (3) Take X = Y = P 1(C ). Look, e.g.,*
* at (1). Using
the fact that P 1(C ) S1, P 1(H ) S3, compute: P1(C )[P 1(C ) P 1(C ) * P 1*
*(C )
S1 * S1 S3 & S3[P 1(H ) S3 * S3 (S3#S3) S3#S3 S3#3S0 4S3#S0
4S3 ) (P 1(C )[P 1(C ))[P 1(H ) 4S3. Similarly, P1 (C )[(P 1(C )[P 1(H )) 2S*
*5. The
singular homology functor H8(_; Z) distinguishes these spaces: H8(4S3; Z) Z, H*
*8(2S5; Z) = 0.
Let f : X ! Y be a pointed continuous function_then the mapping_fiber_seque*
*nce_
associated with f is given by . .!.2Y ! Ef ! X ! Y ! Ef ! X f!Y . Example:
When f = 0, this sequence becomes . .!.2Y ! X x2Y ! X ! Y ! X xY !
X 0!Y .
X? f! Y?
[Note: If the diagram y y commutes in HTOP *and if the vertical *
*arrows
X0 !f0Y 0
are pointed homotopy equivalences, then the mapping fiber sequences of f and f0*
* are
connected by a commutative ladder in HTOP *, all of whose vertical arrows are*
* pointed
homotopy equivalences.]
FACT Let f : X ! Y be a pointed Hurewicz fibration. Assume: The inclusion*
* Xy0 ! X is
nullhomotopic_then Y has the same pointed homotopy type as Xy0x X.
[For ss : Ef ! X is nullhomotopic, thus in HTOP *: Ess Ef x X ) Y Xy0x X.]
REPLICATION THEOREM Let f : X ! Y be a pointed continuous function_then
for any pointed space Z, there is an exact sequence
. .!.[Z; X] ! [Z; Y ] ! [Z; Ef] ! [Z; X] ! [Z; Y ]
in SET *.
If f : X ! Y is a pointed Dold fibration or if f : X ! Y is a Dold fibratio*
*n and Z is
nondegenerate, then in the replication theorem one can replace Ef by Xy0 (cf. p*
*. 318).
This replacement can also be made if f : X ! Y is a Serre fibration provided th*
*at Z is
a CW complex (cf. infra). In particular, when f : X ! Y is either a Dold fibrat*
*ion or a
Serre fibration, there is an exact sequence
. .!.ss2(Y ) ! ss1(Xy0) ! ss1(X) ! ss1(Y ) ! ss0(Xy0) ! ss0(X) ! ss0(Y ):
436
LEMMA Let f : X ! Y be a pointed continuous function. Assume: f is a Serre*
* fibration_then
for every pointed CW complex Z, the arrow [Z; Xy0] ! [Z; Ef] is a pointed bijec*
*tion.
[Proposition 12 is true for Serre fibrations if the "cofibration data" is r*
*estricted to CW complexes.]
Examples:aSupposeethat f : X ! Y is either a Dold fibration or a Serre fib*
*ration,
where XY6=6;=.;(1) If Xy0 is simply connected, then 8 x0 2 Xy0, ss1(X; x0) s*
*s1(Y; y0);
(2) If X is simply connected, then 8 y0 2 f(X), there is a bijection ss1(Y; y0)*
* ! ss0(Xy0);
(3) If X is path connected and if Y is simply connected, then 8 y0 2 Y , ss0(Xy*
*0) = *; (4)
If Y is path connected and Xy0 is path connected, then X is path connected.
LEMMA Let f : X ! Y be a Hurewicz fibration. Fix y0 2 f(X) & x0 2 Xy0 and *
*let (Z; z0) be
wellpointed with {z0} Z closed_then there is a left action ss1(X; x0)x[Z; z0; *
*Xy0; x0] ! [Z; z0; Xy0; x0].
[Represent ff 2 ss1(X; x0) by a loop oe 2 (X; x0). Given OE : (Z; z0) ! (X*
*y0; x0), consider the
i0Z?[ I{z0}F! X? ae
commutative diagram y yf , where F(z; t) = (i O OE)(z)(t =(0)i*
* the inclusion
IZ !h Y oe(t) (z = z0)
Xy0 ! X) and h(z; t) = (f O oe)(t). Proposition 12 says that this diagram has a*
* filler H : IZ ! X. Put
(z) = H(z; 1) to get a pointed continuous function : (Z; z0) ! (Xy0; x0). De*
*finition: ff . [OE] = [ ].]
[Note: There is a left action ss1(X; x0)x[Z; z0; X; x0] ! [Z; z0; X; x0] an*
*d a left action ss1(Xy0; x0)x
[Z; z0; Xy0; x0] ! [Z; z0; Xy0; x0] (cf. p. 318). The arrow [Z; z0; Xy0; x0] !*
* [Z; z0; X; x0] induced by the
inclusion Xy0 ! X is a morphism of ss1(X; x0)sets and the operation of ss1(Xy0*
*; x0) on [Z; z0; Xy0; x0]
coincides with that defined via the homomorphism ss1(Xy0; x0) ! ss1(X; x0).]
EXAMPLE Let f : X ! Y be a Hurewicz fibration. Fix y0 2 f(X) & x0 2 Xy0and*
* n 1_then
there is a left action ss1(X; x0) x ssn(X; x0) ! ssn(X; x0), a left action ss1(*
*X; x0) x ssn(Y; y0) ! ssn(Y; y0),
and a left action ss1(X; x0) x ssn(Xy0; x0) ! ssn(Xy0; x0). All the homomorphis*
*ms in the exact sequence
. .!.ssn+1(Y; y0) ! ssn(Xy0; x0) ! ssn(X; x0) ! ssn(Y; y0) ! . .*
* .
are ss1(X; x0)homomorphisms.
[Note: Suppose that Xy0 is path connected_then there is a left action ss1(Y*
*; y0) x ss*n(Xy0; x0) !
ss*n(Xy0; x0), where ss*n(Xy0; x0) is ssn(Xy0; x0) modulo the (normal) subgroup*
* generated by the ff . 
(ff 2 ss1(Xy0; x0), 2 ssn(Xy0; x0)).]
EXAMPLE Let f : X ! Y be a Hurewicz fibration. Fix y0 2 f(X) & x0 2 Xy0_th*
*en ss1(Y; y0)
operates to the left on ss0(Xy0) and the orbits are the fibers of the arrow ss0*
*(Xy0) ! ss0(X).
FACT Let f : X ! Y be a Hurewicz fibration. Fix y0 2 f(X) & x0 2 Xy0_then*
* 8 n 1,
ss1(Xy0; x0) operates trivially on ker(ssn(Xy0; x0) ! ssn(X; x0)).
437
ae
EXAMPLE (MayerVietoris_Sequence_) Let X; Y; Z be pointed spaces; let f *
*: X ! Zbe point
g *
*: Y ! Z
ed continuous functions_then the projection Wf;g! X x Y is a pointed Hurewicz f*
*ibration (cf. p. 432)
and there is a long exact sequence . .!.ssn+1(Z) ! ssn(Wf;g) ! ssn(X) x ssn(Y )*
* ! ssn(Z) ! . . .
! ss2(Z) ! ss1(Wf;g) ! ss1(X) x ss1(Y ) ! ss1(Z) ! ss0(Wf;g) ! ss0(X x Y ).
[Note: It follows that if X and Y are path connected and if every fl 2 ss1(*
*Z) has the form fl =
f*(ff) . g*(fi) (ff 2 ss1(X); fi 2 ss1(Y )), then Wf;gis path connected.]
If f : X ! Y is either a Dold fibration or a Serre fibration, then the hom*
*otopy
groups of X and Y are related to those of the fibers by a long exact sequence. *
*As for the
homology groups, there is still a connection but it is intricate and best expre*
*ssed in terms
of a spectral sequence.
[Note: In the simplest case, viz. that of a projection Y xT ! Y , the K"unn*
*eth formula
computes the homology of Y x T in terms of the homology of Y and the homology o*
*f T .]
EXAMPLE Let f : X ! Y be a Hurewicz fibration, where X is nonempty and Y *
* is path
connected. Fix y0 2 Y _then 8 q 1, the projection (X; Xy0) ! (Y; y0) induces a*
* bijection ssq(X; Xy0) !
ssq(Y; y0). The analog of this in homology is false. Consider, e.g., the Hopf m*
*ap S2n+1! Pn(C ) with fiber
S1 : Hq(S2n+1; S1) = 0 (2 < q 2n) & H2q(P n(C )) Z (1 < q n). However, a par*
*tial result holds
in that if Xy0 is nconnected and Y is mconnected, then the arrow Hq(X; Xy0) !*
* Hq(Y; y0) induced by
the projection (X; Xy0) ! (Y; y0) is bijective for 1 q < n + m + 2 and surject*
*ive for q = n + m + 2.
Consequently, under these conditions, there is an exact sequence
Hn+m+1 (Xy0) ! Hn+m+1 (X) ! Hn+m+1 (Y ) ! Hn+m (Xy0) ! . . .
! H2(Y ) ! H1(Xy0) ! H1(X) ! H1(Y ):
[One can assume that the inclusion {y0} ! Y is a closed cofibration (pass t*
*o a CW resolution
K ! Y ), hence that the inclusion Xy0 ! X is a closed cofibration (cf. Proposit*
*ion 11). The mapping
cone of the latter is path connected and the mapping fiber of Cy0 ! Y has the s*
*ame homotopy type as
Xy0* Y (cf. p. 432), which is (n + m + 1)connected (cf. p. 340). Thus the ar*
*row Cy0 ! Y is an
(n + m + 2)equivalence, so the Whitehead theorem implies that the induced map *
*Hq(Cy0) ! Hq(Y ) is
bijective for 0 q < n + m + 2 and surjective for q = n + m + 2. But the projec*
*tion Cy0 ! X=Xy0 is a
homotopy equivalence (cf. p. 324) and Hq(X; Xy0) Hq(X=Xy0; *) (cf. p. 38).]
Application: Suppose that X is (n + 1)connected_then Hq(X) Hq1(X) (2 q *
* 2n + 2).
[Note: It is a corollary that if X is nondegenerate and nconnected, then t*
*he arrow of adjunction
e : X ! X induces an isomorphism Hq(X) ! Hq(X) for 0 q 2n + 1. Therefore, by*
* the
Whitehead theorem, the suspension homomorphism ssq(X) ! ssq+1(X) is bijective f*
*or 0 q 2n and
surjective for q = 2n + 1 (Freudenthal).]
438
Let X be a topological space, sinX its singular set_then sinX can be regard*
*ed
ffn
m _________w'
as a category: ') [[^ (ff 2 Mor ([m]; [n])). The objects of [(sinX)*
*OP ; AB ]
X
are called coefficient_systems_on X. Given a coefficient system G, the singular*
* homology
H*(X; G) of X with coefficients in G is by definition the homology of the chain*
* complex
M @ M @ M @
Goe0 Goe1 Goe2 . .;.
oe02sin0X oe12sin1X oe22sin2X
Xn L
where @ = (1)i Gdi.
0 oen2sinnX
[Note: To interpret Gdi, recall that there are arrows di : sinnX ! sinn1X*
* cor
responding to the face operators ffii : [n  1] ! [n] (0 i n). So, 8 oe 2 si*
*nnX,
Gdi: G(n oe!X) ! G(n1 dioe!X).] ae
Example: Fix an abelian group G and define GG by GGGoef=fG _then H*(X; G*
*G ) =
G = idG
H*(X; G), the singular homology of X with coefficients in G.
A coefficient system G is said to be locally_constant_provided that 8 ff, G*
*ffis in
vertible. LCCS X is the full subcategory of [(sinX)OP ; AB ] whose objects ar*
*e the locally
constant coefficient systems on X.
[Note: A coefficient system G is said to be constant_if for some abelian gr*
*oup G, G is
isomorphic to GG .]
Suppose that X is locally path connected and locally simply connected_then *
*the category of locally
constant coefficient systems on X is equivalent to the category of locally cons*
*tant sheaves of abelian groups
on X.
PROPOSITION 25 LCCS X is equivalent to [(X)OP ; AB ].
[We shall define a functor G ! G from LCCS aXe to [(X)OP ; AB ] and a func*
*tor
G ! Gsinfrom [(X)OP ; AB ] to LCCS X such that (G(G)sin G .
sin) G
Definition of G : Given x 2 X, put G x = Goex, where oex 2 sin0X with oex*
*(0) = x.
Givenaaemorphism [oe] : x ! y, put G [oe] = (Gd1) O (Gd0)1, where oe 2 sin1X *
*with
d1oe = x
d0oe = y. In other words,aGe [oe] is the composite Gy ! Goe ! Gx. Note that G*
* [oe] is
0 aed oe0= x = d oe00
welldefined. Indeed, if oeoe002 sin1X with d1 0 1 00and [oe0] = [oe*
*00], then there
ae 0oe = y = d0oe
0
exists a o 2 sin2X such that d1od= oe an00d s0d0oe0= d0o = s0d0oe00.
2o = oe
439
Definition of Gsin: Given oe 2 sinnX, put Gsinoe = G(enoe(0)), where en : s*
*innX !
sin0X is the arrow associatedfwithfthe vertex operator ffln : [0] ! [n] that se*
*nds 0 to n.
m _________w'')n
Given a morphism o [[^oe , put Gsinff= G(oe O ), where : [1] ! [n] is
X
ae
defined by (0)(=1ff(m)):=one O is a path in X which begins at em o(0) and e*
*nds at
enoe(0).]
Because of this result, one can always pass back and forth between locally *
*constant
coefficient systems on X and cofunctors X ! AB . The advantage of dealing with *
*the
latter is that in practice a direct description is sometimes available. For ex*
*ample, fix
n 2 and assign to each x 2 X the homotopy group ssn(X; x)_then every morphism
[oe] : x ! y determines an isomorphism ssn(X; y) ! ssn(X; x) and there is a cof*
*unctor
ssnX : X ! AB .
[Note: Suppose that G is in [(X)OP ; AB ]_then 8 x0 2 X, the fundamental gr*
*oup
ss1(X; x0) operates to the right on Gx0 : Gx0 x ss1(X; x0) ! Gx0. Conversely, i*
*f X is path
connected and if G0 is an abelian group on which ss1(X; x0) operates to the rig*
*ht, then
there exists a G in [(X)OP ; AB ], unique up to isomorphism, with Gx0 = G0 and *
*inducing
the given operation of ss1(X; x0) on G0.]
Application: On a simply connected space, every locally constant coefficien*
*t system
is isomorphic to a constant coefficient system.
EXAMPLE Let f : X ! Y be a Hurewicz fibration_then 8 q 0, there is a cofu*
*nctor Hq(f) :
Y ! AB that assigns to each y 2 Y theasingularehomology group Hq(Xy) of the fi*
*ber Xy. Thus let
[o] : y0 ! y1 be a morphism. Case 1: y0 62 f(X). In this situation, Xy0 & Xy1*
* are empty, hence
y1 ae
Hq(Xy0) = 0 = Hq(Xy1). Definition: Hq(f)[o] is the zero morphism. Case 2: y*
*0 2 f(X). Fix a
ae ae y1
homotopy : IXy0 ! X such that f O (x; t) = o(t)_then the arrow Xy0! Xy1 i*
*s a homotopy
(x; 0) = x x ! (x; 1)
equivalence. Definition: Hq(f)[o] is the inverse of the induced isomorphism Hq(*
*Xy0) ! Hq(Xy1) (it is
independent of the choices).
LEMMA Suppose that X is path connected. Given a locally constant coeffici*
*ent
system G, fix x0 2 X, put G0 = Gx0, and let H0 be the subgroup of G0 generated *
*by the
g  g . ff (g 2 G0; ff 2 ss1(X; x0))_then H0(X; G) G0=H0.
440
Let f : X ! Y and f0 : X0 ! Y 0be a pair of continuous functions. CallaHome*
*(f0; f)
*
*n x X0; X)
the simplicial set specified by taking for Hom (f0; f)n the set of all uv22C(*
*C(n x Y 0; Y )
n x X0? u! X? ae
such that the diagram idxfy0 yf commutes and define disin the obv*
*ious
n x Y 0 !v Y i
way.
Now specialize, putting Y 0= 0, so f0 : X0 ! 0 is the constant map, and wri*
*te
Hom (X0; f) in place of Hom(f0; f). In succession, let X0 = 0; 1; : :t:o obtain*
* a sequence
of simplicial sets and simplicial maps:
Hom(0; f) Hom(1; f) Hom (2; f) . .:.
Here, the arrows come from the face operators [0]!! [1]; [1] !!![2]; . ...This *
*data generates
a double chain complex Koo = {Kn;m :ane 0; m 0} of abelian groups if we write
Kn;m = Fab(Hom (n; f)m ) and define @I@: Kn;m ! Kn1;m as follows.
II : Kn;m ! Kn;m1
!. !.
(@I) The arrows Hom (n; f)m ..Hom(n1; f)m lead to arrows Kn;m ..Kn1*
*;m.
! !
Take for @I their alternating sum multiplied by (1)m .
!. !.
(@II) The arrows Hom(n;f)m ..Hom(n;f)m1 lead to arrows Kn;m ..Kn;m1.
! !
Take for @II their alternating sum.
One can check that @I O @I = 0 = @IIO @II and @I O @II+ @IIO @I = 0. Form t*
*he total
L
chain complex Ko = {Kp} : Kp = Kn;m, where @ = @I + @II_then there are fi*
*rst
n+m=p
quadrant spectral sequences
( 2
IEp;q= IHp(IIHq(Koo)) ) Hp+q(Ko)
:
IIE2p;q= IIHp(IHq(Koo)) ) Hp+q(Ko)
ae
LEMMA IE2p;q Hq(X)0 (p(=p0)>.0)
[From the definitions, sinX = Hom(0; f). On the other hand, each projection*
* n !
0 is a homotopy equivalence and induces an arrow sinX ! Hom(n; f). Since there *
*are
sinX? id! sinX?
n + 1 commutative diagrams y y , passing to homology *
*per
Hom (n; f) ! Hom(n1; f)
@II gives
Hq(X) 0 Hq(X) id Hq(X) 0 . .:.]
(p = 0) (p = 1) (p = 2)
441
Thus the first spectral sequence IE collapses and H*(Ko) H*(X). To explica*
*te the
second spectral sequence IIE, given o 2 sinnY , let Xo be the fiber over o of t*
*he induced
map sinnX ! sinnY , i.e., Xo = {oe : f Ooe = o}. View Xo as a subspace of sinnX*
* (compact
open topology). Put Hq(f)o = Hq(Xo) and 8 ff, let Hq(f)ffbe the homomorphism on
homology defined by the arrow Xo ! XoOff_then Hq(f) is in [(sinY )OP ; AB ] or *
*still, is
a coefficient system on Y .
[Note: 8 y 2 Y , Hq(f)oy = Hq(Xy), where oy 2 sin0Y with oy(0) = y.]
LEMMA IIE2p;q Hp(Y ; Hq(f)).
[IHq(Koo) can be identified with the chain complex on which the homology of*
* Hq(f)
is computed.]
PROPOSITION 26 Suppose that f : X ! Y is a Hurewicz fibration_then Hq(f) is
locally constant. ae
n; X) ! C(m ; X)
[Fix ff 2 Mor ([m]; [n])_then ff determines arrows C(C(n; Y ) ! C(m ; Y )*
* and
C(n;?X) f*!C(n;?Y )
there is a commutative diagram y y . According to Proposi*
*tion
C(m ; X) !f C(m ; Y )
*
5, the horizontal arrows are Hurewicz fibrations. But the vertical arrows are *
*homotopy
equivalences, thus 8 o 2 C(n; Y ) the induced map Xo ! XoOff is a homotopy equi*
*va
lence (cf. p. 425), so Hq(f)ff: Hq(Xo) ! Hq(XoOff) is an isomorphism.]
[Note: Retaining the assumption that f : X ! Y is a Hurewicz fibration, on*
*e may
apply the procedure figuring in the proof of Proposition 25 to the locally cons*
*tant coefficient
system Hq(f). The result is the cofunctor Hq(f) : Y ! AB defined in the exampl*
*e on
p. 439.]
Proposition 26 is also true if f : X ! Y is either a Dold fibration or a Se*
*rre fibration.
Consider first the case when f is Dold_then Proposition 5 still holds and t*
*he validity of the relevant
homotopy theory has already been mentioned (cf. p. 430). As for the case when *
*f is Serre, note that the
arrow C(n; X) ! C(n; Y ) is again Serre (as can be seen from the proof of Propo*
*sition 5). Therefore,
thanks to the Whitehead theorem, the lemma below suffices to complete the argum*
*ent.
X? p! B? ae
LEMMA Suppose given a commutative diagramOyE y in which pare Serr*
*e fibrations
ae Y !q A q
and OE are weak homotopy equivalences_then 8 b 2 B, the induced map Xb ! Y (b)*
*is a weak homotopy
442
equivalence.
[If Xb is empty, then so is Y (b)and the assertion is trivial. Otherwise, l*
*et a = (b) and apply the
five lemma to the commutative diagram
. . .! ssq+1(B)!? ssq(Xb)?! ssq(X)?! ssq(B)?! . . .
y y y y ;
. . .! ssq+1(A)! ssq(Ya)! ssq(Y )! ssq(A) ! . . .
with the usual caveat at the ss0 and ss1 level.]
The coefficient system Hq(f) is defined in terms of the integral singular h*
*omology of
the fibers. Embelish the notation and denote it by Hq(f; Z). One may then repla*
*ce Z by
any abelian group G : Hq(f; G), a coefficient system which is locally constant *
*if f : X ! Y
is either a Dold fibration or a Serre fibration.
FIBRATION SPECTRAL SEQUENCE Let f : X ! Y be either a Dold fibration or
a Serre fibration_then for any abelian group G, there is a first quadrant spect*
*ral sequence
E = {Erp;q; dr} such that E2p;q Hp(Y ; Hq(f; G)) ) Hp+q(X; G) and 8 n; Hn(X; G)
admits an increasing filtration
0 = H1;n+1 H0;n . . .Hn1;1 Hn;0= Hn(X; G)
by subgroups Hi;ni, where E1p;q Hp;q=Hp1;q+1.
X*
*? f! Y?
[Note: The fibration spectral sequence is natural, i.e., if the diagram y*
* y
X*
*0 !f0Y 0
commutes, then there is a morphism : E ! E0 of spectral sequences such that 2p*
*;q
coincides with the homomorphism Hp(Y ; Hq(f; G)) ! Hp(Y 0; Hq(f0; G)) induced b*
*y the
arrow Hq(f; G) ! Hq(f0; G).]
WANG HOMOLOGY SEQUENCE Take Y = Sn+1(n 1) and let f : X ! Y be a Hurewicz
fibration with path connected fibers Xy_then there is an exact sequence
. .!.Hq(X) ! Hqn1(Xy) ! Hq1(Xy) ! Hq1(X) ! . .:.
EXAMPLE Suppose that n 1_then Hkn(Sn+1) Z(k = 0; 1; : :):, while Hq(Sn+1*
*) = 0
otherwise. Moreover, the Pontryagin ring H*(Sn+1) is isomorphic to Z [t], w*
*here t generates
Hn(Sn+1).
443
As formulated, the fibration spectral sequence applies to singular homology*
*. There is also a companion
result in singular cohomology (with additional multiplicative structure when th*
*e coefficient group G is a
commutative ring).
WANG COHOMOLOGY SEQUENCE Take Y = Sn+1 (n 1) and let f : X ! Y be a
Hurewicz fibration with path connected fibers Xy_then there is an exact sequence
. .!.Hq(X) ! Hq(Xy) ! Hqn(Xy) ! Hq+1(X) ! . .:.
[Note: In the graded ring H*(Xy), (ff . fi) = (ff) . fi + (1)nffff . (fi*
*).]
EXAMPLE Suppose that n 1_then : Hkn(Sn+1) ! H(k1)n(Sn+1) (k 1) is an
isomorphism and H0(Sn+1) is the infinite cyclic group generated by 1. Put ff0 =*
* 1 and define ffk (k 1)
inductively through the relation (ffk) = ffk1. Case 1: n even. One has k!ffk =*
* ffk1, therefore H*(Sn+1)
is the divided polynomial algebra generated by ff1; ff2; : :.:Case 2: n odd. On*
*e has ff21= 0, ff1ff2k =
ff2k+1; ff1ff2k+1= 0, and ffk2= k!ff2k, thus ff1 generates an exterior algebra *
*isomorphic to H*(Sn) and
ff2; ff4; : :g:enerate a divided polynomial algebra isomorphic to H*(S2n+1), so*
* H*(Sn+1) H*(Sn)
H*(S2n+1).
In what follows, we shall assume that X is nonempty and Y is path connected.
[Note: If f is Dold, then the Xy have the same homotopy type (cf. p. 431),*
* while if
f is Serre, then the Xy have the same weak homotopy type (cf. Proposition 31).]
(ED H ) Let eH : E1p;0! E2p;0be the edge homomorphism on the horizont*
*al
axis. The arrow of augmentation H0(Xy; G) ! G is independent of y, so there is*
* a ho
momorphism Hp(Y ; H0(f; G)) ! Hp(Y ; G). The composite Hp(X; G) ! Hp;0=Hp1;1
E1p;0eH!E2p;0 Hp(Y ; H0(f; G)) ! Hp(Y ; G) is the homomorphism on homology in*
*duced
by f : X ! Y .
(ED V ) Let eV : E20;q! E10;qbe the edge homomorphism on the vertical *
*axis.
Fix y 2 Y _then there is an arrow Hq(Xy; G) ! H0(Y ; Hq(f; G)). The composite
Hq(Xy; G) ! H0(Y ; Hq(f; G)) E20;qeV!E10;q! Hq(X; G) is the homomorphism on
homology induced by the inclusion Xy ! X.
Keeping to the preceding hypotheses, f : X ! Y is said to be Gorientable_p*
*rovided
that the Xy are path connected and 8 q; Hq(f; G) is constant, so 8 y the right *
*action
Hq(Xy; G) x ss1(Y; y) ! Hq(Xy; G) is trivial.
[Note: If f : X ! Y is Gorientable, then by the universal coefficient *
*theorem,
E2p;q Hp(Y ; Hq(Xy; G)) Hp(Y ) Hq(Xy; G) tor(Hp1(Y ); Hq(Xy; G)).]
444
EXAMPLE Let f : X ! Y be Gorientable. Assume: Hi(Xy0; G) = 0 (0 < i n*
*) and
Hj(Y ; Z) = 0 (0 < j m)_then there is an exact sequence
Hn+m+1 (Xy0; G) ! Hn+m+1 (X; G) ! Hn+m+1 (Y ; G) ! Hn+m (Xy0; G) ! . .*
* .
! H2(Y ; G) ! H1(Xy0; G) ! H1(X; G) ! H1(Y ; G):
[For 2 r < n + m + 2, combine the exact sequence
r
0 ! E1r;0! Err;0d!Er0;r1! E10;r1! 0
with the exact sequence
0 ! E10;r! Hr(X; G) ! E1r;0! 0;
observing that Hr(Y ; G) E2r;0 Err;0and Hr1(Xy0; G) E20;r1 Er0;r1, the arr*
*ow Hr(Y ; G) !
Hr1(Xy0; G) being the transgression.]
[Note: The above assumptions are less stringent than those imposed earlier *
*in the case G = Z (cf.
p. 437).]
EXAMPLE Let f : X ! Y be orientable, where is a principal ideal domain_t*
*hen the arrow
H*(X; ) ! H*(Y ; ) is an isomorphism iff 8 q > 0, Hq(Xy0; ) = 0 and the arrow H*
**(Xy0; ) !
H*(X; ) is an isomorphism iff 8 q > 0, Hq(Y ; ) = 0.
[Note: The formulation is necessarily asymmetric (take Y simply connected a*
*nd consider Y ! Y ).]
FACT Suppose that f : X ! Y is Zorientable_then any two of the following *
*conditions imply
the third: (1) 8 p; Hp(Y ) is finitely generated; (2) 8 q; Hq(Xy0) is finitely *
*generated; (3) 8 n; Hn(X) is
finitely generated.
FACT Suppose that f : X ! Y is Zorientable_then any two of the following *
*conditions imply
the third: (1) 8 p > 0; Hp(Y ) is finite; (2) 8 q > 0; Hq(Xy0) is finite; (3) 8*
* n > 0; Hn(X) is finite.
ae
Given pointed spaces XY, the mapping fiber sequence associated with the i*
*nclusion
f : X _ Y ! X x Y reads: . .!.(X _ Y ) ! (X x Y ) ! X[Y ! X _ Y ! X x Y .
[Note: The homology of (X _ Y ) can be calculated in terms of the homology *
*of X
and Y (AguadeCastellety).]
LEMMA The arrow F : (X x Y ) ! X[Y is nullhomotopic.
_________________________
yCollect. Math. 29 (1978), 36; see also DulaKatz, Pacific J. Math. 86 (198*
*0), 451461.
445
ae__ ae__
[Put X = {oe : oe([1=2; 1]) =_x0}then the inclusions X ! X are p*
*ointed
__Y = {o : o([0; 1=2]) = y0} __Y_! Y
homotopy equivalences, hence the same holds for their product: X x __Y ! X x *
*Y =
__
(X xY ). Use two parameter reversals to see that the composite X x__Y ! (X _Y *
*) !
__ F
X[Y is equal to the composite X x __Y ! (X x Y ) ! X[Y , from which F ' 0.]
GANEANOMURA FORMULA Suppose that X and Y are nondegenerate_then in
HTOP *, (X _ Y ) X x Y x (X#Y ).
[The mapping fiber of 0 : (X xY ) ! X[Y is (X xY )x(X[Y ) and by the lemma,
EF (X x Y ) x (X[Y ). Employing the notation of p. 434, there is a commutative
0
Ess_______wssX[Yu
triangle  h hjF . The vertical arrow is a pointed homotopy equivalen*
*ce, thus
(X x Y )
Ess0 EF or still, (X _ Y ) (X x Y ) x (X[Y ) X x Y x (X#Y ) (cf. p.
434).]
ae
Given pointed spaces XY, the mapping fiber sequence associated with the p*
*rojection
f : X _ Y ! Y reads: . .!.(X _ Y ) ! Y ! Ef ! X _ Y ! Y .
LEMMA The arrow F : Y ! Ef is nullhomotopic.
[Define g : Y ! X _Y by g(y) = (x0; y), so f Og = idY. Let Z be any pointed*
* space_
then in view of the replication theorem, there is an exact sequence [Z; (X _ Y *
*)] !
[Z; Y ] ! [Z; Ef]. Since f has a right inverse, the arrow [Z; (X _ Y )] ! [Z; *
*Y ] is
surjective. This means that the arrow [Z; Y ] ! [Z; Ef] is the zero map, theref*
*ore F is
nullhomotopic.]
GRAYNOMURA FORMULA Suppose that X and Y are nondegenerate_then in
HTOP *, (X _ Y ) Y x (X x Y={x0} x Y ).
[Argue as in the proof of the GaneaNomura formula (Ef is determined on p. *
*434).]
PROPOSITION 27 Let X; Y be pointed spaces_then X x Y={x0} x Y has the
same pointed homotopy type as X _ (X#Y ).
[X x Y={x0} x Y X#Y+ X#(S 0_ Y ) X#(S 0_ Y ) X#(S 1_ Y )
(X#S 1) _ (X#Y ) X _ (X#Y ).]
[Note: Recall that in HTOP *, (X#Y ) X#Y X#Y for arbitrary pointed
X and Y (cf. p. 333).]
446
So, if X is the pointed suspension of a nondegenerate space, then the Gray*
*Nomura
formula can be simplified: (X _ Y ) Y x (X _ (X#Y )). Consequently, for all
nondegenerate X and Y , ae
(X _ Y ) XYxx(Y(_X(Y_#X))(X#Y )) :
ae ae
Suppose that X , Z are wellpointed with {x0} X , {z0} Z closed. Let f*
* : X ! Y be a
Y {y0} Y
pointed continuous function, Cf its pointed mapping cone. Let p : Z ! Cf be a *
*pointed continuous
function, Z0 its fiber over the base point. Assume: p is a Hurewicz fibration_t*
*hen p is a pointed Hurewicz
P ! Z
? ?
fibration. Form the pullback square y y p. Since j O f ' 0, there is a c*
*ommutative triangle
Z Y !j Cf
NNPkp and an induced map e : X ! P.
u
X ___wjOfCf
FACT The pointed mapping cone of the arrow Ce ! Z has the pointed homotopy*
* type of X * Z0.
EXAMPLE Let X be wellpointed with {x0} X closed. The pointed mapping cone*
* of X ! *
X ! X
? ?
is X, the pointed suspension of X. Consider the pullback square y yp*
*1. Here, e : X !
* ! X
X is the arrow of adjunction and the pointed mapping cone of Ce ! X has the sam*
*e pointed
homotopy type as Ce ! *, thus in HTOP *, Ce X * X.
Given a pointed space X, the pointed mapping cone sequence associated with *
*the
arrow of adjunction e : X ! X reads: X e!X ! Ce ! X ! X ! . ...
PROPOSITION 28 Let X be nondegenerate_then X has the same pointed
homotopy type as X _ (X#X).
[Because the evaluation map r : X ! X exhibits X as a retract of X,
the replication theorem of x3 implies that the arrow F : Ce ! X is nullhomotopi*
*c, hence
CF X _ Ce. Reverting to the notation of p. 332, there is a commutative triang*
*le
0
CeA____wjCj
F AC
uin which the vertical arrow is a pointed homotopy equivalence. Accor*
*dingly,
X
Cj0 CF ) X X _ Ce X _ (X#X), the last step by the preceding
example.]
447
Assume: X and Y are nondegenerate. Put X[0]= S0, X[n]= X# . .#.X (n factors*
*).
Starting from the formula (X _Y ) X x(Y _(Y #X)), successive application
of Proposition 28 gives:
_N
(X _ Y ) X x ( Y #X[n]_ (Y #X[N]#X)):
0
ae
FACT Let X be nondegenerate and path connected_then 8 q > 0, ssq(X _ Y )*
* ssq(X)
Y
1W
ssq(( Y #X[n])).
0
[By the above, ssq(X _ Y ) is isomorphic to
_N
ssq(X) ssq(( Y #X[n]_ (Y #X[N]#X))):
0
Since (Y #X[N]#X) is (N + 2)connected (cf. p. 340), it follows that 8 q N + *
*2 : ssq(X _ Y )
NW W
ssq(X) ssq(( Y #X[n])). But ( Y #X[n]) is also (N + 2)connected. Theref*
*ore, 8 q > 0 :
0 1 n>N
W
ssq(X _ Y ) ssq(X) ssq(( Y #X[n])).]
0
A continuous function f : X ! Y is said to be an nequivalence_(n 1)aprov*
*idede
that f induces a onetoone correspondence between the path components of XY *
*and
8 x0 2 X, f* : ssq(X; x0) ! ssq(Y; f(x0)) is bijective for 1 q < n and surject*
*ive for q = n.
Example: A pair (X; A) is nconnected iff the inclusion A ! X is an nequivalen*
*ce.
[Note: f is an nequivalence iff the pair (Mf; i(X)) is nconnected.]
ae
FACT Let X p!B q Y be a 2sink. Suppose that p is an nequivalence_then *
*the projection
__ q is an mequivalence
X _BY ! B is an (n + m + 1)equivalence.
[There is an arrow X ___BY OE!Wp *B Wq that commutesawithethe projections*
* and is a homotopy
equivalence (cf. p. 425), thus one can assume that p are Hurewicz fibrations*
* and work instead with
q
X *B Y (the connectivity of the join is given on p. 340).]
A continuous function f : X ! Y is said to be a weak_homotopy_equivalence_i*
*f f is
an nequivalence 8 n 1. Example: Consider the coreflector k : TOP ! CG _then*
* for
every topological space X, the identity map kX ! X is a weak homotopy equivalen*
*ce.
[Note: When X and Y are path connected, f is a weak homotopy equivalence pr*
*ovided
that at some x0 2 X, f* : ssq(X; x0) ! ssq(Y; f(x0)) is bijective 8 q 1.]
448
f g
X? ! Z?  Y?
Example: Let y y y be a commutative diagram in which the v*
*er
X0 !f0Z0 g0 Y 0
tical arrows are weak homotopy equivalences_then the arrow Wf;g! Wf0;g0is a weak
homotopy equivalence.
[Compare MayerVietoris sequences (use an ad hoc argument to estab*
*lish that
ss0(Wf;g) ss0(Wf0;g0)).]ae
0 X1 . . .
Example: Let XY 0 Y 1 . . .be expanding sequences of topological spaces.*
* As
ae n n+1
sume: 8 n, the inclusions XY !nX! Y n+1are closed cofibrations. Suppose given*
* a sequence
Xn? ! Xn+1?
of continuous functions OEn : Xn ! Y nsuch that 8 n, the diagramyOEn yOEn*
*+1
Y n ! Y n+1
commutes_then OE1 : X1 ! Y 1 is a weak homotopy equivalence if this is the ca*
*se
of the OEn.
telX1? ! X1?
[Consider the commutative diagram telyOE y OE1(cf. p. 312). Since*
* the
telY 1 ! Y 1
horizontal arrows are homotopy equivalences, it suffices to prove thatatelOEeis*
* a weak ho
1 ! [*
*0; 1[
motopy equivalence. To see this, recall that there are projections telXtelY 1*
* ! [0;,1[thus
ae 1 ae 1
a compact subset of telXtelYm1ust lie in telnXtel1(9 n >> 0). But 8 n, the*
* arrow
n Y
telnX1 ! telnY 1 is a weak homotopyaequivalence.]e
0 X1 . . .
[Note: Here is a variant. Let XY 0 Y 1 . . .be expanding sequences of to*
*pological
ae n
spaces. Assume: 8 n, XY n is T 1. Suppose given a sequence of continuous f*
*unctions
Xn? ! Xn+1?
OEn : Xn ! Y n such that 8 n, the diagram OEyn yOEn+1commutes_then OE1*
* :
Y n ! Y n+1
X1 ! Y 1 is a weak homotopy equivalence if this is the case of the OEn.]
EXAMPLE Given pointed spaces X and Y , let X./Y be the double mapping trac*
*k of the 2sink
X ! X _ Y Y . The projection X./Y ! X x Y is a pointed Hurewicz fibration. It*
*s fiber over (x0; y0)
is (X _ Y ) and the composite (X[Y ) ! (X _ Y ) ! X./Y defines a weak homotopy *
*equivalence
(X[Y ) ! X./Y .
Assume: X and Y are nondegenerate_then the argument used to establish that
449
_N
(X _ Y ) X x ( Y #X[n]_ (Y #X[N]#X))
does not explicitly produce a poin0ted homotopy equivalence between either side*
* but such precision is
ae ae
possible. Let X be the inclusions X ! X _ Y . With w0 = Y , inductive*
*ly define w1 =
Y Y ! X _ Y
[w0; X ]; : :;:wn = [wn1; X ], the bracket being the Whitehead product, so w*
*1 : (Y #X) ! X _
NW
Y; : :;:wn : (Y #X[n]) ! X _ Y . Write (X ) + ( wn _ [wN ; X O r]) for the c*
*omposite
_N 0
X x ( Y #X[n]_ (Y #X[N]#X)) ! (X _ Y ) x (X _ Y ) +!(X _ Y ):
0 NW
Then Spencery has shown that (X ) + ( wn _ [wN ; X O r]) is a pointed homoto*
*py equivalence.
ae 0
EXAMPLE Let X be nondegenerate and path connected_then the map
Y _1 _1
(X ) + ( wn) : X x ( Y #X[n]) ! (X _ Y )
is a weak homotopy equivale0nce. 0
Let L be the free Lie algebra over Z on two generators t1; t2. The basic co*
*mmutators of weight one
are t1 and t2. Put e(t1) = 0, e(t2) = 0. Proceeding inductively, suppose that t*
*he basic commutators of
weight less than n have been defined and ordered as t1; : :;:tp and that a func*
*tion e from {1; : :;:p} to the
nonnegative integers has been defined: 8 i; e(i) < i. Take for the basic commut*
*ators of weight n the [ti; tj],
where weight ti+ weight tj = n and e(i) j < i. Order these commutators in any *
*way and label them
tp+1; : :;:tp+q. Complete the construction by setting e([ti; tj]) = j. Let B be*
* the set of basic commutators
thus obtained_then B is an additive basis for L, the Hall_basis_.
ae
EXAMPLE (HiltonMilnor_Formula_) Let X be nondegenerate and path conne*
*cted. Put
ae ae Y
Z(t1) = Xand let i1 : Z(t1) ! X _ Y be the inclusions. For t 2 B of weight*
* n > 1, write
Z(t2) = Y i2 : Z(t2) ! X _ Y
uniquely t = [ti; tj], where weight ti+ weight tj = n. Via recursionaonethe wei*
*ght, put Z(t) = Z(ti)#Z(tj)
and let it: Z(t) ! X _ Y be the Whitehead product [ii; ij], where ii: Z(ti) !*
* X _ Y . The
P Q ij : Z(tj) *
*! X _ Y
it combine to define a continuous function i = it from (w) Z(t) (cf. p. 1*
*36) to (X _ Y ).
t2B t2B
Claim: i is a weak homotopy equivalence. To see this, attach to each N = 1; 2;*
* : :;:a "remainder"
W W
RN = Z(ti). Applying the preceding example to (Z(tN ) _ Z(ti)), it f*
*ollows that the
iN i>N
e(i)N(i)N i=1
_________________________
yJ. London Math. Soc. 4 (1971), 291303.
450
is a weak homotopy equivalence. To finish, let N ! 1 (justified, since the conn*
*ectivity of RN+1 tends to
1 with N).
[Note: The isomorphism i* : t2Bss*(Z(t)) ! ss*((X _ Y )) depends on the cho*
*ice of the Hall
basis B. Consult Goerssy for an intrinsic description.]
A nonempty path connected topological space X is said to be homotopically_t*
*rivial_if
X is nconnected for all n, i.e., provided that 8 q > 0, ssq(X) = 0. Example: A*
* contractible
space is homotopically trivial.
Example: Let X f!Z gY be a 2sink. Assume: X & Z are homotopically trivia*
*l_
then the arrow Wf;g! Y is a weak homotopy equivalence.
EXAMPLE A homotopy equivalence is a weak homotopy equivalence but the conv*
*erse is false.
(1) (The_Wedge_of_the_Broom_) Consider the subspace X of R 2consisting*
* of the line seg
ments joining (0; 1) to (0; 0) & (1=n; 0) (n = 1; 2; : :):_then X is contractib*
*le, thus it and its base point
(0; 0) have the same homotopy type. But in the pointed homotopy category, (X; (*
*0; 0)) and ({(0; 0)}; (0; 0))
areanoteequivalent. Consider X _ X, the subspace of R 2 consisting of the line*
* segments joining
(0; 1) to(0; 0) & (1=n;(0)n = 1; 2; : :):_then X _ X is path connected and ho*
*motopically triv
(0; 1) to(0; 0) & (1=n; 0)
ial. However, X _ X is not contractible, so the map that sends X _ X to (0; 0)*
* is a weak homotopy
equivalence but not a homotopy equivalence.
8 (2) (The_Warsaw_Circle_) Consider the subspace X of R2 consisting of t*
*he union of {(x; y) :
< x = 0; 2 y 1
and {(x; y) : 0 < x 1; y = sin(2ss=x)}_then X is path connect*
*ed and homo
: 0 x 1; y = 2
x = 1; 2 y 0
topically trivial. However, X is not contractible, so the map that sends X to (*
*0; 0) is a weak homotopy
equivalence but not a homotopy equivalence.
FACT Let p : X ! B be a Hurewicz fibration, where X and B are path connect*
*ed and X is
nonempty. Suppose that [p] is both a monomorphism and an epimorphism in HTOP _*
*then p is a weak
homotopy equivalence.
A continuous function f : (X; A) ! (Y; B) is said to be a relative_nequiv*
*alence_
(n 1) provided that the sequence * ! ss0(X; A) ! ss0(Y; B) is exact and 8 x0 2*
* A,
f* : ssq(X; A; x0) ! ssq(Y; B; f(x0)) is bijective for 1 q < n and surjective *
*for q = n.
ae ae ae
PROPOSITION 29 Suppose that X1X & Y1 are open subspaces of X with
2 Y2 Y
_________________________
yQuart. J. Math. 44 (1993), 4385.
451
ae ae
X = X1 [ X2 X1 = f1 *
*(Y1)
Y = Y1 [ Y2 . Let f : X ! Y be a continuous function such that X2 = f1 *
*(Y2).
Fix n 1. Assume: f : (Xi; X1\X2) ! (Yi; Y1\Y2) is a relative nequivalence (i *
*= 1; 2)_
then f : (X; Xi) ! (Y; Yi) is a relative nequivalence (i = 1; 2).
[This is the content of the result on p. 346.]
A continuous function f : (X; A) ! (Y; B) is said to be a relative_weak_hom*
*otopy_
equivalence_if f is a relative nequivalence 8 n 1. Example: Let p : X ! B b*
*e a
Serre fibration, where B is path connected and X is nonempty_then 8 b 2 B, the *
*arrow
(X; Xb) ! (B; b) is a relative weak homotopy equivalence.
LEMMA Let f : (X; A) ! (Y; B) be a continuous function. Assume: f : A ! B
and f : X ! Y are weak homotopy equivalences_then f : (X; A) ! (Y; B) is a rela*
*tive
weak homotopy equivalence.
ae ae ae
PROPOSITION 30 Suppose that X1X & Y1 are open subspaces of X with
ae 2 Y2 ae Y
X = X1 [ X2 X1 = f1 *
*(Y1)
Y = Y1a[eY2 . Let f : X ! Y be a continuous function such that X2 = f1 *
*(Y2).
Assume: ff::X1X! Y1 & f : X1 \ X2 ! Y1 \ Y2 are weak homotopy equivalences_th*
*en
2 ! Y2
f : X ! Y is a weak homotopy equivalence.
[The lemma implies that f : (Xi; X1\X2) ! (Yi; Y1\Y2) is a relative weak ho*
*motopy
equivalence (i = 1; 2). Therefore, on the basis of Proposition 29, f : (X; Xi) *
*! (Y; Yi) is
a relative weak homotopy equivalence (i = 1; 2). Since a given x 2 X belongs to*
* at least
one of the Xi, this suffices (modulo low dimensional details).]
X? f Z? g! Y?
Application: Let y y y be a commutative diagram in which *
*the
X0 f0 Z0 !g0 Y 0
vertical arrows are weak homotopy equivalences_then the arrow Mf;g! Mf0;g0is a *
*weak
homotopy equivalence. ae
[Note: If in addition ff0are closed cofibrations, then the arrow X tg Y !*
* X0tg0Y 0
is a weak homotopy equivalence (cf. x3, Proposition 18).]
ae
FACT Let X be topological spaces and let f : X ! Y be a continuous funct*
*ion. Assume: V =
Y
{V } is an open covering of Y which is closed under finite intersections such t*
*hat 8 V 2 V, f : f1(V ) ! V
is a weak homotopy equivalence_then f : X ! Y is a weak homotopy equivalence.
452
[Use Zorn on the collection of subspaces B of Y that have the following pro*
*perties: B is a union of
elements of V, f : f1(B) ! B is a weak homotopy equivalence, and 8 V 2 V, f : *
*f1(B \ V ) ! B \ V
is a weak homotopy equivalence. Order this collection by inclusion and fix a ma*
*ximal element B0. Claim:
B0 = Y . If not, choose V 2 V : V 6 B0 and consider B0[ V .]
ae n1
SUBLEMMA Let f 2 C(X; Y ) and suppose given continuous functions OE : S *
* ! Xwith fO
: Dn !*
* Y ae__
OE = Sn1_then there exists a neighborhood U of Sn1in Dn and continuous func*
*tions OE:_U ! X
__ __ __ __ *
* : Dn ! Y
such that OESn1 = OE and f O OE= U, where ' relSn1. *
* ae
[Let U = {x : 1=2 < kxk 1} and put __OE(x) = OE(x=kxk) (x 2 U). Write v(x)*
* = x (kxk 1).
__ *
* x=kxk (kxk 1)
Define H : ID n! Y by H(x; t) = (v((1 + t)x)) and take = H O i1.]
ae ae ae ae
LEMMA Suppose that X1 & Y1 are subspaces of X with X = intX1[ int*
*X2. Let
X2 Y2 ae Y Ya=eintY1[ int*
*Y2
f : X ! Y be a continuous function such that f(X1) Y1. Assume: f : X1 ! Y1*
*& f : X1\ X2 !
f(X2) Y2 f : X2 ! Y2
Y1\ Y2 are weak homotopy equivalences_then f : X ! Y is a weak homotopy equival*
*ence.ae
*
* _Iq! X
[In the notation employed at the end of x3, given continuous functions OE*
* : such that
*
*: Iq ! Y
f OOE = _Iq, it is enough to find a continuous function : Iq ! X suchathate*
*_Iq= OE and_f_O_'___rel_Iq.
1(X *
* intX1) [ 1(Y  Y1)
This can be done by a subdivision argument. The trick is to consider OE *
* __________.
OE1(X *
* intX2) [ 1(Y  Y2)
These sets are closed. However, they need not be disjoint and the point of the *
*sublemma is to provide an
escape for this difficulty.]
EXAMPLE In the usual topology, take Y = R, Y1 = Q, Y2 = P; in the discrete*
* topology, take
X = R, X1 = Q,aX2e= P_then the identity map X ! Y is not a weak homotopy equiva*
*lence, yet the
restrictions X1 ! Y1, X1\ X2 ! Y1\ Y2 are weak homotopy equivalences.
X2 ! Y2
ae
FACT Let X be topological spaces and let f : X ! Y be a continuous funct*
*ion. Suppose that
ae Y ae
U = {Ui: i 2 I}are open coverings of X such that 8 i : f(U ) V . Assume: F*
*or every nonempty
V = {Vi: i 2 I} T Y T i i
finite subset F I, the induced map Ui! Viis a weak homotopy equivalence_*
*then f is a weak
i2F i2F
homotopy equivalence.
ae
Topological spaces XY are said to have the same weak_homotopy_type_if the*
*re exists
ae
a topological space Z and weak homotopy equivalences fg::ZZ!!XY. The relatio*
*n of
having the same weak homotopy type is an equivalence relation.
453
ae
[Note: One can always replace Z by a CW resolution K ! Z, hence XY have t*
*he
ae
same weak homotopy type iff they admit CW resolutions KK!!XY by the same CW
complex K.]
Transitivity is the only issue. For this, let X1; X2; X3 be topological spa*
*ces, let K; L be CW com
K' L' aef1 aeg2
plexes, and consider the diagram f[[^1 ')f2 [[^g2 ')g3 , where , a*
*re weak homo
X1 X2 X3 f2 g3
topy equivalences. Since (K; f2) and (L; g2) are both CW resolutions of X2, the*
*re is a homotopy equivalence
OE : K ! L such that f2 ' g2O OE (cf. p. 518). Thus g3O OE : K ! X3 is a weak *
*homotopy equivalence, so
X1 and X3 have the same weak homotopy type.
EXAMPLE Two aspherical spaces having isomorphic fundamental groups have th*
*e same weak
homotopy type.
[Note: A path connected topological space X is said to be aspherical_provid*
*ed that 8 q > 1, ssq(X) =
0. Example: If X is path connected and metrizable with dimX = 1, then X is asph*
*erical.]
Let X be in TOP =B. Assume that the projection p : X ! B is surjective_the*
*n p
is said to be a quasifibration_if 8 b 2 B, the arrow (X; Xb) ! (B; b) is a rela*
*tive weak
homotopy equivalence. If p : X ! B is a quasifibration, then 8 b0 2 B; 8 x0 2 X*
*b0, there
is an exact sequence
. .!.ss2(B) ! ss1(Xb0) ! ss1(X) ! ss1(B) ! ss0(Xb0) ! ss0(X) ! ss0(B):
LEMMA Let p : X ! B be a Serre fibration. Suppose that B is path connected*
* and
X is nonempty_then p is a quasifibration.
EXAMPLE Take X = ([1; 0] x {1}) [ ({0} x [0; 1]) [ ([0; 1] x {0}), B = [*
*1; 1], and let p be the
vertical projection_then p is a quasifibration (X and B are contractible, as ar*
*e all the fibers) but p is
neither a Serre fibration nor a Dold fibration.
[Note: The pullback of a Serre fibration is a Serre fibration, i.e., Propos*
*ition 4 is valid with "Hurewicz"
replacedabye"Serre". This fails for quasifibrations. Let B0 = [0; 1] and define*
* 0 : B0 ! B by (t) =
t sin(1=t)(t >_0)then the projection p0: X0! B0is not a quasifibration (consi*
*der ss ).]
0 (t = 0) *
* 0
PROPOSITION 31 Let p : X ! B be a quasifibration, where B is path connecte*
*d_
then the fibers of p have the same weak homotopy type.
454
[Using the mapping track Wp, factor p as q O fl and note that 8 b 2 B, fl i*
*nduces a
weak homotopy equivalence Xb ! q1 (b). But q : Wp ! B is a Hurewicz fibration *
*and
since B is path connected, the fibers of q have the same homotopy type (cf. p. *
*413).]
EXAMPLE Let B = [0; 1]n (n 1). Put X = B x B  B and let p be the vertica*
*l projection_
then p is not a quasifibration (cf. p. 48).
LEMMA Let p : X ! B be a continuous function. Suppose that O B and
pO : XO ! O is a quasifibration_then the arrow (X; XO ) ! (B; O) is a relative *
*weak
homotopy equivalence iff 8 b 2 O, the arrow (X; Xb) ! (B; b) is a relative weak*
* homotopy
equivalence.
ae
PROPOSITION 32 Let X be in TOP =B. Suppose that O1O are open subspaces
ae 2
of B with B = O1 [ O2. Assume: pO1p: XO1 ! O1 & pO1\O2 : XO1\O2 ! O1 \ O2 are
O2 : XO2 ! O2
quasifibrations_then p : X ! B is a quasifibration.
[From the lemma, the arrows (XOi; XO1\O2 ) ! (Oi; O1 \ O2) are relative wea*
*k ho
motopy equivalences (i = 1; 2). Therefore the arrow (X; XOi) ! (B; Oi) is a rel*
*ative weak
homotopy equivalence (i = 1; 2) (cf. Proposition 29). Since p is clearly surjec*
*tive, another
appeal to the lemma completes the proof.]
Application: Let X be in TOP =B. Suppose that O = {Oi : i 2 I} is an open
covering of B which is closed under finite intersections. Assume: 8 i, pOi : XO*
*i ! Oi is a
quasifibration_then p : X ! B is a quasifibration.
[The argument is the same as that indicated on p. 452 for weak homotopy eq*
*uiva
lences.]
[Note: This is the localglobal principle for quasifibrations. Here, nume*
*rability is
irrelevant.]
EXAMPLEaeLet X be R2 equippedawithethe following topology: Basic neighborho*
*ods of (x; y),
where x 0 & 1 < y < 1 or 0 < x < 1 & y >,0are the usual neighborhoods but*
* the basic neigh
x 1 0 < x < 1 & y < 0
borhoods of (x; 0), where 0 < x < 1, are the open semicircles centered at (x; 0*
*) of radius < min{x; 1x} that
lie in the closed upper half plane. Take B = R2(usual topology)_then the identi*
*tyamapep : X ! B isanotea
quasifibration (since ss1(B) = 0, ss1(X) 6= 0 and the fibers are points). Put *
* O1 = {(x; y) : x:> 0}O1
ae *
* O2a=e{(x; y) : x < 1}O2
are open subspaces of B with B = O1[ O2. Moreover, XO1 are contractible, thus*
* pO1 : XO1 ! O1
XO2 *
* pO2 : XO2 ! O2
are quasifibrations. However, pO1\O2 : XO1\O2 ! O1\ O2 is not a quasifibration.
455
FACT Let p : X ! B be a surjective continuous function, where B = colimBn *
*is T1. Assume:
8 n, p1(Bn) ! Bn is a quasifibration_then p is a quasifibration.
Let A be a subspace of X, i : A ! X the inclusion.
(WDR) A is said to be a weak_deformation_retract_of X if there is a ho*
*motopy
H : IX ! X such that H O i0 = idX, H O it(A) A (0 t 1), and H O i1(X) A.
[Note: Define r : X ! A by i O r = H O i1_then i O r ' idX and r O i ' idA.]
A strong deformation retract is a weak deformation retract. The comb is a *
*weak
deformation retract of [0; 1]2 (consider the homotopy H((x; y); t) = (x; (1  t*
*)y)) but the
comb is not a retract of [0; 1]2.
[Note: A pointed space (X; x0) is contractible to x0 in TOP * iff {x0} is *
*a weak (or
strong) deformation retract of X. The broom with base point (0; 0) is an examp*
*le of a
pointed space which is contractible in TOP but not in TOP *. Therefore a def*
*ormation
retract need not be a weak deformation retract.]
On a subspace A of X such that the inclusion A ! X is a cofibration, "stron*
*g"="weak".
PROPOSITION 33 Let p : X ! B be a surjective continuous function. Suppose*
*ae
that O is a subspace of B for which pO : XO ! O is a quasifibration and OX i*
*s a weak
ae ae O
deformation retract of BX, say aer::BX!!OX. Assume: p O r = ae O p and 8 b *
*2 B, rXb
O
is a weak homotopy equivalence Xb ! Xae(b)_then p : X ! B is a quasifibration.
[Given b 2 B, r : (X; Xb) ! (XO ; Xae(b)), as a map of pairs, is a relative*
* weak homotopy
(X; Xb) ! (XO ; Xae(b))
? ?
equivalence and, by assumption, the diagram y y commutes*
*.]
(B; b) ! (O; ae(b))
X? f Z? g! Y?
Application: Let y y y be a commutative diagram in which *
*the
X0 f0 Z0 !g0 Y 0
ae
vertical arrows are quasifibrations. Assume: 8 z0 2 Z0, fZz0gZis a weak h*
*omotopy
ae z0
equivalence Zz0!ZXf0(z0)_then the arrow Mf;g! Mf0;g0is a quasifibration.
z0! Yg0(z0)
456
X? f Z? g! Y?
PROPOSITION 34 Let y y y be a commutative diagram in
X0 f0 Z0 !g0Y 0
which the left vertical arrow is a surjective Hurewicz fibration and the right *
*vertical arrow
X? f Z? ae
is a quasifibration. Assume: y y is a pullback square, ff0are clo*
*sed cofi
X0 f0 Z0
brations, and 8 z0 2 Z0, gZz0 is a weak homotopy equivalence Zz0 ! Yg0(z0)_the*
*n the
induced map X tg Y ! X0tg0Y 0is a quasifibration.
Mf;g? ! Mf0;g0? ae
[Consider the commutative diagram OyE y OE0 . Since ff0are*
* cofi
X tg Y ! X0tg0Y 0
ae
brations, OEOE0are homotopy equivalences (cf. x3, Proposition 18) and, by the*
* above, is
a quasifibration. Thus it need only be shown that 8 m0 2 Mf0;g0, the arrow 1(m*
*0) !
1 (OE0(m0)) is a weak homotopy equivalence, which can be done by examining cas*
*es.]
The conclusion of Proposition 34 cannot be strengthened to "Hurewicz fibrat*
*ion".
To see this, take X = [1; 0]axe[0; 1], Yae= [0; 2] x [0; 2], Z = {0} x [0; 1],*
* X0 = [1; 0],
0 : Z0 ! X0
Y 0= [0; 2], Z0 = {0}, let fg::ZZ!!XY, fg0: Z0 ! Y 0be the inclusions, and *
*let X ! X0,
Z ! Z0, Y ! Y 0be the vertical projections_then X tg Y = X [ Y , X0tg0Y 0= X0[ *
*Y 0,
and the induced map X [ Y ! X0[ Y 0is the vertical projection. But it is not a *
*Hurewicz
fibration since it fails to have the slicing structure property (cf. p. 414).
EXAMPLE (Cone_Construction_) Fix nonempty topological spaces X; Y and let*
* OE : X x Y ! Y
X x Y *
*OE!Y
? *
* ?
be a continuous function. Define E by the pushout square y *
* y .
X x Y *
*! E
Assume: 8 x 2 X, OEx : {x} x Y ! Y is a weak homotopy equivalence. Consider th*
*e commutative
X x Y  X x Y OE! Y
? ? ?
diagram y y y . Since the arrows X ! X, X x Y ! X x Y ar*
*e closed
X  X ! *
cofibrations, all the hypotheses of Proposition 34 are met. Therefore the indu*
*ced map E ! X is a
quasifibration.
[Note: The same construction can be made in the pointed category provided t*
*hat (X; x0) is well
457
pointed with {x0} X closed.]
EXAMPLE (DoldLashof_Construction_) Let G be a topological semigroup with*
* unit in which
the operations of left and rightatranslationeare homotopy equivalences. Let p :*
* X !aBebe a quasifibration.
Assume: There is a right action X x G ! X such that p(x.g) = p(x) and the arr*
*ow G ! Xp(x)is a
(x; g) ! x . g *
* g ! x . g
X x G ! X
? ? *
* __
weak homotopy equivalence. Define __Xby the pushout square y y and *
*put B = Cp. Since
X x G ! __X
X x G  X x G ! X
? ? ? _*
* __ __
the diagram y y y commutes, Proposition 34 implies that p*
*: X ! B is a
X  X ! B __ __
quasifibration. Represent a generic point of X (B ) by the symbol [x;at;eg]_([x*
*;_t]) (with the obvious under
standing at t = 0 or t = 1), so _p[x; t; g] = [x; t]. The assignment X x G ! *
*X is unambiguous
__ ([x; t; *
*g]; h) ! [x; t; gh]
and satisfies the algebraic conditions for a right action of G on X but it is n*
*ot necessarily continuous.
The resolution is to place a smaller topology on __X. Let t : __X! [0; 1] be th*
*e function [x; t; g] ! t; let
x : t1(]0; 1[) ! X be the function [x; t; g] ! x; let g : t1([0; 1[) ! G be t*
*he function [x; t; g] ! g; let
x . g : t1(]0; 1]) ! X be the function [x; t; g] ! x . g. Definition:aTheecoor*
*dinate_topology_on___Xis the
initial topology determined by t; x; g; x.g. The injection X ! X is an emb*
*edding, as is the injection
ae __ x ! [x; 1; e] *
* ae __
G ! X (t 6= 0; 1). Moreover, G acts continuously and 8 _x2 __X, the arrow*
* G ! X_p(_x)is a weak
g ! [x; t; g] __ *
* g !__x._g_
homotopy equivalence. Now equip B with its coordinate topology (cf.ap.e33)_the*
*n _p: X ! B is contin
uous and remains a quasifibration (apply Propositions 32 and 33 to O1 = {[x; *
*t] : 0 <)t. 1}In other
O2 = {[x; *
*t] : 0 t < 1} __
*
* X ! X
*
* ? ?
words, (__X; __B) satisfies the same conditions as (X; B) and there is a commut*
*ative diagram y y ,
ae __ *
* B ! __B
where X ! __Xis inessential (consider H : IX ! X ).
(x;_t)_! [x; t; e]
Example: Let G be a topological group_then G (coordinate topology) is homeo*
*morphic to G *cG
(coarse join).
Let G beaaetopological group, X a topological space. Suppose that X is a r*
*ight
Gspace: (Xxx;Gg!)X! x_.tghen the projection X ! X=G is an open map and X=G is
Hausdorff iff X xX=G X is closed in X xX. The continuous function : X xG ! X x*
*X=G X
defined by (x; g) ! (x; x.g) is surjective. It is injective iff the action is f*
*ree, i.e., iff 8 x 2 X,
the stabilizer Gx = {g : x . g = x} of x in G is trivial. A free right Gspace*
* X is said
458
to beaprincipal_providedethat is a homeomorphism or still, that the division f*
*unction
d : X(xX=GxX;!xG. g)i!sgcontinuous.
Let X be in TOP =B_then X is said to be a principal_Gspace_over_Bif X is a
principal Gspace, B is a trivial Gspace, the projection p : X ! B is open, su*
*rjective, and
equivariant, and G operates transitively on the fibers. There is a commutative*
* triangle
X
[[^ u and the arrow X=G ! B is a homeomorphism. PRIN B;G is the catego*
*ry
X=G ___w B
whose objects are the principal Gspaces over B and whose morphisms are the equ*
*iv
ariant continuous functions over B. If 0 2 C(B0; B), then for every X in PRIN *
* B;G
0
X0? f! X?
there is a pullback square y y with X0 = B0xB X in PRIN B0;Gand f0 eq*
*uiv
B0 !0 B
ariant.
LEMMA Every morphism in PRIN B;G is an isomorphism.
[Note: The objects in PRIN B;G which are isomorphic to B x G (product topo*
*logy)
are said to be trivial_. It follows from the lemma that the trivial objects are*
* precisely those
that admit a section.]
ae 0 ae
Application: Let XX be in PRINPRB0;GIN; let f0 2 C(X0; X); 0 2 C(B0; B*
*). As
B;G
0
X0? f*
*! X?
sume: f0 is equivariant and p O f0 = 0O p0_then the commutative diagram y *
* y
B0 *
*!0 B
is a pullback square.
[Compare this diagram with the pullback square defining the fiber product.]
Let X be in TOP =B_then X is said to be a Gbundle_over_B_if X is a free r*
*ight G
space, B is a trivial Gspace, the projection p : X ! B is open, surjective, an*
*d equivariant,
and there exists an open covering O = {Oi: i 2 I} of B such that 8 i, XOi is eq*
*uivariantly
homeomorphic to Oix G over Oi. Since the division function is necessarily cont*
*inuous
and G operates transitively on the fibers, X is a principal Gspace over B. If *
*O can be
chosen numerable, then X is said to be a numerable_Gbundle_over_B_(a condition*
* that is
automatic when B is a paracompact Hausdorff space, e.g., a CW complex). BUN B;*
*G is the
full subcategory of PRIN B;G whose objects are the numerable Gbundles over B.*
* Each X
in BUN B;G is numerably locally trivial with fiber G and the localglobal prin*
*ciple implies
459
that the projection X ! B is a Hurewicz fibration. There is a functor I : BUN *
* B;G !
BUN IB;Gthat sends p : X ! B to Ip : IX ! IB, where (x; t) . g = (x . g; t).
EXAMPLE A Gbundle over B need not be numerable. For instance, take G = R_*
*then every
object in BUN B;R admits a section (R being contractible), hence is trivial. Le*
*t now X be the subset of R3
defined by the equation x1x3+x22= 1 and let R act on X via (x1; x2; x3).t = (x1*
*; x2+tx1; x32tx2t2x1).
X is an Rbundle over X=R but it is not numerable. For if it were, then there *
*would exist a section
X=R ! X, an impossibility since X=R is not Hausdorff.
FACT Suppose that X is a Gbundle over B_then the projection p : X ! B is *
*a Serre fibration
(cf. p. 411) which is Zorientable if B and G are path connected.
ae 0 ae
Let XX be in BUNBUB0;GN. Write X0xG X for the orbit space (X0x X)=G_then
B;G
X0x?X ! X0?
there is a commutative diagram y y which is a pullback square. A*
*s an
X0xG X ! B0
object in TOP =B0, X0xG X is numerably locally trivial with fiber X so, e.g., *
*has the SEP
if X is contractible. The s0 2 secB0(X0 xG X) correspond bijectively to the eq*
*uivariant
f0 2 C(X0; X). As an object in TOP =B0 x B, X0 xG X is numerably locally triv*
*ial
with fiber G. Given 0 2 C(B0; B), there exists an equivariant f0 2 C(X0; X) ren*
*dering
0
X0? f! X? ae 0 0
the diagram y y commutative iff the arrow Bb!0B!x(Bb0; 0(b0))admits *
*a lifting
B0 !0 B
X0xG X
""] u .
B0 _____wB0x B
ae 0 ae
COVERING HOMOTOPY THEOREM Let XX be in BUNBUB0;GN. Suppose that
B;G
f0 : X0 ! X is an equivariant continuous function and h : IB0 ! B is a homotopy*
* with
p O f0 = h O i0 O p0_then there exists an equivariant homotopy H : IX0 ! X such*
* that
IX0? H! X?
H O i0 = f0 and for which the diagram y y commutes.
IB0 !h B
X0xG X
[Take 0 = h O i0 to get a lifting ""] u and a commutative diagram
B0 _____wB0x B
460
B0? ! IX0xG X
i0y ?y . The projection IX0 xG X ! IB0x B is a Hurewicz fibratio*
*n,
IB0 ! IB0x B
thus the diagram has a filler IB0! IX0xG X and this guarantees the existence of*
* H.]
ae0
Application: Let X be in BUN B;G. Suppose that 10 2 C(B0; B) are homotop*
*ic_
ae 2
0
then X1X0are isomorphic in BUN B0;G.
2
FACT The functor I : BUN B;G ! BUN IB;G has a representative image.
The relation "isomorphic to" is an equivalence relation on Ob BUN B;G. Cal*
*l kG B the
"class" of equivalence classes arising therefrom_then kG B is a "set" (see belo*
*w). Since for
any 02 C(B0; B) and each X in BUN B;G, the isomorphism class [X0] of X0 in BUN*
* B0;G
depends only on the homotopy class [0] of 0, kG is a cofunctor HTOP ! SET *
*. A
topological space BG is said to be a classifying_space_for G if BG represents k*
*G , i.e., if there
exists a natural isomorphism : [_ ; BG ] ! kG , an XG in BG (idBG) being a uni*
*versal_
numerable Gbundle over BG . From the definitions, 8 2 C(B; BG ), B [] = [X], *
*where
X? ! XG?
X is defined by the pullback square y y and is the classifying_map_.
B ! BG
ae0 0
(UN) Assume that 0:0[_:;[BG_];!BkG00are natural isomorphisms_then t*
*here
Ga]e!0kG0 00 ae 0 *
* 00
exist mutually inverse homotopy equivalences 0:0BG:!BBG00 0such that kG [0]*
*([XG0])0=
G ! BG kG [ ]*
*([XG ]) =
[X0G]
[X00G].
Recall that the members of a class are sets, therefore kG B is not a class *
*but rather a conglomerate.
Still, BUN B;G has a small skeleton _____BUNB;G. Indeed, any X in BUN B;G is is*
*omorphic to B x G. Here,
the topology on B x G depends on X and is in general not the product topology b*
*ut the action is the
same ((b; g) . h = (b; gh)). Thus one can modify the definition of kG and inste*
*ad take for kG B the set
Ob _____BUNB;G.
PROPOSITION 35 Suppose that there exists a BG in TOP and an XG in BUN BG*
*;G
such that XG is contractible_then kG is representable.
[Define a natural transformation : [_ ; BG ] ! kG by assigning to a given *
*homotopy
class [] ( 2 C(B; BG )) the isomorphism class [X] of the numerable Gbundle X o*
*ver B
461
X? ! XG?
defined by the pullback square y y . The claim is that 8 B, B : [B; BG*
* ] ! kG B
B ! BG
is bijective.
Surjectivity: Take any X in BUN B;G and form X xG XG . Since XG is contrac*
*tible,
X xG XG has the SEP, thus secB(X xG XG ) is nonempty, so there exists an equiva*
*riant
X? f! XG?
f 2 C(X; XG ). Determine 2 C(B; BG ) from the commutative diagram y y *
* _
B ! BG
then B [] = [X].
Injectivity: Let 0, 002 C(B; BG ) and assume that B [0] = B [00], say [X0] =
0 *
* f0!X
X0f_____wOElfflX00 X? *
* G?
[X00], where , ADwith OE equivariant. There are pullback squares y *
* y ,
B B *
* !0 BG
00
X00? f! XG
y ?y . Put B0 = B x ([0; 1=2[[]1=2; 1]) and define H0 : IX0B0 ! XG *
* by
B !00BG
ae 0 0
H0(x0; t) = ff(x0)0O OE(t(H1=2)0is equivariant, hence corresponds*
* to a section s0
of (IX0xG XG )B0. Since B0 is a halo of i0B [ i1B in IB and since IX0xG XG has*
* the
SEP, 9 s 2 secIB(IX0xG XG ) : sB x ({0} [ {1}) = s0B x ({0} [ {1}). Translate*
*d, this
means that there exists an equivariant homotopy H : IX0 ! XG . Determine h : IB*
* ! BG
IX0? H! XG? ae 0
from the commutative diagram y y _then hhOOi0i= 00) [0] = [00].]
1 =
IB !h BG
The converse of Proposition 35 is also true: In order that kG be representa*
*ble, it is necessary that XG
be contractible. Thus let X1Gbe the numerable Gbundle over B1Gproduced by the *
*Milnor construction_
then X1G is contractible, so 1 is a natural isomorphism. As the same holds for *
* by assumption, there
1
XG f! X1G X1G f! XG
? ? ? ?
are pullback squares y y , y y and 1 O ' idBG. Owing to th*
*e covering
BG ! B1G B1G !1 BG
462
OE
XGaeaeo_*
*_wXG
homotopy theorem, f1 O f is equivariantly homotopic to an isomorphism *
*NNQ . But OE is
BG
necessarily inessential, X1G being contractible.
EXAMPLE Let E be an infinite dimensional Hilbert space_then its general li*
*near group GL (E)
is contractible (cf. p. 610). Any compact Lie group G can be embedded as a clo*
*sed subgroup of GL (E).
So, if XG = GL (E), BG = GL (E)=G, then BG is a classifying space for G and XG *
*is universal.
[BG is a paracompact Hausdorff space. Local triviality of XG is a consequen*
*ce of a generality due
to Gleason, viz: Suppose that G is a compact Lie group and X is a Hausdorff pri*
*ncipal Gspace which is
completely regular_then X, as an object in TOP =B (B = X=G), is a Gbundle.]
EXAMPLE Let G be a noncompact connected semisimple Lie group with finite c*
*enter, K G a
maximal compact subgroup. The coset space K\G is contractible, being diffeomorp*
*hic to some Rn. Let
be a discrete subgroup of G. Assume: is cocompact and torsion free_then oper*
*ates on K\G by
right translation and K\G is a numerable bundle over K\G=. So, if X = K\G, B *
* = K\G=, then
B is a classifying space for and X is universal.
[Note: B is a compact riemannian manifold. Its universal covering space is*
* X , thus B is aspherical
and of homotopy type (; 1).]
MILNOR CONSTRUCTION Let G be a topological group. Consider the subset of
P
([0; 1] x G)! made up of the strings {(ti; gi)} for which ti = 1 & #{i : ti *
*6= 0} < !.
i
Write {(t0i; g0i)} ~ {(t00i; g00i)} iff 8 i; t0i= t00iand at those i such that *
*t0i= t00iis positive,
g0i= g00i. Call X1Gatheeresulting set ofaequivalenceeclasses. Define coordina*
*te functions
1 ! [0; 1] t1(]0; 1]) ! G
ti and gi by ti : XGx ! t and gi : i , where x = [(ti(x); g*
*i(x))].
i(x) x ! gi(x)
The Milnor_topology_on X1G is the initialatopologyedetermined by the ti and gi.*
* Thus
1 x G ! X1
topologized, X1G is a right Gspace: XG(x; g) ! Gx ..g Here, ti(x . g) = ti(*
*x) and
gi(x . g) = gi(x)g. Let B1G be the orbit space X1G=G.
[Note: Put X0G= G, XnG= G *c . .*.cG, the (n + 1)fold coarse join of G wit*
*h itself.
One can identify XnGwith {x : 8 i n + 1; ti(x) = 0}. Each XnGis a zero set in *
*X1G and
there is an equivariant embedding XnG! Xn+1G. So, X0G X1G . . .is an expanding
sequence of topological spaces and the colimit in TOP associated with this da*
*ta is X1G
equipped with the final topology determined by the inclusions XnG! X1G. The co*
*limit
topology is finer than the Milnor topology and in general, there is no guarante*
*e that the
Gaction (x; g) ! x . g remains continuous.]
(M) X1G is a numerable Gbundle over B1G.
463
[It is clear that X1G is a principal Gspace. Write Oi for the image of t1*
*i(]0; 1]) under
the projection X1G ! B1G_then {Oi} is a countable cozero set covering of B1G, h*
*ence is
numerable (cf. p. 125). On the other hand, 8 i; secOi(X1GOi) is nonempty. To *
*see this,
define a continuous fiber preserving function fi: X1GOi! X1GOi by fi(x) = x .*
* gi(x)1 :
8 g 2 G, fi(x . g) = fi(x). Consequently, fi drops to a section si: Oi! X1GOi,*
* therefore
X1GOi is trivial.]
(D) X1G is contractible.
[Let 1G be the subset of X1G consisting of those x such that gi(x) = e if t*
*i(x) > 0_
then 1G is contractible, so one need only construct a homotopy H : IX1G ! X1G s*
*uch
that H O i0 = idX1Gand H O i1(X1G) 1G. Put Uk = o1k(]0; 1]) and Ak = o1k(1),*
* where
P
ok = ti. Define H0k: IUk ! Uk by
ik
8
< t_+_(1__t)ok(x)_t (x)(i k)
ti(H0k(x; t)) = : ok(x) i
(1  t)ti(x) (i > k)
and gi(H0k(x; t)) = gi(x) when ti(H0k(x; t)) > 0. Note that H0k(x; 0) = x, H0k(*
*x; 1) 2 Ak,
and x 2 1G ) H0k(x; t) 2 1G (0 t 1). Define H00k: IAk ! Ak+1 by
8
< (1  t)ti(x)(i k)
ti(H00k(x; t)) = : t (i = k + 1)
0 (i > k + 1)
ae
and gi(H00k(x; t)) = gi(x)e(i(ik)= k +w1)hen ti(H00k(x; t)) > 0. Note that H0*
*0k(x; 0) = x,
ae 0
H00k(x; 1) 2 1G, and x 2 1G ) H00k(x; t) 2 1G (0 t 1). Combine HkH00and
k
obtain a homotopy Hk : IUk ! Uk+1 such that Hk(x; 0) = x, Hk(x; 1) 2 1G, and
x 2 1G ) Hk(x; t) 2 1G (0 t 1). Proceeding recursively, write G1 = H1 and
8
> Hk+1(Gk(x; t); 2t(2  3ok(x)))(1=2 ok(x) 2=3)
: Hk+1(Gk(x; 2t(3ok(x)  1)); t)(1=3 ok(x) 1=2)
Hk+1(x; t) (0 ok(x) 1=3)
to get a sequence of homotopies Gk : IUk ! Uk+1 such that Gk+1Io1k(]2=3; 1]) =
GkIo1k(]2=3; 1]) and Gk(x; 0) = x, Gk(x; 1) 2 1G. Take for H the homotopy IX1*
*G !
X1G that agrees on Io1k(]2=3; 1]) with Gk.]
[Note: The argument shows that 1G is a weak deformation retract of X1G.]
464
*
* f
*
* X ! X1G
*
* ? ?
FACT (Borel_Construction_) Let X be in BUN B;G. There is a pullback squar*
*e y y
*
* B ! B1G
ae 1
and since f is equivariant, the continuous function X ! X x XG induces a map*
* B ! X xG X1G,
x ! (x; f(x))
which is a homotopy equivalence (cf. p. 325).
FACT Let ff : G ! K be a continuous homomorphism_then ff determines a cont*
*inuous function
*
* X1G fff!X1K
*
* ? ?
fff: X1G ! X1K such that fff(x . g) = fff(x) . ff(g). There is a commutative di*
*agram y y
*
* B1G ! B1K
*
* ff
and ffis a homotopy equivalence iff ff is a homotopy equivalence.
CLASSIFICATION THEOREM For any topological group G, the functor kG is rep
resentable.
[This follows from Proposition 35 and the Milnor construction.]
The isomorphism classes of numerable Gbundles over B are therefore in a on*
*eto
one correspondence with the elements of [B; B1G]. By comparison, recall that on*
* general
grounds the isomorphism classes of Gbundles over B are in a onetoone corresp*
*ondence
with the elements of the cohomology set H1(B; G) (G the sheaf of Gvalued cont*
*inuous
functions on B).
LEMMA Suppose that G is metrizable_then the Milnor topology on X1G is metr*
*izable.
[Fix a metric dG on G : dG 1. Define a metric d on X1G by
d(x; y)P=min{ti(x); ti(y)}dG (gi(x); gi(y))P+m(1in{ti(x); ti(y)}):
i i
To check the triangle inequality, consider 1_2ti(x)ti(y)+min{ti(x); ti(y)}dG*
* (gi(x); gi(y)) and distinguish
two cases: ti(z) min{ti(x); ti(y)} & ti(z) < min{ti(x); ti(y)}. In the metric *
*topology, the coordinate
functions are continuous, thus the metric topology is finer than the Milnor top*
*ology. To go the other
way, let {xn} be a net in X1G such that xn ! x in the Milnor topology. Claim: x*
*n ! x in the metric
P NP ffl
topology. Fix ffl > 0. Since ti(x) = 1, 9 N : ti(x) > 1  _. Choose n0 : 8 *
*n n0 & 1 i N,
ffl i 1 4 f*
*fl
ti(xn)  ti(x) < ___4Nand ti(x) > 0 ) ti(xn) > 0 with dG (gi(xn); gi(x)) < __*
*_4N, from which
NP NP ffl *
* ffl
d(xn; x) min{ti(xn); ti(x)}dG (gi(xn); gi(x))m+i(1n{ti(xn); ti(x)}) _+ 1 *
* (1  _) < ffl:]
1 1 4 *
* 2
[Note: B1G is also metrizable. For this, it need only be shown that B1G is *
*locally metrizable and
paracompact (cf. p. 119). Local metrizability follows from the fact that X1G*
*Oi is homeomorphic to
465
OixG. Since a metrizable space is paracompact and since {Oi} is numerable, B1Ga*
*dmits a neighborhood
finite closed covering by paracompact subspaces, hence is a paracompact Hausdor*
*ff space (cf. p. 54).]
EXAMPLE X1G in the colimit topology is contractible. This is because 8 n, *
*the inclusion XnG!
Xn+1Gis a cofibration (cf. p. 34) and inessential, thus the result on p. 320 *
*can be applied. Consequently,
if the underlying topology on G is locally compact and Hausdorff (e.g., if G is*
* Lie), then colim(XnGx G) =
(colimXnG) x G, so X1G in the colimit topology is a right Gspace. As such, it *
*is a numerable Gbundle
over B1G, which is therefore a classifying space for G (cf. Proposition 35). Wh*
*ile the topology on B1G
arising in this fashion is finer than that produced by the Milnor construction,*
* it has the advantage of
being "computable". For example, let G be S0, S1, or S3, the multiplicative gro*
*up of elements of norm
one in R, C, or H _then XnG= Sn, S2n+1, or S4n+3, hence X1G = S1 and factoring *
*in the action,
B1G= P1 (R ); P1 (C ), or P1 (H ). As a colimit of the Sn, S1 is not first coun*
*table. However, the three
topologies on its underlying set coming from the Milnor construction are metriz*
*able, in particular first
countable.
[Note: Here is another model for XG and BG when G = S0; S1, or S3. Take an *
*infinite dimensional
Banach space E over R, C, or H and let S be its unit sphere_then S is an AR (cf*
*. p. 613), hence
contractible (cf. p. 614), so XG = S is universal and BG = S=G is classifying.]
Let G be a compact Lie group_then Notbohmyhas shown that the homotopy type *
*of B1Gdetermines
the Lie group isomorphism class of G.
Consider G as a pointed space with base point e. Let x1G = [(1; e); (0; e);*
* : :]:be the
base point in X1G, b1G = x1G. G the base pointaineB1G_then 8 q 0, ssq(G) ssq+*
*1(B1G).
1
Choose a homotopy H : IX1G ! X1G such that H(x;H0)(=xxG;.1)T=axking adjoints *
*and
X1G4___________wB1G
projecting leads to a map X1G ! B1G. The triangle 46 hhkp1 commutes,
B1G
thus there is an arrow G ! B1G.
PROPOSITION 36 The arrow G ! B1G is a homotopy equivalence.
[The map X1G ! B1G is a homotopy equivalence (by contractibility). But the
projections X1G ! B1G; B1G!p1B1G are Hurewicz fibrations. Therefore the map X1G*
* !
B1G is a fiber homotopy equivalence (cf. Proposition 15).]
_________________________
yJ. London Math. Soc. 52 (1995), 185198.
466
EXAMPLE Take B = Sn (n 1)_then kG Sn [Sn; B1G] ss1(B1G; b1G)\[Sn; sn; B1*
*G; b1G]
ss1(B1G; b1G)\ssn(B1G; b1G) ss0(G; e)\ssn1(G; e), i.e., in brief: kG Sn ss0(G*
*)\ssn1(G).
LEMMA Suppose that G is an ANR_then X1G and B1G are ANRs (cf. p. 645) and*
* the arrow
G ! B1G is a pointed homotopyaequivalence.e
1 ; x1 )
[Being ANRs, (G; e) & (XG G are wellpointed (cf. p. 614). Therefore X*
*1G is contractible
(B1G; b1G)
to x1G in TOP * and the arrow G ! B1G is a pointed map. But (B1G; j(b1G)) is we*
*llpointed (cf. p.
317) (actually B1G is an ANR (cf. x6, Proposition 7)), so the arrow G ! B1G is*
* a pointed homotopy
equivalence (cf. p. 319).]
EXAMPLE Let G be a Lie group_then G is an ANR (cf. p. 628). Consider kG B*
*, where (B; b0)
is nondegenerate and B is the pointed suspension. Thus kG B [B; B1G] ss1(B1G*
*; b1G)\[B; b0;
B1G; j(b1G)] ss0(G; e)\[B; b0; G; e], which, when G is path connected, simplif*
*ies to [B; b0; G; e] or still,
[B; G] (the action of ss1(G; e) on [B; b0; G; e] is trivial).
[Note: Suppose that G is an arbitrary path connected topological group_the*
*n again kG B
[B; b0; B1G; j(b1G)]. However, B1Gis a path connected H group, hence [B; b0; B1*
*G; j(b1G)] [B; B1G]
and, by Proposition 36, [B; B1G] [B; G].]
51
x5. VERTEX SCHEMES AND CW COMPLEXES
Vertex schemes and CW complexes pervade algebraic topology. What follows i*
*s an
account of their basic properties. All the relevant facts will be stated with p*
*recision but I
shall only provide proofs for those that are not readily available in the stand*
*ard treatments.
A vertex_scheme_K is a pair (V; ) consisting of a set V = {v} and a subset*
* =
{oe} 2V subject to: (1) 8 oe : oe 6= ; & #(oe) < !; (2) 8 oe : ; 6= o oe ) *
*o 2 ;
(3) 8 v : {v} 2 . The elements v of V are called the vertexes_of K and the el*
*ements
oe of are called the simplexes_of K, the nonempty o oe being termed the faces*
*_of oe.
A vertex_map_f : K1 = (V1; 1) ! K2 = (V2; 2) is a function f : V1 ! V2 such that
8 oe1 2 1; f(oe1) 2 2. VSCH is the category whose objects are the vertex scheme*
*s and
whose morphisms are the vertex maps.
EXAMPLE Let X be a set; let S = {S} be a collection of subsets of X_then t*
*he nerve_of S,
written N(S), is the vertex scheme whose vertexes are the nonempty elements of *
*S and whose simplexes
are the nonempty finite subsets of S with nonempty intersection.
Let K = (V; ) be a vertex scheme. If #() < ! ( !), then K is said to be fin*
*ite_
(countable_). If 8 v; #{oe : v 2 oe} < !, then Kaisesaid to be locally_finite_.*
* A subscheme_
0 V
of K is a vertex scheme K0 = (V 0; 0) such that V0 . An nsimplex_is a si*
*mplex of
cardinality n + 1 (n 0). The nskeleton_of K is the subscheme K(n)= (V (n); (n*
*)) of K
defined by putting V (n)= V and letting (n) be the set of msimplexes of K with
m n. The combinatorial_dimension_of K, written dim K, is 1 if K is empty, oth*
*erwise
is n if K contains an nsimplex but no (n + 1)simplex and is 1 if K contains n*
*simplexes
for all n 0. If K is finite, then dim K is finite. The converse is trivially f*
*alse.
EXAMPLE In the plane, take V = {(0; 0)} [ {(1; 1=n) : n 1}. Let K = (V; )*
* be any vertex
scheme having for its 1simplexes the sets oen = {(0; 0); (1; 1=n)} (n 1)_then*
* K is not locally finite.
Given a vertex scheme K = (V; ), let K be the set of all functionsaOEe: V*
* ! [0; 1] such
P = {OE 2 *
*K : OE1(]0; 1])
that OE1(]0; 1]) 2 & OE(v) = 1. Assign to each oe the sets *
* 1
v oe = {OE 2 *
*K : OE (]0; 1])
= oe} S
oe}. So, 8 oe : oe and K = oe, a disjoint union. Traditionally,*
* there are two
ways to topologize K.
(WT) If oe is an nsimplex, then oe can be viewed as a compact Hausd*
*orff space:
oe $ n. This said, the Whitehead_topology_on K is the final topology deter*
*mined
52
by the inclusionsaoee! K. K is a perfectly normalaparacompacteHausdorff *
*space.
Moreover, K is compactlocally compactiff K is finitelocally.finite
ae K
(BT) There is a map Vv!![0;b1] . The bv are called the baryc*
*entric_
v : bv(OE) = OE(v)
coordinates_, the initial topology on K determined by them being the barycent*
*ric_topology_,
P
a topology that is actually metrizable: d(OE; ) = bv(OE)  bv( ).
v
To keep things straight, denote by Kb the set K equipped with the baryc*
*entric
topology_then the identity map i : K ! Kb is continuous, thus the Whitehead*
* topology
is finer than the barycentric topology. The two agree iff K is locally finite.
ae [Note: A vertex map f : K1 = (V1; 1) ! K2 = (V2; 2) induces a map f :
K1 ! K2 P
OE1 ! OE2 , where OE2(v2) = f(v1)=v2OE1(v1). Topologically, f is continuou*
*s in either
the Whitehead topology or the barycentric topology. Consequently, there are two*
* functors
from VSCH to TOP , connected by the obvious natural transformation.]
EXAMPLE Let E be a vector space over R. Let V be a basis for E; let be th*
*e set of nonempty
finite subsets of V . Call K(E) the associated vertex scheme. Equip E with th*
*e finite topology_then
K(E) can be identified with the convex hull of V in E. But K(E) and K(E)b*
* are homeomorphic iff
E is finite dimensional.
[Note: Let K = (V; ) be a vertex scheme. Take for E the free Rmodule on V *
*, equipped with the
finite topology_then K can be embedded in K(E).]
PROPOSITION 1 The identity map i : K ! Kb is a homotopy equivalence.
[The collection {b1v(]0; 1])} is an open covering of Kb, hence has a pre*
*cise neighbor
hood finite open refinement {Uv}. Choose a partition of unity {v}aoneKb subor*
*dinate to
{Uv}. Let j : Kb ! K be the map that sends to the function Vv!![0; 1]. *
*Consider
ae ae v( )
the homotopies HG::IKI!KKdefined by H(OE; t) = tOE + (1  t)j O.i(OE)]
b ! Kb G( ; t) = t + (1  t)i O j( )
ae
Let X be a topological space_then two continuous functions f : X ! Kare *
*said to be contiguous_
g : X ! K
if 8 x 2 X 9 oe 2 : {f(x); g(x)} oe.
ae
FACT Suppose that f : X ! Kare contiguous_then f ' g.
g : X ! K
[Define a homotopy H : IX ! Kb between i O f and i O g by writing bv(H(x;*
* t)) = (1  t)bv(f(x)) +
tbv(g(x)) and apply Proposition 1.]
53
EXAMPLE Let X be a topological space; let U = {U} be a numerable open cove*
*ring of X_then
a Umap_is a continuous function f : X ! N(U) such that 8 U 2 U : (bU O f)1(*
*]0; 1]) U. Every
partition of unity on X subordinate to U defines a Umap and any two Umaps are*
* contiguous, hence
homotopic.
ae
FACT Let X be a topological space. Suppose that f : X ! Kare two conti*
*nuous functions
g : X ! K
such that 8 x 2 X 9 v 2 V : {f(x); g(x)} b1v(]0; 1])_then f ' g.
ADJUNCTION THEOREM Let K and L0be vertex schemes. Let K0be a subscheme
of K and let f : K0 ! L0be a vertex map_then there exists a vertex scheme L con*
*taining
L0 as a subscheme and a homeomorphism K tfL0 ! L whose restriction to *
*L0 is
the identity map.
A topological space X is said to be a polyhedron_if there exists a vertex s*
*cheme K and
a homeomorphism f : K ! X (K in the Whitehead topology). The ordered pair (*
*K; f)
is called a triangulation_of X. Put fv = bv O f1 _then the collection TK = {f*
*1v(]0; 1])}
is a numerable open covering of X and Whitehead'sy "Theorem 35" says: For any o*
*pen
covering U of X, there exists a triangulation (K; f) of X such that TK refines*
* U.
Every polyhedron is a perfectly normal paracompact Hausdorff space. A polyh*
*edron
is metrizable iff it is locally compact. Every open subset of a polyhedron is a*
* polyhedron.
Let X be a topological space_then a closure preserving closed covering A = *
*{Aj : j 2 J} of X is
S
said to be absolute_if for every subset I J, the subspace XI = Aihas the fin*
*al topology with respect
i
to the inclusions Ai! XI. Example: Every neighborhood finite closed covering of*
* X is absolute.
[Note: Let K be a vertex scheme_then {oe} is an absolute closure preservi*
*ng closed covering of K
but, in general, is only a closure preserving closed covering of Kb.]
EXAMPLE Take X = [0; 1], put X1 = [0; 1], Xn = {0} [ [1=n; 1] (n > 1)_then*
* {Xn} is a closure
S
preserving closed covering of X but {Xn} is not absolute since X = Xn does n*
*ot have the final
n>1
topology with respect to the inclusions Xn ! X (n > 1).
LEMMA Let A = {Aj : j 2 J} be an absolute closure preserving closed coveri*
*ng of X_then for
any compact Hausdorff space K, A x K = {Aj x K : j 2 J} is an absolute closure *
*preserving closed
covering of X x K.
_________________________
yProc. London Math. Soc. 45 (1939), 243327.
54
FACT If X is a topological space and if A = {Aj : j 2 J} is an absolute cl*
*osure preserving closed
covering of X such that each Aj is a normal (normal and countably paracompact, *
*perfectly normal, collec
tionwise normal, paracompact) Hausdorff space, then X is a normal (normal and c*
*ountably paracompact,
perfectly normal, collectionwise normal, paracompact) Hausdorff space.
[In every case, X is T1. And: T1+ normal ) Hausdorff.
(Normal) Let A be a closed subset of X, take an f 2 C(A; [0; 1]), an*
*d let F be the set of
*
* S
continuous functions F that are extensions of f and have domains of the form A *
*[ XI, where XI = Ai
*
* i
(I J). Order F by writing F0 F00iff F00is an extension of F0. Every chain in *
*F has an upper bound,
so by Zorn, F has a maximal element F0. But the domain of F0 is necessarily all*
* of X and F0A = f.
(Normal and Countably Paracompact) First recall that a normal Hausdo*
*rff space is count
ably paracompact iff its product with [0; 1] is normal. Since A x [0; 1] = {Ajx*
* [0; 1] : j 2 J} is an absolute
closure preserving closed covering of X x [0; 1], it follows that X x [0; 1] is*
* normal, thus X is countably
paracompact.
(Perfectly Normal) Fix a closed subset A of X. To prove that A is a *
*zero set in X, equip
S
J with a well ordering <. Given j 2 J, put X(j) = Ai. Inductively construct *
*continuous functions
ij
fj : X(j) ! [0; 1] such that fj00X(j0) = fj0if j0< j00and Z(fj) = A \ X(j).
(Collectionwise Normal) Let A be a closed subset of X, E any Banach *
*space_then it suffices
to show that every f 2 C(A; E) admits an extension F 2 C(X; E) (cf. p. 637). T*
*his can be done by
imitating the argument used to establish normality.
(Paracompact) Tamano's theorem says that a normal Hausdorff space X *
*is paracompact iff
X x fiX is normal, which enables one to proceed as in the proof of countable pa*
*racompactness.]
EXAMPLE The ordinal space [0; [ is not paracompact but {[0; ff] : ff < } i*
*s a covering of [0; [
by compact Hausdorff spaces and [0; [ has the final topology with respect to th*
*e inclusions [0; ff] ! [0; [.
FACT Let X be a topological space; let A = {Aj : j 2 J} be an absolute clo*
*sure preserving closed
covering of X. Suppose that each Aj can be embedded as a closed subspace of a p*
*olyhedron_then X can
be embedded as a closed subspace of a polyhedron.
[For every j there is a vertex scheme Kj, a vector space Ej over R, and a c*
*losed embedding fj : Aj !
Kj ( Ej). Write E for the direct sum of the Ej and give E the finite topology*
*. Let EI stand for the
direct sum of the Ei(i 2 I) and put KI = K(EI)_then KI K(E). Here, as abov*
*e, I is a subset of J.
Consider the set P of all pairs (I; fI), where fI : XI ! KI is a closed embed*
*ding. Order P by stipulating
that (I0; fI0) (I00; fI00) iff I0 I00and (1) fI00XI0= fI0& (2) fI00(XI00 XI0*
*) \ KI0 = ;. Every chain
in P has an upper bound, so by Zorn, P has a maximal element (I0; fI0). Verify *
*that XI0= X.]
Application: Let X be a paracompact Hausdorff space. Suppose that X admits *
*a covering U by open
55
sets U, each of which is homeomorphic to a closed subspace of a polyhedron_then*
* X is homeomorphic to
a closed subspace of a polyhedron.
The embedding theorem of dimension theory implies that every second countab*
*le compact Hausdorff
space of finite topological dimension can be embedded in some euclidean space (*
*cf. p. 1928). It there
fore follows that if a topological space X has an absolute closure preserving c*
*losed covering made up of
metrizable compacta of finite topological dimension, then X can be embedded as *
*a closed subspace of a
polyhedron. This setup is realized, e.g., by the CW complexes (cf. p. 512).
The product X x Y of polyhedrons X and Y need not be a polyhedron (cf. p. 5*
*14),
although this will be the case if one of the factors is locally compact.
FACT Let X and Y be polyhedrons_thenaXex Y has the homotopy type of a poly*
*hedron.
[Consider a product K x L. Since K & Kbhave the same homotopy typ*
*e, it need only be
L & Lb ae
shown that Kbx Lb has the homotopy type of a polyhedron. Let U be the coz*
*ero set covering of
ae ae V
Kb associated with the barycentric coordinates_then K can be identified w*
*ith the corresponding
Lbae L
nerve N(U) . Put U x V = {U x V : U 2 U; V 2 V}. Claim: There is a homoto*
*py equivalence
N(V) ae
N(U x V)b ! N(U)bx N(V)b. Indeed, the projections U x V ! U (U x V ! U*
*)define vertex
ae U x V ! V (U x V ! V*
* )
maps pU : N(U x V) ! N(U), from which p : N(U x V)b ! N(U)bx N(V)b, whe*
*re p = pU x pV.
pV : N(U x V) ! N(V)
A homotopy inverse q : N(U)bx N(V)b ! N(U x V)b to p is given in terms of*
* barycentric coordinates
by bUxV (q(OE; )) = bU (OE)bV ( ).]
Let X be a topological space; let A be a closed subspace of X_then X is sai*
*d to
be obtained from A by attaching_ncells_if there exists an indexed collection o*
*f continuous
` *
* n
functions fi: Sn1 ! A such that X is homeomorphic to the adjunction space ( *
*D ) tf
` ` ` i
A (f = fi). When this is so, X  A is homeomorphic to (D n  Sn1) = B n*
*, a
i i i
decomposition that displays its path components as a collection of ncells.
EXAMPLE Put sn = (1; 0; : :;:0) 2 R n+1(n 1). Let I be a set indexing a *
*collection of
W
copies of the pointed space (Sn; sn)_then the wedge Snis a pointed space with*
* basepoint *. Since the
W I
quotient Dn=Sn1 can be identified with Sn, Sn is obtained from * by attachin*
*g ncells.
I
56
Let X be a topological8space_then a CW_structure_on X is a sequence X(0); X*
*(1); : : :
< X = 1SX(n)
of closed subspaces X(n): : 0 and subject to:
X(n) X(n+1)
(CW 1) X(0)is discrete.
(CW 2) X(n)is obtained from X(n1) by attaching ncells (n > 0).
(CW 3) X has the final topology determined by the inclusions X(n)! X.
A CW_complex__is a topological space X equipped with a CW structure. Just *
*as a
polyhedron may have more than one triangulation, a CW complex may have more than
one CW structure. Every CW complex is a perfectly normal paracompact Hausdorff *
*space.
[Note: Let K be a vertex scheme. Consider K (Whitehead topology)_then K(*
*0)
is discrete and K(n) is obtained from K(n1) by attaching ncells (n > 0) :*
* oe  !
K(n1), oe an nsimplex. Since K has the final topology determined by the i*
*nclusions
K(n) ! K, it follows that the sequence {K(n)} is a CW structure on K.]
CW is the full subcategory of TOP whose objects are the CW complexes and H*
*CW
is the associated homotopy category.
1S
EXAMPLE Equip R1 with the finite topology. Let S1 = Snand give it the in*
*duced topology
0
or, what amounts to the same, the final topology determined by the inclusions S*
*n ! S1. The sequence
{Sn} is a CW structure on S1 . Indeed, Sn is obtained from Sn1 by attaching t*
*wo ncells (n > 0)
(seal the upper and lower hemispheres at the equator). On the other hand, R ni*
*s not obtained from
R n1by attaching ncells. Therefore the sequence {R n} is not a CW structure *
*on R1 . But R1 is
obviously a polyhedron. A less apparent aspect is this. Put s1 = (1; 0; : :):_t*
*hen it can be shown that
S1 and S1 {s1 } are homeomorphic. Since stereographic projection from s1 defin*
*es a homeomorphism
S1  {s1 } ! R1 , the conclusion is that S1 and R1 are actually1homeomorphic.
S
[Note: The sequence {D n} is not a CW structure for D 1 = D n. However, *
*D n[ Sn can be
0
obtained from D n1[ Sn1 by attaching four ncells (n > 0), so the sequence {D*
* n[ Sn} is a CW
structure for D1 .]
Let X be a CW complex with CW structure {X(n)} : X(n)is the nskeleton_of X*
*. The
inclusion X(n)! X is a closed cofibration (cf. p. 35) and 8 n 1, the pair (X;*
* X(n)) is
nconnected. Put E0 = X(0)and denote by En the set of path components of X(n)X*
*(n1)
1S
(n > 0). Let E = En_then an element e of E is said to be a cell_in X, e being*
* termed
0
an ncell_if e 2 En. Set theoretically, X is the disjoint union of its cells. O*
*n the basis of
the definitions, for every e 2 En, there exists a continuous function e : D n! *
*e [ X(n1),
the characteristic_map_of e, such that eB n is an embedding and (i) e(B n) = e*
*; (ii)
e(S n1) X(n1); (iii) e(D n) = _e. X has the final topology determined by th*
*e e.
57
A subspace A X is called a subcomplex_if there exists a subset EA E : A = [EA
& 8 e 2 EA \ En, e(D n) A. A subcomplex A of X is itself a CW complex with
CW structure {A(n)= A \ X(n)}. The inclusion A ! X is a closed cofibration and *
*for
every open U A there exists an open V A with V U such that A is a strong
deformation retract of V . If E0 E, then [E0 is a subcomplex iff [E0 is closed*
*. Arbitrary
unions and intersections of subcomplexes are subcomplexes. In general, the _ea*
*re not
subcomplexes, although this will be the case if all the characteristic maps are*
* embeddings.
The combinatorial_dimension_of X, written dim X, is 1 if X is empty, otherwise*
* is the
smallest value of n such that X = X(n)(or 1 if there is no such n). It is a fac*
*t that dim X
is equal to the topological dimension of X (cf. p. 1921), therefore is indepen*
*dent of the
CW structure.
Let X be a CW complex_then the collection __E= {_e: e 2 E} is a closed cove*
*ring of X and X has
the final topology determined by the inclusions _e! X but __Eneed not be closur*
*e preserving.
EXAMPLE (Simplicial_Sets_) Let X be a simplicial set_then its geometric r*
*ealization X is a
CW complex with CW structure {X(n)}. In fact, X(0) is discrete and, using t*
*he notation of p. 018,
X#n. _[n] ! X(n1)
the commutative diagram ?y ?y is a pushout square in SISET. Sin*
*ce the geomet
X#n. [n] ! X(n)
ric realization functor ? is a left adjoint, it preserves colimits. Therefor*
*e the commutative diagram
X#n. _n ! X(n1)
?y ?y is a pushout square in TOP , which means that X(n) is o*
*btained from
X#n. n ! X(n)
X(n1) by attaching ncells (n > 0). Moreover, X = colimX(n)) X = colimX(*
*n), so X has
the final topology determined by the inclusions X(n) ! X. Denoting now by G*
* the identity component
of the homeomorphism group of [0; 1], there is a left action G x X ! X and *
*the orbits of G are the cells
of X.
[Note: If Y is a simplicial subset of X, then Y  is a subcomplex of X, *
*thus the inclusion Y  ! X
is a closed cofibration.]
It is true but not obvious that if X is a simplicial set, then X is actua*
*lly a polyhedron (cf. p. 1311).
A CW_pair_is a pair (X; A), where X is a CW complex and A X is a subcomple*
*x.
CW 2 is the full subcategory of TOP 2whose objects are the CW pairs and HCW *
* 2is the
associated homotopy category.
58
A pointed_CW_complex_ is a pair (X; x0), where X is a CW complex and x0 2 X*
*(0).
CW * is the full subcategory of TOP * whose objects are the pointed CW comple*
*xes and
HCW * is the associated homotopy category.
[Note: If (X; x0) is a pointed CW complex, then 8 q 1, ssq(X; x0) co*
*lim
ssq(X(n); x0).]
Let X be a CW complex_then 8 x0 2 X, the inclusion {x0} ! X is a cofibratio*
*n (cf. p. 317),
thus (X; x0) is wellpointed. Of course, a given x0 need not be in X(0)but there*
* always exists some CW
structure on X having x0 as a 0cell.
Let X be a topological space, A X a closed subspace_then a relative_CW_str*
*ucture_
on8 (X; A) is a sequence (X; A)(0); (X; A)(1); : : :of closed subspaces *
*(X; A)(n) :
< X = 1S(X; A)(n)
: (X; 0A)(n) (X; A)(n+1) and subject to:
(RCW 1) (X; A)(0)is obtained from A by attaching 0cells.
(RCW 2) (X; A)(n)is obtained from (X; A)(n1)by attaching ncells (n*
* > 0).
(RCW 3) X has the final topology determined by the inclusions (X; A)*
*(n)! X.
[Note: (X; A)(0)is the coproduct of A and a discrete space, so when A = ; *
*the
definition reduces to that of a CW structure.]
A relative_CW_complex_is a topological space X and a closed subspace A equi*
*pped
with a relative CW structure.
[Note: If (X; A) is a relative CW complex, then the inclusion A ! X is a c*
*losed
cofibration and X=A is a CW complex. On the other hand, if X is a CW complex an*
*d if
A X is a subcomplex, then (X; A) is a relative CW complex.]
Example: Suppose that (X; A) is a relative CW complex_then (IX; IA) is a re*
*lative
CW complex, where (IX; IA)(n)= i0(X; A)(n)[ (I(X; A)(n1)[ IA) [ i1(X; A)(n).
Let (X; A) be a relative CW complex with relative CW structure {(X; A)(n)} *
*: (X; A)(n)
is the nskeleton_of X relative to A. The inclusion (X; A)(n)! X is a closed co*
*fibration (cf.
p. 35) and 8 n 1, the pair (X; (X; A)(n)) is nconnected. The relative_combi*
*natorial_
dimension_of (X; A), written dim (X; A), is 1 if X is empty, otherwise is the *
*smallest
value of n such that X = (X; A)(n)(or 1 if there is no such n). Obviously, dim(*
*X; A) =
dim (X=A) provided that X is nonempty.
LEMMA Let (X; A) be a relative CW complex_then for every compact subset K
X there exists an index n such that K (X; A)(n).
[Consider the image of K under the projection X ! X=A, bearing in mind that*
* X=A
is a CW complex.]
59
Application: Let (X; A; x0) be a pointed pair. Assume: (X; A) is a relat*
*ive CW
complex_then 8 q 1, ssq(X; x0) colimssq((X; A)(n); x0).
HOPF EXTENSION THEOREM Let (X; A) be a relative CW complex with
dim (X; A) n + 1 (n 1). Suppose that f 2 C(A; Sn)_then 9 F 2 C(X; Sn) : F A *
*= f
iff f*(Hn (S n)) i*(Hn (X)), i : A ! X the inclusion.
HOPF CLASSIFICATION THEOREM Let (X; A) be a relative CW complex with
dim (X; A) n (n 1). Fix a generator 2 Hn (S n; sn; Z)_then the assignment [f*
*] ! f*
defines a bijection [X; A; Sn; sn] ! Hn (X; A; Z).
EXAMPLE The unit tangent bundle of S2ncan be identified with the Stiefel m*
*anifold V2n+1;2.
It is (2n  2)connected with euclidean dimension 4n  1. One has Hq(V 2n+1;2)*
* Z (q = 0; 4n 
1), H2n1(V 2n+1;2) Z=2Z, and Hq(V 2n+1;2) = 0 otherwise. By the Hopf classif*
*ication theorem,
[V 2n+1;2; S4n1] H4n1(V 2n+1;2), so there is a map f : V2n+1;2! S4n1such th*
*at f* induces an iso
morphism H4n1(S4n1) ! H4n1(V 2n+1;2). Consequently, under f*, H*(V 2n+1;2; Q*
*) H*(S4n1; Q),
thus the mapping fiber Ef of f is rationally acyclic, i.e., eH*(Ef; Q) = 0 (cf.*
* p. 444).
ae ae (n)
Let XY be CW complexes with CW structures {X{Y }(n)}_then a skeletal_ma*
*p_is a
continuous function f : X ! Y such that 8 n : f(X(n)) Y (n).
[Note: A CW complex is filtered by its skeletons, so the term "skeletal map*
*" is just
the name used for "filtered map" in the CW context.]
EXAMPLE (Simplicial_Sets_) If f : X ! Y is a simplicial map, then f : *
*X ! Y  is a skeletal
map and transforms cells of X onto cells of Y .
SKELETAL APPROXIMATION THEOREM Let X and Y be CW complexes. Sup
pose that A is a subcomplex of X_then for any continuous function f : X ! Y suc*
*h that
fA is skeletal there exists a skeletal map g : X ! Y such that fA = gA and f*
* ' g relA.
[Note: In particular, every continuous function f : X ! Y is homotopic to a*
* skeletal
map g : X ! Y .]
ae ae *
* (n)
Let (X;(A)Y;bB)e relative CW complexes with relative CW structures {(X;*
*{A)(Y;}B)(n)}_
then a relative_skeletal_map_is a continuous function f : (X; A) ! (Y; B) such *
*that 8 n :
f((X; A)(n)) (Y; B)(n).
510
RELATIVE SKELETAL APPROXIMATION THEOREM Let (X; A) and (Y; B) be
relative CW complexes_then every continuous function f : (X; A) ! (Y; B) is hom*
*otopic
relA to a relative skeletal map g : (X; A) ! (Y; B).
Here is a summary of the main topological properties of CW complexes.
(TCW 1) Every CW complex is compactly generated.
(TCW 2) Every CW complex is stratifiable, hence is hereditarily para*
*compact.
(TCW 3) Every CW complex is uniformly locally contractible, therefor*
*e locally
contractible.
(TCW 4) Every CW complex is numerably contractible.
(TCW 5) Every CW complex is locally path connected.
(TCW 6) Every CW complex is the coproduct of its path components and*
* these
are subcomplexes.
(TCW 7) Every connected CW complex is path connected.
(TCW 8) Every connected CW complex has a universal covering space.
[Note: If X is a connected CW complex with CW structure {X(n)} and if p : e*
*X! X
is a covering projection, then the sequence {Xe(n)= p1(X(n))} is a CW structur*
*e on Xe
with respect to which p is skeletal.]
If (X; A) is a relative CW complex, then certain topological properties of *
*A are au
tomatically transmitted to X. For example, if A is in CG , CG , or CGH , *
*then the
same holds for X. Analogous remarks apply to a Hausdorff A which is normal, per*
*fectly
normal, paracompact, etc.
(F) A CW complex X is said to be finite_if #(E) < !. Every finite CW c*
*omplex
is compact and conversely. A compact subset of a CW complex is contained in a *
*finite
subcomplex.
(C) A CW complex X is said to be countable_if #(E) !. A CW complex is
countable iff it does not contain an uncountable discrete set. Every countable *
*CW complex
is Lindel"of and conversely.
[Note: The homotopy groups of a countable connected CW complex are countabl*
*e.]
(LF) A CW complex X is said to be locally_finite_if each x 2 X has a n*
*eigh
borhood U such that U is contained in a finite subcomplex of X. Every locally f*
*inite CW
complex is locally compact and conversely. Every locally finite CW complex is m*
*etrizable
and conversely. A locally finite connected CW complex is countable.
What spaces carry a CW structure? There is no known characterization but th*
*e foregoing conditions
impose a priori limitations. For example, a nonmetrizable LCH space cannot be e*
*quipped with a CW
511
structure. On the other hand, the Cantor set and the Hilbert cube are metrizab*
*le compact Hausdorff
spaces but neither supports a CW structure.
[Note: Every compact differentiable manifold can be triangulated but exampl*
*es are known of compact
topological manifolds that cannot be triangulated, i.e., that are not polyhedro*
*ns (DavisJanuszkiewiczy).]
EXAMPLE (The_Sorgenfrey_Line_) Topologize X = R by choosing for the basic*
* neighborhoods
of a given x all sets of the form [x; y[ (x < y). In this topology, the line is*
* a perfectly normal paracompact
Hausdorff space but it is not locally compact. While not second countable, X i*
*s first countable (and
separable), therefore is compactly generated. However, X is not locally connect*
*ed, thus carries no CW
structure.
[Note: The square of the Sorgenfrey line is not normal (apply Jones' lemma)*
*.]
EXAMPLE (The_Niemytzki_Plane_) Let X be the closed upper half plane in R *
*2. Topologize
X as follows: The basic neighborhoods of (x; y) (y > 0) are as usual but the ba*
*sic neighborhoods of
(x; 0) are the {(x; 0)} [ B, where B is an open disk in the upper half plane wi*
*th horizontal tangent
at (x; 0). X is a compactly generated CRH space. In addition, X is Moore, hen*
*ce is perfect. And
Xaiseconnected, locally path connected, and even contractible (consider the hom*
*otopy H((x; y); t) =
(x; y) + t(0; 1) (0 t 1=2)). However, X is not normal, thus carries no CW *
*structure.
t(0; 1) + 2(1  t)(x;(y)1=2 t 1)
[Note: X is neither countably paracompact nor metacompact but is countably *
*metacompact.]
EXAMPLE An open subset of a polyhedron is a polyhedron but an open subset *
*of a CW complex
need not be a CW complex. To see this, fix an enumerationa{qn}eof Q \ ]0; 1[. C*
*onsider the CW complex
X defined as follows: X(0)= {0; 1}; X(1)= [0; 1] 0 ! 0 and at each point qn at*
*tach a 2cell by taking
1 ! 1
for fn : S1! X(1)the constant map fn = qn. Choose a point xn 2 en (2 E2) and pu*
*t A = {xn}_then
A is closed and U = X  A carries no CW structure.
[Otherwise: (a) [0; 1] U(1); (b) 8 n; U(1)\ en 6= ;; (c) 8 n; qn 2 U(0).]
PROPOSITION 2 Every CW complex has the homotopy type of a polyhedron.
[Let X be a CW complex with CW structure {X(n)} : X = colimX(n). Taking
into account x3, Proposition 15, it will be enough to construct a sequence of v*
*ertex
schemes K(n) such that 8 n, K(n1) is a subscheme of K(n) and a sequence of hom*
*o
topy equivalences OEn : X(n) ! K(n) such that 8 n, OEnX(n1) = OEn1. Proce*
*eding by
induction, make the obvious choices when n = 0 and then assume that K(0); : :;:*
*K(n1)
and OE0; : :;:OEn1 have been defined. At level n there is an index set In and*
* a pushout
_________________________
yJ. Differential Geom. 34 (1991), 347388.
512
In . _n f! X(n1) `
square ?y ?y (f = fi). Given i 2 In, use the simplicial appro*
*xima
i
In . n ! X(n)
tion theorem to produce a vertex scheme Ki and a vertex map gi : Ki ! K(n1) wi*
*th
`
Ki = n_ and gi ' OEn1 O fi. Combine the Ki and put g = gi. The a*
*d
i
junction theorem implies that there exists a vertex scheme K(n) containing K(n*
*1) as
a subscheme and a homeomorphism In . n tgK(n1) ! K(n) whose restriction *
*to
In . _n _____wf4X(n1)
K(n1) is the identity map. The triangle g46 uOEn1is homoto*
*py commuta
K(n1)
tive: g ' OEn1 O f. Since OEn1 is a homotopy equivalence, one can find a *
*homotopy
equivalence OEn : In . n tf X(n1)! In . n tgK(n1) such that OEnX(n1)= O*
*En1 (cf.
p. 324), which completes the induction.]
[Note: Similar methods lead to the expected analogs in CW 2 or CW *. Co*
*nsider,
e.g., a CW pair (X; A) with relative CW structure {(X; A)(n)} : (X; A)(n)= X(n)*
*[ A.
Choose a vertex scheme L and a homotopy equivalence OE : A ! L_then there is a
vertex scheme K(0)containing L as a subscheme and a homotopy equivalence of pai*
*rs
((X; A)(0); A) ! (K(0); L) so, arguing as above, there is a vertex scheme K*
* containing L
as a subscheme and a homotopy equivalence : X ! K such that A = OE. Conclus*
*ion:
In HTOP 2; (X; A) (K; L) (cf. x3, Proposition 14).]
PROPOSITION 3 Let X be a CW complex. Assume: (i) X is finite (countable) or
(ii) dim X n_then there exists a vertex scheme K such that X has the homotopy *
*type
of K, where (i) K is finite (countable) or (ii) dim K n.
[This is implicit in the proof of the preceding proposition.]
Let X be a CW complex; let A be the collection of finite subcomplexes of X_*
*then A is an absolute
closure preserving closed covering of X. Since every finite subcomplex of X is *
*a second countable compact
Hausdorff space of finite topological dimension, it follows that X can be embed*
*ded as a closed subspace
of a polyhedron (cf. p. 55).
FACT Every CW complex is the retract of a polyhedron, hence every open sub*
*set of a CW complex
is the retract of a polyhedron.
EXAMPLE Every polyhedron is a CW complex but there exist CW complexes that*
* cannot be
triangulated. Thus let f(t) = t sin(ss=2t) (0 < t 1) and set f(0) = 0. Denot*
*e by m the absolute
513
minimum of f on [0; 1] (so 1 < m < 0). Take for X the image of the square [0; *
*1] x [0; 1] under the map
(u; v) ! (u; uv; f(v)). The following subspaces constitute a CW structure on X:
X(0)= {(0; 0; 0); (1; 0; 0); (0; 0; 1); (1; 1; 1); (0; 0; *
*m)};
ae
X(1)= {(u; 0; 0) : 0 u 1}[{(u; u; 1) : 0 u 1}[ {(0; 0; v) : m v[{0}(1; v;*
* f(v)) : 0 v 1},
{(0; 0; v) : 0 v 1}
and X(2)= X. Using the fact that f has a sequence {Mn} of relative maxima: M1 >*
* M2 > . .(.1 > M1),
look at the (0; 0; Mn) and deduce that X is not a polyhedron.
FACT Let X be a CW complex. Suppose that all the characteristic maps are e*
*mbeddings_then
X is a polyhedron.
There are two other issues.ae ae
(n)}
(Products) Let XY be CW complexes with CW structures {X{Y (n)}. *
*Put
S
(X xk Y )(n)= X(p)xk Y (q). Consider X xk Y _then the sequence {(X xk Y )(*
*n)}
p+q=n
satisfies CW 1, CW 2, and CW 3 above, meaning that it is a CW structure on X xk*
* Y . When
can "xk" be replaced by "x"? Useful sufficient conditions to ensure this are th*
*at one of
the factors be locally finite or that both of the factors be countable (necessa*
*ry conditions
have been discussed by Tanakay).
EXAMPLE (Dowker's_Product_) Suppose that X and Y are CW complexes_then th*
*e product
X x Y need not be compactly generated, hence, when this happens, X x Y is not a*
* CW complex. Here is
an illustration.aDefinitioneof X: Put X(0)= NN [ {0} (discrete topology), let f*
*s : {0; 1} ! X(0)be the
map 0 ! 0(s 2 NN ), write X(1)for the space thereby obtained from X(0)by atta*
*ching 1cells, and
1 ! s
take X = X(0)[aX(1).eDefinition of Y : Put Y (0)= N [ {0} (discrete topology), *
*let fn : {0; 1} ! Y (0)
be the map 0 ! 0(n 2 N), write Y (1)for the space thereby obtained from Y (0)*
*by attaching 1cells,
1 ! n
and take Y = Y (0)[ Y (1). Let s (n) be the characteristic map of the 1cell co*
*rresponding to s 2 NN
(n 2 N ). Consider the following subset of X x Y : K = {(s(1=sn); n(1=sn)) : (s*
*; n) 2 N N x N}.
Evidently K is a closed subset of X xk Y . But K is not a closed subset of X x*
* Y . For if it were,
X x Y  K would be open and since the point (0; 0) 2 X x Y  K, there would be *
*a basic neighborhood
U x V : (0; 0) 2 U x V X x Y  K. Given s 2 N N, 9 a real number as : 0 < as *
* 1 such that
U {s(p) : p < as} and given n 2 N, 9 a real number bn : 0 < bn 1 such that V *
* {n(q) : q < bn}.
Define _s2 NN by _sn= 1 + [max{n; 1=bn}] (so _sn> n & _sn> 1=bn); define __n2 *
*N by __n= 1 + [1=a_s] (so
_________________________
yProc. Amer. Math. Soc. 86 (1982), 503507.
514
__
n > 1=a_s)_then the pair (_s(1=_s_n),a__n(1=_s_n))eis in both U x V and K. Cont*
*radiction. Incidentally, one
can show that the projections X xk Y ! X are not Hurewicz fibrations (althoug*
*h, of course, they are
X xk Y ! Y
CG fibrations).
[Note: This construction has an obvious interpretation in terms of cones. O*
*bserve too that X and
Y are polyhedrons. Corollary: The square of a polyhedron need not be a polyhedr*
*on.]
FACT Every countable CW complex has the homotopy type of a locally finite *
*countable CW
complex.
[Let X be a countable CW complex. Fix an enumeration {ek} of its cells. Giv*
*en ek, denote by X(ek)
the intersectionnof all subcomplexes of X containing ek_then X(ek) is a finite *
*subcomplex of X. Put
S
Xn = X(ek) : X0 X1 . .i.s an expanding sequence of topological spaces with *
*X1 = X. The
0
telescope telX1 of X1 has the same homotopy type as X1 = X (cf. p. 312) and*
* is a CW complex.
In fact, telX1 is the subcomplex of Xxk [0; 1[= X x [0; 1[ made up of the cell*
*s e x {n}, ex]n; n + 1[,
where e is a cell of Xm (m n), a description which makes it clear that telX1 *
*is locally finite.]
[Note: Suppose that X is a locally finite countable CW complex_then there e*
*xists a sequence of
S
finite subcomplexes Xn such that 8 n; Xn intXn+1, with X = Xn.]
n
ae ae (n)
(Adjunctions) Let XY be CW complexes with CW structures {X{Y }(n)*
*}. Sup
pose that A is a subcomplex of X. Let f : A ! Y be a skeletal map_then the adju*
*nction
space X tf Y is a CW complex, the CW structure being {X(n)tf(n)Y (n)}(f(n)= fA*
*(n)).
Examples: (1) If X is a CW complex and if A X is a subcomplex, then the quotie*
*nt
X=A is a CW complex; (2) If X is a CW complex, then its cone X and its suspensi*
*on
X are CW complexes; (3) If X and Y are CW complexes and if f : X ! Y is a skele*
*tal
map, then the mapping cylinder Mf of f is a CW complex, containing both X and Y*
* as
embedded subcomplexes; (4) If X and Y are CW complexes and if f : X ! Y is a sk*
*eletal
map, then the mapping cone Cf of f is a CW complex containing Y as an embedded
subcomplex. ae
[Note: There are also pointed analogs of these results. For example, if (*
*X;(x0)Y;ayre
*
* 0)
pointed CW complexes, then the smash product X#kY is a pointed CW complex.]
Let X and Y be CW complexes. Let A be a subcomplex of X and let f : A ! Y b*
*e a continuous
function_then X tf Y has the homotopy type of a CW complex. Proof: By the skele*
*tal approximation
theorem, there exists a skeletal map g : A ! Y such that f ' g, so X tf Y has t*
*he same homotopy type
as X tg Y (cf. p. 324).
FACT A CW complex is path connected iff its 1skeleton is path connected.
515
EXAMPLE (Trees_) Let X be a nonempty connected CW complex_then a tree_in *
*X is a
nonempty simply connected subcomplex T of X with dimT 1. Every tree in X is c*
*ontractible and
contained in a maximal tree. A tree is maximal iff it contains X(0). If T is a *
*maximal tree in X, then
X=T is a connected CW complex with exactly one 0cell and the projection X ! X=*
*T is a homotopy
equivalence (cf. p. 324).
ae
WHE CRITERION Let XY be topological spaces, f : X ! Y a continuous
function_then f is a weak homotopy equivalence if for any finite CW pair (K; L)*
* and
L? OE! X?
any diagram y yf , where f O OE = L, there exists a : K ! X such th*
*at
K ! Y
L = OE and f O ' relL.
sn? ! X? Sn? ! X?
[Indeed, diagrams of the form y y f, y yf evidently suf*
*fice.]
S n ! Y D n+1 ! Y
LEMMA Suppose that f : X ! Y is an nequivalence_then in any diagram
Sn1? OE!X
y ?yf, where f O OE ' on Sn1 by h : IS n1! Y , there exists a :*
* D n! X
D n ! Y
such that S n1= OE and an H : ID n ! Y such that HIS n1= h and f O ' on *
*D n
by H.
HOMOTOPY EXTENSION LIFTING PROPERTY Suppose that f : X ! Y is a
weak homotopy equivalence. Let (K; L) be a relative CW complex_then in any diag*
*ram
L? OE!X
y ?yf, where f O OE ' on L by h : IL ! Y , there exists a : K ! X su*
*ch that
K ! Y
L = OE and an H : IK ! Y such that HIL = h and f O ' on K by H.
Application: Let f : X ! Y be a weak homotopy equivalence_then for any CW
complex K, the arrow f* : [K; X] ! [K; Y ] is bijective.
[To see that f* is surjective (injective), apply the homotopy extension lif*
*ting property
to (K; ;) ((IK; i0K [ i1K)).]
[Note: The condition is also characteristic. Thus first take K = * and re*
*duce to
516
ae
W 1
when XY are path connected. Next, take K = S (I a suitable index set) to g*
*et that
I
8 x 2 X, f* : ss1(X; x) ! ss1(Y; f(x)) is surjective. Finish by taking K = Sn (*
*cf. p. 318).]
ae
EXAMPLE Let (X; x0)be pointed connected CW complexes. Suppose that f 2 C(*
*X; x0; Y; y0)
(Y; y0)
has the property that 8 n > 1, f* : ssn(X; x0) ! ssn(Y; y0) is bijective_then f*
*or any pointed simply
connected CW complex (K; k0), the arrow f* : [K; k0; X; x0] ! [K; k0; Y; y0] is*
* bijective.
FACT Let p : X ! B be a continuous function_then p is both a weak homotopy*
* equivalence
L *
* OE!X
? *
* ?
and a Serre fibration iff for any relative CW complex (K; L) and any diagram y *
* y p , where
K *
* ! B
p O OE = L, there exists a : K ! X such that L = OE and p O = .
[Note: The characterization can be simplified: A continuous function p : X *
*! B is both a weak
Sn1 *
* ! X
? *
* ?
homotopy equivalence and a Serre fibration iff every commutative diagram y *
* y (n 0)
D n *
* ! B
admits a filler Dn ! X.]
X0 ! X
? ?
Application: Let py0 yp be a pullback square. Suppose that p is a Se*
*rre fibration and a
B0 ! B
weak homotopy equivalence_then p0is a Serre fibration and a weak homotopy equiv*
*alence.
A continuous function f : (X; A) ! (Y; B) is said to be a weak_homotopy_equ*
*ivalence_
of_pairs_provided that f : X ! Y and f : A ! B are weak homotopy equivalences.
[Note: A weak homotopy equivalence of pairs is a relative weak homotopy equ*
*ivalence
(cf. p. 451) but not conversely.]
Application: Let f : (X; A) ! (Y; B) be a weak homotopy equivalence of pair*
*s_then
for any CW pair (K; L), the arrow f* : [K; L; X; A] ! [K; L; Y; B] is bijective.
[Note: The condition is also characteristic. For [K; ;; X; A] [K; ;; Y; B]*
* ) [K; X]
[K; Y ] and [IK; i0K; X; A] [IK; i0K; Y; B] ) [K; A] [K; B].]
REALIZATION THEOREM Suppose that X and Y are CW complexes. Let f :
X ! Y be a weak homotopy equivalence_then f is a homotopy equivalence.
[Note: It is a corollary that the result remains true when X and Y have the*
* homotopy
type of CW complexes.]
517
Application: A connected CW complex is contractible iff it is homotopically*
* trivial.
EXAMPLE Let X and Y be CW complexes_then the identity map X xk Y ! X x Y i*
*s a
homotopy equivalence.
[A priori, the identity map X xk Y ! X x Y is a weak homotopy equivalence. *
*However, X and Y
each have the homotopy type of a polyhedron (cf. Proposition 2), thus the same *
*holds for their product
X x Y (cf. p. 55).]
EXAMPLE (H_Groups_) Let (X; x0) be a nondegenerateahomotopyeassociative H*
* space. Assume:
X is path connected_then the shearing map sh: X x X ! X x X is a weak homot*
*opy equivalence,
(x; y) ! (x; xy)
thus X is an H group if X carries a CW structure (cf. p. 427).
ae
The pointed version of the realization theorem says that if XY are CW com*
*plexes and
if f : X ! Y is a weakahomotopyeequivalence, then f is a pointed homotopy equiv*
*alence
for any choice of x0y2 X with f(x0) = y0. Proof: By the realization theorem*
*, f is a
0 2 Y ae
homotopy equivalence, so f is actually a pointed homotopy equivalence, (X;(x0*
*)Y;byeing
*
* 0)
wellpointed (cf. p. 319).
RELATIVE REALIZATION THEOREM Suppose that (X; A) and (Y; B) are CW
pairs. Let f : (X; A) ! (Y; B) be a weak homotopy equivalence of pairs_then f *
*is a
homotopy equivalence of pairs.
[Note: This result need not be true if one merely assumes that f is a relat*
*ive weak
homotopy equivalence. Example: Take X path connected, fix a point a0 2 A, and c*
*onsider
the projection (X x A; a0 x A) ! (X; a0). It is a relative weak homotopy equiva*
*lence but
the induced map on relative singular homology is not necessarily an isomorphism*
*.]
The relative realization theorem is a consequence of the following assertio*
*n. Suppose that (X; A) and
(Y; B) are relative CW complexes. Let f : (X; A) ! (Y; B) be a weak homotopy eq*
*uivalence of pairs with
fA : A ! B a homotopy equivalence_then f is a homotopy equivalence of pairs.
EXAMPLE Let (K; L) be a relative CW complex. Assume: The inclusion L ! K *
*is a weak
homotopy equivalence_then the inclusion L ! K is a homotopy equivalence. Proof:*
* Consider the arrow
(L; L) ! (K; L).
PROPOSITION 4 Let (Y; B) and (Y 0; B0) be pairs and let h : (Y; B) ! (Y 0;*
* B0) be
a continuous function; let (X; A) and (X0; A0) be CW pairs and let f : (X; A) !*
* (Y; B)
518
& f0 : (X0; A0) ! (Y 0; B0) be continuous functions. Assume: f0 is a weak hom*
*otopy
equivalence of pairs_then there exists a continuous function g : (X; A) ! (X0; *
*A0), unique
(X;?A) g! (X0;?A0)
up to homotopy of pairs, such that the diagram fy y f0 commutes *
*up to
(Y; B) !h (Y 0; B0)
homotopy of pairs.
[The arrow f0*: [X; A; X0; A0] ! [X; A; Y 0; B0] is bijective.]
Given a topological space X, a CW_resolution_for X is an ordered pair (K; f*
*), where
K is a CW complex and f : K ! X is a weak homotopy equivalence. The homotopy ty*
*pe
of a CW resolution is unique. Proof: Let f : K ! X & f0 : K0 ! X be CW resoluti*
*ons
of X_then by Proposition 4, there exists a continuous function g : K ! K0 such *
*that
K g! K0
the diagram f?y ?yf0is homotopy commutative: f ' f0O g. Theref*
*ore
X =======================X
g is a weak homotopy equivalence, hence is a homotopy equivalence (via the real*
*ization
theorem).
RESOLUTION THEOREM Every topological space X admits a CW resolution f :
K ! X.
[Note: If X is path connected (nconnected), then one can choose K path con*
*nected
with K(0)(K(n)) a singleton.]
Application: Suppose that X is homotopically trivial_then for any CW comple*
*x K,
the elements of C(K; X) are inessential.
Given a pair (X; A), a relative_CW_resolution_for (X; A) is an ordered pair*
* ((K; L); f),
where (K; L) is a CW pair and f : (K; L) ! (X; A) is a weak homotopy equivalenc*
*e of
pairs. A relative CW resolution is unique up to homotopy of pairs (cf. Proposit*
*ion 4).
RELATIVE RESOLUTION THEOREM Every pair (X; A) admits a relative CW
resolution f : (K; L) ! (X;aA).e
[Fix CW resolutions OE ::LK!!AXand let i : A ! X be the inclusion. Using *
*Propo
sition 4, choose a g : L ! K such that O g ' i O OE. Owing to the skeletal ap*
*proximation
theorem, one can assume that g is skeletal, thus its mapping cylinder Mg is a C*
*W complex
containing L and K as embedded subcomplexes. If r : Mg ! K is the usual retract*
*ion,
then r is a homotopy equivalence and O rL ' i O OE. Since the inclusion L ! *
*Mg is a
519
cofibration, O r is homotopic to a map f : Mg ! X such that fL = i O OE. Cha*
*nge the
notation to conclude the proof.]
[Note: If (X; A) is nconnected, then one can choose K with K(n) L.]
It follows from the proof of the relative resolution theorem that given (X;*
* A) and a CW resolution
g : L ! A, there exists a relative CW resolution f : (K; L) ! (X; A) extending *
*g.
Let X and Y be topological spaces_thenaXeis said to be dominated_in_homotop*
*y_by
Y if there exist continuous functions fg::XY!!YX such that g O f ' idX. Exa*
*mple: A
topological space is contractible iff it is dominated in homotopy by a one poin*
*t space.
[Note: Let f : X ! Y be a continuous function, Mf its mapping cylinder_the*
*n f
admits a left homotopy inverse g : Y ! X iff i(X) is a retract of Mf. By compar*
*ison, f is
a homotopy equivalence iff i(X) is a strong deformation retract of Mf (cf. x3, *
*Proposition
17).]
EXAMPLE Let X be a topological space which is dominated in homotopy by a c*
*ompact connected
nmanifold Y . Assume: Hn(X; Z2) 6= 0_then Kwasiky has shown that X and Y have *
*the same homotopy
type.
FACT If X is dominated in homotopy by a CW complex, then the path componen*
*ts of X are open.
DOMINATION THEOREM Let X be a topological space_then X has the homotopy
type of a CW complex iff X is dominated in homotopy by a CW complex.ae
[Suppose that X is dominated in homotopy by a CW complex Y : fg::XY!!YX &
g O f ' idX. Fix a CW resolution h : K ! X. Using Proposition 4, choose continu*
*ous
f0 g0
ae 0 K? ! Y ! K?
functions fg0::KY!!YKsuch that the diagram hy k y h is homot*
*opy
X !f Y !g X
commutative. Claim: h is a homotopy equivalence with homotopy inverse g0O f. In*
* fact:
(g O f) O h ' g O f0 ' h O (g0O f0) & (g O f) O h ' h O idK ) g0O f0 ' idK (cf.*
* Proposition
4), so (g0O f) O h ' g0O f0 ' idK & h O (g0O f) ' g O f ' idX.]
Application: Every retract of a CW complex has the homotopy type of a CW co*
*mplex.
_________________________
yCanad. Math. Bull. 27 (1984), 448451.
520
[Note: Consequently, every open subset of a CW complex has the homotopy typ*
*e of
a CW complex (cf. p. 512).]
COUNTABLE DOMINATION THEOREM Let X be a topological space_then X
has the homotopy type of a countable CW complex iff X is dominated in homotopy *
*by a
countable CW complex. ae
[Suppose that X is dominated in homotopy by a countable CW complex Y : fg*
*::XY!!
Y *
* 0
X & g O f ' idX. Using the notation of the preceding proof, consider the image*
* g (Y ) of
Y in K. Claim: g0(Y ) is contained in a countable subcomplex L0 of K. Indeed, f*
*or any
cell e of Y , g0(_e) is compact, thus is contained in a finite subcomplex of K *
*and a countable
union of finite subcomplexes is a countable subcomplex. Fix a homotopy H : IK *
*! K
between g0O f O h and idK. Since IL0 is a countable CW complex, there exists a *
*countable
subcomplex L1 K : H(IL0) L1. Iteration then gives a sequence {Ln} of countab*
*le
S
subcomplexes Ln of K : 8 n; H(ILn) Ln+1. The union L = Ln is a countable CW
n
complex whose homotopy type is that of X.]
Application: Every Lindel"of space having the homotopy type of a CW complex*
* has
the homotopy type of a countable CW complex.
[The subcomplex generated by a Lindel"of subspace of a CW complex is necess*
*arily
countable.]
Is it true that if X is dominated in homotopy by a finite CW complex, then *
*X has the homotopy
type of a finite CW complex? The answer is "no" in general but "yes" under cert*
*ain assumptions.
Notation: Given a group G, let Z[G] be its integral group ring and write eK*
*0(G) for the reduced
Grothendieck group attached to the category of finitely generated projective Z[*
*G]modules.
The following results are due to Wally.
OBSTRUCTION THEOREM Suppose that X is path connected and dominated in homo*
*topy
by a finite CW complex_then there exists an element ew(X) 2 eK0(ss1(X)) such th*
*at ew(X) = 0 iff X has
the homotopy type of a finite CW complex.
One calls ew(X) Wall's_obstruction_to_finiteness_. Example: If X is simply *
*connected and dominated
in homotopy by a finite CW complex, then X has the homotopy type of a finite CW*
* complex.
_________________________
yAnn. of Math. 81 (1965), 5669.
521
FULFILLMENT LEMMA Let G be a finitely presented group_then given any ff 2 *
*eK0(G),
there exists a connected CW complex Xffwhich is dominated in homotopy by a fini*
*te CW complex such
that ss1(Xff) = G and ew(X) = ff.
Let A be a Dedekind domain, e.g., the ring of algebraic integers in an alge*
*braic number field_then the
reduced Grothendieck group of A is isomorphic to the ideal class group of A. Th*
*is fact, in conjunction with
the fulfillment lemma, can be used to generate examples. Thus fix a prime p, pu*
*t !p = exp(2ssp ___1=p),
and consider Z[!p], the ring of algebraic integers in Q(!p). It is known that *
*eK0(Z=pZ) is isomorphic
to the reduced Grothendieck group of Z[!p]. But the ideal class group of Z[!p] *
*is nontrivial for p > 19
(Montgomery). Moral: There exist connected CW complexes which are dominated in *
*homotopy by a finite
CW complex, yet do not have the homotopy type of a finite CW complex.
EXAMPLE Every path connected compact Hausdorff space X which is dominated *
*in homotopy
by a CW complex is automatically dominated in homotopy by a finite CW complex. *
*Is ew(X) = 0? Every
connected compact ANR (in particular, every connected compact topological manif*
*old) has the homotopy
type of a CW complex (cf. p. 619), thus is dominated in homotopy by a finite C*
*W complex and one can
prove that its Wall obstruction to finiteness must vanish, so such an X does ha*
*ve the homotopy type of
a finite CW complex. Still, some restriction on X is necessary. This is because*
* Ferryy has shown that
any Hausdorff space which is dominated in homotopy by a second countable compac*
*t Hausdorff space
must itself have the homotopy type of a second countable compact Hausdorff spac*
*e and since there exist
connected CW complexes with a nonzero Wall obstruction to finiteness, it follow*
*s that there exist path
connected metrizable compacta which are dominated in homotopy by a finite CW co*
*mplex, yet do not
have the homotopy type of a finite CW complex.
EXAMPLE Suppose that X is path connected and dominated in homotopy by a fi*
*nite CW
complex_then Gerstenz has shown that for any connected CW complex K of zero Eul*
*er characteris
tic, the product X x K has the homotopy type of a finite CW complex, i.e., mult*
*iplication by K kills
Wall's obstruction to finiteness. For example, one can take K = S2n+1. In parti*
*cular: X xS 1is homotopy
equivalent to a finite CW complex Y , say f : X x S1! Y . Since X is homotopy e*
*quivalent to X x R and
X x R is the covering space of X x S1determined by ss1(X) ss1(X x S1), it foll*
*ows that X is homotopy
equivalent to the covering space eYof Y determined by the subgroup f*(ss1(X)) o*
*f ss1(Y ). Conclusion: X
has the homotopy type of a finite dimensional CW complex.
A (pointed) topological space is said to be a (pointed)_CW_space_if it has *
*the (pointed)
_________________________
yTopology 19 (1980), 101110; see also SLN 870 (1981), 15 and 7381.
zAmer. J. Math. 88 (1966), 337346; see also Kwasik, Comment. Math. Helv. 58*
* (1983), 503508.
522
homotopy type of a (pointed) CW complex. CWSP (CWSP *) is the full subcate*
*gory
of TOP (TOP *) whose objects are the CW spaces (pointed CW spaces) and HCWSP
(HCWSP *) is the associated homotopy category. Example: Suppose that (X; A*
*) is a
relative CW complex, where A is a CW space_then X is a CW space.
[Note: If (X; x0) is a pointed CW space, then (X; x0) is nondegenerate (cf.*
* p. 335).]
Every CW space is numerably contractible (cf. p. 313). Every connected CW *
*space
is path connected. Every totally disconnected CW space is discrete. Every homot*
*opically
trivial CW space is contractible (cf. p. 517).
[Note: A CW space need not be locallyapatheconnected.]
The product X x Y of CW spaces XY is a CW space. Proof: There exist CW
ae ae
complexes KL such that in HTOP , XY K L ) X xY K xL K xk L (cf. p. 517)
and K xk L is a CW complex.
A CW space need not be compactly generated. Example: Suppose that X is not *
*in
CG _then X is not in CG but X is a CW space. However, for any CW space X, the
identity map kX ! X is a homotopy equivalence.
PROPOSITION 5 Let X be a connected CW space_then X has a simply connected
covering space eX which is universal. Moreover, every simply connected covering*
* space of
X is homeomorphic over X to eX.
[Fix a CW complex K and a homotopy equivalence OE : X ! K. Let eKbe a unive*
*rsal
Xe eOE!eK
covering space of K and define eXby the pullback square ?y ?y. Since the*
* covering
X !OEK
projection eK ! K is a Hurewicz fibration (cf. p. 47), eOEis a homotopy equiva*
*lence (cf.
p. 424), so eX is a simply connected covering space of X. To see that eX is un*
*iversal, let
Xe0be some other connected covering space of X_then the claim is that there is *
*an arrow
Xe_________wfeX0
Xe!f eX0and a commutative triangle '') [^[ . For this, form the pullback *
*square
e X
eK0 ! Xe0
?y ?y
, a homotopy inverse for OE. Due to the universality of Ke, th*
*ere is an
K ! X
Ke'_________wgeK0
arrow Ke g!eK0and a commutative triangle ') [[^ . Consider the diagr*
*am
K
523
Xe eOE!eK g! Ke0 e! Xe0
?yp ?y ?y ?y
p0:
X !OEK =======================K!X
From the definitions, p0O e O g O eOE= O OE O p ' p, thus 9 f 2 CX (Xe; eX0) *
*: f ' e O g O eOE.
Finally, if Xe0is simply connected, then eK0is simply connected and one can ass*
*ume that
g is a homeomorphism. Therefore f is a fiber homotopy equivalence (cf. x4, Prop*
*osition
15). Because the fibers are discrete, it follows that f is also an open bijecti*
*on, hence is a
homeomorphism.]
ae EXAMPLE The Cantor set is not a CW space. The topologist's sine curve C = *
*A [ B, where
A = {(0; y) : 1 y 1} , is not a CW space. The wedge of the broom is not a*
* CW space but
B = {(x; sin(2ss=x)) : 0 < x 1}
*
* 1Q
the broom, being contractible, is a CW space, although it carries no CW structu*
*re. The product Sn is
*
* 1
not a CW space.
FACT Suppose that X is a connected CW space. Assume: ss1(X) is finite and *
*8 q > 1, ssq(X) is
finitely generated_then there exists a homotopy equivalence f : K ! X, where K *
*is a CW complex such
that 8 n; K(n)is finite.
Dydaky has shown that the full subcategory of HCWSP * whose objects are th*
*e pointed connected
CW spaces is balanced.
Every open subset of a CW complex is a CW space (cf. p. 520). Every open s*
*ubset
of a metrizable topological manifold is a CW space (cf. p. 628).
PROPOSITION 6 Let U be an open subset of a normed linear space E_then U is
a CW space.
[Fix a countable neighborhood basis at zero in E consisting of convex balan*
*ced sets Un
such that Un+1 Un. Assuming that U is nonempty, for each x 2 U, there exists a*
*n index
n(x): x + 2Un(x) U. Since U is paracompact, the open covering {x + Un(x): x 2 U*
*} has
a neighborhood finite open refinement O = {O}. So, 8 O 2 O 9 xO 2 U : O xO + U*
*n(O)
(n(O) = n(xO )). Let {O : O 2 O} be a partition of unity on U subordinate to *
*O.
_________________________
yProc. Amer. Math. Soc. 116 (1992), 11711173; see also DyerRoitberg, Topol*
*ogy Appl. 46 (1992),
119124.
524
Consider N(O), the nerve of O. If {O1; : :;:Ok} is a simplex of N(O) and if n(O*
*1) . . .
n(Ok), then the convex hull of {xO1; : :;:xOk}8is containedPin xO1 + 2Un(O1) U.*
* Define
ae < f(x) = O (x)OO
continuous functions fg::U!NN(O)(O)b!yU: g(OE) =OP OE(O)x and put H(x; t*
*) = tx +
O
P O
(1  t) O (x)xO to get a homotopy H : IU ! U between g O f and idU. This sho*
*ws
O
that U is dominated in homotopy by N(O), hence, by the domination theorem, ha*
*s the
homotopy type of a CW complex.]
[Note: If E is second countable, then U has the homotopy type of a countabl*
*e CW
complex. Reason: Every open covering of a second countable metrizable space h*
*as a
countable star finite refinement (cf. p. 125).]
FACT Let E be a normed linear space. Suppose that E0 is a dense linear sub*
*space of E. Equip
E0 with the finite topology_then for every open subset U of E, the inclusion U *
*\ E0 ! U is a weak
homotopy equivalence.
FACT Let E be a normed linear space. Suppose that E0 E1 . .i.s an increa*
*sing sequence
of finite dimensional linear subspaces of E whose union is dense in E. Given an*
* open subset U of E, put
Un = U \ En_then U0 U1 . .i.s an expanding sequence of topological spaces and*
* the inclusion
U1 ! U is a homotopy equivalence.
PROPOSITION 7 Let A ! X be a closed cofibration and let f : A ! Y be a
continuous function. Assume: A, X, and Y are CW spaces_then X tf Y is a CW spac*
*e.
K?  L? g! Y
[There is a CW pair (K; L) and a commutative diagram y y k ,
X  A !f Y
where the vertical arrows are homotopy equivalences and g is the composite. Acc*
*ordingly,
K tg Y X tf Y in HTOP (cf. p. 324 ff.) and K tg Y is a CW space (cf. p. 51*
*4).]
Application: Let X f Z g!Y be a 2source. Assume: X, Y , and Z are CW space*
*s_
then Mf;gis a CW space.
[Note: One can establish an analogous result for the double mapping track o*
*f a 2sink
in CWSP (cf. x6, Proposition 8). For example, given a nonempty CW space X, 8 x0*
* 2 X,
(X; x0) is a CW space (consider the 2sink * ! X *).]
EXAMPLE Suppose that X and Y are CW spaces_then their join X * Y is a CW s*
*pace.
525
[Note: The double mapping cylinder of X X x Y ! Y defines the join. If X *
*and Y are CW
complexes, then X * Y is a CW complex provided that X x Y = X xk Y . Otherwise,*
* consider X *k Y ,
the double mapping cylinder of X X xk Y ! Y .]
LEMMA Let X0 X1 . .b.e an expanding sequence of topological spaces. As
sume: 8 n; Xn is a CW complex containing Xn1 as a subcomplex_then X1 is a CW
complex containing Xn as a subcomplex.
ae
EXAMPLE (The_Mapping_Telescope_) Let (X ; f)be objects in FIL(TOP ). *
*Suppose that
(Y ; g) f
Xn? n*
*! Xn+1?
OE : (X ; f) ! (Y ; g) is a homotopy morphism, i.e., 8 n, the diagram OEyn *
* y OEn+1is homo
Yn *
*!g Yn+1
*
* n
topy commutative_then there is an arrow telOE : tel(X ; f) ! tel(Y ; g) such th*
*at 8 n, the diagram
Xn?  teln(X?; f)! tel(X?; f)
y y y is homotopy commutative and telOE is a homotopy*
* equivalence
Yn  teln(Y ; g)! tel(Y ; g)
if each OEn is a homotopy equivalence. Thanks to the skeletal approximation the*
*orem and the lemma, it
then follows that for any object (X ; f) in FIL(CW ), there exists another obj*
*ect (X ; g) in FIL(CW ) such
that tel(X ; f) and tel(X ; g) have the same homotopy type and tel(X ; g) is a *
*CW complex.
[The mapping telescope is a double mapping cylinder (cf. p. 323). Use the *
*fact that a homotopy
X? f Z? g! *
*Y?
morphism of 2sources, i.e., a homotopy commutative diagram y y *
*y , gives rise to
X0 f0Z0 !g0 *
*Y 0
an arrow Mf;g! Mf0;g0which is a homotopy equivalence if this is the case of the*
* vertical arrows (cf.
p. 324).]
PROPOSITION 8 Let X0 X1 . . .be an expanding sequence of topological
spaces. Assume: 8 n, Xn is a CW space and the inclusion Xn ! Xn+1 is a cofibrat*
*ion_
then X1 is a CW space.
K0? ! K1? ! . . .
[There is a commutative ladder y y , where the vertic*
*al ar
X0 ! X1 ! . . .
rows Kn ! Xn are homotopy equivalences and K0 K1 . .i.s an expanding sequence
of CW complexes such that 8 n, (Kn ; Kn1 ) is a CW pair. The induced map K1 !*
* X1
is a homotopy equivalence (cf. x3, Proposition 15) and, by the lemma, K1 is a*
* CW
complex.]
526
Application: Let (X ; f) be an object in FIL (TOP ). Assume: 8 n; Xn is*
* a CW
space_then tel(X ; f) is a CW space.
FACT Let X be a topological space. Suppose that U = {Ui: i 2 I} is a numer*
*able covering of X
T
with the property that for every nonempty finite subset F I, Ui is a CW spa*
*ce_then X is a CW
space. i2F
[In the notation of the SegalStasheff construction, show that BU is a CW s*
*pace.]
Application: Let X be a topological space. Suppose that U = {Ui: i 2 I} is *
*a numerable covering of
T
X with the property that for every nonempty finite subset F I, Uiis either *
*empty or contractible_
i2F
then X is a CW space.
[Note: One can be more precise: X and N(U) have the same homotopy type. E*
*xample: Every
paracompact open subset of a locally convex topological vector space is a CW sp*
*ace (cf. Proposition 6).]
EXAMPLE Let X be the Cantor set. In X, let U1 be the image of X x [0; 2=3*
*[ and let U2
be the image of Xx]1=3; 1]_then {U1; U2} is a numerable open covering of X. Bot*
*h U1 and U2 are
contractible, hence are CW spaces. But X is not a CW space. In this connection,*
* observe that U1\ U2
has the same homotopy type as X, thus is not a CW space.
A sequence of groups ssn (n 1) is said to be a homotopy_system_if 8 n > 1 *
*: ssn is
abelian and there is a left action ss1 x ssn ! ssn.
HOMOTOPY SYSTEM THEOREM Let {ssn : n 1} be a homotopy system_
then there exists a pointed connected CW complex (X; x0) and 8 n 1, an isomorp*
*hism
ssn(X; x0) ! ssn such that the action of ss1(X; x0) on ssn(X; x0) corresponds t*
*o the action
of ss1 on ssn.
[Note: One can take X locally finite if all the ssn are countable.]
Let ss be a group and let n be an integer 1, where ss is abelian if n > 1_*
*then a
pointed path connected space (X; x0) is said to have homotopy_type_(ss;_n)_if s*
*sn(X; x0) is
isomorphic to ss and ssq(X; x0) = 0 (q 6= n). An EilenbergMacLane_space_of typ*
*e (ss; n)
is a pointed connected CW space (X; x0) of homotopy type (ss; n). Notation: (X;*
* x0) =
(K(ss; n); kss;n). Two spaces of homotopy type (ss; n) have the same weak homot*
*opy type
and two EilenbergMacLane spaces of type (ss; n) have the same pointed homotopy*
* type.
Every EilenbergMacLane space is nondegenerate, therefore the same is true of i*
*ts loop
space which, moreover, is a pointed CW space (cf. p. 624). Example: K(ss; n + *
*1) =
K(ss; n), ss abelian.
527
EXAMPLE A model for K(G; 1), G a discrete topological group, is B1G (cf. p*
*. 625).
Upon specializing the homotopy system theorem, it follows that for every ss*
*, (K(ss; n),
kss;n) exists as a pointed CW complex. If in addition ss is abelian, then (K(s*
*s; n); kss;n)
carries the structure of a homotopy commutative H group, unique up to homotopy,*
* and
the assignment (X; A) ! [X; A; K(ss; n); kss;n] defines a cofunctor TOP 2! AB .
EXAMPLE A model for K(Zn ; 1) is Tn.
[Note: Suppose that X is a homotopy commutative H space with the pointed ho*
*motopy type of a
finite connected CW complex_then Hubbucky has shown that in HTOP *, X Tn for *
*some n 0.]
EXAMPLE A model for K(Z=nZ; 1) is the orbit space S1 =, where is the subg*
*roup of S1
generated by a primitive nthroot of unity.
[Note: Recall that S1 is contractible (cf. p. 320).]
EXAMPLE A model for K(Q ; 1) is the pointed mapping telescope of the seque*
*nce S1! S1! . .,.
the kthmap having degree k.
[Note: Shelahz has shown that if X is a compact metrizable space which is p*
*ath connected and locally
path connected, then ss1(X) cannot be isomorphic to Q.]
QN QN
The homotopy type of K(Z; 2q) or K(Z=nZ; 2q) admits an interpretation*
* in terms of the
q=1 q=1
theory of algebraic cycles (Lawsonk).
(ss; 1) Suppose that (X; x0) has homotopy type (ss; 1)_then for any p*
*ointed
connected CW complex (K; k0), the assignment [f] ! f* defines a bijection [K; k*
*0; X; x0] !
Hom (ss1(K; k0); ss1(X; x0)). Since (K; k0) is wellpointed, the orbit *
*space ss1(X; x0)\
[K; k0; X; x0] can be identified with [K; X] (cf. p. 318), thus there is a bij*
*ection [K; X] !
ss1(X; x0)\Hom (ss1(K; k0); ss1(X; x0)), the set of conjugacy classes of homo*
*morphisms
ss1(K; k0) ! ss1(X; x0). If ss is abelian, then Hom (ss1(K; k0), ss1(X; x0)) H*
*om (H1(K; k0),
ss1(X; x0)) H1(K; k0; ss1(X; x0)) and the forgetful function [K; k0; X; x0] ! *
*[K; X] is bi
jective.
Example: Fix a pointed connected CW complex (K; k0)_then the functor GR !
SET that sends ss to [K; k0; K(ss; 1); kss;1] is represented by ss1(K; k0).
_________________________
yTopology 8 (1969), 119126.
zProc. Amer. Math. Soc. 103 (1988), 627632.
kAnn. of Math. 129 (1989), 253291.
528
EXAMPLE Take X = K(ss; 1), x0 = kss;nand realize (X; x0) as a pointed CW c*
*omplex. Assume:
X is locally finite and finite dimensional. Write HE(X; x0) (HE(X)) for the spa*
*ce of homotopy equiv
alences of (X; x0) (X) equipped with the compact open topology_then ss0(HE(X; x*
*0)) (ss0(HE(X)))
is the isomorphism group of (X; x0) (X) viewed as an objectaineHTOP * (HTOP )*
*. By the above,
ss0(HE(X; x0)) Autss (ss0(HE(X)) Outss). The evaluation HE(X) ! X is a Hu*
*rewicz fibra
f ! f(x0)
tion (cf. x4, Proposition 6) and its fiber over x0 is HE(X; x0). With idX as *
*the base point, one has
ssq(HE(X; x0); idX) = 0 (q > 0), ssq(HE(X); idX) = 0 (q > 1), and ss1(HE(X); id*
*X) Censs, the center
of ss. The homotopy sequence of the evaluation thus reduces to 1 ! ss1(HE(X); i*
*dX) ! ss1(X; x0) !
ss0(HE(X; x0); idX) ! ss0(HE(X); idX) ! 1, i.e., to 1 ! Censs ! ss ! Autss ! Ou*
*tss ! 1.
EXAMPLE Let p : X ! B be a Hurewicz fibration, where B = K(G; 1). Suppose *
*that 8 b 2 B,
Xbis a K(ss; 1) (ss abelian)_then the only nontrivial part of the homotopy sequ*
*ence for p is the short exact
sequence 1 ! ss ! ss1(X) ! G ! 1. Therefore ss1(X) is an extension of ss by G a*
*nd X is a K(ss1(X); 1)
(cf. x6, Proposition 11). Algebraically, there is a left action G x ss ! ss and*
* geometrically, there is a left
action G x ss ! ss. These two actions are identical.
EXAMPLE Consider a 2source ss0 G ! ss00in GR , where the arrows are mon*
*omorphisms.
G? ! ss00?
Define ss by the pushout square y y , i.e., ss = ss0*G ss00_then there e*
*xists a pointed CW
ss0 ! ss ae
0= K(ss0; 1)
complex X = K(ss; 1) and pointed subcomplexes X , Y = K(G; 1) such *
*that X = X0[ X00
X00= K(ss00; 1)
and Y = X0\ X00.
EXAMPLE Let X and Y be connected CW complexes. Suppose that f : X ! Y is a*
* contin
uous function such that for every finite connected CW complex K, the induced ma*
*p [K; X] ! [K; Y ] is
bijective_then f is a homotopy equivalence iff 8 x 2 X, f* : ss1(X; x) ! ss1(Y;*
* f(x)) is surjective (cf.
p. 318) but this condition is not automatic. To construct an example, let S1 *
*be the subgroup of the
symmetric group of N consisting of those permutationsathatehave finite support.*
* Each injection : N ! N
determines a homomorphism 1 : S1 ! S1 , viz. 1 (oe)(N  (N )) =,idand on any*
* finite product,
Q Q Q 1 (oe)(N ) = O oe OQ1
1 : S1 \ S1 ! S1 \ S1 is bijective. Here the action of S1 on S1 is by c*
*onjugation. Choose
OE : K(S1 ; 1) ! K(S1 ; 1) such that OE* = 1 on S1 _then for every finite conne*
*cted CW complex K, the
induced map [K; K(S1 ; 1)] ! [K; K(S1 ; 1)] is bijective (consider first a fini*
*te wedge of circles). However,
OE is not a homotopy equivalence unless is surjective.
[Note: There are various conditions on ss1(X) (or ss1(Y )) which guarantee *
*that f* is surjective (under
the given assumptions). For example, any of the following will do: (1) ss1(X) (*
*or ss1(Y )) nilpotent; (2)
ss1(X) (or ss1(Y )) finitely generated; (3) ss1(X) (or ss1(Y )) free.]
529
EXAMPLE Let ss be a group_then K(ss; 1) can be realized by a path connecte*
*d metrizable
topological manifold (cf. p. 628) iff ss is countable and has finite cohomolog*
*ical dimension (Johnsony).
[Note: Under these circumstances, the cohomological dimension of ss cannot *
*exceed the euclidean
dimension of K(ss; 1), there being equality iff K(ss; 1) is compact.]
EXAMPLE The homotopy type of an aspherical compact topological manifold is*
* completely de
termined by its fundamental group. Question: If X and Y are aspherical compact *
*topological manifolds
and if ss1(X) ss1(Y ), is it then true that X and Y are homeomorphic? Borel ha*
*s conjectured that the
answer is "yes". To get an idea of the difficulty of this problem, a positive r*
*esolution easily leads to a
proof of the Poincare conjecture (modulo a result of Milnor). Additional inform*
*ation and references can
be found in FarrellJonesz.
(ss; n) Suppose that (X; x0) has homotopy type (ss; n), where ss is a*
*belian. Let
2 Hn (X; x0; ssn(X; x0)) be the fundamental class_then for any pointed connect*
*ed CW
complex (K; k0), the assignment [f] ! f* defines a bijection [K; k0; X; x0] ! H*
*n (K; k0;
ssn(X; x0)).
Assuming that ss0 and ss00are abelian, [K(ss0; n); kss0;n; K(ss00; n); kss0*
*0;n] [K(ss0; n),
K(ss00; n)] Hom (ss0; ss00). Example: Suppose that 0 ! ss0! ss ! ss00! 0 is a *
*short exact
sequence of abelian groups_then (1) The mapping fiber of the arrow K(ss; n) ! K*
*(ss00; n)
is a K(ss0; n); (2) The mapping fiber of the arrow K(ss0; n + 1) ! K(ss; n + 1)*
* is a K(ss00; n);
(3) The mapping fiber of the arrow K(ss00; n) ! K(ss0; n + 1) is a K(ss; n).
[Note: CWSP * is closed under the formation of mapping fibers (cf. x6, Pr*
*oposition
8).]
EXAMPLE A model for K(Z; 2) is P1 (C ). Fix n > 1 and choose a map P1 (C )*
* ! K(Z; 2n)
representing a generator of H2n(P 1(C ); Z) Z. Put Y = P1 (C ) and define X by*
* the pullback square
X? ! K(Z;?2n)
y y . The fiber Xy0 is a K(Z; 2n  1). Since 2n  1 3, there i*
*s an isomorphism
Y ! K(Z; 2n)
ss2n1(Xy0) ss2n1(X) but the corresponding arrow in homology H2n1(Xy0) ! H2n*
*1(X) is not
even onetoone.
Let (X; A) be a relative CW complex_then for any abelian group ss, there is*
* a bijection
[X; A; K(ss; n); kss;n] ! Hn (X; A; ss) which, in fact, is an isomorphism of ab*
*elian groups,
_________________________
yProc. Camb. Phil. Soc. 70 (1971), 387393.
zCBMS Regional Conference 75 (1990), 154; see also ConnerRaymond, Bull. Am*
*er. Math. Soc.
83 (1977), 3685.
530
natural in (X; A). This applies in particular when A = ;, thus there is an iso*
*morphism
[X; K(ss; n)] ! Hn (X; ss) of abelian groups, natural in X. So, on HCW the *
*cofunctor
Hn (_ ; ss) is representable by K(ss; n). But on HTOP itself, this is no long*
*er true in that
the relation [X; K(ss; n)] Hn (X; ss) can fail if X is not a CW complex.
EXAMPLE Let X be the Warsaw circle and take ss = Z_then H1(X; Z) = 0, whil*
*e [X; K(Z; 1)]
Z or still, [X; K(Z; 1)] H1(X; Z).
In general, for an arbitrary abelian group ss and an arbitrary pair (X; A),*
* there is a
natural isomorphism [X; A; K(ss; n); kss;n] ! H (X; A; ss) (cf. p. 201). Moral*
*: It is Cech
cohomology rather than singular cohomology that is the representable theory.
Suppose that (X; x0) is a pointed connected CW complex. Equip C(X; K(ss; n)*
*) with the compact
open topology_then [X; K(ss; n)] = ss0(C(X; K(ss; n))), X being a compactly gen*
*erated Hausdorff space.
Because the forgetful function [X; x0; K(ss; n); kss;n] ! [X; K(ss; n)] is surj*
*ective, every path component of
C(X; K(ss; n)) contains a pointed map f0 : f0(x0) = kss;n.
EXAMPLE Let (X; x0) be a pointed connected CW complex.aAssume:eX is locall*
*y finite_then
nq(X; ss)(1 q n)
for any abelian group ss; ssq(C(X; K(ss; n)); f0) H .
0 (q > n)
[Since K(ss; n) is an H group, all the path components of C(X; K(ss; n)) ha*
*ve the same homo
topy type. Let f0 be the constant map X ! kss;n; C0(X; K(ss; n)) its path comp*
*onent. To compute
ssq(C0(X; K(ss; n)); f0), consider the Hurewicz fibration C0(X; K(ss; n)) ! K(s*
*s; n) which sends f to f(x0)
(cf. x4, Proposition 6), bearing in mind that ss1(C0(X; K(ss; n)); f0) is abeli*
*an.]
[Note: Suppose in addition that X is finite_then C(X; K(ss; n)) (compact op*
*enntopology) is a CW
L
space (cf. p. 623) and there is a decomposition Hn(C(X; K(ss; n)) x X; ss) *
* Hq(C(X; K(ss; n));
q*
*=0
Hnq(X; ss)). Let ev : C(X; K(ss; n))nx X ! K(ss; n) be the evaluation. Take *
*the fundamental class
L
2 Hn(K(ss; n); ss) and write ev* = q, where q 2 Hq(C(X; K(ss; n)); Hnq(X; *
*ss)). Let [fq] 2
q=0
[C(X; K(ss; n)); K(Hnq(X; ss); q)] correspond to q(conventionally,nK(Hn(X; ss)*
*; 0) is Hn(X; ss) (discrete
Q
topology)). The fq determine an arrow C(X; K(ss; n)) ! K(Hnq(X; ss); q). It*
* is a weak homotopy
q=0
equivalence, hence, by the realization theorem, a homotopy equivalence.]
EXAMPLE Let (X; x0) be a pointed connected CW complex. Assume:aXeis locall*
*y finite and fi
nite dimensional_then for any group ss, ssq(C(X; K(ss; 1)); f0) Cen(ss; f0)(*
*q =.1)Here, Cen(ss; f0)
0 (q *
*> 1)
is the centralizer of (f0)*(ss1(X; x0)) in ss1(K(ss; 1); kss;1) ss. Special ca*
*se: Suppose that (X; x0) is a
spherical, let ss = ss1(X; x0), take f0 = idX, and conclude that the path compo*
*nent of the identity in
531
C(X; X) has homotopy type (Cen ss; 1); Censs the center of ss. Example: Cen ss *
*is trivial if X is a compact
connected riemannian manifold whose sectional curvatures are < 0.
[Reduce to when X(0)= {x0} (cf. p. 515), observe that ssq(C(X; K(ss; 1)); *
*f0) ssq(C(X(1); K(ss; 1));
f0X(1)), and use the fact that X(1)is a wedge of circles.]
[Note: It can happen that ss is finitely generated but Cen(ss; f0) is infin*
*itely generated even if X = S1
(Hanseny).]
A compactly_generated_group_is a group G equipped with a compactly generated
topology in which inversion G ! G is continuous and multiplication G xk G ! G is
continuous. Since multiplication is not required to be continuous on G x G (pr*
*oduct
topology), a compactly generated group is not necessarily a topological group, *
*although
this will be the case if G is a LCH space or if G is first countable. Example: *
*Let G be a
simplicial group_then its geometric realization G is a compactly generated gr*
*oup (cf. p.
132).
[Note: If G is a topological group, then kG is a compactly generated group *
*but kG
need not be a topological group (cf. p. 136). A compactly generated group is*
* T 0iff
it is separated. Therefore any separated compactly generated group which is *
*not
Hausdorff cannot be a topological group.]
Suppose that ss is abelian_then it is always possible to realize K(ss; n) a*
*s a pointed
CW complex carrying the structure of an abelian compactly generated group on wh*
*ich
Aut ss operates to the right by base point preserving skeletal homeomorphisms s*
*uch that
ssn(K(ss;?n)) ss?
8 OE 2 Aut ss, there is a commutative square OyE* yOE (AdemMilgra*
*mz)
ssn(K(ss; n)) ss
(0 = kss;n). With this understanding, let G be a group, assume that ss is a rig*
*ht Gmodule,
and denote by O : G ! Aut ss the associated homomorphism. Calling eK(G; 1) the *
*universal
covering space of K(G; 1), form the product eK(G; 1)xK(ss; n), and write K(ss; *
*n; O) for the
orbit space (Ke(G; 1) x K(ss; n))=G. As an object in TOP =K(G; 1); K(ss; n; O*
*) is locally
trivial with fiber K(ss; n), thus the projection pO : K(ss; n; O) ! K(G; 1) is *
*a Hurewicz
fibration (localglobal principle) and K(ss; n; O) is a CW space (cf. x6, Propo*
*sition 11). The
inclusion eK(G; 1) x {0} ! eK(G; 1) x K(ss; n) defines a section sO : K(G; 1) !*
* K(ss; n; O),
so K(ss; n; O) is an object in TOP (K(G; 1)) (cf. p. 03). Example: Take G *
*= Aut ss :
_________________________
yCompositio Math. 28 (1974), 3336.
zCohomology of Finite Groups, Springer Verlag (1994), 51.
532
ae
ss x Aut ss ! ss
(ff; OE) ! OE1(ff)_then the associated homomorphism Aut ss ! Aut ss is idAut*
*ss Oss.
ae
[Note: Given G, consider the trivial action ss x G ! ss, where O : Gg!!Au*
*tissd. In
*
* ss
this case, K(ss; n; O) reduces to the product K(G; 1) x K(ss; n).]
Example: Take ss = Z, G = Z=2Z and let O : G ! Autss be the nontr*
*ivial
homomorphism_then K(Z ; 2; O) "is" BO (2).
EXAMPLE The homotopy sequence for pO breaks up into a collection of split *
*short exact se
quencesa0e! ssq(K(ss; n)) ! ssq(K(ss; n; O)) ! ssq(K(G; 1)) ! 0. Case 1: n 2. *
*Here, ssq(K(ss; n; O))
ss (q = n)and ss (K(ss; n; O)) = 0 otherwise. The algebraic right action ss x*
* G ! ss corresponds to
G (q = 1) q
an algebraic left action G x ss ! ss and this is the same as the geometric left*
* action G x ss ! ss. Case 2:
n = 1. In this situation, ss1(K(ss; n; O)) is a split extension of ss by G and *
*the higher homotopy groups are
trivial. If s;pK(ss; n; O) is the subspace of PK(ss; n; O) made up of those oe *
*such that oe(0) 2 sO(K(G; 1))
and pO(oe(t)) = pO(oe(0)) (0 t 1), then the projection s;pK(ss; n; O) ! K(ss;*
* n; O) sending oe to oe(1)
is a Hurewicz fibration whose fiber over the base point is K(ss; n). Specialize*
* and take G = Autss (so
O = Oss). Let B be a connected CW complex. The "class" of fiber homotopy classe*
*s of Hurewicz fibrations
X ! B with fiber K(ss; n) is a "set" (cf. p. 428 ff.). As such, it is in a one*
*toone correspondence with the
set of homotopy classes [B; K(ss; n+1; Oss)] : [X] $ []; : B ! K(ss; n+1; Oss)*
* the classifying_map, where
X? ! s;pK(ss;?n + 1; Oss)
X is defined by the pullback square y y . For example, if X*
* is a connected
B ! K(ss; n + 1; Oss)
CW space with two nonzero homotopy groups ss1(X) = G and ssn(X) = ss (n > 1), t*
*hen the geometry
furnishes a right action ss x G ! ss and an associated homomorphism O : G ! Aut*
*ss. To construct X up
to homotopy, fix a map f : X ! K(G; 1) which induces the identity on G, pass to*
* the mapping track Wf,
and consider the Hurewicz fibration Wf ! K(G; 1). There is an arrow : K(G; 1) *
*! K(ss; n + 1; Oss)
such that O = * : G ! Autss and [Wf] $ [].
[Note: Suppose that B is a pointed simply connected CW complex_then the set*
* of fiber homo
topy classes of Hurewicz fibrations X ! B with fiber K(ss; n) is in a onetoon*
*e correspondence with
Autss\Hn+1(B; ss). Proof: The set of homotopy classes [B; K(ss; n+1; Oss)] can *
*be identified with the set of
pointed homotopy classes [B; K(ss; n+1; Oss)] mod ss1(K(ss; n+1; Oss)), i.e., w*
*ith the set of pointed homo
topy classes [B; K(ss; n+1; Oss)] mod Aut ss, i.e., with the set of pointed hom*
*otopy classes [B; K(ss; n+1)]
mod Autss (cf. p. 516), i.e., with Autss\Hn+1(B; ss). Translated, this means t*
*hat in the simply connected
K(ss;?n + 1)
case, one can use y to carry out the classification but then it is als*
*o necessary to build in the
K(ss; n + 1)
action of Autss.]
533
ae 0 0 ae*
* 0
EXAMPLE Let G be a group; let O : G ! Autss be homomorphisms, where *
*ss are
O00: G! Autss00 *
*ss00
abelian_then [K(ss0; n + 1; O0), K(ss00; n + 1; O00)]G HomG (ss0; ss00), [ ; *
* ]G standing for homotopy in
TOP (K(G; 1)).
Notation: Given X in TOP =B and OE 2 C(E; B), let lifOE(E; X) be the set o*
*f liftings
: E ! X of OE. Relative to a choice of base points b0 2 B, x0 2 Xb0, and e0 2*
* E,
where OE(e0) = b0, let lifOE(E; e0; X; x0) be the subset of lifOE(E; X) consist*
*ing of those
such that (e0) = x0. Write [E; X]OEfor the set of fiber homotopy classes in li*
*fOE(E; X)
and [E; e0; X; x0]OEfor the set of pointed fiber homotopy classes in lifOE(E; e*
*0; X; x0).
LEMMA If (B; b0), (E; e0) are wellpointed with {b0} B, {e0} E closed, th*
*en
the fundamental group ss1(Xb0; x0) operates to the left on [E; e0; X; x0]OEand *
*the for
getful function [E; e0; X; x0]OE! [E; X]OEpasses to the quotient to define an i*
*njection
ss1(Xb0; x0)\[E; e0; X; x0]OE! [E; X]OEwhich, when Xb0 is path connected, is a *
*bijection.
Let G and ss be groups. Given O 2 Hom(G; Autss), denote by HomO (G; ss) the*
* set of crossed_homo_
morphisms_per O, so f : G ! ss is in Hom O(G; ss) iff f(g0g00) = f(g0)(O(g0)f(g*
*00)). There is a left action
ss x HomO(G; ss) ! HomO(G; ss), viz. (ff . f)(g) = fff(g)(O(g)ff1).
[Note: The elements of Hom O(G; ss) correspond bijectively to the sections *
*s : G ! sso OG, where
sso OG is the semidirect product (cf. p. 556).]
EXAMPLE Suppose that B is a connected CW complex. Fix a group ss and a Hur*
*ewicz fibration
p : X ! B with fiber K(ss; 1). Assume: secB(X) 6= ;, say s 2 secB(X). Choose b0*
* 2 B and put x0 = s(b0).
Let (E; e0) be a pointed connected CW complex, OE : E ! B a pointed continuous *
*function. There is a
split short exact sequence 1 ! ss1(Xb0; x0) ! ss1(X; x0) ! ss1(B; b0) ! 1, from*
* which a left action of
G = ss1(E; e0) on ss = ss1(Xb0; x0) or still, a homomorphism O : G ! Autss, O(g*
*) thus being conjugation
by (s O OE)*(g). Attach to 2 lifOE(E; e0; X; x0) an element f 2 HomO(G; ss) v*
*ia the prescription f (g) =
*(g)(s O OE)*(g)1_then the assignment ! f induces a bijection [E; e0; X; x0]*
*OE! Hom O(G; ss), so
[E; X]OE ss\[E; e0; X; x0]OE ss\Hom O(G; ss).
[Note: The considerations on p. 527 are recovered by taking B = * and X = *
*K(ss; 1).]
(Locally Constant Coefficients) Let (X; x0) be a pointed connected CW*
* com
plex. Assume given a homomorphism OG : ss1(X; x0) ! G and a homomorphism O : G !
Aut ss, where ss is abelian. Let G : X ! AB be the cofunctor determined by the*
* composite
O O OG (cf. p. 439). Choose a pointed continuous function fG : X ! K(G; 1) cor*
*respond
ing to OG and put kss;n;O= sO(kG;1)_then [X; x0; K(ss; n; O); kss;n;O]fG Hn (X*
*; x0; G).
So, if n = 1, H1(X; x0; G) Hom OOOG(ss1(X; x0); ss) (see the preceding exampl*
*e) )
534
H1(X; G) ss\H1(X; x0; G) ss\Hom OOOG(ss1(X; x0); ss) [X; K(ss; 1; O)]fG but*
* if n > 1,
Hn (X; x0; G) Hn (X; G) [X; K(ss; n; O)]fG.
[Note: The cohomology of any cofunctor G : X ! AB fits into this scheme. S*
*imply
take ss = Gx0, G = Aut ss, O = Oss, and let OG : ss1(X; x0) ! Aut ss be the hom*
*omorphism
derived from the right action ss x ss1(X; x0) ! ss (of course, H0(X; G) is fixO*
*G(ss), the
subgroup of ss whose elements are fixed by OG). When OG is trivial, one can cho*
*ose fG as the
map to the base point of K(Aut ss; 1) and recover the fact that [X; K(ss; n)] *
*Hn (X; ss).]
LEMMA Fix a set of representatives fi for [X; x0; K(G; 1); kG;1]_the*
*n [X; x0;
S
K(ss; n; O); kss;n;O] is in a onetoone correspondence with the union [X; x0*
*; K(ss; n; O),
i
kss;n;O]fi(which is necessarily disjoint).
Application: There is a onetoone correspondence between the set of pointe*
*d homo
topy classes of pointed continuous functions f : X ! K(ss; n; O) such that ss1(*
*f) = OG and
the elements of Hn (X; G) (n > 1).
ae
FACT Let (X; x0)be pointed connected CW complexes; let f 2 C(X; x0; Y; y*
*0). Assume given
(Y; y0)
a homomorphism OG : ss1(Y; y0) ! G and a homomorphism O : G ! Autss. Put Of*G =*
* OG O ss1(f)
and suppose that f* : [Y; y0; K(ss; n; O); kss;n;O] ! [X; x0; K(ss; n; O); kss;*
*n;O] is bijective_then Hn(Y ; G)
Hn(X; f*G).
The singular homology and cohomology groups of an EilenbergMacLane space o*
*f type
(ss; n) with coefficients in G depend only on (ss; n) and G. Notation: Hq(ss; n*
*; G); Hq(ss; n; G)
(or Hq(ss; n); Hq(ss; n) if G = Z). Example: Hn(ss; n) ss=[ss; ss].
[Note: There are isomorphisms H*ss H*(ss; 1) (H*ss H*(ss; 1)), where H*ss*
* (H*ss)
is the homology (cohomology) of ss. In general, if G is a right ssmodule and *
*if G is the
locally constant coefficient system on K(ss; 1) associated with G, then H*(ss; *
*G) (H*(ss; G))
is isomorphic to H*(K(ss; 1); G) (H*(K(ss; 1); G)).]
EXAMPLE If ss is abelian, then 8 n 2; Hn+1(ss; n) = 0 but this can fail i*
*f n = 1 since, e.g.,
H2(Z=2Z Z =2Z; 1) H1(Z=2Z; 1)H1(Z=2Z; 1) Z=2Z. When does H2(ss; 1) vanish? To*
* formulate the
answer, let 0 ! sstor! ss ! ! 0 be the short exact sequence in which sstoris t*
*he torsion subgroup of ss
and denote by sstor(p) the pprimary component of sstor_then Varadarajany has s*
*hown that H2(ss; 1) = 0
iff rank 1 plus 8 p : (p1) sstor(p) = 0 & (p2) sstor(p) is the direct sum of*
* a divisible group and a
_________________________
yAnn. of Math. 84 (1966), 368371.
535
cyclic group. Example: Assume that ss is finite_then H2(ss; 1) = 0 iff ss is cy*
*clic. Other examples include
ss = Z, ss = Q, and ss = Z=p1 Z (the pprimary component of Q=Z).
EXAMPLE Let (X; x0) be a pointed path connected space. Denote by hurn(X) *
*the image in
Hn(X) of ssn(X) under the Hurewicz homomorphism.
(ss; 1) Set ss = ss1(X) and assume that ssq(X) = 0 for 1 < q < n_then*
* Hq(X) Hq(ss; 1)
(q < n) and Hn(X)=hurn(X) Hn(ss; 1).
[Note: In particular, there is an exact sequence ss2(X) ! H2(X) ! H2(ss; 1)*
* ! 0.]
(ss; n) Set ss = ssn(X) (n > 1) and assume that ssq(X) = 0 for 1 q <*
* n & ssq(X) = 0 for
n < q < N_then Hq(X) Hq(ss; n) (q < N) and HN (X)=hurN(X) HN (ss; n).
[Note: Take N = n + 1 to see that under the stated conditions the Hurewicz*
* homomorphism
ssn+1(X) ! Hn+1(X) is surjective.]
EXAMPLE Let ss be a finitely generated (finite) abelian group_then 8 q 1,*
* Hq(ss; n) isafinitelye
generated (finite). The Hq(ss; 1) are handled by computation. Simply note that *
*Hq(Z; 1) = Z (q = 1)
ae *
* 0 (q > 1)
& Hq(Z=kZ ; 1) = Z=kZ (q odd)and use the K"unneth formula. To pass inductive*
*ly from n to n + 1,
0 (q even)
apply the generalities on p. 444 to the Zorientable Hurewicz fibration K(ss; *
*n + 1) ! K(ss; n + 1). One
can, of course, say much more. Indeed, Cartany has explicitly calculated the Hq*
*(ss; n; G), Hq(ss; n; G) for
any finitely generated abelian G. However, there are occasions when a qualitati*
*ve description suffices. To
illustrate, recall that H*(Z; n; Q) is an exterior algebra on one generator of *
*degree n if n is odd and a
polynomial algebra on one generator of degree n if n is even. Therefore, if n i*
*s odd, then Hq(Z; n; Q) = Q
for q = 0 & q = n with Hq(Z; n; Q) = 0 otherwise and if n is even, then Hq(Z; n*
*; Q) = Q for q = kn
(k = 0; 1; : :):with Hq(Z; n; Q) = 0 otherwise. So, by the above, if n is odd, *
*then Hq(Z; n) is finite for
q 6= 0 & q 6= n and if n is even, then Hq(Z; n) is finite unless q = kn (k = 0;*
* 1; : :):, Hkn(Z; n) being the
direct sum of a finite group and an infinite cyclic group.
EXAMPLE If ss0 and ss00are finitely generated abelian groups and if F is a*
* field, then the al
gebra H*(ss0 ss00; n; F) is isomorphic to the tensor product over F of the alge*
*bras H*(ss0; n; F) and
H*(ss00; n; F). Specialize and take F = F2_then for ss a finitely generated abe*
*lian group, the determi
nation of H*(ss; n; F2) reduces to the determination of H*(ss; n; F2) when ss =*
* Z=2kZ, ss = Z=plZ (p =
odd prime), or ss = Z. The second possibility is easily dispensed with: Hq(Z=pl*
*Z; n; F2) = 0 8 q > 0,
so H*(Z=plZ; n; F2) = F2. The outcome in the other cases involves the Steenrod *
*squares Sqi and their
iterates SqI. To review the definitions, a sequence I = (i1; : :;:ir) of positi*
*ve integers is termed admissible
_________________________
yCollected Works, vol. III, Springer Verlag (1979), 13001394; see also Moo*
*re, Asterisque 3233
(1976), 173212.
536
provided that i1 2i2; : :;:ir1 2ir, its excess e(I) being the difference (i1*
*2i2)+. .+.(ir12ir)+ir.
SqI is the composite Sqi1O . .O.Sqir(SqI = idif e(I) = 0).
(ss = Z=2kZ) Let un be the unique nonzero element of Hn(Z=2kZ; n; F2).
(k=1) H*(Z=2Z; 1; F2) = F2[u1], the polynomial algebra with generator u1. *
*For n > 1, H*(Z=2Z; n;
F 2) = F2[(SqIun)], the polynomial algebra with generators the SqIun, where I r*
*uns through all admissible
sequences of excess e(I) < n.
V
(k>1) H*(Z=2kZ; 1; F2) = (u1)F 2[v2], the tensor product of the exterior *
*algebra with generator
u1 and the polynomial algebra with generator v2. Here, v2 is the image of the f*
*undamental class under
the Bockstein operator H1(Z=2kZ; 1; F2) ! H2(Z=2kZ; 1; F2) corresponding to the*
* exact sequence 0 !
Z=2Z ! Z=2k+1Z ! Z=2kZ ! 0. Using this, extend the definition and let vn be th*
*e image of the
fundamental class under the Bockstein operator Hn(Z=2kZ; n; Z=2kZ) ! Hn+1(Z=2kZ*
*; n; F2). Write
__SqIu __I i
n = SqIun if ir > 1 and Sq un = Sqi1O. .O.Sq r1vn if ir = 1_then for n >*
* 1, H*(Z=2kZ; n; F2) =
F 2[(__SqIun)], the polynomial algebra with generators the __SqIun, where I run*
*s through all admissible
sequences of excess e(I) < n.
*
* V
(ss = Z) Let un be the unique nonzero element of Hn(Z; n; F2)_then H**
*(Z; 1; F2) = (u1),
the exterior algebra with generator u1, and for n > 1, H*(Z; n; F2) = F2[(SqIun*
*)], the polynomial algebra
with generators the SqIun, where I runs through all admissible sequences of exc*
*ess e(I) < n and ir > 1.
Let ss be a finitely generated abelian group_then, as vector spaces over F2*
*, the Hq(ss; n; F2) are finite
dimensional,1 so it makes sense to consider the associated Poincare se*
*ries: P(ss; n; t) =
P
dim(Hq(ss; n; F2)) . tq. Obviously, P(ss0 ss00; n; t) = P(ss0; n; t) . P*
*(ss00; n; t). Examples: (1)
q=0 1
P
P(Z=2Z; 1; t) = tq; (2) P(Z; 1; t) = 1 + t.
0
(PS1) P(ss; n; t) converges in the interval 0 t < 1.
[It suffices to treat the cases ss = Z=2kZ, ss = Z=plZ (p = odd prime), ss *
*= Z. The second case is
trivial: P(Z=plZ; n; t) = 1.
(ss = Z=2kZ) In view of what has been said above, H*(Z=2kZ; n; F2) an*
*d H*(Z=2Z; n; F2)
are isomorphic as vector spaces over F2, thus one need only examine the situati*
*on when k = 1 and
n > 1. Given an admissible I, let I = i1 + . .+.ir () e(I) = 2i1  I)_then*
* P(Z=2Z; n; t) =
Q ___1____
. Since the number of admissible I with e(I) < n such that n + I*
* = N is equal to the
e(I) 1, the extra condition ir > 1 is incorpora*
*ted by the requirement
537
hn1 = hn2. Consequently, P(Z; n; t) = P(Z=2Z; n  1; t)=P(Z; n  1; t) or sti*
*ll,
P(Z; n; t) = P(Z=2Z;_n__1;_t)_._P(Z=2Z;_n_P3;(t)Z.=.2.Z; *
*n  2; t) . P(Z=2Z; n  4; t) . . .
via iteration of the data.]
Put (ss; n; x) = log2P(ss; n; 1  2x) (0 x < 1).
(PS2) Suppose that ss is the direct sum of cyclic groups of order a *
*powernof 2, a finite group
of odd order, and cyclic groups of infinite order_then: (i) 1 ) (ss; n; x) ~*
* x___n!; (ii) = 0 &
n1
1 ) (ss; n; x) ~ x_____(n;(1)!iii) = 0 & = 0 ) (ss; n; x) = 0.
n
[The essential point is the asymptotic relation (Z=2Z; n; x) ~ x__n!, every*
*thing else being a corollary.
Observe first that P(Z=2Z; 1; t) = _1__1)(tZ=2Z; 1; x) = x. Proceeding by indu*
*ction on n, introduce
the abbreviations Pn(t) = P(Z=2Z; n; t), n(x) = (Z=2Z; n; x), and the auxiliary*
* functions Qn(t) =
Q _______1_______
, n(x) = log2Qn(1  2x)_then Qn(t)=Pn1(t) Pn(t) *
*Qn(t)
0h1...hn1 1  t2h1+...+2hn1
n1
(0 t < 1) ) n(x)  n1(x) n(x) n(x) (0 x < 1). Because n1(x) ~ _x____(n(i*
*1)!nduction
n
hypothesis), one need only show that n(x) ~ x__n!. But from the definitions, Qn*
*(t)=Pn1(t) = Qn(t2),
hence n(x) = n1(x) + n(x  1  log2(1  2x1)). So, 8 ffl > 0, 9 xffl> 0 : 8 *
*x > xffl,
n(x  1) + (1__ffl)_(nxn1)!1 n(x) n(x  1 + ffl) + (1_+_ffl)_*
*(nxn1)!1:
*
* n
Claim: Given A and n 1, there exists a polynomial Fn(x) of degree n with l*
*eading term Ax__n!such
n1
that Fn(x) = Fn(x  1) + Ax____(n. 1)!
n Pn (1)k
[Use induction on n: Put F1(x) = Ax and consider Fn(x) = Ax__n!+ _____Fn*
*k+1(x).]
k=2 k!
n1 *
* Axn1
Claim: Let f 2 C([0; 1[). Assume: f(x) f(x1)+ Ax____(nf(1)!x) f(x  1) *
*+ ______(n_t1)!hen
there exists a constant C0(C00) such that f(x) Fn(x) + C0(f(x) Fn(x) + C00).
[Let C0 = max{f(x)  Fn(x) : 0 xn1}1: f(x) Fn(x) + C0 (0 xn1)1and by i*
*nduction on
N : N x N + 1 ) f(x) f(x  1) + Ax____(n F1)!n(x  1) + C0+ Ax____(n=F1)!n*
*(x) + C0.]
These generalities allow one to say that 8 ffl > 0, there exist polynomials*
* R0ffland R00fflof degree
< n : 8 x >> 0, n i j n
(1  ffl)x__n!+ R0ffl(x) n(x) 1_+_ffl1x_ffln!+ R00ffl(x*
*):
n
Since ffl is arbitrary, this means that n(x) ~ x__n!.]
LEMMA Suppose that A is path connected_then 8 n 1 there exists a path
connected space X A which is obtained from A by attaching (n + 1)cells such t*
*hat
ssn(X) = 0 and, under the inclusion A ! X, ssq(A) ssq(X) (q < n).
538
[Let {ff} be a set of generators for ssn(A). Represent ff by fff: Sn ! A a*
*nd put
` n+1 `
X = ( D ) tf A (f = fff).]
ff ff
Let X be a pointed path connected space. Fix n 0_then an nth_Postnikov_app*
*roxi_
mate_to X is a pointed path connected space X[n] X, where (X[n]; X) is a relat*
*ive CW
complex whose cells in X[n]  X have dimension > n + 1, such that ssq(X[n]) = 0*
* (q > n)
and, under the inclusion X ! X[n], ssq(X) ssq(X[n]) (q n).
[Note: X[0] is homotopically trivial and X[1] has homotopy type (ss1(X); 1)*
*.]
PROPOSITION 9 Every pointed path connected space X admits an nth Postnikov
approximate X[n].
[Using the lemma, construct a sequence X = X0 X1 . .o.f pointed path conn*
*ected
spaces Xk such that 8 k > 0, Xk is obtained from Xk1 by attaching (n + k + 1)*
*cells,
ssn+k (Xk) = 0, and, under the inclusion Xk1 ! Xk, ssq(Xk1) ssq(Xk) (q < n +*
* k).
Consider X[n] = colimXk.]
[Note: If X is a pointed connected CW space, then the X[n] are pointed conn*
*ected
CW spaces.]
EXAMPLE Let ss be a group and let n be an integer 1, where ss is abelian *
*if n > 1_then
a pointed connected CWaspaceeX is said to be a Moore_space_of type (ss; n) prov*
*ided that ssn(X) is
isomorphic to ss and ssq(X) = 0(q < n). Notation: X = M(ss; n). If n = 1, t*
*hen M(ss; n) exists
Hq(X) = 0 (q > n)
iff H2(ss; 1) = 0 but if n > 1, then M(ss; n) always exists. If n = 1 and H2(ss*
*; 1) = 0, then the pointed
homotopy type of M(ss; 1) is not necessarily unique (e.g., when ss = Z) but if *
*n > 1, then the pointed
homotopy type of M(ss; n) is unique. In any event, M(ss; n)[n] = K(ss; n).
FACT Suppose that X is a pointed path connected space. Fix n 1_then there*
* exists a pointed
nconnected space eXnin TOP =X such that the projection eXn! X is a pointed Hur*
*ewicz fibration and
induces an isomorphism ssq(eXn) ! ssq(X) 8 q > n.
[Consider the mapping fiber of the inclusion X ! X[n].]
EXAMPLE Take X = S3_thenatheefibers of the projection eX3! X have homotopy*
* type (Z; 2)
and 8 q 1, Hq(eX3) = 0 (q odd).
Z=(q=2)Z (q even)
[Use the Wang cohomology sequence and the fact that H*(Z; 2) is the polynom*
*ial algebra over Z
generated by an element of degree 2.]
[Note: Given a prime p, let C be the class of finite abelian groups with or*
*der prime to p_then from
the above, Hn(eX3) 2 C (0 < n < 2p), so by the mod C Hurewicz theorem, ssn(eX3)*
* 2 C (0 < n < 2p) and
539
the Hurewicz homomorphism ss2p(eX3) ! H2p(eX3) is Cbijective. Therefore the p*
*primary component of
ssn(S3) is 0 if n < 2p and is Z=pZ if n = 2p.]
Put W1 = eX1. Let W2 be the mapping fiber of the inclusion eX1! eX1[2]_then*
* the mapping fiber
of the projection W2 ! W1 has homotopy type (ss2(X); 1). Iterate: The result is*
* a sequence of pointed
Hurewicz fibrations Wn ! Wn1, where the mapping fiber has homotopy type (ssn(X*
*); n  1) and Wn
is nconnected with ssq(Wn) ssq(X) (8 q > n). The diagram X = W0 W1 . .i.s*
* called "the"
Whitehead_tower_of X.
[Note: If X is a pointed connected CW space, then the Wn are pointed connec*
*ted CW spaces and
the mapping fiber of the projection Wn ! Wn1 is a K(ssn(X); n  1).]
EXAMPLE Let X be a pointed simply connected CW complex which is finite and*
* noncontractible.
Assume: 9 i > 0 such that Hi(X; F2) 6= 0_then ssq(X) contains a subgroup isomor*
*phic to Z or Z=2Z for
infinitely many q.
[Because the Hq(X) are finitely generated 8 q, the same is true of the ssq(*
*X) (cf. p. 544). The set
of positive integers n such that ssn(X) Z=2Z 6= 0 is nonempty. To get a contra*
*diction,1suppose that
P
there is a largest such N. Working with the Whitehead tower of X, let Pn(t) = *
* dim(Hq(Wn; F2)) . tq,
q*
*=0
the mod 2 Poincare series of H*(Wn; F2) (meaningful, the Hq(Wn; F2) being finit*
*e dimensional over F2).
In particular: PN (t) = 1, PN1 (t) = P(ssN (X); N; t), P1(t) = PX (t), the Poi*
*ncare series of H*(X; F2).
On general grounds, there is a majorization Pn(t) Pn1(t) . P(ssn(X); n  1; t*
*), where the symbol
means that each coefficient of the formal power series on the left is the cor*
*responding coefficient
of the formal power series on the right. So, starting with n = N  1 and multi*
*plying out, one finds
Q
that P(ssN (X); N; t) PX (t) . P(ssi(X); i  1; t). Since PX (t) is a pol*
*ynomial, hence is bounded
1 0 : P(ssN (X); N; t) C . P(ssi(X); i  1; t) or still, i*
*n the notation of p. 537,
P 1 0, Hi(X; Fp) = 0, then the arr*
*ow X ! * is a homology
equivalence (cf. p. 88), thus by the Whitehead theorem, X is contractible.]
LEMMA Let (X; A; x0) be a pointed pair. Assume: (X; A) is a relative CW co*
*mplex
_________________________
yComment. Math. Helv. 27 (1953), 198232.
zProc. Japan Acad. 35 (1959), 563566; see also McGibbonNeisendorfer, Comme*
*nt. Math. Helv.
59 (1984), 253257.
540
whose cells in X  A have dimension > n + 1. Suppose that (Y; y0) is a pointed*
* space
such that ssq(Y; y0) = 0 8 q > n_then every pointed continuous function f : A !*
* Y has a
pointed continuous extension F : X ! Y .
It follows from the lemma that if X and Y are pointed path connected spaces*
* and if f :
X ! Y is a pointed continuous function, then for m n there exists a pointed co*
*ntinuous
X? f! Y?
function fn;m : X[n] ! Y [m] rendering the diagram y y commutative*
*, any
X[n] f! Y [m]
n;m
two such being homotopic relX. Proof: Let F : X ! Y [m] be the composite X f!Y !
X[n]ffi
u ffiffi
Y [m]. To establish the existence of fn;m, consider any filler for  *
*ffiffl and
X ________*
*_wFY [m]
to establish the uniqueness of fn;marelX,etake two extensions f0n;m& f00n;m, de*
*fine :
0 (x)
i0X[n] [ IX [ i1X[n] ! Y [m] by (x;(0)x=;fn;m1) =;f00(x; t) = F (x), and cons*
*ider any
n;m(x)
IX[n] i i
u i i
filler for  i iij .
i0X[n] [ IX [ i1X[n] _______wY [m]
Application: Let X0[n] and X00[n] be nth Postnikov approximates to X_then *
*in
HTOP 2, (X0[n]; X) (X00[n]; X).
EXAMPLE Let X and Y be pointed connected CW spaces_then it can happen that*
* X[n] and
Y [n] have the same pointed homotopy type for all n, yet X and Y are not homoto*
*py equivalent. To
construct an example, let K be1a pointed simply connected CW complex. Assume: *
* K is finite and
Q
noncontractible. Put X = (w) K[n], Y = X x K_then 8 n, X[n] Y [n] in HTOP **
*. However, it
0
is not true that X Y in HTOP . For if so, K would be dominated in homotopy b*
*y X or still, by
K[0] x . .x.K[n] (9 n), thus 8 q, ssq(K) would be a direct summand of ssq(K[0] *
*x . .x.K[n]). But this is
impossible: The ssq(K) are nonzero for infinitely many q (cf. p. 539).
[Note: This subject has its theoretical aspects as well (McGibbonMollery).]
Let X be a pointed path connected space. Given a sequence X[0]; X[1]; : :o:*
*f Post
_________________________
yTopology 31 (1992), 177201; see also DrorDwyerKan, Proc. Amer. Math. Soc*
*. 74 (1979), 183
186.
541
nikov approximates to X, 8 n 1 there is a pointed continuous function fn : X[n*
*] !
X[
X[n  1] such that the triangle aaeeAE [] commutes. Put P0X = X[0]*
*, let
X[n] _________wfnX[n  1]
X[1]? f1!*
*X[0]?
s0 be the identity map, and denote by P1X the mapping track of f1: s1y *
* y s0.
P1X !p *
*P0X
1
Recall that s1 is a pointed homotopy equivalence, while p1 is the usual pointed*
* Hurewicz
fibration associated with this setup. Repeat the procedure, taking for P2X the*
* map
X[2]? f2!X[1]?
ping track of s1 O f2: sy2 ys1. The upshot is that the fn can be con*
*verted
P2X !p P1X
2
to pointed Hurewicz fibrations pn, where at each stage there is a commutative t*
*riangle
X[
ae []
aeAE . The diagram P0X P1X . .o.f pointed Hurewicz fibratio*
*ns
PnX _________wpnPn1X
is called "the" Postnikov_tower_of X. Obviously, ssq(PnX) = 0 (q > n), ssq(X) *
*ssq(PnX)
(q n), and ssq(PnX) ssq(Pn1X) (q 6= n). Therefore the mapping fiber of pn h*
*as
homotopy type (ssn(X); n).
[Note: If X is a pointed connected CW space, then the PnX are pointed conne*
*cted
CW spaces, so the mapping fiber of pn is a K(ssn(X); n).]
EXAMPLE Let X be a pointed path connected space. Fix n > 1_then ssn(X) def*
*ines a locally
constant coefficient system on Pn1X and there is an exact sequence
Hn+2(PnX) ! Hn+2(Pn1X) ! H1(Pn1X; ssn(X)) ! Hn+1(PnX) ! Hn+1(Pn1X)
! H0(Pn1X; ssn(X)) ! Hn(PnX) ! Hn(Pn1X) ! 0:]
[Work with the fibration spectral sequence of pn : PnX ! Pn1X, noting that*
* Erp;q= 0 if 0 < q < n
or q = n + 1.]
A nonempty path connected topological space X is said to be abelian_if ss1(*
*X) is
abelian and if 8 n > 1, ss1(X) operates trivially on ssn(X). Every simply conne*
*cted space
is abelian as is every path connected H space or every path connected compactly*
* generated
semigroup with unit (obvious definition).
[Note: If X is abelian, then 8 x0 2 X, the forgetful function [S n; sn; X; *
*x0] ! [S n; X]
is bijective (cf. p. 318).]
542
EXAMPLE Pn(R ) is abelian iff n is odd.
Let X be a pointed connected CW space. Assume: X is abelian. There is a com*
*mu
X[
[]
tative triangle aaeeAE and an embedding IX ! Mfn+1. Define bX[n*
*] by
X[n + 1] _________wfn+1X[n]
IX? p! X?
the pushout square y y _then bX[n] contains X[n] as a strong defo*
*rmation
Mfn+1 ! Xb[n]
retract, hence ssq(Xb[n]) ssq(X[n]) (q 1). Using the exact sequence
. .!.ssq+1(X[n + 1])! ssq+1(Xb[n])! ssq+1(Xb[n]; X[n + 1])! ssq(X[n + 1])! ssq*
*(Xb[n])! . .;.
one finds that ssq(Xb[n]; X[n+1]) = 0 (q 6= n+2) and ssn+2(Xb[n]; X[n+1]) ssn+*
*1(X[n + 1])
ssn+1(X). Thus the relative Hurewicz homomorphism hur : ssn+2(Xb[n]; X[n + 1*
*])
! Hn+2(Xb[n]; X[n + 1]) is bijective, so the composite n+2 : Hn+2(Xb[n]; X[n +*
* 1])
hur1!ss b b
n+2(X [n]; X[n+1]) ! ssn+1(X) is an isomorphism. Since Hn+1(X [n]; X[n+*
*1]) = 0,
the universal coefficient theorem implies that Hn+2 (Xb[n]; X[n + 1]; ssn+1(X))*
* can be
identified with Hom (Hn+2(Xb[n]; X[n + 1]); ssn+1(X)), therefore n+2 correspond*
*s to a
cohomology class in Hn+2 (Xb[n]; X[n + 1]; ssn+1(X)) whose image kn+2 (= kn+2 (*
*X)) in
Hn+2 (X[n]; ssn+1(X)) is the Postnikov_invariant_of X in dimension n + 2. Put K*
*n+2 =
K(ssn+1(X); n + 2), let kn+2 : X[n] ! Kn+2 be the arrow associated with kn+2, a*
*nd define
W [n?+ 1] ! Kn+2?
W [n + 1] by the pullback square y y _then W [n + 1] is a CW*
* space
X[n] k! Kn+2
n+2
W [n + 1]
n+1
(cf. x6, Proposition 9) and there is a lifting aeaeo u of fn+1 wh*
*ich is a
X[n + 1] _____wfn+1X[n]
weak homotopy equivalence or still, a homotopy equivalence (realization theorem*
*). The
restriction of n+1 to X is an embedding and n+1 : (X[n + 1]; X) ! (W [n + 1]; X*
*) is a
homotopy equivalence of pairs.
[Note: n+1 is constructed by considering a specific factorization of kn+2 a*
*s a com
posite X[n] ! bX[n]=X[n + 1] ! Kn+2 (kn+2 is determined only up to homotopy).]
ae ae
INVARIANCE THEOREM Let XY be pointed CW spaces. Assume: XY are
abelian. Suppose that OE : X ! Y is a pointed continuous function. Fix point*
*ed OEn :
543
X? OE! Y?
X[n] ! Y [n] such that the diagram y y commutes_then 8 n, OE*nkn+2*
*(Y ) =
X[n] !OE Y [n]
n
OEcokn+2(X) in Hn+2 (X[n]; ssn+1(Y )).
[Note: Here, OEco is the coefficient group homomorphism Hn+2 (X[n]; ssn+1(*
*X)) !
Hn+2 (X[n]; ssn+1(Y )).]
NULLITY THEOREM Let X be a pointed CW space. Assume: X is abelian_then
kn+1 = 0 iff the Hurewicz homomorphism ssn(X) ! Hn(X) is split injective.
EXAMPLE Suppose that kn+1 = 0_then W[n] is fiber homotopy equivalent to X[*
*n  1] x
K(ssn(X); n) (cf. p. 424), hence X[n]1 X[n  1] x K(ssn(X); n). Therefore X ha*
*s the same pointed
Q
homotopy type as the weak product (w) K(ssn(X); n) provided that the Hurewicz*
* homomorphism
1
ssn(X) ! Hn(X) is split injective for all n. This condition can be realized. In*
* fact, Puppey has shown
that if G is a path connected abelian compactly generated semigroup1with unit, *
*then 8 n, the Hurewicz
Q
homomorphism ssn(G) ! Hn(G) is split injective, thus G (w) K(ssn(G); n) when*
* G is in addition a
1
CW space.
[Note: Analogous remarks apply if G is a path connected abelian topological*
* semigroup with unit.
Reason: The identity map kG ! G is a weak homotopy equivalence.]
ABELIAN OBSTRUCTION THEOREM Let (X; A) be a relative CW complex; let Y be a
pointed abelian CW space. Suppose that 8 n > 0, Hn+1(X; A; ssn(Y )) = 0_then e*
*very f 2 C(A; Y )
admits an extension F 2 C(X; Y ), any two such being homotopic relA provided th*
*at 8 n > 0, Hn(X; A;
ssn(Y )) = 0.
EXAMPLE Let (X; x0) be a pointed CW complex; let (Y; y0) be a pointed simp*
*ly connected CW
complex. Assume: 8 n > 0, Hn(X; ssn(Y )) = 0_then [X; x0; Y; y0] = *.
[In fact, Hn(X; x0; ssn(Y; y0)) Hn(X; ssn(Y )) = 0 ) [X; x0; Y; y0] = *() *
*[X; Y ] = * (cf. p.
318)).]
PROPOSITION 10 Let X be a pointed abelian CW space. Assume: The Hq(X) are
finitely generated 8 q_then 8 n, the Hq(X[n]) are finitely generated 8 q.
[The assertion is trivial if n = 0. Next, X[1] is a K(ss1(X); 1), hence ss1*
*(X) H1(X),
which is finitely generated. For q > 1, Hq(X[1]) Hq(ss1(X); 1) and these too *
*are
_________________________
yMath. Zeit. 68 (1958), 367421.
544
finitely generated (cf. p. 535). Proceeding by induction, suppose that the H*
*q(X[n])
are finitely generated 8 q_then the Hq(X[n]; X) are finitely generated 8 q. In*
* particu
lar, Hn+2(X[n]; X) is finitely generated. Since ssn+1(X[n]) = ssn+2(X[n]) = 0, *
*the arrow
ssn+2(X[n]; X) ! ssn+1(X) is an isomorphism. But X is abelian, so from the rel*
*ative
Hurewicz theorem, ssn+2(X[n]; X) Hn+2(X[n]; X). Therefore ssn+1(X) is finitely*
* gener
ated. Consider now the mapping track Wn+2 of kn+2 : X[n] ! Kn+2. The fiber of*
* the
Z orientable Hurewicz fibration Wn+2 ! Kn+2 over the base point is homeomorphi*
*c to
W [n+1] (parameter reversal). The Hq(Kn+2) = Hq(ssn+1(X); n+2) are finitely gen*
*erated
8 q (cf. p. 535), as are the Hq(Wn+2) (induction hypothesis), thus the Hq(W [n*
* + 1]) are
finitely generated 8 q (cf. p. 444). Because X[n+1] and W [n+1] have the same *
*homotopy
type, this completes the passage from n to n + 1.]
Application: Let X be a pointed abelian CW space. Assume: The Hq(X) are fin*
*itely
generated 8 q_then the ssq(X) are finitely generated 8 q.
[Note: This result need not be true for a nonabelian X. Example: Take X = S*
*1_S 2_
then the Hq(X) are finitely generated 8 q and ss1(X) Z. On the other hand, ss2*
*(X)
H2(Xe), eXthe universal covering space of X, i.e., the real line with a copy of*
* S2 attached
at each integral point. Therefore ss2(X) is free abelian on countably many gene*
*rators.]
PROPOSITION 11 Let X be a pointed abelian CW space. Assume: The Hq(X) are
finite 8 q > 0_then 8 n, the Hq(X[n]) are finite 8 q > 0.
Application: Let X be a pointed abelian CW space. Assume: The Hq(X) are fin*
*ite
8 q > 0_then the ssq(X) are finite 8 q > 0.
EXAMPLE (Homotopy_Groups_of_Spheres_) The ssq(S2n+1) of the odd dimension*
*al sphere are
finite for q > 2n + 1 and the ssq(S2n) of the even dimensional sphere are finit*
*e for q > 2n except that
ss4n1(S2n) is the direct sum of Z and a finite group. Here are the details.
(2n+1) Fix a map f : S2n+1! K(Z; 2n+1) classifying a generator of H2n*
*+1(S2n+1)_then
f* induces an isomorphism H*(S2n+1; Q) ! H*(K(Z; 2n+1); Q) (cf. p. 535), so 8 *
*q > 0, Hq(Ef; Q) = 0
(cf. p. 444). Accordingly, 8 q > 0, Hq(Ef) is finite (being finitely generated*
*). Therefore all the homotopy
groups of Ef are finite. But ssq(Ef) ssq(S2n+1) if q > 2n + 1.
(2n) The even dimensional case requires a double application of the o*
*dd dimensional case.
First, consider the Stiefel manifold V2n+1;2and the map f : V2n+1;2! S4n1defin*
*ed on p. 59. As
noted there, 8 q > 0, Hq(Ef; Q) = 0, hence the ssq(Ef) are finite and this mean*
*s that the ssq(V 2n+1;2)
are finite save for ss4n1(V 2n+1;2) which is the direct sum of Z and a finite *
*group. Second, examine the
homotopy sequence of the Hurewicz fibration V2n+1;2! S2n, noting that its fiber*
* is S2n1.
545
Given a category C , the tower_category_TOW (C ) of C is the functor cate*
*gory
[[N ]OP ; C]. Example: The Postnikov tower of a pointed path connected space is*
* an object
in TOW (TOP *).
Take C = AB _then an object (G ; f) in TOW (AB ) is a sequence {Gn; fn :*
* Gn+1 !
Gn}, where Gn is an abelian group and fn : Gn+1 ! Gn is a homomorphism, a morph*
*ism
OE : (G 0; f0) ! (G 00; f00) in TOW (AB ) being a sequence {OEn}, where OEn *
*: G0n! G00nis a
homomorphism and OEn O f0n= f00nO OEn+1. TOW (AB ) is an abelian category. A*
*s such, it
has enough injectives.
[Note: Equip [N ] with the topology determined by , i.e., regard [N ] as an*
* A space_
then TOW (AB ) is equivalent to the category of sheaves of abelian groups on*
* [N ].]
The functor lim : TOW (AB ) ! AB that sends G to limG is left exact (*
*being a
right adjoint) but it need not be exact. The right derived functors limiof lim*
* live only
in dimensions 0 and 1, i.e., the limi(i > 1) necessarily vanish. To compute lim*
*1G , form
Q
G = Gn and define d : G ! G by d(x0; x1; : :):= (x0  f0(x1); x1  f1(x2); : *
*:):_then
n
kerd = limG and cokerd = lim1G . Example: Suppose that 8 n, Gn is finite_then
lim1G = 0.
[Note: Translated to sheaves, limicorresponds to the ith right derived func*
*tor of the
global section functor.]
The fact that the limi(i > 1) vanish is peculiar to the case at hand. Indee*
*d, if (I; ) is a directed set
and if Iis the associated filtered category, then for a suitable choice of I, o*
*ne can exhibit a G in [IOP; AB]
such that limiG6= 0 8 i > 0 (Jenseny).
EXAMPLE Let 6=aebe relatively prime natural numbers > 1. Define G () in *
*TOW (AB )
by G()n = Z 8 n & G()n+1 ! G()n and OE 2 Mor(G (); G()) by OEn(1) = _then th*
*e cokernel
1 !
of OE is isomorphic to the constant tower on [N ] with value Z=Z . Applying lim*
*to the exact sequence
0 ! G() OE!G() ! cokerOE ! 0 and noting that limG() = 0, one obtains a sequence*
* 0 ! 0 ! 0 !
*
* 1OE
Z=Z ! 0 which is not exact. On the other hand, the sequence 0 ! Z=Z ! lim1G() l*
*im!G () ! 0
is exact, so lim1G() contains a copy of Z=Z 8 : (; ) = 1.
To extend the applicability of the preceding considerations, replace AB by*
* gr. Again,
there is a functor lim : TOWae (gr) ! gr that sends G to limG . As for lim1G *
*, it is
Q x0 = {x0}
the quotient Gn= ~, where 00 n00 are equivalent iff 9 x = {xn} such *
*that
n x = {xn}
_________________________
ySLN 254 (1972), 5152.
546
8 n : x00n= xnx0nfn(x1n+1). While not necessarily a group, lim1G is a pointe*
*d set with
base point the equivalence class of {en} and it is clear that lim1: TOW (gr) *
*! SET * is a
functor.
Q
[Note: Put X = Gn_then the assignment ((g0; g1; : :):; (x0; x1; : :):)*
* ! (g0x0
n Q
f0(g11); g1x1f1(g12); : :):defines a left action of the group Gn on the poi*
*nted set X.
Q n
The stabilizer of the base point is limG and the orbit space Gn\X is lim1G .*
* For the
n
definition and properties of lim1"in general", consult BousfieldKany.]
LEMMA Let * ! G 0! G ! G 00! * be an exact sequence in TOW (gr)_then
there is a natural exact sequence of groups and pointed sets
* ! limG 0! limG ! limG 00! lim1G 0! lim1G ! lim1G 00! *:
[Note: Specifically, the assumption is that 8 n, the sequence * ! G0n! Gn !*
* G00n! *
is exact in gr.]
EXAMPLE Suppose that {Gn} is a tower of finitely generated abelian groups_*
*then lim1Gn
is isomorphic to a group of the form Ext(G; Z), where G is countable and torsio*
*n free. To see this,
write G0nfor the torsion subgroup of Gn and call G00nthe quotient Gn=G0n. Sinc*
*e each G0nis finite,
lim1G0n= * ) lim1Gn lim1G00n. Assume, therefore, that the Gn are torsion free*
*. Let Kn =
L
Gi = Gn Kn1 and define Kn ! Kn1 by Gn ! Gn1 ! Kn1 on the first factor a*
*nd
in
by the identity on the second factor. So, 8 n, Kn ! Kn1 is surjective, thus t*
*he sequence 0 !
limGn ! limKn ! limKn=Gn ! lim1Gn ! 0 is exact. Because Gn; Kn, and Kn=Gn are *
*free
abelian, the sequence 0 ! Hom(Kn=Gn; Z) ! Hom(Kn; Z) ! Hom(Gn; Z) ! 0 is exact *
*) the sequence
0 ! colimHom(Kn=Gn; Z) ! colimHom(Kn; Z) ! colimHom(Gn; Z) ! 0 is exact ) the s*
*equence
0 ! Hom (colimHom(Gn; Z); Z) ! Hom (colimHom(Kn; Z); Z) ! Hom (colimHom(Kn=Gn; *
*Z); Z) !
Ext(colimHom(Gn; Z); Z) ! Ext(colimHom(Kn; Z); Z) is exact ) the sequence 0 ! l*
*imGn ! limKn !
L
limKn=Gn ! Ext(colimHom(Gn; Z); Z) ! 0 is exact (for colimHom(Kn; Z) Hom (Gn*
*; Z), which
n
is free). Consequently, lim1Gn Ext(colimHom(Gn; Z); Z), where colimHom(Gn; Z) *
*is countable and
torsion free.
[Note: It follows that lim1Gn is divisible, hence if lim1Gn 6= *, then on g*
*eneral grounds, there exist
L
cardinals ff and fl(p) (p 2 ) : lim1Gn ff . Q fl(p) . (Z=p1 Z). But here o*
*ne can say more, viz.
p
ff = 2! and 8 p, fl(p) is finite or 2!.]
_________________________
ySLN 304 (1972), 305308.
547
HuberWarfieldy have shown that an abelian G is isomorphic to a lim1Gfor so*
*me G in TOW (AB )
iff Ext(Q ; G) = 0.
When is lim1G = *? An obvious sufficient condition is that the fn : Gn+1 *
*! Gn
be surjective for every n. More generally, G is said to be MittagLeffler_if 8*
* n 9 n0 n :
8 n00 n0, im(Gn0! Gn) = im(Gn00! Gn).
MITTAGLEFFLER CRITERION Suppose that G is MittagLeffler_then lim1G = *.
[Note: There is a partial converse, viz. if lim1G = * and if the Gn are c*
*ountable,
then G is MittagLeffler (DydakSegalz).]
Q
EXAMPLE Fix a sequence 0 < 1. .o.f natural numbers (0 > 1). Put Gn = Z=*
*kZ and
kn
let Gn+1 ! Gn be the inclusion_then G is not MittagLeffler, yet lim1G= *.
FACT Assume: lim1G 6= * and the Gn are countable_then lim1G is uncountable.
EXAMPLE Let X be a CW complex. Suppose that X0 X1 . .i.s an expanding se*
*quence
S
of subcomplexes of X such that X = Xn. Fix a cofunctor G : X ! AB and put G*
*n = GXn_
n
then 8 q 1, there is an exact sequence 0 ! lim1Hq1(Xn; Gn) ! Hq(X; G) ! limHq*
*(Xn; Gn) ! 0
of abelian groups (Whiteheadk). To illustrate, take X = K(Q ; 1) (realized as *
*on p. 527) and let
G : X ! AB be the cofunctor corresponding to the usual action of Q on Q[Q ] (cf*
*. p. 439). This data
generates a short exact sequence 0 ! lim1H1(Z; Q[Q ]) ! H2(Q ; Q[Q ]) ! limH2(Z*
*; Q[Q ]) ! 0. The
tower H1(Z; Q[Q ]) H1(Z; Q[Q ]) . .i.s not MittagLeffler but H1(Z; Q[Q ]) *
*is countable, therefore
lim1H1(Z; Q[Q ]) is uncountable. In particular: H2(Q ; Q[Q ]) 6= 0.
FACT Let {Gn} be a tower of nilpotent groups. Assume: 8 n, #(Gn) !_then l*
*im1Gn = * iff
lim1Gn=[Gn; Gn] = *.
[For as noted above, in the presence of countability, lim1Gn=[Gn; Gn] = * )*
* {Gn=[Gn; Gn]} is
MittagLeffler.]
ae
PROPOSITION 12 Let {Xn}{Ybe two sequences of pointed spaces. Suppose giv*
*en
ae n}
pointed continuous functions OEn : Xn ! Xn+1. Assume: The OEn are closed cofi*
*brations
n : Yn+1 ! Yn
and the n are pointed Hurewicz fibrations_then there is an exact sequence
_________________________
yArch. Math. 33 (1979), 430436.
zSLN 688 (1978), 7880.
kElements of Homotopy Theory, Springer Verlag (1978), 273274.
548
* ! lim1[Xn; Yn] ! [colimXn; limYn] ! lim[Xn; Yn] ! *
in SET * and is an injection. ae
[Write X1 = colimXn & Y1 = limYn. Embedded in the data are arrows n : X*
*n !
ae n : *
*Y1 !
X1 n+1 O OEn = n
Yn with n O n+1 = n and 8 n, an arrow [Xn+1; Yn+1] ! [Xn; Yn], viz. [f*
*] !
[ n O f O OEn].
Define n : [X1 ; Y1 ] ! [Xn; Yn] by n([f]) = [n O f O n]. Because the coll*
*ec
tion {n : [X1 ; Y1 ] ! [Xn; Yn]} is a natural source, there exists a unique poi*
*nted map
[X1 ; Y1 ] _____w1lim*
*[Xn; Yn]
1 : [X1 ; Y1 ] ! lim[Xn; Yn] such that 8 n, the triangle n *
* ucom
[Xn; Y*
*n]
mutes. To_prove_that 1 is surjective, take {[fn]} 2 lim[Xn;_Yn]_then 8 n, _nOfn*
*+1_OOEn '
fn. Set f0 = f0 and, proceeding_inductively,_assume that f1 2 [f1]; : :;:fn 2 *
*[fn] have
been found with ak1eO fk O OEk1 = fk1 (1 k n). Choose a pointed homotopy
hn : IXn ! Yn : hnhO i0 = n_O_fn+1 O OEn. Since n is a pointed Hurewicz fi*
*bration,
n O i1 = fn
Xn? fn+1OOEn!Yn+1?
the commutative diagram iy0 y n admits a pointed filler Hn : IXn ! *
*Yn+1.
IXn !h Yn
n
Fix a retraction rn : IXn+1 ! i0Xn+1 [ IOEn(Xn) (cf. x3, Proposition 1) and sp*
*ecify
aapointedecontinuous function Fn+1 : i0Xn+1 [ IOEn(Xn) ! Yn+1 by the prescript*
*ion
Fn+1(xn+1; 0) = fn+1(xn+1) __
Fn+1(OEn(xn); t) = Hn(xn; t). Put hn = n O Fn+1 O rn to get a commutative di*
*agram
i0Xn+1 [?IOEn(Xn) Fn+1!Yn+1
y ?y n. Bearing in mind that OEn is a closed cofibrat*
*ion, this
IXn+1 !_ Yn
hn __
diagram has a pointed filler H n+1_: IXn+1 ! Yn+1 (cf. x4, Proposition 12). Fin*
*ally, to
__
push the induction forward,_let fn+1 = H n+1 O i1._ Conclusion: There exists a*
* pointed
continuous function f1 : X1 ! Y1 such that 1 ([f1 ]) = {[fn]}, i.e., 1 is su*
*rjective.
As for the kernel of 1 , it consists of those [f] : 8 n, n Of On is nullhom*
*otopic. Thus
there are pointed homotopies n : IXn ! Yn such that n Oi0 = 0n & n Oi1 = n Of On
with n O n+1 O IOEn O i0 = 0n & n O n+1 O IOEn O i1 = n O f O n, where 0n is *
*the zero
morphism Xn ! Yn. To define j1 : ker1 ! lim1[Xn; Yn], let oen;f: Xn ! Yn be t*
*he
pointed continuous function given by
ae
oen;f(xn; t) = n(xn; 2t) (0 t 1=2) :
n O n+1(OEn(xn); 2 (2t)1=2 t 1)
549
Q
The oen;f determine a string in [Xn; Yn] or still, an element of lim1[Xn; Yn]*
*, call
n
it [oef]. Definition: j1 ([f]) = [oef]. One can check that j1 does not depe*
*nd on the
choice of the n and is independent of the choice of f 2 [f]. Claim: j1 is bi*
*jective.
To verify, e.g., injectivity, suppose that j1 ([f0]) = j1 ([f00])_then there ex*
*ists a string
Q
{[oen]} 2 [Xn; Yn] : 8 n,
n
8
< oen(xn; 3t) (0 t 1=3)
0(xn; 3t  1) (1=3 t 2=3)
: oen;f
n O oen+1(OEn(xn);(32=3t)3 t 1)
represents oen;f00. In addition, the formulas
8
< 0n(xn; 1  3t)(0 t 1=3)
: oen(xn;020 3t)(1=3 t 2=3)
n(xn; 3t(22)=3 t 1)
define a pointed homotopy Hn : IXn ! Yn having the property that Hn O i0 = n O *
*f0O n
__
& Hn Oi1 = n Of00On. Arguing as before, construct pointed homotopies H n: IXn !*
* Yn
__ __ __ __
such that H nO i0 = n O f0 O n & H nO i1 = n O f00O n with n O H n+1O IOEn = H*
* n.
__ __
The H n combine and induce a pointed homotopy H 1 : IX1 ! Y1 between f0 and f*
*00,
i.e., j1 is injective.]
Application: Let {Xn} be a sequence of pointed spaces. Suppose given pointe*
*d con
tinuous functions OEn : Xn ! Xn+1 such that 8 n, OEn is a closed cofibration_th*
*en for any
pointed space Y , there is an exact sequence
* ! lim1[Xn; Y ] ! [colimXn; Y ] ! lim[Xn; Y ] ! *
in SET * and is an injection.
EXAMPLE Fix an abelian group ss. Let (X; x0) be a pointed CW complex. Supp*
*ose that x0 2
S
X0 X1 . .i.s an expanding sequence of subcomplexes of X such that X = Xn_th*
*en 8 q 1,
n
there is an exact sequence 0 ! lim1eHq1(Xn; ss) ! eHq(X; ss) ! limeHq(Xn; ss) *
*! 0 of abelian groups.
Example: 8 q 1, Hq(Z=p1 Z; n) limHq(Z=pkZ; n).
[In the above, substitute Y = K(ss; q).]
LEMMA Let X be a pointed finite CW complex. Let K be a pointed connected C*
*W complex.
Assume: The homotopy groups of K are finite_then the pointed set [X; K] is fini*
*te.
[This result is contained in obstruction theory but one can also give a dir*
*ect inductive proof.]
550
EXAMPLE Let (X; x0) be a pointed CW complex. Suppose that x0 2 X0 X1 . .*
*i.s an
S
expanding sequence of finite subcomplexes of X such that X = Xn. Let K be a p*
*ointed connected CW
n
complex. Assume: The homotopy groups of K are finite_then the natural map ssX :*
* [X; K] ! lim[Xn; K]
is bijective. In fact, surjectivity is automatic, so injectivity is what's at i*
*ssue. For this, consider the natural
map ssIX : [IX; K] ! lim[i0X [ IXn [ i1X; K] and the obvious arrows i0; i1 : li*
*m[i0X [ IXn [ i1X; K] !
[X; K]. Since i0 O ssIX = i1 O ssIX and since ssIX is surjective, i0 = i1. That*
* ssX is injective is thus a
consequence of the following claim. *
* ae
Claim: If ssX ([f0]) = ssX ([f1]), then there exists [F] 2 lim[i0X [ IXn [ *
*i1X; K] : [f0] = i0([F]).
*
* [f1] = i1([F])
[Let in0; in1: [i0X [ IXn [ i1X;aK]e! [X; K] be the obvious arrows. For eac*
*h n, there is at least
n([Fn])
one [Fn] 2 [i0X [ IXn [ i1X; K] : [f0] = i0 . Denote by In the subset of [i0*
*X [ IXn [ i1X; K]
[f1] = in1([Fn])
consisting of all such [Fn]_then, from the lemma, In is finite, hence limIn 6= *
*;.]
[Note: The Xn are finite CW complexes, therefore the [Xn; K] are finite gro*
*ups, so lim1[Xn; K] =
*. But this only means that the kernel of ssX is [0].]
Application: Let {Yn} be a sequence of pointed spaces. Suppose given pointe*
*d con
tinuous functions n : Yn+1 ! Yn such that 8 n, n is a pointed Hurewicz fibrat*
*ion_then
for any pointed space X, there is an exact sequence
* ! lim1[X; Yn] ! [X; limYn] ! lim[X; Yn] ! *
in SET * and is an injection.
[Note: The exact sequence * ! lim1ssq+1(Yn) ! ssq(limYn) ! limssq(Yn) ! * *
*of
pointed sets is a special case (take X = Sq).]
*
*ae 1 1
EXAMPLE For each n, put Yn = S1and let n : Yn+1 ! Yn be the squaring map *
* S ! S _
*
* s ! s2
then limss1(Yn) = 0 but lim1ss1(Yn) bZ2=Z, the 2adic integers mod Z.
EXAMPLE Let ssssss= {ssn} be a tower of abelian groups. Assume: ssssssis *
*MittagLeffler_then
8 q 1, K(limssssss; q) = limK(ssn; q), so for any pointed CW complex (X; x0), *
*there is an exact sequence
0 ! lim1eHq1(X; ssn) ! eHq(X; limssssss) ! limeHq(X; ssn) ! 0 of abelian group*
*s.
Given a pointed path connected space X, let P1 X = lim PnX_then 8 q 0,
ssq(P1 X) limssq(PnX) ssq(PqX). Proof: The relevant lim1term vanishes.
PROPOSITION 13 The canonical arrow X ! P1 X is a weak homotopy equivalence.
551
[For each n, there is an inclusion X ! X[n], a projection P1 X ! PnX, and a
pointed homotopy equivalence X[n] ! PnX. Consider the associated commutative d*
*ia
X? ! P1?X
gram y y , recalling that ssn(X) ssn(X[n]).]
X[n] ! PnX
FACT Let {Xn; fn : Xn+1 ! Xn} be a tower in TOP . Assume: The Xn are CW sp*
*aces and the
fn are Hurewicz fibrations_then limXn is a CW space iff all but finitely many o*
*f the fn are homotopy
equivalences.
[Necessity: If infinitely many of the fn are not homotopy equivalences, the*
*n limXn is not numerably
contractible.
Sufficiency: If all of the fn are homotopy equivalences, then X0 and limXn *
*have the same homotopy
type (cf. p. 417).]
Application: Suppose that X is a pointed connected CW space_then the canoni*
*cal arrow X ! P1 X
is a homotopy equivalence iff X has finitely many nontrivial homotopy groups.
WHITEHEAD THEOREM Suppose that X and Y are path connected topological
spaces.
(1) Let f : X ! Y be an nequivalence_then f* : Hq(X) ! Hq(Y ) is bij*
*ective
for 1 q < n and surjective for q = n.
(2) Suppose in addition that X and Y are simply connected. Let f : X *
*! Y
be a continuous function such that f* : Hq(X) ! Hq(Y ) is bijective for 1 q < *
*n and
surjective for q = n_then f is an nequivalence.
[The condition on f* amounts to requiring that Hq(Mf; i(X)) = 0 for q n, t*
*hus the
result follows from the relative Hurewicz theorem.]
EXAMPLE Let X be a pointed connected CW space_then the inclusion X ! X[n] *
*is an (n +
1)equivalence, hence there are bijections Hq(X) Hq(X[n]) (q n) and a surject*
*ion Hn+1(X) !
Hn+1(X[n]). So, if X is abelian and if the ssq(X) are finitely generated 8 q, t*
*hen the Hq(X) are finitely
generated 8 q (cf. p. 544).
EXAMPLE (Suspension_Theorem_) Suppose that X is nondegenerate and nconne*
*cted. Let K
be a pointed CW complex_then the suspension map [K; X] ! [K; X] is bijective if*
* dimK 2n and
surjective if dimK 2n + 1. In fact the arrow of adjunction e : X ! X induces a*
*n isomorphism
Hq(X) ! Hq(X) for 0 q 2n + 1 (cf. p. 437), therefore by the Whitehead theor*
*em e is a
(2n + 1)equivalence. So, if dimK is finite and if n 2 + dimK, then [nK; nX] *
*[n+1K; n+1X].
552
A continuous function f : X ! Y is said to be a homology_equivalence_if 8 n
0, f* : Hn(X) ! Hn(Y ) is an isomorphism. Example: Consider the coreflector k*
* :
TOP ! CG _then for every topological space X, the identity map kX ! X is a ho*
*mology
equivalence.
EXAMPLE A homology equivalence f : X ! Y need not be a weak homotopy equiv*
*alence. One
can take, e.g., X to be Poincare's homology 3sphere S3=SL (2; 5) and Y = S3. *
*There is a homology
equivalence f : X ! Y obtained by collapsing the 2skeleton of X to a point whi*
*ch, though, is not a weak
homotopy equivalence, the fundamental group of X being SL(2; 5). Eight differen*
*t descriptions of X have
been examined by KirbyScharlemanny.
WHITEHEAD THEOREM (bis) Suppose that X and Y are path connected topo
logical spaces.
(1) Let f : X ! Y be a weak homotopy equivalence_then f is a homology
equivalence.
[Note: It is a corollary that in general a weak homotopy equivalence is a h*
*omology
equivalence.]
(2) Suppose in addition that X and Y are simply connected. Let f : X *
*! Y
be a homology equivalence_then f is a weak homotopy equivalence.
Consequently, if X and Y are simply connected topological spaces that are d*
*ominated
in homotopy by CW complexes, then a continuous function f : X ! Y is a homotopy
equivalence iff it is a homology equivalence.
The following familiar remarks serve to place this result in perspective.
(1) There exist path connected topological spaces X and Y such that 8*
* n : ssn(X) is isomor
phic to ssn(Y ) but 9 n : Hn(X) is not isomorphic to Hn(Y ).
(2) There exist simply connected topological spaces X and Y such that*
* 8 n : Hn(X) is
isomorphic to Hn(Y ) but 9 n : ssn(X) is not isomorphic to ssn(Y ).
(3) There exist path connected topological spaces X and Y admitting a*
* homology equivalence
f : X ! Y with the property that f* : ss1(X) ! ss1(Y ) is an isomorphism, yet f*
* is not a weak homotopy
equivalence.
[Note: Recall too that there exist topological spaces X and Y such that 8 n*
* : Hn(X) is isomorphic
to Hn(Y ) and 8 n : ssn(X; x0) isaisomorphiceto ssn(Y; y0) (8 x0 2 X; 8 y0 2 Y *
*), yet X and Y do not have
the same homotopy type. Example: X = {0} [ {1=n : n .1}]
Y = {0} [ {n : n 1}
_________________________
yIn: Geometric Topology, J. Cantrell (ed.), Academic Press (1979), 113146.
553
EXAMPLE There exists a sequence X1; X2; : :o:f simply connected CW complex*
*es Xn having
isomorphic integral singular cohomology rings such that 8 n06= n00, the homotop*
*y types of Xn0 & Xn00
are distinct (BodyDouglasy).
EXAMPLE Let X be a pointed connected CW space_then X is contractible iff H*
*1(ss; 1) = 0 =
H2(ss; 1) (ss = ss1(X)) and Hq(X) = 0 (q 2).
EXAMPLE (Stable_Splitting_) Let G be a finite abelian group_then there ex*
*ist positive integers
T and t such that TK(G; 1) has the pointed homotopy type of a wedge X1_ . ._.Xt*
*, where the Xiare
pointed simply connected CW spaces. For let G = G(p1) . . .G(pn) be the primar*
*y decomposition of
G. Since the arrow K(G(p1); 1) _ . ._.K(G(pn); 1) ! K(G(p1); 1) x . .x.K(G(pn);*
* 1) = K(G; 1) is a
homology equivalence, its suspension is a pointed homotopyrequivalence, thus on*
*e can assume that G is
Q
pprimary, say G = Z=pe1Z . . .Z=perZ, so K(G; 1) = K(Z=peiZ; 1). Accordingl*
*y, thanks to the
1
Puppe formula and the fact that (X#Y ) X#Y X#Y , it suffices to consider K(Z=*
*peZ; 1).
Claim: There exist pointed simply connected CW spaces X1; : :;:Xp1 and a p*
*ointed homotopy
equivalence K(Z=peZ; 1) ! X1_ . ._.Xp1.
[A generator of the multiplicative group of units in Z=pZ defines a pointed*
* homotopy equivalence
K(Z=peZ; 1) ! K(Z=peZ; 1).]
ae
The rather restrictive assumption that ss1(X)s=s0is not necessary in orde*
*r to guar
1(Y ) = 0
antee thataaehomology equivalence f : X ! Y is a weak homotopy equivalence. F*
*or
example, XY abelian will do and in fact one can get away with considerably le*
*ss.
Notation: Given a group G, let Z[G] be its integral group ring and I[G] Z[*
*G] the
augmentation ideal. Given a Gmodule M, let MG be its group of coinvariants, i.*
*e., the
quotient M=I[G] . M or still, H0(G; M).
[Note: In this context, "Gmodule" means left Gmodule. If K is a normal su*
*bgroup
of G, then the action of G on M induces an action of G=K on MK and MG (MK )G=K*
* .]
FUNDAMENTAL EXACT SEQUENCE Fix a Gmodule M. Let K be a normal
subgroup of G_then there is an exact sequence
H2(G; M) ! H2(G=K; MK ) ! H1(K; M)G=K ! H1(G; M) ! H1(G=K; MK ) ! 0:
[The LHS spectral sequence reads: E2p;q Hp(G=K; Hq(K; M)) ) Hp+q(G; M). Ex
2
plicate the associated five term exact sequence H2(G; M) ! E22;0d!E20;1! H1(G; *
*M) !
E21;0! 0:]
_________________________
yTopology 13 (1974), 209214.
554
Application: Let K be a normal subgroup of G_then there is an exact sequen*
*ce
H2(G) ! H2(G=K) ! K=[G; K] ! H1(G) ! H1(G=K) ! 0.
[Specialize the fundamentalaexactesequence and take M = Z (trivial Gaction*
*). Ob
serve that the arrows H1(G)H! H1(G=K) are induced by the projection G ! G=*
*K.]
2(G) ! H2(G=K)
Using a superscript to denote the "invariants" functor, the fundamental exa*
*ct sequence in cohomology
is 0 ! H1(G=K; MK ) ! H1(G; M) ! H1(K; M)G=K ! H2(G=K; MK ) ! H2(G; M).
Notation: Given a group G, let 0(G) 1(G) . .b.e its descending central se*
*ries,
so i+1(G) = [G; i(G)]. In particular: 0(G) = G, 1(G) = [G; G] and G is nilpoten*
*t_if
there exists a d : d(G) = {1}, the smallest such d being its degree_of_nilpoten*
*cy_: nilG.
FACT Let G be a nilpotent group_then G is finitely generated iff G=[G; G] *
*is finitely generated.
EXAMPLE Let G be a nilpotent group_then G is finitely generated iff 8 q 1*
*, Hq(G) is finitely
generated. For suppose that G is finitely generated. Case 1: nilG 1. In this s*
*ituation, G is abelian
and the assertion is true (cf. p. 535). Case 2: nilG > 1. Argue by induction, *
*using the LHS spectral
sequence E2p;q Hp(G=i(G); Hq(i(G)=i+1(G))) ) Hp+q(G=i+1(G)). To discuss the con*
*verse, note
that H1(G) G=[G; G] and quote the preceding result.
It is false in general that a subgroup of a finitely generated group is fin*
*itely generated. Example: Let
G be the free group on two symbols and consider [G; G].
FACT Suppose that G is a finitely generated nilpotent group_then every sub*
*group of G is finitely
generated.
FACT Suppose that G is a finitely generated nilpotent group_then G is fini*
*tely presented.
[The class of finitely presented groups is closed with respect to the forma*
*tion of extensions.]
Notation: Given a group G, Gtoris its subset of elements of finite order.
[Note: Gtorneed not be a subgroup of G (consider G = Z=2Z * Z=2Z) but will *
*be if G is nilpotent
(since nilG d and ym = e ) (xy)md = xmd ).]
FACT Suppose that G is a finitely generated nilpotent group. Assume: G is *
*torsion_then G is
finite.
Application: If G is a finitely generated nilpotent group, then Gtoris a f*
*inite nilpotent normal
subgroup.
555
PROPOSITION 14 Let f : G ! K be a homomorphism of groups. Assume: (i)
f* : H1(G) ! H1(K) is bijective and (ii) f* : H2(G) ! H2(K) is surjective_then *
*8 i 0,
the induced map G=i(G) ! K=i(K) is an isomorphism.
[The assertion is trivial if i = 0 and holds by assumption if i = 1. Fix i*
* > 1 and
proceed by induction. There is a commutative diagram
H2(G) ! H2(G=i(G)) ! i(G)=i+1(G) ! H1(G) ! H1(G=i(G)) ! 0
?y ?y ?y ?y ?y ?y
H2(K) ! H2(K=i(K)) ! i(K)=i+1(K) ! H1(K) ! H1(K=i(K)) ! 0
with exact rows, hence, by the five lemma, i(G)=i+1(G) i(K)=i+1(K). But then
from
1 ! i(G)=i+1(G) ! G=i+1(G) ! G=i(G) ! 1
?y ?y ?y
;
1 ! i(K)=i+1(K) ! K=i+1(K) ! K=i(K) ! 1
one concludes that G=i+1(G) K=i+1(K).]
Application: Let f : G ! K be a homomorphism of nilpotent groups. Assume: (*
*i)
f* : H1(G) ! H1(K) is bijective and (ii) f* : H2(G) ! H2(K) is surjective_then *
*f is an
isomorphism.
Let G and ss be groups. Suppose that G operates on ss, i.e., suppose given *
*a homomor
phism O : G ! Aut ss. Put 0O(ss) = ss and, via recursion, write i+1O(ss) for th*
*e subgroup
of ss generated by the ff(O(g)ffi)ff1ff1i(ff 2 ss; ffi 2 iO(ss)), where g 2 G*
*_then iO(ss)
is a Gstable normal subgroup of ss containing i+1O(ss). The quotient iO(ss)=i*
*+1O(ss) is
abelian and the induced action of G is trivial. One says that G operates nilpo*
*tently on
ss or that ss is Onilpotent_if there exists a d : dO(ss) = {1}, the smallest s*
*uch d being
its degree_of_nilpotency_: nilOss. Example: Take G = ss and let O : ss ! Aut*
* ss be the
representation of ss by inner automorphisms_then ss is Onilpotent iff ss is ni*
*lpotent.
[Note: From the definitions, for any O, i(ss) iO(ss), thus if ss is Onilp*
*otent, then
ss must be nilpotent.]
Let and ss be groups, where Aut ss. Suppose that ss = ss0 ss1 . . .ssd*
* = {1}
is a finite filtration of ss by stable normal subgroups such that operates tr*
*ivially on the
ssi=ssi+1_then there is a lemma in group theory that says must be nilpotent (S*
*uzukiy).
So, given O : G ! Aut ss; imO is nilpotent provided that ss is Onilpotent.
_________________________
yGroup Theory, vol. II, Springer Verlag (1986), 1920.
556
FACT Given a homomorphism O : G ! Autss, consider the semidirect product s*
*sxOG, i.e., the set
of all ordered pairs (ff; g) 2 ss x G with law of composition (ff0; g0)(ff00; g*
*00) = (ff0(O(g0)ff00); g0g00)_then
ssxOG is nilpotent iff ss is Onilpotent and G is nilpotent.
EXAMPLE Every finite pgroup is nilpotent. Since the semidirect product of*
* two finite pgroups
is a finite pgroup, it follows that if G and ss are finite pgroups and if G o*
*perates on ss, then G actually
operates nilpotently on ss.
FACT Suppose that G operates on ss_then G operates nilpotently on ss iff s*
*s is nilpotent and G
operates nilpotently on ss=[ss; ss].
EXAMPLEaeLet 1 ! G0! G ! G00! 1 be a short exact sequence of groups. Obviou*
*sly: G nilpo
0
tent ) G nilpotent. The converse is false (consider A3 S3). However, there *
*is a characterization:
G00 ae
0
G is nilpotent iff G are nilpotent and the action of G00on G0=[G0; G0] is ni*
*lpotent.
G00
Example: Suppose that ss = M is a Gmodule. Since M is abelian, it is nilp*
*otent
but it needn't be Onilpotent. In fact, iO(M) = (I[G])i. M, therefore M is Oni*
*lpotent iff
(I[G])d . M = 0 for some d. When this is so, M is referred to as a nilpotent_G*
*module_.
EXAMPLE Let ss be a nilpotent Gmodule. Fix n 1_then 8 q 0, Hq(ss; n) is*
* a nilpotent
Gmodule.
[G operates nilpotently on the iO(ss) and 8 i, there is a short exact seque*
*nce 0 ! i+1O(ss) ! iO(ss) !
iO(ss)=i+1O(ss) ! 0 of Gmodules, the action of G on iO(ss)=i+1O(ss) being triv*
*ial. The mapping fiber
of the arrow K(iO(ss); n) ! K(iO(ss)=i+1O(ss); n) is a K(i+1O(ss); n). Consider*
* the associated fibration
spectral sequence, noting that by induction, G operates nilpotently on E2p;q.]
FACT Suppose that G is a finitely generated nilpotent group. Let M be a ni*
*lpotent Gmodule.
(1) If M is finitely generated, then 8 q 0, Hq(G; M) is finitely gen*
*erated.
(2) If M is not finitely generated, then H0(G; M) is not finitely gen*
*erated.
A nonempty path connected topological space X is said to be nilpotent_if ss*
*1(X) is
nilpotent and if 8 n > 1, ss1(X) operates nilpotently on ssn(X). Examples: (1*
*) Every
abelian topological space is nilpotent; (2) Every path connected topological sp*
*ace whose
homotopy groups are finite pgroups is nilpotent (cf. supra); (3) Take for X t*
*he Klein
bottle_then ss1(X) is not a nilpotent group; (4) Take for X the real projective*
* plane_
then ss1(X) Z=2Z , ss2(X) Z and the action of ss1(X) on ss2(X) is the inversi*
*on n ! n,
557
thus ss1(X) does not operate nilpotently on ss2(X); (5) Take for X the torus S1*
*xS 1_then
X is nilpotent but its 1skeleton X(1)= S1_ S1 is not nilpotent.
EXAMPLE Let G be a topological group with base point e and denote by G0 th*
*e path component
of e_then ss0(G) = G=G0 can be identified with ss1(B1G) and ssn(G) = ssn(G0) ca*
*n be identified with
ssn+1(B1G) (cf. p. 465). These identifications are compatible in that the homo*
*morphism On : ss0(G) !
Autssn(G0) arising from the operation of G on itself by inner automorphisms cor*
*responds to the action of
ss1(B1G) on ssn+1(B1G). Accordingly, B1Gis a nilpotent topological space iff ss*
*0(G) is a nilpotent group and
8 n 1, ssn(G0) is Onnilpotent or still, 8 n 1, the semidirect product ssn(G0*
*)xOnss0(G) is nilpotent
(cf. p. 556). The forgetful function [Sn; sn; G0; e] ! [Sn; G0] is bijective*
*, hence [Sn; G0] ssn(G0).
In addition, [Sn; G] is isomorphic to ssn(G0)xOnss0(G). To see this, let f : *
*Sn ! G be a continuous
function. Choose gf 2 G : f(Sn) G0gf, put f0 = f .g1fand consider the assignm*
*ent [f] ! ([f0]; gfG0).
It therefore follows that B1G is a nilpotent topological space iff 8 n 1, [Sn;*
* G] is a nilpotent group.
Example: B1O(2n+1)is nilpotent but B1O(2n)is not nilpotent.
[Note: Here is another illustration. The higher homotopy groups of a connec*
*ted nilpotent Lie group
are trivial. So, if G is an arbitrary nilpotent Lie group, then B1G is a nilpot*
*ent topological space.]
FACT Let G be a topological group. Assume: 8 n 1, [Sn; G] is a nilpotent *
*group_then for any
finite CW complex K, [K; G] is a nilpotent group.
[Take K connected and argue by induction on the number of cells.]
EXAMPLE Let X be a nilpotent CW space_then Misliny has shown that X is dom*
*inated in
homotopy by a finite CW complex iff the Hq(X) are finitely generated 8 q and th*
*ere exists q0 : 8 q > q0,
Hq(X) = 0. Moreover, under these conditions, Wall's obstruction to finiteness i*
*s zero provided that ss1(X)
is infinite but this can fail if ss1(X) is finite (Mislinz).
DROR'S WHITEHEAD THEOREM Suppose that X and Y are nilpotent topological
spaces. Let f : X ! Y be a homology equivalence_then f is a weak homotopy equiv*
*alence.
[To prove that f is a weak homotopy equivalence amounts to proving that for*
* every
n, the pair (Mf; i(X)) is nconnected, where, a priori, H*(Mf; i(X)) = 0. Cons*
*ider the
X? f! Y?
commutative diagram y y . Since the vertical arrows are 2equivalen*
*ces, f1;1
X[1] !f Y [1]
1;1
_________________________
yAnn. of Math. 103 (1976), 547556.
zTopology 14 (1975), 311317.
558
inducesaaebijection H1(X[1])a!eH1(Y [1]) andaaesurjection H2(X[1]) ! H2(Y [1]).*
* But
X[1] (ss1(X); 1) ss1(X)
Y [1] has homotopy type (ss1(Y ); 1)and ss1(Y )are nilpotent groups, thus*
* f* :
ss1(X) ! ss1(Y ) is an isomorphism (cf. p. 555) and so (Mf; i(X)) is 1connect*
*ed. Noting
that here ss2(Mf; i(X)) is abelian, fix n > 1 and assume inductively that ssq(M*
*f; i(X)) =
0 for q < n_then, from the relative Hurewicz theorem, ssn(Mf; i(X))ss1(X)= 0, i*
*.e.,
ssn(Mf; i(X)) = I[ss1(X)] . ssn(Mf; i(X)). On the other hand, there is an exact*
* sequence
ssn(Mf) ! ssn(Mf; i(X)) ! ssn1(i(X)) of ss1(X)modules. Because the flanking *
*terms
are, by hypothesis, nilpotent ss1(X)modules, the same must be true of ssn(Mf; *
*i(X)).
Conclusion: ssn(Mf; i(X)) = 0.]
PROPOSITION 15 Let f : X ! Y be a Hurewicz fibration, where X and Y are
path connected. Assume: X is nilpotent_then 8 y0 2 Y , the path components of X*
*y0 are
nilpotent.
[Fix x0 2 Xy0 and take Xy0 path connected. The homomorphisms in the homotopy
sequence
. .!.ssn+1(Y; y0) ! ssn(Xy0; x0) ! ssn(X; x0) ! ssn(Y; y0) ! . . .
of f are ss1(X; x0)homomorphisms (cf. p. 436). Of course, ss1(X; x0) oper*
*ates on
ssn(Y; y0) through f* and if i : Xy0 ! X is the inclusion, then ff . = (i*ff) *
*. (ff 2
ss1(Xy0; x0); 2 ssn(Xy0; x0)). Since the base points will play no further role*
*, drop them
from the notation.
(n = 1) To see that ss1(Xy0) is nilpotent, consider the short exact se*
*quence as
sociated with the exact sequence ss2(Y ) @!ss1(Xy0) i*!ss1(X), noting that im@ *
*is contained
in the center of ss1(Xy0).
(n > 1) There is an exact sequence ssn+1(Y ) @!ssn(Xy0) i*!ssn(X) and *
*by as
sumption, 9 d : (I[ss1(X)])d . ssn(X) = 0. Claim: (I[ss1(Xy0)])d+1 . ssn(Xy0) =*
* 0. For let
ff 2 (I[ss1(Xy0)])d, 2 ssn(Xy0) : i*(ff . ) = i*ff . i* = 0 ) ff . = @j (j 2 *
*ssn+1(Y )). And:
8 fi 2 ss1(Xy0), (i*fi  1) . j = (f*i*fi  1) . j = 0, so 0 = @((i*fi  1) . j*
*) = (i*fi  1) . @j =
((fi  1)ff) . . Hence the claim.]
Application: Let X and Y be pointed path connected spaces. Assume: X is nil*
*potent_
then for every pointed continuous function f : X ! Y , the path components of t*
*he map
ping fiber Ef of f are nilpotent.
EXAMPLE Let (K; k0) be a pointed connected CW complex. Assume: K is finit*
*e_then for
any pointed path connected space (X; x0), the path components of C(K; k0; X; x0*
*) are nilpotent. In
559
particular, the fundamental group of the path component of the constant map K !*
* x0 is nilpotent, thus
[K; k0; X; j(x0)] is a nilpotent group. Observe that the base points play a rol*
*e here: [S1; P 2(R )] is a
group but it is not nilpotent.
FACT Let f : X ! B be a Hurewicz fibration. Given 02 C(B0; B), define X0 b*
*y the pullback
X0 ! X ae
? ? X
square y yf. Assume: & B0are nilpotent_then the path components of *
*X0are nilpotent.
B
B0 !0 B
[Work with the MayerVietoris sequence (cf. p. 437).]
EXAMPLE The preceding result implies that nilpotency behaves well with res*
*pect to pullbacks
but the situation for pushouts is not as satisfactory since nilpotency is not o*
*rdinarily inherited (consider
S1 _ S2). For example, suppose that f : X ! Y is a continuous function, where X*
* and Y are nonempty
path connected CW spaces. Assume: Y is nilpotent_then Raoy has shown that the m*
*apping cone Cf
of f is nilpotent iff one of the following conditions is satisfied: (i) f* : ss*
*1(X) ! ss1(Y ) is surjective; (ii)
8 q > 0, Hq(X) = 0; (iii) 9 a prime p such that ss1(Cf) is a finite pgroup and*
* 8 q > 0, Hq(X) is a pgroup
of finite exponent. Example: If f : X ! Y is a closed cofibration, then under (*
*i), (ii), or (iii), Y=f(X) is
nilpotent (cf. p. 324). Moreover, under (ii), the projection Y ! Y=f(X) is a h*
*omology equivalence (cf.
p. 38), hence by Dror's Whitehead theorem is a homotopy equivalence.
ae
Let XY be pointed connected CW spaces. Suppose that f : X ! Y is a poin*
*ted
continuous function_then f is said to admit a principal_refinement_of_order_nif*
* f can be
written as a composite X ! WN qN!WN1 ! . .!.W1 q1!W0 = Y , where is a pointed
homotopy equivalence and each qi: Wi! Wi1 is a pointed Hurewicz fibration for *
*which
there is an abelian group ssi and a pointed continuous function i1 : Wi1 ! K(*
*ssi; n+1)
Wi? ! K(ssi;?n + 1)
such that the diagram qyi y is a pullback square.
Wi1 ! K(ssi; n + 1)
i1
[Note: Wi is a pointed connected CW space homeomorphic to Ei1 (parameter
reversal).]
Example: If X is a pointed abelian CW space, then 8 n, the arrow fn : X[n] *
*! X[n1]
W [n]
admits a principal refinement of order n: ' ' '') u (cf. p. 542), wit*
*h N = 1.
X[n] _____wX[n  1]
_________________________
yProc. Amer. Math. Soc. 87 (1983), 335341.
560
EXAMPLE (Central_Extensions_) Let ss and G be groups, where ss is abelian*
*_then the isomor
phism classes of central extensions 1 ! ss ! ! G ! 1 of ss by G are in a onet*
*oone correspondence
with the elements of H2(G; 1; ss) or still, with the elements of [K(G; 1); K(ss*
*; 2)]. Therefore G is nilpotent
iff the constant map K(G; 1) ! * admits a principal refinement of order 1.
[Any nilpotent G generates a finite sequence of central extensions 1 ! i(G)*
*=i+1(G) ! G=i+1(G)
! G=i(G) ! 1.]
Let X be a pointed connected CW space_then, in view of the preceding exampl*
*e,
the arrow f1 : X[1] ! X[0] admits a principal refinement of order 1 iff ss1(X) *
*is nilpotent.
PROPOSITION 16 Let X be a pointed connected CW space. Fix n > 1_then the
arrow fn : X[n] ! X[n  1] admits a principal refinement of order n iff ss1(X) *
*operates
nilpotently on ssn(X).
[Necessity: Suppose that fn factors as a composite X[n] ! WN qN!WN1 ! . *
*.!.
W1 q1!W0 = X[n  1], where and the qi are as in the definition. Obviously, ss*
*1(X)
ss1(Wi) for all i. Since ssn(W0) = ssn(X[n  1]) = 0, ss1(X) operates nilpotent*
*ly on ssn(W0).
___
Claim: ss1(X) operates nilpotently on_ssn(W1)._Thus let W0 be the mapping trac*
*k of 0
W1 ! K(ss1; n + 1)
___ ? ?
and define W1 by the pullback square y y _then there is a p*
*ointed
___
___ W0 ! K(ss1; n + 1)
homotopy equivalence W1 ! W1 and, from the proof of the "n > 1" part of Proposi*
*tion 15,
___
ss1(X) operates nilpotently on ssn(W1 ). Iterate to conclude that ss1(X) operat*
*es nilpotently
on ssn(WN ) ssn(X).
Sufficiency: One can copy the argument employed in the abelian case to con*
*struct
the Postnikov invariant (cf. p. 542). At the first stage, the only difference *
*is that after
replacing n by n  1, the coefficient group for cohomology is not ssn(X) but ss*
*n(X)ss1(X)=
W1
H0(ss1(X); ssn(X)). Because the initial lifting '1' '')uq1 of fn is a po*
*inted homo
X[n] _____wfnX[n  1]
topy equivalence iff I[ss1(X)].ssn(X) = 0, it is in general necessary to repeat*
* the procedure,
which will then terminate after finitely many steps.]
Application: Let X be a pointed connected CW space_then X is nilpotent iff *
*8 n,
the arrow fn : X[n] ! X[n  1] admits a principal refinement of order n.
[Note: If X is nilpotent and if On : ss1(X) ! Aut ssn(X) is the homomorphis*
*m corre
sponding to the action of ss1(X) on ssn(X), then a choice for the abelian group*
*s figuring in
the principal refinement of the arrow X[n] ! X[n  1] are the iOn(ssn(X))=i+1On*
*(ssn(X)).]
561
EXAMPLE Let K be a finite CW complex_then for any pointed nilpotent CW spa*
*ce X, the
path components of C(K; X) are nilpotent.
[Bearing in mind x4, Proposition 5, use Proposition 15 and induction to sho*
*w that 8 n, the path
components of C(K; X[n]) are nilpotent.]
EXAMPLE Let (K; k0) be a pointed CW complex. Assume: K is finite_then for *
*any pointed
nilpotent CW space (X; x0), the path components of C(K; k0; X; x0) are nilpoten*
*t. Indeed, C(K; k0; X; x0)
= C(K0; k0; X; x0) x C(K1; X) x . .x.C(Kn; X), where K0; K1; : :;:Kn are the pa*
*th components of K
and k0 2 K0.
NILPOTENT OBSTRUCTION THEOREM Let (X; A) be a relative CW complex; let Y be
a pointed nilpotent CW space. Suppose that 8 n > 0 & 8 i 0, Hn+1(X; A; iOn(ssn*
*(Y ))=i+1On(ssn(Y ))) =
0_then every f 2 C(A; Y ) admits an extension F 2 C(X; Y ), any two such being *
*homotopic relA provided
that 8 n > 0 & 8 i 0, Hn(X; A; iOn(ssn(Y ))=i+1On(ssn(Y ))) = 0.
PROPOSITION 17 Let X be a pointed connected CW space, eXits universal cove*
*ring
space. Assume: ss1(X) is nilpotent_then X is nilpotent iff 8 n 1, ss1(X) ope*
*rates
nilpotently on Hn(Xe).
[Xe exists and is a pointed connected CW space (cf. Proposition 5).
Necessity: Consider the Postnikov tower of Xe, so epn: PnXe ! Pn1Xe. Sup*
*pose
inductively that ss1(X) operates nilpotently on the homology of Pn1Xe. Since X*
* is nilpo
tent, the Hq(ssn(X); n) are nilpotent ss1(X)modules (cf. p. 556), i.e., ss1(*
*X) operates
nilpotently on the homology of the mapping fiber of epn. Therefore, by the univ*
*ersal co
efficient theorem, the E2p;q Hp(Pn1Xe; Hq(ssn(X); n)) in the fibration spectra*
*l sequence
of epnare nilpotent ss1(X)modules, thus the same is true of the Hi(PnXe). But *
*the arrow
Xe ! PnXe induces an isomorphism of ss1(X)modules Hi(Xe) ! Hi(PnXe) for i n.
Sufficiency: Introduce the Whitehead tower of eXand argue as above.]
PROPOSITION 18 Let X be a pointed connected CW space. Assume: X is nilpote*
*nt_
then the ssq(X) are finitely generated 8 q iff the Hq(X) are finitely generated*
* 8 q.
[Suppose that the ssq(X) are finitely generated 8 q_then, eXbeing simply co*
*nnected,
hence abelian, the Hq(Xe) are finitely generated 8 q (cf. p. 551). On the ot*
*her hand,
according to Proposition 17, ss1(X) operates nilpotently on the Hq(Xe). Consequ*
*ently, the
Hp(ss1(X); Hq(Xe)) are finitely generated (cf. p. 556). However, these terms a*
*re precisely
the E2p;qin the spectral sequence of the covering projection Xe ! X (see below)*
*, so 8 i,
Hi(X) is finitely generated.
562
Suppose that the Hq(X) are finitely generated 8 q_then, since ss1(X)=[ss1(X*
*); ss1(X)]
H1(X), the nilpotent group ss1(X) is finitely generated (cf. p. 554). As for *
*the ssq(X)
(q > 1), their finite generation will follow if it can be shown that the Hq(Xe)*
* are finitely
generated (cf. p. 544). Proceeding by contradiction, fix an i0 such that Hi0*
*(Xe) is not
finitely generated and take i0 minimal. The E2p;q Hp(ss1(X); Hq(Xe)) are finite*
*ly gener
ated if q < i0 but E20;i0 H0(ss1(X); Hi0(Xe)) is not finitely generated (cf. p.*
* 556), thus
E10;i0is not finitely generated. Therefore Hi0(X) contains a subgroup which is *
*not finitely
generated.]
[Note: A finitely generated nilpotent group is finitely presented and its i*
*ntegral group
ring is (left and right) noetherian. This said, it then follows that under the*
* equivalent
conditions of the proposition, X necessarily has the pointed homotopy type of a*
* pointed
CW complex with a finite nskeleton 8 n (Wally).]
The spectral sequence E2p;q Hp(ss1(X); Hq(Xe)) ) Hp+q(X) of the covering pr*
*ojec
tion eX! X is an instance of a fibration spectral sequence. In fact, consider t*
*he inclusion i :
X ! X[1] = K(ss1(X); 1) and pass to its mapping track Wi! K(ss1(X); 1)_then Ei *
*has
the same pointed homotopy type as eX. Moreover, Hp(ss1(X); Hq(Xe)) Hp(K(ss1(X)*
*; 1);
Hq(Xe)), where Hq(Xe) is the locally constant coefficient system on K(ss1(X); 1*
*) determined
by Hq(Xe) (cf. p. 534).
ae
FACT Suppose that X are pointed connected CW spaces. Let f : X ! Y be a *
*pointed Hurewicz
Y
fibration with ss0(Xy0) = *_then ss1(X) operates nilpotently on the ssq(Xy0) 8 *
*q iff Xy0 is nilpotent and
ss1(Y ) operates nilpotently on the Hq(Xy0) 8 q.
ae
EXAMPLE Suppose that X are pointed connected CW spaces. Let f : X ! Y be*
* a pointed
Y
Hurewicz fibration with ss0(Xy0) = *_then any two of the following conditions i*
*mply the third and the
third implies that Xy0is nilpotent: (i) X is nilpotent; (ii) Y is nilpotent; (i*
*ii) ss1(X) operates nilpotently on
the ssq(Xy0) 8 q. Assume now that ss1(Y ) operates nilpotently on the Hq(Xy0) 8*
* q. Claim: X is nilpotent
iff both Y and Xy0are nilpotent. For X nilpotent ) Xy0nilpotent (cf. Propositio*
*n 15) ) ss1(X) operates
nilpotently on the ssq(Xy0) 8 q ) Y nilpotent, and conversely.
ae ae 0
HILTONROITBERGz COMPARISON THEOREM Suppose that X & X are pointed
Y Y 0
connected CW spaces. Let f : X ! Y and f0 : X0 ! Y 0be pointed Hurewicz fibrati*
*ons such that Ef
_________________________
yAnn. of Math. 81 (1965), 5669.
zQuart. J. Math. 27 (1976), 433444; see also Sch"on, Quart. J. Math. 32 (19*
*81), 235237.
563
ae ae
and Ef0are path connected and ss1(Y )operates nilpotently on the Hq(Ef) 8 q*
*. Suppose there is
ss1(Y 0) Hq(Ef0)
X? f! Y?
a commutative diagram y y , where ss1(Y ) ss1(Y 0) or ss1(Y ) & ss1(Y *
*0) are nilpotent_then,
X0 !f0 Y 0
assuming that all isomorphisms are induced, any two of the following conditions*
* imply the third: (1)
8 p; Hp(Y ) Hp(Y 0); (2) 8 q; Hq(Ef) Hq(Ef0); (3) 8 n; Hn(X) Hn(X0).
A nonempty path connected topological space X is said to be acyclic_provide*
*d that
8 q > 0, Hq(X) = 0. So: X acyclic ) ss = [ss; ss] and H1(ss; 1) = 0 = H2(ss; 1)*
* (cf. p. 535),
where ss = ss1(X). Example: Every nilpotent acyclic space is homotopically triv*
*ial (quote
Dror's Whitehead theorem).
EXAMPLE (Acyclic_Groups_) A group G is said to be acyclic_if 8 n > 0, Hn(*
*G) = 0 or, equiv
alently, if K(G; 1) is an acyclic space. Nontrivial finite groups are never acy*
*clic (Swany). However, there
are plenty of concretely defined infinite acyclic groups. A list of examples ha*
*s been compiled by Harpe
McDuffz. They include: (1) The symmetric group on an infinite set; (2) The grou*
*p of invertible linear
transformations of an infinite dimensional vector space; (3) The group of inver*
*tible bounded linear trans
formations of an infinite dimensional Hilbert space; (4) The automorphism group*
* of the measure algebra
of the unit interval; (5) The group of compactly supported homeomorphisms of Rn.
FACT Let G be a group which is the colimit of subgroups Gn (n 2 N) with th*
*e property that 8 n,
there exists a nontrivial gn 2 Gn+1 and a homomorphism OEn : Gn ! CenGn+1(Gn) s*
*uch that 8 g 2 Gn,
g = [gn; OEn(g)]_then G is acyclic.
[It suffices to work with coefficients in an arbitrary field k. Since H*(G;*
* k) colimH*(Gn; k), one
need only show that 8 n 1 & 8 N 1, the morphism Hq(Gn; k) ! Hq(Gn+N ; k) indu*
*ced by the
inclusion Gn ! Gn+N is trivial when 1 q < 2N . For this, fix n and use inducti*
*on on N. Recall that
conjugation induces the identity on homology and apply the K"unneth formula.]
[Note: It is clear that OEn is injective () gn 2 Gn+1  Gn). Observe too th*
*at it is not necessary to
assume that OEn(Gn) is contained in the centralizer of Gn in Gn+1 as this is im*
*plied by the other condition.
Proof: 8 g; h 2 Gn : [gn; OEn(gh)] = [gn; OEn(g)] . [OEn(g); [gn; OEn(h)]] . [g*
*n; OEn(h)] ) gh = g[OEn(g); h]h ) e =
[OEn(g); h].]
_________________________
yProc. Amer. Math. Soc. 11 (1960), 885887.
zComment. Math. Helv. 58 (1983), 4871; see also Berrick, In: Group Theory, *
*K. Cheng and Y.
Leong (ed.), Walter de Gruyter (1989), 253266.
564
EXAMPLE Let Hc(Q ) be the set of bijections of Q that are the identity out*
*side some finite
interval. Given a group G, let Fc(Q ; G) be the set of functions Q ! G that sen*
*d all elements outside some
finite interval to the identity. Both Hc(Q ) and Fc(Q ; G) are groups and there*
* is a homomorphism O :
Hc(Q ) ! AutFc(Q ; G), viz. O(fi)ff(q) =aff(fi1(q)).eThe cone_ofaGeis the asso*
*ciated semidirect product:
G = Fc(Q ; G)xOHc(Q ). The assignment G ! G : ffg(q) = g (q = 0)is a mon*
*omorphism of
g ! ffg e (q 6= 0)
groups and G is acyclic.
[Let Gn = {(ff; fi) : sptff [ sptfi [n; n]} and construct a homomorphism *
*OEn : Gn !
CenGn+1 (Gn) in terms of a bijection fin 2 Hc(Q ) : sptfin [n1; n+1] & 8 k :*
* fikn[n; n]\[n; n] =
;.]
FACT Every group can be embedded in an acyclic simple group.
[By the above, every group can be embedded in an acyclic group. On the othe*
*r hand, every group
can be embedded in a simple group (Robinsony). So given G, there is a sequence *
*G G1 G2 . .,.
S
where Gn is acyclic if n is odd and simple if n is even. Consider Gn.]
n
Recall that a group G is said to be perfect_if G = [G; G]. Examples: (1) Ev*
*ery acyclic
group is perfect; (2) Every nonabelian simple group is perfect.
[Note: The fundamental group of an acyclic space is perfect.]
The homomorphic image of a perfect group is perfect. Therefore, if G is per*
*fect and
ss is nilpotent, then G operates nilpotently on ss iff G operates trivially on *
*ss (cf. p. 555).
Proof: A perfect nilpotent group is trivial.
Every group G has a unique maximal perfect subgroup Gper, the perfect_radic*
*al_of G.
The automorphisms of G stabilize Gper, thus Gperis normal.
(P 1) Let f : G ! K be a homomorphism of groups_then f(Gper) Kper.
(P 2) Let f : G ! K be a homomorphism of groups, where Kper= {1}_then
Gper kerf.
FACT A locally free group is acyclic iff it is perfect.
[Note: A group is said to be locally_free_if its finitely generated subgrou*
*ps are free.]
LEMMA Let f : G ! K be an epimorphism of groups. Put N = kerf_then
f(Gper) = Kperprovided that 9 n : N(n) Gper.
[Note: N(n) is the nth derived group of N : N(0)= N; N(i+1)= [N(i); N(i)]. *
*Obvi
ously, N(0) Gperif N is perfect and N(1) Gperif N is central.]
_________________________
yFiniteness Conditions and Generalized Soluble Groups, vol. I, Springer Verl*
*ag (1972), 144.
565
Application: Let N be a perfect normal subgroup of G_then the perfect radic*
*al of
G=N is the quotient Gper=N, hence the perfect radical of G=N is trivial iff N =*
* Gper.
EXAMPLE Let A be a ring with unit. Agreeing to employ the usual notation o*
*f algebraic K
theory, denote by GL (A) the infinite general linear group of A and write E(A) *
*for the subgroup of GL (A)
consisting of the elementary matrices_then, according to the Whitehead lemma, E*
*(A) = [E (A); E(A)] =
[GL (A); GL(A)], thus E(A) is the perfect radical of GL (A). Let now ST(A) be t*
*he Steinberg group of
A : ST(A) is perfect and there is an epimorphism ST(A) ! E(A) of groups whose k*
*ernel is the center of
ST (A).
[Note: On occasion, it is necessary to consider rings which may not have a *
*unit (pseudorings). Given
a pseudoring A, let __Abe the set of all functions X : N x N ! A such that #{(i*
*; j) : Xij6= 0} < !_then
__Ais again a pseudoring (matrix operations). The law of composition X ? Y = X *
*+ Y + X x Y equips __A
with the structure of a semigroup with unit. Definition: ___GL(A) is the group *
*of units of (__A; ?). Therefore,
using obvious notation, __E(A) = [__E(A); __E(A)] = [___GL(A); ___GL(A)]. Every*
* bijection OE : N ! N x N defines
an isomorphismaofepseudorings:_A= __A, hence ___GL(__A) ___GL(A). In the event*
* that A has a unit, the
assignment GL(A) ! GL (A)is an isomorphism of groups () ___GL(__A) GL (A)).]
X ! X + I
EXAMPLE (Universal_Central_Extensions_) Let G be a group_then a central e*
*xtension 1 !
N ! U ! G ! 1 is said to be universal_if for any other central extension 1 ! ss*
* ! ! G ! 1 there
U[______w]
is a unique homomorphism aeAEover G. A central extension 1 ! N ! U ! G ! *
*1 is universal
G
iff H1(U) = 0 = H2(U). On the other hand, a universal central extension 1 ! N *
*! U ! G ! 1
exists iff G is perfect. To identify N in terms of G, use a portion of the fun*
*damental exact sequence:
H2(U) ! H2(G) ! N=[U; N] ! H1(U) or still, 0 ! H2(G) ! N=[U; N] ! 0 ) H2(G) N.*
* Example:
Take G = E(A)_then H1(ST (A)) = 0 = H2(ST (A)) and there is a universal central*
* extension 1 !
H2(E (A)) ! ST(A) ! E(A) ! 1.
EXAMPLE Let ACYGR be the full subcategory of grwhose objects are the acy*
*clic groups_then
Berricky has defined a functor ff : AB ! ACYGR such that 8 G, the center of f*
*fG is naturally isomorphic
to G. The quotient fiG = ffG=Cen G is a perfect group and the central extension*
* 1 ! G ! ffG ! fiG ! 1
is universal, so G H2(fiG).
[Note: By contrast, the cone construction defines a functor : gr! ACYGR .]
ae
FACT Let G1 be groups_then the perfect radical of G1x G2 is (G1)perx (G2*
*)per.
G2
_________________________
yJ. Pure Appl. Algebra 44 (1987), 3543.
566
ae
FACT Let G1 be groups with trivial perfect radicals_then the perfect rad*
*ical of their free
G2
product G1* G2 is trivial.
[A theorem of Kurosch says that any subgroup G of G1 * G2 has the form F * *
*(* Gi), where F is a
*
* i W
free group and 8 i, Gi is isomorphic to a subgroup of either G1 or G2. Put X = *
*K(F; 1) _ K(Gi; 1) :
L *
* i
ss1(X) G. If G is perfect, then 0 = H1(X) H1(F) H1(Gi), and it follows tha*
*t F and the Giare
i
perfect, hence trivial.]
ae
Let XY be pointed connected CW spaces. Suppose that f : X ! Y is a poin*
*ted
continuous function_then f is said to be acyclic_if its mapping fiber Ef is acy*
*clic. For
this, it is therefore necessary that ss0(Ef) = *.
[Note: Using the mapping cylinder Mf, write f = r O i (cf. p. 321)_then (M*
*f; i(x0))
is nondegenerate, thus r : Mf ! Y is a pointed homotopy equivalence (cf. p. 33*
*5) which
implies that the arrow Ei ! ErOi= Ef is a pointed homotopy equivalence (cf. p. *
*433).
Conclusion: f : X ! Y is acyclic iff i : X ! Mf is acyclic.]
Observation: Suppose that f : X ! Y is acyclic_then f* : ss1(X) ! ss1(Y ) i*
*s sur
jective and its kernel is a perfect normal subgroup of ss1(X).
[Inspect the exact sequence ss2(Y ) ! ss1(Ef) ! ss1(X) ! ss1(Y ) ! ss0(Ef).]
ae
PROPOSITION 19 Let XY be pointed connected CW spaces, f : X ! Y a pointed
continuous function_then f is a pointed homotopy equivalence iff f is acyclic a*
*nd f* :
ss1(X) ! ss1(Y ) is an isomorphism.
[The necessity is clear. As for the sufficiency, the arrow ss2(Y ) ! ss1(Ef*
*) is surjective,
hence ss1(Ef) is both abelian and perfect. But this means that ss1(Ef) must be *
*trivial, so,
being a pointed connected CW space, Ef is contractible.]
Let P be a set of primes. Fix an abelian group G_then G is said to be Ppr*
*imary_if 8 g 2 G,
Q Q
9 F P (#(F) < !) & n 2 N : ( p)ng = 0 ( = 1) and G is said to be uniquely_*
*Pdivisible_if
p2F ;
8 g 2 G, 8 p 2 P, 9! h 2 G : ph = g.
[Note: If P is empty, then the only Pprimary abelian group is the trivial *
*group and every abelian
group is uniquely Pdivisible.]
LEMMA Let C be a class of abelian groups containing 0. Assume: C is closed*
* under the formation
of direct sums andafiveeterm exact sequences, i.e., for any exact sequence G1 !*
* G2 ! G3 ! G4 ! G5 of
abelian groups: G1; G22 C ) G3 2 C_then there exists a set of primes P such t*
*hat C is either the
G4; G5
class of Pprimary abelian groups or the class of uniquely Pdivisible abelian *
*groups.
567
[The hypotheses imply that C is colimit closed. Given a set P of primes, it*
* follows that if Z=pZ 2
C 8 p 2 P, then every Pprimary abelian group is in C or if Q 2 C and Z=pZ 2 C *
*8 p 62 P, then
every uniquely Pdivisible abelian group is in C. On the other hand, if some G*
* 2 C is not uniquely
Pdivisible, then Z=pZ 2 C (consider G p!G) and if some G 2 C is not torsion, t*
*hen Q 2 C (consider
Q G = colim(. .!.G n!G ! . .).). To summarize: (1) If Q 62 C and if Z=pZ 2 C e*
*xactly for p 2 P,
then C consists of the Pprimary abelian groups; (2) If Q 2 C and if Z=pZ 2 C e*
*xactly for p 62 P, then C
consists of the uniquely Pdivisible abelian groups.]
ae
Application: Fix abelian groups A _then AB = 0 = Tor(A; B) iff there exis*
*ts a set P of primes
B
such that one of the groups is Pprimary and the other is uniquely Pdivisible.
[Supposing that A B = 0 = Tor(A; B), the class of abelian groups G for whi*
*ch G B = 0 =
Tor(G; B) satisfies the assumptions of the lemma.]
ae
EXAMPLE Given a 2sink X p!B q Y , where X & B are pointed connected CW*
* spaces,
__ __ Y
form X _BY (cf. p. 425). Let r : X _BY ! B be the projection_then the f*
*ollowing conditions
are8equivalent:L(i) r is a pointed homotopy equivalence; (ii) Er is acyclic; (i*
*ii) 9 P such that one of
< He*(Ep) = Hei(Ep)
Li
: He*(Eq) = Hej(Eq)
j
is Pprimary and the other is uniquely Pdivisible. To see this, recall that Er*
* Ep * Eq (cf. p. 432)
L L
and, on general grounds, eHk+1(Ep * Eq) eHi(Ep) eHj(Eq) Tor(eHi(E*
*p); eHj(Eq)). In
i+j=k i+j=k1
particular: Er acyclic ) 0 = eH1(Er) = eH0(Ep)He0(Eq), so at least one of Ep an*
*d Eq is path connected,
thus Ep * Eq is simply connected (cf. p. 340) or still, Er is contractible and*
* r is a pointed homotopy
equivalence. Therefore (i) and (ii) are equivalent. To check (ii) , (iii), use *
*the algebra developed above.
ae
EXAMPLE Let X be pointed connected CW spaces, f : X ! Y a pointed contin*
*uous function.
Y
Denote by Cssthe mapping cone of the pointed Hurewicz fibration ss : Ef ! X_the*
*n, specializing the
preceding8example,Lthe projection Css! Y is a pointed homotopy equivalence iff *
*9 P such that one of
< eH*(Ef) = Hei(Ef)
L i is Pprimary and the other is uniquely Pdivisible. To ill*
*ustrate the situation
: He*(Y ) = Hej(Y )
when P is thejset of all primes, consider the short exact sequence 0 ! Z ! Q ! *
*Q =Z ! 0_then
the mapping fiber of the arrow K(Z; n + 1) ! K(Q ; n + 1) is a K(Q =Z; n) (cf. *
*p. 529). Furthermore,
K(Q ; n + 1) = K(Q ; n) and eH*(Q ; n) is a uniquely divisible abelian group (b*
*eing a vector space over
Q ), while eH*(Q =Z; n) is a torsion abelian group (cf. p. 79). When P = ;, th*
*ere are two possibilities:
(1) eH*(Ef) = 0; (2) eH*(Y ) = 0. In the first case, f is acyclic and in the se*
*cond case, Y is contractible
568
and ss : Ef ! X is a pointed homotopy equivalence. Consequently, if ss1(Y ) 6= *
*0, then f is acyclic iff the
projection Css! Y is a pointed homotopy equivalence.
[Note: A priori, Cssis calculated in TOP but is viewed as an object in TOP*
* *. As such, it has the
same pointed homotopy type as the pointed mapping cone of ss.]
FACT Suppose that f : X ! Y is acyclic. Let Z be any pointed space_then th*
*e arrow [Y; Z] !
[X; Z] is injective.
[The orbits of the action of [Ef; Z] on [Css; Z] are the fibers of the arro*
*w [Css; Z] ! [X; Z] (cf.
p. 333). But Ef is contractible in TOP *, hence [Ef; Z] is the trivial group a*
*nd, as noted above, one
can replace Cssby Y .]
ae
PROPOSITION 20 Let XY be pointed connected CW spaces. Suppose that f :
X ! Y is a pointed continuous function with ss0(Ef) = *_then f is acyclic iff *
*f is a
homology equivalence and ss1(Y ) operates nilpotently on the Hq(Ef) 8 q.
Wf? ! Yfifi
[Consider the commutative diagram y fifiand apply the HiltonRoitb*
*erg
Y == Y
comparison theorem.]
EXAMPLE Take X = S3=SL (2; 5); Y = S3_then the arrow X ! Y is an acyclic *
*map (cf.
p. 552).
ae
FACT Let X be pointed connected CW spaces, f : X ! Y a pointed continuo*
*us function.
Y
Denote by Cf its mapping cone_then f acyclic ) Cf contractible and Cf contracti*
*ble ) f acyclic
provided that ss1(Y ) = 0.
[If Cf is contractible and Y is simply connected, then f is a homology equi*
*valence (cf. p. 322) and
ss1(Y ) operates trivially on the Hq(Ef) 8 q, so Proposition 20 can be cited.]
ae
FACT Let X be pointed connected CW spaces, f : X ! Y a pointed continuo*
*us function.
Y
Assume: X is acyclic and f* : ss1(X) ! ss1(Y ) is trivial_then f is nullhomotop*
*ic.
[Take X to be a pointed connected CW complex, consider a lifting ef: X ! eY*
*of f, and show that
eY! Cefis an acyclic map.]
[Note: It is a corollary that if X is acyclic and Hom(ss1(X; x0); ss1(Y; y0*
*)) = *, then C(X; x0; Y; y0)
is homotopically trivial.]
ae
Application: Let X & Y be pointed connected CW spaces. Suppose that f : *
*X ! Y & f0 :
Y 0
X ! Y 0are pointed continuous functions with f acyclic_then there exists a poin*
*ted continuous function
g : Y ! Y 0such that g O f ' f0 iff kerss1(f) kerss1(f0).
569
[Note: Up to pointed homotopy, g is unique.]
ae
PROPOSITION 21 Let XY be pointed connected CW spaces. Suppose that f :
X ! Y is a pointed continuous function with ss0(Ef) = *_then f is a pointed ho*
*mo
topy equivalence iff f is a homology equivalence and ss1(X) operates nilpotentl*
*y on the
ssq(Ef) 8 q.
[The stated condition on ss1(X) implies that ss1(Y ) operates nilpotently o*
*n the Hq(Ef)
8 q (cf. p. 562), thus, by Proposition 20, Ef is acyclic. But Ef is also nilpo*
*tent. Therefore
Ef is contractible and f : X ! Y is a pointed homotopy equivalence.]
It will be convenient to insert here a technical addendum to the fibration *
*spectral
sequence.
Notation: A continuous function f : X ! Y induces a functor f* : LCCS Y*
* !
LCCS X or still, a functor f* : [(Y )OP ; AB ] ! [(X)OP ; AB ] (cf. x4, Propo*
*sition 25).
If X is a subspace of Y and f is the inclusion, oneawriteseGX instead of f*G.
Let f : X ! Y be a Hurewicz fibration, where XY and the Xy are path conne*
*cted.
Fix a cofunctor G : Y ! AB _then 8 y 2 Y , the projection Xy ! Y is inessential*
*, hence
f*GXy is constant. So, 8 q 0, there is a cofunctor Hq(f; G) : Y ! AB that as*
*signs
to each y 2 Y the singular homology group Hq(Xy; f*GXy) and the fibration spe*
*ctral
sequence assumes the form E2p;q Hp(Y ; Hq(f; G)) ) Hp+q(X; f*G).
[Note: A morphism [o] : y0 ! y1 determines a homotopy equivalence Xy0 ! Xy1
(cf. p. 439) and an isomorphism G[o] : Gy1 ! Gy0, thus Hq(f; G)[o] is the co*
*mposite
Hq(Xy1; Gy1) ! Hq(Xy0; Gy1) ! Hq(Xy0; Gy0).]
ae
PROPOSITION 22 Let XY be pointed connected CW spaces, f : X ! Y a pointed
continuous function_then f is acyclic iff for every locally constant coefficien*
*t system G on
Y , the induced map f* : H*(X; f*G) ! H*(Y ; G) is an isomorphism.
[Upon passing to the mapping track, one can assume that f is a pointed Hure*
*wicz
fibration.
Necessity: 8 y 2 Y , Xy is acyclic, thus from the universal coefficient the*
*orem, 8 q > 0,
Hq(Xy; f*GXy) = 0. Accordingly, the edge homomorphism eH : E1p;0! E2p;0is an
isomorphism, so 8 p 0, Hp(X; f*G) Hp(Y ; G).
Sufficiency: The integral group ring Z[ss1(Y )] is a right ss1(Y )module. *
*Viewed as a
locally constant coefficient system on Y , its homology is that of eY. Form th*
*e pullback
570
f0
X xY?eY ! eY?
square y y _then H*(X xY eY) H*(X; f*(Z [ss1(Y )])) and f0*: H*(X*
* xY
X !f Y
Ye) ! H*(Ye) is the composite H*(X xY eY) ! H*(X; f*(Z [ss1(Y )])) f*!H*(Y ; Z[*
*ss1(Y )]) !
H*(Ye). By hypothesis, f* is an isomorphism, hence f0*is too. Since eY is sim*
*ply0con
X xY?eY f!*
* eY?
nected, Ef0 is path connected. Consider the commutative diagram f0y *
* y id.
eY ! *
* eY
id
Owing to the HiltonRoitberg comparison theorem, the projection Ef0! * is a hom*
*ology
equivalence. Therefore Ef is acyclic.]
ae
Application: Let X; Y; Z be pointed connected CW spaces. Suppose that fg:*
*:XY!!YZ
are pointed continuous functions. Assume: f is acyclic_then g is acyclic iff g *
*Of is acyclic.
ae
FACT Let X f Z g!Y be a pointed 2source, where X & Z are pointed connec*
*ted CW spaces.
g Y
Z? ! Y?
Consider the pushout square yf yj. Assume: f is a cofibration_then f (or *
*g) acyclic ) j (or
X ! P
) acyclic.
PLUS CONSTRUCTION Fix a pointed connected CW space X. Let N be a perfect
normal subgroup of ss1(X)_then there exists a pointed connected CW space X+N and
an acyclic map f+N : X ! X+N such that ker ss1(f+N) = N () ss1(X+N) ss1(X)=N).
Moreover, the pointed homotopy type of X+Nis unique, i.e., if g+N: X ! YN+ is a*
*cyclic and
if kerss1(g+N) = N, then there is a pointed homotopy equivalence OE : X+N! YN+ *
*such that
OE O f+N' g+N.
[Existence: We shall first deal with the case when N = ss1(X). Thus let {f*
*f} be a
` 2
set of generators for ss1(X). Represent ff by fff: S1 ! X and put X1 = ( D )*
* tf X
` ff
(f = fff) to obtain a relative CW complex (X1; X) with ss1(X1) = 0 (cf. p. 5*
*37).
Consider the exact sequence H2(X1) ! H2(X1; X) ! H1(X): (a) ss2(X1) H2(X1);
(b) H2(X1; X) is free abelian on generators wff, say; (c) H1(X) = 0. Given ff, *
*choose a
continuous function gff: S2 ! X1 such that the homotopy class [gff] maps to wff*
*under the
` 3 `
composite ss2(X1) ! H2(X1) ! H2(X1; X). Put X+N= ( D ) tg X1 (g = gff)_then
ff ff
the pair (X+N; X1) is a relative CW complex with ss1(X+N) = 0. The inclusion X *
*! X+Nis
a closed cofibration. In addition, it is a homology equivalence (for H*(X+N; X)*
* = 0), hence
571
is an acyclic map (cf. Proposition 20). Turning to the general case, let eXN be*
* the covering
space of X corresponding to N (so ss1(XeN) N). Apply the foregoing procedure t*
*o XeN
to get an acyclic closed cofibration ef+N: eXN ! eX+N, where eX+Nis simply conn*
*ected. Define
+
XeN efN!eX+N
? ? +
X+N by the pushout square y y . Thanks to Proposition 7, XN is a p*
*ointed
X ! X+N
f+N
connected CW space. And: f+N is an acyclic closed cofibration (cf. p. 570). Fi*
*nally, the
Van Kampen theorem implies thatass1(X+N)e ss1(X)=N.
+ )
Uniqueness: Since N = kerkss1(fNerss, t+here exists a pointed continuou*
*s function
1(gN ) ae
+
OE : X+N ! YN+ such that OE O f+N ' g+N (cf. p. 568). But fNg+acyclic ) OE a*
*cyclic and
N
OE* : ss1(X+N) ! ss1(YN+) is necessarily an isomorphism. Therefore OE is a poin*
*ted homotopy
equivalence (cf. Proposition 19).
[Note: X+N is called the plus_construction_with respect to N. Like an Eil*
*enberg
MacLane space, X+Nis really a pointed homotopy type, thus, while a given repres*
*entative
may have a certain property, it need not be true that all representatives do. A*
*s for OE, if
f+Nis an acyclic closed cofibration and if g+Nis another such, then matters can*
* be arranged
so that there is commutativity on the nose: OE O f+N= g+N. This in turn means t*
*hat OE is a
homotopy equivalence in X\TOP (cf. x3, Proposition 13).]
One can interpret X+Nas a representing object of the functor on the homotop*
*y category of pointed
connected CW spaces which assigns to each Y the set of all [f] 2 [X; Y ] : kers*
*s1(f) N.
Different notation is used when N = ss1(X)per, the perfect radical of ss1(X*
*) : X+N is
replaced by X+ and f+N : X ! X+N is replaced by i+ : X ! X+ . Example: X acyc*
*lic
) X+ contractible.
[Note: The perfectaradicaleof ss1(X+ ) is trivial (cf. p. 565).]
Examples: Let XY be pointed connected CW spaces_then (1) X+ x Y + is a mo*
*del
for (X x Y )+ ; (2) X+ _ Y + is a model for (X _ Y )+ ; (3) X+ #Y + is a model *
*for (X#Y )+ .
EXAMPLE (Homology_Spheres_)aeFix n > 1. Suppose that X is a pointed connec*
*ted CW space
such that eHq(X) = Z(q = n)_then ss1(X) is perfect and X+ has the same pointe*
*d homotopy type as
0 (q 6= n)
Sn.
FACT Let X be a pointed connected CW space_then for any pointed acyclic CW*
* space Z, the
arrow [Z; Ei+] ! [Z; X] is bijective.
572
[Note: The central extension 1 ! imss2(X+ ) ! ss1(Ei+) ! ss1(X)per! 1 is un*
*iversal.]
Convention: Henceforth it will be assumed that i+ : X ! X+ is an acyclic c*
*losed
cofibration.
ae
LEMMA Let XY be pointed connected CW spaces. Suppose that f : X ! Y is
a pointed continuous function_then there is a pointed continuous function f+ :*
* X+ !
X? f! Y?
Y + rendering the diagram y y commutative, f+ being unique up to po*
*inted
X+ ! Y +
f+
homotopy.
ae
Application: Let XY be pointed connected CW spaces. Assume: X and Y have *
*the
same pointed homotopy type_then X+ and Y + have the same pointed homotopy type.
PROPOSITION 23 Let X be a pointed connected CW space. Denote by XeN the
covering space of X corresponding to N, where N is a normal subgroup of ss1(X) *
*containing
ss1(X)per_then Xe+Nhas the same pointed homotopy type as the covering space of *
*X+
corresponding to the normal subgroup N=ss1(X)perof ss1(X+ ) ss1(X)=ss1(X)per.
[The pointed homotopy type of XeN can be calculated as the mapping fiber of*
* the
composite X ! X[1] = K(ss1(X); 1) ! K(ss1(X)=N; 1). This arrow factors through *
*X+
and ss1(X)=N (ss1(X)=ss1(X)per)=(N=ss1(X)per).]
Notation: Given a group G, put BG = K(G; 1).
EXAMPLE BGperis the covering space of BG corresponding to Gper. There is a*
*n arrow BG+per!
BG+ and BG+per"is" the universal covering space of BG+.
EXAMPLE Let A be a ring with unit_then the fundamental group of the mappin*
*g fiber of
BGL (A) ! BGL (A)+ is isomorphic to ST(A).
ae
PROPOSITION 24 Let XY be pointed connected CW spaces. Suppose that f :
X ! Y is a pointed continuous function with ss0(Ef) = *_then ss0(Ef+) = * and *
*the
perfect radical of ss1(Ef+) is trivial.
Ef ______wE+f
[Note: It follows that there is a commutative triangle u [[[^ .]
Ef+
573
ae
FACT Let X be pointed connected CW spaces. Suppose that f : X ! Y is a p*
*ointed continuous
Y
function with ss0(Ef) = *_then the arrow E+f! Ef+ is a pointed homotopy equival*
*ence if ss1(Y )peris
trivial or if E+fis nilpotent and ss1(Y )peroperates nilpotently on the Hq(Ef) *
*8 q.
[Note: ss1(Y )peroperates nilpotently on the Hq(Ef) 8 q iff ss1(Y )peroper*
*ates trivially on the
Hq(Ef) 8 q (cf. p. 564).]
EXAMPLE (Central_Extensions_) Let ss and G be groups, where ss is abelian*
*. Consider a central
extension 1 ! ss ! ! G ! 1_then Bss can be identified with the mapping fiber o*
*f the arrow
B+ ! BG+.
[Since ss is abelian, Bss = Bss+ and G (= ss1(BG)) operates trivially on ss*
*, hence operates trivially
on the Hq(Bss) 8 q.]
EXAMPLE Let G be an abelian group_then there is a universal central extens*
*ion 1 ! G !
ffG ! fiG ! 1 (cf. p. 565). Specializing the preceding example, the mapping *
*fiber of the arrow
K(ffG; 1)+ ! K(fiG; 1)+ is a K(G; 1) and K(fiG; 1)+ is a K(G; 2).
[Recall that ffG is acyclic, thus K(ffG; 1)+ is contractible.]
ae
PROPOSITION 25 Let XY be pointed connected CW spaces. Suppose that f :
X ! Y is a pointed continuous function for which the normal closure of f*(ss1(X*
*)per) is
ss1(Y )per_then the adjunction space X+ tf Y represents Y +.
[Since i+ : X ! X+ is an acyclic closed cofibration, the same is true of th*
*e inclusion
Y ! X+ tf Y (cf. p. 570). On the other hand, by Van Kampen, the fundamental gr*
*oup
of X+ tf Y is isomorphic to ss1(Y ) modulo the normal closure of f*(ss1(X)per)*
*, i.e., to
ss1(Y )=ss1(Y )per.]
EXAMPLE (Algebraic_KTheory_) Let A be a ring with unit_then by definitio*
*n, K0(A) is the
Grothendieck group attached to the category of finitely generated projective A*
*modules and for n 1,
Kn(A) is taken to be the homotopy group ssn(BGL (A)+). While it is immediate th*
*at K0 is a functor
from RG to AB , the plus construction requires some choices, so to guarantee t*
*hat Kn is a functor one
has to fix the data. Thus first construct BGL (Z)+. This done, define BGL (A)+ *
*by the pushout square
BGL (Z) ! BGL (A)
?y ?y
. Here, Proposition 25 comes in (the normal closure of *
*im(E (Z) ! E(A))
BGL (Z)+ ! BGL (A)+
is E(A)). Observe that the Kn preserve products: Kn(A0x A00) Kn(A0) x Kn(A00).
(n = 1) K1(A) = ss1(BGL (A)+) ss1(BGL (A))=ss1(BGL (A))per GL (A)=[GL*
* (A); GL(A)]
= H1(GL (A)).
(n = 2) K2(A) = ss2(BGL (A)+) ss2(BE (A)+) H2(BE (A)+) H2(BE (A)) =*
* H2(E (A)).
574
[Note: The central extension 1 ! K2(A) ! ST (A) ! E(A) ! 1 is universal (cf*
*. p. 565) and
BK2(A) can be identified with the mapping fiber of the arrow B ST(A)+ ! BE (A)+*
*.]
(n = 3) K3(A) = ss3(BGL (A)+) ss3(BE (A)+) ss3(B ST(A)+) H3(B ST(A)*
*+)
H3(B ST(A)) = H3(ST (A)).
There is no known homological interpretation of K4 and beyond.
EXAMPLE (Relative_Algebraic_KTheory_) Let A be a ring with unit, I A a *
*two sided ideal.
Write cGL(A=I) for the image of GL (A) in GL (A=I)_then cGL(A=I) E(A=I), thus *
*cGL(A=I) is normal
and GA;I = GL (A=I)=GcL(A=I) is abelian. Since BGcL(A=I)+ can be identified wi*
*th the mapping
fiber of the arrow BGL (A=I)+ ! BG+A;I(= BGA;I) (cf. p. 573), it follows that *
*ssn(BGcL(A=I)+)
ssn(BGL (A=I)+) (n > 1) but ss1(BGcL(A=I)+) im(K1(A) ! K1(A=I)) and there is a*
* short exact
sequence 0 ! ss1(BGcL(A=I)+) ! K1(A=I) ! GA;I! 0. If K(A; I) is the mapping fib*
*er of the arrow
BGL (A)+ ! BGcL(A=I)+, then K(A; I) is path connected, so letting Kn(A; I) = ss*
*n(K(A; I)) (n 1),
one obtains a functorial long exact sequence . .!.Kn+1(A=I) ! Kn(A; I) ! Kn(A) *
*! Kn(A=I) !
. .!.K1(A; I) ! K1(A) ! K1(A=I).
PROPOSITION 26 Let X be a pointed connected CW space. Put ss = ss1(X) and
denote by eXperthe mapping fiber of the composite X ! K(ss; 1) ! K(ss=ssper; 1)*
*. Assume:
ss=ssper is nilpotent and ss=ssper operates nilpotently on the Hq(Xeper) 8 q_th*
*en X+ is
nilpotent.
[Since (ss=ssper)peris trivial (cf. p. 565), eX+percan be identified with *
*the mapping fiber
of the composite X+ ! K(ss; 1)+ ! K(ss=ssper; 1)+ (cf. p. 573). By constructio*
*n, eX+peris
simply connected (cf. Proposition 23), hence nilpotent. But K(ss=ssper; 1)+ = K*
*(ss=ssper; 1)
is also nilpotent. Therefore, bearing in mind that the inclusion eXper! eX+peri*
*s a homology
equivalence, it follows that X+ is nilpotent (cf. p. 562).]
FACT Let G be a group. Fix OE 2 AutG. Assume: Given g1; : :;:gn 2 G, 9 g 2*
* G : OE(gi) = ggig1
(1 i n)_then OE* : H*(G) ! H*(G) is the identity.
Application: Let G be a group. Let K be a normal subgroup of G which is the*
* colimit of subgroups
Kn (n 2 N) such that 8 n, G = K . CenG(Kn)_then G operates trivially on H*(K).
EXAMPLE Let A be a ring with unit_then BGL (A)+ is nilpotent. To see this,*
* consider the
short exact sequence 1 ! E(A) ! GL (A) ! GL (A)=E (A) ! 1. Here, E(A) = GL (A)p*
*erand BE (A) is
the mapping fiber of the arrow BGL (A) ! K(GL (A)=E (A); 1). The quotient GL (A*
*)=E (A) is abelian,
hence nilpotent. On the other hand, if E(n; A) is the subgroup of GL (n; A) con*
*sisting of the elementary
matrices, then E(A) = colimE(n; A) and 8 n; GL(A) = E(A) . CenGL(A)(E (n; A)), *
*so GL (A) operates
trivially on H*(E (A)). That BGL (A)+ is nilpotent is therefore a consequence o*
*f Proposition 26.
575
[Note: More is true.aThusedefine a homomorphism : GL (A) x GL(A) ! GL (A) *
*by (X; Y ) !
X Y , where (X Y )ij= xkl (i = 2k  1; j = 2l&01)otherwise_then Lodayy has*
* shown that
ykl (i = 2k; j = 2l)
the composite BGL (A)+ x BGL (A)+ ! B(GL (A) x GL(A))+ ! BGL (A)+ serves to equ*
*ip BGL (A)+
with the structure of a homotopy commutative H group. In particular: BGL (A)+ i*
*s abelian.]
EXAMPLE Let A be a ring with unit. Write UT (A) for the ring of upper tri*
*angular 2by2
matrices with entries in A_then the projection p : UT (A) ! A x A (p a1 a =*
* (a1; a2)) induces an
0 a2
epimorphism p : GL (UT (A)) ! GL (A x A). Its kernel is not perfect, therefore *
*Bp : BGL (UT (A)) !
BGL (AxA) is not acyclic. Nevertheless, Bp is a homology equivalence. Consider *
*now the commutative di
BGL (UT (A)) ! BGL (UT (A))+
? ?
agram Bpy yBp+ . Since the horizontal arrows are homolog*
*y equivalences,
BGL (A x A) ! BGL (A x A)+
Bp+ is a pointed homotopy equivalence, so 8 n 1, Kn(UT (A)) Kn(A) x Kn(A).
[Note: Bp+ is acyclic (cf. Proposition 19), thus the composite BGL (UT (A))*
* Bp!BGL (A x A) !
BGL (A x A)+ is acyclic even though Bp is not.]
FACT Let G be a group. Assume:
ae () aThereeis a homomorphism : G x G ! G such that for any finite set *
*{g1; : :;:gn} G,
1 = gi
9 u 2 G : u(gi e)u (i = 1; : :;:n).
v v(e gi)v1 = gi
(") There is a homomorphism " : G ! G such that for any finite set {g*
*1; : :;:gn} G,
9 ae 2 G : ae(gi "gi)ae1 = gi(i = 1; : :;:n).
Then G is acyclic.
[Fix a field of coefficients k. Let : G ! G x G be the diagonal map_then "*
* and O (idx ") O
operate in the same way on homology. Since H1(G; k) = 0, one can take n > 1 and*
* assume inductively
that Hq(G; k) = 0 (0 < q < n). Let x 2 Hn(G; k) : "*(x) = (O(idx")O)*(x) = *(x1*
*+1"*(x)) =
x + "*(x) ) x = 0.]
EXAMPLE (Delooping_Algebraic_KTheory_) Let A be a ring with unit. Denote*
* by A the set
of all functions X : NxN ! A such that 8 i; #{j : Xij6= 0} < ! and 8 j; #{i : X*
*ij6= 0} < !_then A is
a ring with unit containing __Aas a two sided ideal. A is called the cone_of A *
*and the quotient A = A=__A
is called theasuspension_ofeA. Define a homomorphism : A x A ! A by (X; Y ) ! *
*X Y , where
(X Y )ij= xkl (i = 2k  1; j = 2l&01)otherwise and define a homomorphism " *
*: A ! A
ykl (ia=e2k; j = 2l)
k(2m  1)
by "(X)ij= Xmn if i = 2 for some k; m; n & 0 otherwise. Evidently, X "*
*X = "X for all
j = 2k(2n  1)
_________________________
yAnn. Sci. Ecole Norm. Sup. 9 (1976), 309377.
576
ae
X 2 A and induce homomorphisms : GL (A) x GL(A) ! GL (A) & " : GL (A) ! GL*
* (A)
"
satisfying the preceding assumptions. Therefore GL (A) is acyclic, so GL (A) = *
*E(A). Taking into
account the exact sequences 1 ! ___GL(__A) ! ___GL(A) ! ___GL(A); __E(A) ! __E(*
*A) ! 1, it follows that
there is an exact sequence 1 ! GL (A) ! GL (A) ! E(A) ! 1. The mapping fiber o*
*f the arrow
BGL (A)+ ! BE (A)+ is BGL (A)+. Since BGL (A)+ is contractible, this means that*
* in HTOP *,
BGL (A)+ BE (A)+. Consequently, 8 n 1; Kn(A) = ssn(BGL (A)+) ssn(BE (A)+)
ssn+1(BE (A)+) ssn+1(BGL (A)+) = Kn+1(A). It is also true that K0(A) K1(A) (*
*Farrell
Wagonery). Let 0BGL (A)+ be the path component of BGL (A)+ containing the const*
*ant loop_
then in HTOP *, BE (A)+ 0BGL (A)+ (cf. p. 572). But ss1(BGL (A)+) = K1(A), h*
*ence
K0(A) x BGL (A)+ BGL (A)+.
[Note: Additional information can be found in Wagonerz. There it is shown *
*that by fixing the
data, the pointed homotopy equivalence K0(A) x BGL (A)+ BGL (A)+ can be made n*
*atural, i.e.,
K0(A0) x BGL (A0)+ BG*
*L (A0)+
? *
* ?
if f : A0! A00is a morphism of rings, then the diagram y *
* y is
K0(A00) x BGL (A00)+ BG*
*L (A00)+
pointed homotopy commutative.]
EXAMPLE Let A be a ring with unit_then UT (A) UT (A) ) K0(UT (A)) K1(UT *
*(A))
K1(UT (A)) K1(A) x K1(A) K0(A) x K0(A).
KANTHURSTON THEOREM Let X be a pointed connected CW space_then there
exists a group GX and an acyclic map X : K(GX ; 1) ! X.
[Because of Proposition 2, one can take for X a pointed connected CW complex
with all characteristic maps embeddings. Moreover, it will be enough to deal wi*
*th finite
X, the transition to infinite X being straightforward (given the naturality bui*
*lt into the
argument). Since dim X 1 ) X aspherical, we shall assume that dim X > 1 and pr*
*oceed
by induction on #(E), supposing that the construction has been carried out in s*
*uch a way
that if X0 is a connected subcomplex of X, then K(GX0; 1) = 1X(X0) and GX0 ! G*
*X is
Sn1? *
*! X?
injective. To execute the inductive step, consider the pushout square y *
* y (n
ae D n *
*! Y
n1! X)
2), where the horizontal arrows are embeddings and X0Y= im(S n are co*
*nnected
0 = im(D ! Y )
_________________________
yComment. Math. Helv. 47 (1972), 474501.
zTopology 11 (1972), 349370.
577
ae X0? ! X?
subcomplexes of XY, so y y is a pushout square. Recalling that the*
*re is a
Y0 ! Y
monomorphism GX0 ! GX0 of groups (cf. p. 564), define GY by the pushout square
GX0? ! GX K(GX0; 1) ! K(GX*
* ; 1)
y ?y and realize K(GY ; 1) by the pushout square ?y *
*?y
GX0 ! GY K(GX0; 1) ! K(GY*
* ; 1)
(cf. p. 528). Extend X : K(GX ; 1) ! X to Y : K(GY ; 1) ! Y in the obvious
K(GX?; 1) X! X?
way (thus Y K(GX0; 1) Y0 and the diagram y y commutes). The
K(GY ; 1) ! Y
Y
induction hypothesis implies that X and X0 are acyclic. In addition, K(GX0; 1)*
* is an
acyclic space and Y0 is contractible, hence Y K(GX0; 1) is acyclic (cf. Propos*
*ition 20).
Therefore, by comparing MayerVietoris sequences and applying the five lemma, i*
*t follows
that Y is acyclic (cf. Proposition 22). Finally, the condition on connected sub*
*complexes
passes on to Y .]
[Note: Put N = kerss1(X )_then X is a model for K(GX ; 1)+N.]
Application: Every nonempty path connected topological space has the homolo*
*gy of
a K(G; 1).
EXAMPLE Suppose given two sequences ssn (n 2) & Gq (q 1) of abelian grou*
*ps_then there
exists a pointed connected CW space Z such that 8 n 2 : ssn(Z) ssn & 8 q 11 :*
* Hq(Z) Gq.
W
Thus choose X : ssn+1(X) ssn (n 2) (homotopy system theorem) and put Y = M(*
*Gq; q) (cf.
ae 1
p. 538): Hq(Y ) Gq (q 1). Using KanThurston, form X : K(GX ; 1) ! Xand co*
*nsider Z =
Y : K(GY ; 1) ! Y
EX xK(GY ; 1), the mapping fiber of the arrow K(GX xGY ; 1) = K(GX ; 1)xK(GY ;*
* 1) ! X. Example:
If Gq (q 1) is any sequence of abelian groups, then there exists a group G suc*
*h that 8 q 1 : Hq(G) Gq.
[Note: Z also has the property that ss1(Z) operates trivially on ssn(Z) 8 n*
* 2.]
The homotopy categories of algebraic topology are not complete (or cocomple*
*te),
a circumstance that precludes application of the representable functor theorem *
*and the
general adjoint functor theorem (or their duals). However, there is still a cer*
*tain amount
of structure. For instance, consider HTOP . It has products and the double *
*mapping
track furnishes weak pullbacks. Therefore HTOP is weakly complete, i.e., ever*
*y diagram
: I! HTOP has a weak limit (meaning: "existence without uniqueness"). HTOP *
* is
also weakly cocomplete. In fact, HTOP has coproducts, while weak pushouts are*
* furnished
578
by the double mapping cylinder. Example: Let (X ; f) be an object in FIL (HTOP *
* )_then
tel(X ; f) is a weak colimit of (X ; f).
[Note: The discussion of HTOP *is analogous. Example: Let f : X ! Y be a *
*pointed
continuous function, Cf its pointed mapping cone_then Cf is a weak cokernel of *
*[f].]
EXAMPLE For each n, put Yn = S3and let [ n] : Yn+1 ! Yn be the homotopy cl*
*ass of maps
of degree 2_then Y = limYn does not exist in HTOP . To see this, assume the c*
*ontrary, thus 8 X,
[X; Y ] lim[X; Yn], so, in particular, Y must be 3connected. Form the adjunct*
*ion space D3tfS 2, where
f : S2! S2is skeletal of degree 3. Since dim(D 3tfS2) 3, of necessity [D 3tfS2*
*; Y ] = *. But according
to the Hopf classification theorem, [D 3tf S2; S3] H3(D 3tf S2; Z), which is Z*
*=3Z, and in the limit,
[D 3tf S2; Y ] Z=3Z.
EXAMPLE Working in HTOP *, let f : X ! Y be a pointed Hurewicz fibration,*
* where X and
Y are path connected. Suppose that K = ker[f] exists, say [] : K ! X. If ss is *
*the projection Ef ! X,
then f O ss ' 0, so there exists a pointed continuous function OE : Ef ! K such*
* that O OE ' ss and, by
construction, f O ' 0, so there exists a pointed continuous function : K ! E*
*f such that ' ss O .
Thus O OE O ' ) OE O ' idK; [] being a monomorphism in HTOP *. Take now*
* X = SO (3),
Y = SO (3)=SO (2), and let f : X ! Y be the canonical map_then ss1(Ef) Z, ss1(*
*K) Z=2Z and
Z=2Z is not a direct summand of Z.
[Note: Similar examples show that cokernels do not exist in HTOP *.]
Let C be a category with products and weak pullbacks_then every diagram in*
* C
has a weak limit. Any functor F : C ! SET that preserves products and weak pul*
*lbacks
necessarily preserves weak limits.
PROPOSITION 27 Let C be a category with products and weak pullbacks. Assum*
*e:
Ob C contains a set U = {U} with the following properties.
(U1) A morphism f : X ! Y is an isomorphism provided that 8 U 2 U, the
arrow Mor (Y; U) ! Mor (X; U) is bijective.
(U2) Each object (X ; f) in TOW (C ) has a weak limit X1 such that*
* 8 U 2 U,
the arrow colimMor (Xn; U) ! Mor (X1 ; U) is bijective.
Then a functor F : C ! SET is representable iff it preserves products an*
*d weak
pullbacks.
[The condition is certainly necessary. As for the sufficiency, introduce t*
*he comma
category *; F . Recall that an object of *; F  is a pair (x; X) (x 2 F X; X*
* 2 ObC ), while
a morphism (x; X) ! (y; Y ) is an arrow f : X ! Y such that (F f)x = y. The ass*
*umptions
imply that *; F  has products and weak pullbacks, hence is weakly complete, a*
*nd F is
579
representable iff *; F  has an initial object. Let UF be the subset of Ob*; *
*F  consisting of
the pairs (u; U) (u 2 F U; U 2 U).
__ __ __
Claim: 8 (x; X) 2 Ob*; F  9 (__x; X) 2 Ob*; F  and a morphism (x; X) ! *
*(x; X) such
__
that 8 (u; U) 2 UF there is a unique morphism (__x; X) ! (u; U).
*
* Q
[Define an object (X ; f) in TOW (*; F ) by setting (x0; X0) = (x; X) x*
* (u; U) and
inductively choose (xn+1; Xn+1) ! (xn; Xn) to equalize all pairs of morphisms (*
*xn; Xn)!!
*
* __
(u; U) ((u; U) 2 UF ). Any weak limit of (X ; f) created via U2 is a candidate *
*for (__x; X).]
The existence of an initial object in *; F  is then a consequence of obse*
*rving that
__ __ __
for all (x; X) & (y; Y ): (i) Every morphism (__x; X) ! (y; Y) is an isomorphi*
*sm (apply
__
the_claim_and_U1); (ii) There is at least one morphism (__x; X) ! (y; Y ) (the *
*composite
__ __ __ __ __
(__x; X) x (y;!Y()x; X) x (y; Y ) ! (x; X) is an isomorphism); (iii) There is a*
*t most one
__ __ __
morphism (__x; X) ! (y; Y ) (form the equalizer (z; Z) of (x; X)!!(y; Y ) and c*
*onsider the
__ __ __
composite (__z; Z) ! (z; Z) ! (x; X)).]
[Note: Proposition 27 can also be formulated in terms of a category C tha*
*t has
coproducts and weak pushouts together with a set U = {U} of objects satisfying*
* the
following conditions.
(U1) A morphism f : X ! Y is an isomorphism provided that 8 U 2 U, the
arrow Mor (U; X) ! Mor (U; Y ) is bijective.
(U2) Each object (X ; f) in FIL (C ) has a weak colimit X1 such that*
* 8 U 2 U,
the arrow colimMor (U; Xn) ! Mor (U; X1 ) is bijective.
Under these hypotheses, the conclusion is that a cofunctor F : C ! SET is*
* repre
sentable iff it converts coproducts into products and weak pushouts into weak p*
*ullbacks.]
EXAMPLE Let C be a category with coproducts and weak pushouts whose repres*
*entable cofunc
tors are precisely those that convert coproducts into products and weak pushout*
*s into weak pullbacks.
Suppose that T = (T; m; ffl) is an idempotent triple in C and let S MorC be t*
*he class consisting
of those f such that Tf is an isomorphism_then (1) S admits a calculus of left *
*fractions; (2) S is
saturated; (3) S satisfies the solution set condition; (4) S is coproduct close*
*d, i.e., si : Xi ! Yi in
` ` `
S 8 i 2 I ) si : Xi ! Yi in S. Conversely, any class S MorC with proper*
*ties (1)(4) is
i i i
generated by an idempotent triple, thus S? is the object class of a reflective *
*subcategory of C.
[The functor LS : C ! S1C preserves coproducts and weak pushouts. So, for *
*fixed Y 2 ObS1C ,
Mor (LS_ ; Y ) is a cofunctor C ! SET which converts coproducts into products a*
*nd weak pushouts into
weak pullbacks, hence is representable: Mor (LSX; Y ) Mor(X; YS). Use the assi*
*gnment Y ! YS to
define a functor S1C ! C and take for T the composite C ! S1C ! C. Let fflX*
* 2 Mor(X; TX)
correspond to idLSXunder the bijection Mor(LSX; LSX) Mor(X; TX)_then ffl : idC*
*! T is a natural
580
transformation, fflT = Tffl is a natural isomorphism, and Tf is an isomorphism *
*iff f 2 S.]
Notation: CONCW *is the full subcategory of CW * whose objects are the*
* pointed
connected CW complexes and HCONCW * is the associated homotopy category.
LEMMA HCONCW *has coproducts and weak pushouts.
[If X f Z g!Y is a 2source in CONCW *, then using the skeletal approx*
*imation
theorem, one can always arrange that Mf;gremains in CONCW *.]
BROWN REPRESENTABILITY THEOREM A cofunctor F : HCONCW * ! SET
is representable iff it converts coproducts into products and weak pushouts int*
*o weak pull
backs.
[Take for U the set {(S n; sn) : n 2 N }_then U1 holds since in CONCW * *
*a pointed
continuous function f : X ! Y is a pointed homotopy equivalence iff it is a wea*
*k homotopy
equivalence (cf. p. 517) and U2 holds since one can take for a weak colimit of*
* an object
(X ; f) in FIL (HCONCW *) the pointed mapping telescope constructed using *
*pointed
skeletal maps (cf. p. 525).]
[Note: Since F converts coproducts into products, F takes an initial obj*
*ect to a
terminal object: F * = * and X ! * ) * = F * ! F X, thus F X has a natural base*
* point.]
Spelled out, here are the conditions on F figuring in the Brown representab*
*ility the
orem.
W
(Wedge Condition) For any collection {Xi : i 2 I} in CONCW *; F ( *
* Xi)
Q i
F Xi.
i g
Z? ! Y?
(MayerVietoris Condition) For any weak pushout square fy yj *
* in
X ! P
F?P Fj!F?Y ae
HCONCW *, F y y Fg is a weak pullback square in SET , so 8 xy2*
*2FFXY :
F X !FfF Z
ae
(F f)x = (F g)y, 9 p 2 F P : (F()pF=jx)p.= y
[Note: It is not necessary to make the verification for an arbitrary weak *
*pushout
square. In fact, it is sufficient to consider pointed double mapping cylinders*
* calculated
relative to skeletal maps, thus it is actually enough to consider diagrams of t*
*he form
581
C? ! B ae
y ?y, where X is a pointed connected CW complex and A & C are pointed
B
A ! X
connected subcomplexes such that X = A [ B; C = A \ B.]
Examples: (1) Fix a pointed path connected space (X; x0)_then [_ ; X; x0] *
*is a
cofunctor on HCONCW * satisfying the wedge and MayerVietoris conditions, *
*hence
there exists a pointed connected CW complex (K; k0) and a natural isomorphism :
[_ ; K; k0] ! [_ ; X; x0], each f 2 K;k0([idK ]) being a weak homotopy equivale*
*nce K !
X, thus the Brown representability theorem implies the resolution theorem; (2) *
*Fix n 2 N
and an abelian group ss_then the cofunctor Hn (_ ; ss) (singular cohomology) sa*
*tisfies the
wedge and MayerVietoris conditions, hence there exists a pointed connected CW *
*com
plex (K(ss; n); kss;n) and a natural isomorphism : [_ ; K(ss; n); kss;n] ! Hn *
*(_ ; ss), thus
the Brown representability theorem implies the existence of EilenbergMacLane s*
*paces of
type (ss; n) (ss abelian); (3) Fix a group ss_then the cofunctor that assigns t*
*o a pointed
connected CW complex (K; k0) the set of homomorphisms ss1(K; k0) ! ss satisfies*
* the
wedge and MayerVietoris conditions, hence there exists a pointed connected CW *
*complex
(K(ss; 1); kss;1) and a natural isomorphism : [_ ; K(ss; 1); kss;1] ! Hom (ss1*
*_ ; ss), thus the
Brown representability theorem implies the existence of EilenbergMacLane space*
*s of type
(ss; 1) (ss arbitrary).
[Note: Both HCW * and HCW have coproducts and weak pushouts but Brown
representability can fail. Indeed, Matveevy has given an example of a nonrepre*
*sentable
cofunctor F : HCW *! SET which converts coproducts into products and weak pu*
*shouts
into weak pullbacks and Hellerz has given an example of a nonrepresentable cofu*
*nctor
F : HCW ! SET which converts coproducts into products and weak pushouts into*
* weak
pullbacks.]
EXAMPLE Let U : gr! SET be the forgetful functor.
(HCW *) Suppose that F : HCW *! gris a cofunctor such that U O F co*
*nverts coproducts
into products and weak pushouts into weak pullbacks_then U O F is representable.
[Represent the composite HCONCW * ! HCW *! gr! SET by K. Put G = FS 0and*
* equip it
with the discrete topology.
Claim: For any X in CONCW *, U O F(X+) [X+; K x G].
[There is a split short exact sequence 1 ! FX ! FX+ ! FS 0! 1, hence U O F(*
*X+) U O F(X) x
_________________________
yMath. Notes 39 (1986), 471474.
zJ. London Math. Soc. 23 (1981), 551562.
582
G [X; K] x G or, reinstating the base points: U O F(X+) [X; x0; K; k0] x G. A*
*nd: [X; x0; K; k0]
[X; K] ) [X; x0; K; k0] x G [X; K] x G [X; K] x [X; G] [X; K x G] [X+; K x *
*G].]
Given (X; x0) in CW *, let Xi0; Xi (i 2 I) be its set of path components, *
*where x0 2 Xi0_
W Q Q
then X = Xi0_ Xi+, so U O F(X) U O F(Xi0) x U O F(Xi+) [Xi0; K] x [Xi+;*
* K x G]
iQ i i
[Xi0; K x G] x [Xi+; K x G] [X; K x G].]
i
(HCW) Suppose that F : HCW ! gris a cofunctor such that U O F conver*
*ts coproducts
into products and weak pushouts into weak pullbacks_then U O F is representable.
[Let F* be the composite HCONCW * ! HCONCW ! HCW ! gr! SET .
Claim: If F* = *, then F* is representable.
[The assumption on F implies that FA = * for any discrete topological space*
* A. To check that
`
F* satisfies the wedge condition, put X = Xi and let A X be the set made up *
*of the base points
W iW Q
xi2 Xi_then F(X=A) FX. But X=A = Xi) F*( Xi) U O F(X) F*Xi. As F* neces*
*sarily
i i i
satisfies the MayerVietoris condition, F* is representable: [_ ; K*] F*.]
Claim: If F* = *, then U O F is representable.
`
[If X is in CW and if X = Xi is its decomposition into path components, *
*then U O F(X)
Q Q Q i`
U O F(Xi) F*Xi [Xi; K*] [ Xi; K*] [X; K*].]
i i i i
Given X in CW , view ss0(X) as a discrete topological space_then U O F O s*
*s0 is represented by
F* (discrete topology). On the other hand, F is the semidirect product of F O *
*ss0 and the kernel F0
of F ! F O ss0 induced by the embedding ss0(X) ! X. Moreover, U O F U O F0 x U*
* O F O ss0 and
F0* = * ) U O F0 is representable.]
Given a small, full subcategory C0 of HCW *, denote by __C0the full subcat*
*egory of HCW *whose
objects are those Y such that g : Y ! Z is an isomorphism (= pointed homotopy e*
*quivalence) if g* :
[X0; Y ] ! [X0; Z] is bijective for all X0 2 ObC 0.
FACT Suppose that F : HCW *! SET is a cofunctor which converts coproducts*
* into products
and weak pushouts into weak pullbacks_then there exists an object XF in HCW *a*
*nd a natural trans
formation : [_ ; XF ] ! F such that 8 X0 2 ObC 0, X0 : [X0; XF ] ! FX0 is bije*
*ctive.
FACT Suppose that F : HCW *! SET is a cofunctor which converts coproducts*
* into products
and weak pushouts into weak pullbacks_then F is representable if for some C0; X*
*F 2 Ob__C0.
[With as above, put xF = XF ([idXF]), so that 8 X 2 ObHCW *, X ([f]) = F[*
*f]xF ([f] 2
[X; XF ]).
Surjectivity: Given X 2 ObHCW *, call C00the full subcategory of HCW *obt*
*ained by adding X
and XF to C0. Determine X0Fand 0: [_ ; X0F] ! F accordingly. In particular, 0XF*
*: [XF ; X0F] ! FXF
is surjective, thus 9 [f] 2 [XF ; X0F] : xF = F[f]x0F. From the definitions, 8 *
*X0 2 ObC 0, f* : [X0; XF ] !
583
[X0; X0F] is bijective. Therefore f is an isomorphism. Let x 2 FX and choose *
*[g] 2 [X; X0F] : 0X
([g]) = x_then X ([f1] O [g]) = F([f1] O [g])xF = F[g](F[f1]xF ) = F[g]x0F= *
*x.
Injectivity: Given X 2 ObHCW *, let u; v : X ! XF be a pair of morphisms: *
*X ([u]) = X ([v]),
i.e., F[u]xF = F[v]xF . Fix a weak coequalizer f : XF ! Z of u, v and choose z *
*2 FZ : F[f]z = xF . Since
Z : [Z; XF ] ! FZ is surjective, 9 g : Z ! XF such that Z([g]) = z, hence xF = *
*F[g O f]xF . From the
definitions, 8 X0 2 ObC 0, (g O f)* : [X0; XF ] ! [X0; XF ] is bijective. There*
*fore g O f is an isomorphism.
Finally, f O u ' f O v ) g O f O u ' g O f O v ) u ' v.]
Application: Let C 0be the full subcategory of HCW * consisting of the (S*
*n; sn) (n 0), so
__C
0= HCONCW *_then a cofunctor F : HCW *! SET which converts coproducts into*
* products and
weak pushouts into weak pullbacks is representable provided that #(FS 0) = 1.
[In fact, ss0(XF ) = [S0; XF ] = FS 0, thus XF is connected.]
EXAMPLE Fix a nonempty topological space F. Given a CW complex B, let kF B*
* be the set
Ob ____FIBB;F, where ____FIBB;Fis the skeleton of FIBB;F(cf. p. 428)_then kF i*
*s a cofunctor HCW ! SET
which converts coproducts into products and weak pushouts into weak pullbacks (*
*cf. p. 419). However,
kF is not automatically representable since Brown representability can fail in *
*HCW . To get around this
difficulty, one employs a subterfuge. Thus given a pointed CW complex (B; b0),*
* let FIBB;F;*be the
category whose objects are the pairs (p; i), where p : X ! B is a Hurewicz fibr*
*ation such that 8 b 2 B,
Xb has the homotopy type of F and i : F ! p1(b0) is a homotopy equivalence, an*
*d whose morphisms
(p; i) ! (q; j) are the fiber homotopy classes [f] : X ! Y and the homotopy cla*
*sses [OE] : F ! F such
that fb0O i ' j O OE. As in the unpointed case, FIBB;F;*has a small skeleton an*
*d there is a cofunctor
kF;*: HCW *! SET which converts coproducts into products and weak pushouts int*
*o weak pullbacks.
Since #(kF;*S0) = 1, it follows from the above that kF;*is representable: [_ ; *
*BF ; bF ] kF;*, (BF ; bF ) a
pointed connected CW complex. If now B is a CW complex, then the functor FIBB;F*
* ! FIBB+;F;*that
assigns to p : X ! B the pair (p q c; idF) (c : F ! *) induces a bijection Ob__*
*__FIBB;F! Ob____FIBB+;F;*,
so kF B kF;*B+ [B+; *; BF ; bF ] [B; BF ], i.e., BF represents kF . Example:*
* Take F = K(ss; n) (ss
abelian)_then BF has the same pointed homotopy type as K(ss; n + 1; Oss) (cf. p*
*. 532) (K(ss; n + 1; Oss)
is not necessarily a CW complex).
Example: Consider the Hurewicz fibration p1 : Sn ! Sn (n 2). Let i : Sn *
*! Sn be
the identity and : Sn ! Sn the inversion_then the pairs (p1; i) and (p1; ) are*
* not isomorphic in
FIB Sn; Sn; *.
Let G be a topological group_then in the notation of p. 460, the restricti*
*on kG HCW is a cofunctor
HCW ! SET which converts coproducts into products and weak pushouts into weak*
* pullbacks. To ensure
that it is representable, one can introduce the pointed analog of BUN B;G, say *
*BUN B;G;*, and proceed
584
as above. The upshot is that the classifying space BG is now a CW complex but t*
*his need not be true
*
* XG ! X1G
*
* ? ?
of the universal space XG . To clarify the situation, consider the pullback squ*
*are y y . Since
*
* BG ! B1G
for any CW complex B; [B; BG ] kG B [B; B1G], the arrow BG ! B1Gis a weak hom*
*otopy equivalence
(cf. p. 515 ff.). Therefore the arrow XG ! X1Gis a weak homotopy equivalence, *
*so XG is homotopically
trivial (X1G being contractible).
LEMMA XG is contractible iff G is a CW space.
[Necessity: For then XG is a CW space and because the fibers of the Hurewic*
*z fibration XG ! BG
are homeomorphic to G, it follows that G is a CW space (cf. p. 625).
Sufficiency: Due to x6, Proposition 11, XG is a CW space. But a homotopical*
*ly trivial CW space is
contractible.]
Moral: When G is a CW space, kG can be represented by a CW complex (cf. x4,*
* Proposition 35).
[Note: Under these conditions, BG and B1Ghave the same homotopy type (repre*
*senting objects are
isomorphic), thus B1G is a CW space (see p. 625 for another argument).]
Notation: FCONCW * is the full subcategory of CONCW * whose objects *
*are the
pointed finite connected CW complexes and HFCONCW * is the associated homo*
*topy
category.
_____________
[Note: Any skeleton HFCONCW *of HFCONCW *is countable (cf. p. 62*
*8).]
A cofunctor F : HFCONCW * ! SET is said to be representable_in_the_l*
*arge_if
there exists a pointed connected CW complex X and a natural isomorphism [_ ; X]*
* ! F .
[Note: In this context, [_ ; X] stands for the restriction to HFCONCW *
* * of the
representable cofunctor determined by X. Observe that in general it is meaning*
*less to
consider F X.]
Example: The restriction to HFCONCW *of any cofunctor HCONCW * ! S*
*ET
satisfying the wedge and MayerVietoris conditions is representable in the larg*
*e.
Let F : HCONCW * ! SET be a cofunctor. g
Z? !*
* Y?
(Finite MayerVietoris Condition) For any weak pushout square fy *
* yj
X !*
* P
F?P Fj!F?Y
in HCONCW *, where Z is finite, F y yFg is a weak pullback squar*
*e in SET ,
F X !FfF Z
585
ae ae
so 8 xy22FFXY: (F f)x = (F g)y, 9 p 2 F P : (F()pF=jx)p.= y
(Limit Condition) For any pointed connected CW complex X and for any
collection {Xi : i 2 I} of pointed connected subcomplexes of X such that X = co*
*limXi,
where I is directed and the Xi are ordered by inclusion, the arrow F X ! limF X*
*i is
bijective.
SUBLEMMA Let F : HCONCW * ! SET be a cofunctor satisfying the wedge a*
*nd
finite MayerVietoris conditions. Fix an X in CONCW * and choose xa2eF X. Su*
*ppose
that X f K g!X is a pointed 2source, where K is in FCONCW *and fgare ske*
*letal
with (F f)x = (F g)x_then there is a Y in CONCW *containing X as an embedd*
*ed
pointed subcomplex, say i : X ! Y , such that i O f ' i O g and a y 2 F Y such*
* that
(F i)y = x.
K _?K f_g!X?
[Consider the weak pushout square rK y y , where Y is the poin*
*ted
K ! Y
double mapping cylinder of the foldingamaperK and the wedge f _ g. By construc*
*tion,
Y is a pointed weak coequalizer of fg and the existence of y 2 F Y follows *
*from the
assumptions.]
LEMMA Let F : HCONCW * ! SET be a cofunctor satisfying the wedge, fin*
*ite
MayerVietoris, and limit conditions. Fix an X in CONCW *and choose x 2 F *
*X_
then there is a Y in CONCW * containing X as an embedded pointed subcomplex,*
* say
i : X ! Y , such thataieO f ' i O g for any pointed 2source X f K g!X, where K*
* is
in FCONCW * and fg are skeletal with (F f)x = (F g)x, and a y 2 F Y such*
* that
(F i)y = x.
_____________
[Since it is enough to let K run over the objects in HFCONCW *, one ne*
*ed only deal
with a set {X fsKs!gsX : s 2 S} of pointed 2sources. Given anyWT S, proceed a*
*s in
(Kt_ Kt) ! X
t ? *
* ?
the proof of the sublemma and form the weak pushout square y *
* yiT ,
W
Kt ! *
*YT
t
X ae
so for T 0 T 00there is a commutative triangle NNNQ aeaeo . Consider the*
* set T
YT0 _________wjYT00
of pairs (T; yT ) (yT 2 F YT ) : (F iT )yT = x. Order T by writing (T 0; yT0) *
* (T 00; yT00) iff
T 0 T 00and (F j)yT00= yT0_then the limit condition implies that every chain in*
* T has an
586
upper bound, thus T has a maximal element (T0; yT0) (Zorn). Thanks to the subl*
*emma,
T0 = S, therefore one can take Y = YS; y = yS.]
PROPOSITION 28 Let F : HCONCW * ! SET be a cofunctor satisfying the
wedge, finite MayerVietoris, and limit conditions_then the restriction *
* of F to
HFCONCW * is representable in the large.
W _____________
[Put X0 = K, where K runs over the objects in HFCONCW *and for each*
* K,
K;k
k runs over F K. Using the wedge condition, choose x0 2 F X0 such that the asso*
*ciated
natural transformation 0 : [_ ; X0] ! F has the property that 0K : [K; X0] ! F K
is surjective for all K. Per the lemma, construct X0 X1 & x1 2 F X1 and conti*
*nue
by induction to obtain an expanding sequence X0 X1 . . .of topological spaces
and elements x0 2 F X0; x1 2 F X1; : :s:uch that 8 n, Xn is a pointed connected*
* CW
complex containing Xn1 as a pointed subcomplex and xn ! xn1 under Xn1 ! Xn .
Put X = X1 _then X is a pointed connected CW complex containing Xn as a pointed
subcomplex (cf. p. 525). Let x 2 F X be the element corresponding to {xn} via *
*the limit
condition and let : [_ ; X] ! F be the associated natural transformation. That*
* K is
surjective for all K is automatic. But K is also injective for all K : K ([f]) *
*= K ([g]), i.e.,
(F f)x = (F g)x (f; g skeletal) ) (F f)xn = (F g)xn (9 n) ) i O f ' i O g (i : *
*Xn ! Xn+1 ).]
__
Given a cofunctor F : HFCONCW * ! SET , for X in CONCW *, let F *
*X =
limF Xk, where Xk runs over the pointed finite connected subcomplexes of X orde*
*red by
__
inclusion_then F is the object function of a cofunctor HCONCW * ! SET whos*
*e re
striction to HFCONCW *is (naturally isomorphic to) F . On the basis of the*
* definitions,
__ __
F satisfies the limit condition. Moreover, F satisfies the wedge condition p*
*rovided that
F converts finite coproducts into finite products so, in order to conclude that*
* F is repre
__
sentable in the large, it need only be shown that F satisfies the finite Mayer*
*Vietoris con
dition (cf. Proposition 28). Assume, therefore, that F converts weak pushouts i*
*nto weak
C? ! B?
pullbacks. Consider a diagram y y , where X is a pointed connected CW c*
*omplex
ae A ! X
and AB & C are pointed connected subcomplexes such that X = A [ B, C = A \ B *
*with
__ __
F?X ! F?B ae
C finite. To prove that y y is a weak pullback square, let Ki run *
*over the
__ __ Lj
F A ! F C ae
pointed finite connected subcomplexes of ABwhich contain C and using obvious *
*notation,
587
ae__ __ _ *
*__ ae__ __
let a2_FAb2 __FB: __aC = bC_then the question is whether there exists __x2 *
*FX : xA_=xaB.= _b
ae__ __
For this, note first that FA_=FlimFBKi= limFaLnd FX = limF Xij(Xij= Ki[ Lj). *
*Represent
ae__ ae j ae
a_ {ai} (ai2 F Ki) xijKi= ai
b by {bj} (bj 2 F Lj)and let Sijbe the set of xij2 F Xij: xijLj = bj. *
*Since
Sij is nonempty and limSij is a subset of limF Xij, it suffices to prove that l*
*imSij is
nonempty as any __x2 limSij will work. However, this is a subtle point that ha*
*s been
resolved only by placing restrictions on the range of F .
EXAMPLE Let U : CPTHAUS ! SET be the forgetful functor. Suppose that F : *
*HFCONCW *
! CPTHAUS is a cofunctor such that U O F converts finite coproducts into fini*
*te products and weak
pushouts into weak pullbacks_then U O F is representable in the large. In fact,*
* if Tijis the subspace of
FXijsuch that UTij= Sij, then Tijis closed and limTijis calculated over a cofil*
*tered category, hence
limTijis a nonempty compact Hausdorff space. But U preserves limits, therefore *
*limSij= U(limTij) is
also nonempty.
[Note: More is true: _____UsOaFtisfies the MayerVietoris condition, hence *
*is representable. Example:
If Y is a pointed connected CW complex whose homotopy groups are finite, then f*
*or every pointed finite
connected CW complex X, [X; Y ] is finite (cf. p. 549), thus is a compact Ha*
*usdorff space (discrete
_____
topology) and so [_ ; Yi]s representable.]
ae
REPLICATION THEOREM Let f : K ! L be a pointed skeletal map, where KL
are in FCONCW *_then for any cofunctor F : HFCONCW *! SET which conve*
*rts
finite coproducts into finite products and weak pushouts into weak pullbacks, t*
*here is an
exact sequence
. .!.F L ! F K ! F Cf ! F L ! F K
in SET *.
[Note: F takes (abelian) cogroup objects to (abelian) group objects, so all*
* the arrows
to the left of F K are homomorphisms of groups. In addition, F K operates to th*
*e left
on F Cf and the orbits are the fibers of the arrow F Cf ! F L (cf. p. 333).]
Application: There is an exact sequence
F Kix F Lj ! F C ! F Xij! F Kix F Lj
in SET *.
[The pointed mapping cone of the arrow Ki_Lj ! Xijhas the same pointed homo*
*topy
type as C.]
588
Let (I; ) be a nonempty directed set, Ithe associated filtered category. Su*
*ppose that
: IOP ! SET is a diagram, where 8 i 2 Ob I, i 6= ; and 8 ffi 2 Mor I, ffiaise*
*surjective.
In I, write i ~ j iff there exists a bijective map f : i ! j and a k with ij *
*k such
k'
that the triangle [[[^ '') commutes.
i __________wfj
LEMMA If #(I=~) !, then lim is nonempty.
ADAMS REPRESENTABILITY THEOREM Let U : GR ! SET be the forgetful
functor. Suppose that F : HFCONCW *! GR is a cofunctor such that U O F co*
*nverts
finite coproducts into finite products and weak pushouts into weak pullbacks_th*
*en U O F
is representable in the large. ae
[The arrow Si0j0! Sijis surjective if KiLKi0 . This is because F C acts*
* transi
ae j Lj0
tively to the left on Si0j0Sand Si0j0! Sijis equivariant. Claim: #({ij}=~) !*
*. For one
ij
can check that ij ~ i0j0 iff F Kix F Lj ! F C & F Ki0x F Lj0! F C have the
same image, of which there are at most a countable number of possibilities. The*
* lemma
thus implies that limSijis nonempty.]
Working in CONCW *, two pointed continuous functions f; g : X ! Y are *
*said
to be prehomotopic_if for any pointed finite connected CW complex K and any poi*
*nted
continuous function OE : K ! X, f O OE ' g O OE. Homotopic maps are prehomotop*
*ic but
the converse is false since, e.g., there are phantom maps that are not nullhomo*
*topic (see
below).
Notation: PREHCONCW * is the quotient category of CONCW * defined *
*by the
congruence of prehomotopy, [X; Y ]prebeing the set of morphisms from X to Y .
__
If F : HFCONCW *! SET is a cofunctor, then F can be viewed as a cofun*
*ctor
*
* __
PREHCONCW * ! SET . Given X in CONCW *, there is a bijection Nat([_ ; *
*X]pre; F)
__ __
! F X (Yoneda). On the other hand, there is a bijection Nat([_ ; X]; F ) !_F_X*
*,_viz.
! {Xk ([ik])}, ik : Xk ! X the inclusion. Example: Take F = [_ ; X], so [X; X*
*]=
lim[Xk; X], and put X = {[ik]}_then id[_ ;X]$ X .
______
PROPOSITION 29 Let Y be in CONCW *. Assume: [_ ; Ys]atisfies the fi*
*nite
MayerVietoris condition_then for all X in CONCW *, the natural map [X; Y ]*
*pre !
lim[Xk; Y ] is bijective.
589
____[Injectivity_is_immediate._Turning to surjectivity,_note that by definition*
* lim[Xk; Y ] =
[X; Y ]. Fix_x0_2_[X;_Y_]and_let y0 = Y (2 [Y; Y)]. Put Z0 = X _ Y and writ*
*e z0 =
(x0; y0) 2 [Z0; Y ][X; Yx][Y; Y.]Imitating the argument used in the proof of Pr*
*oposition
28, construct a_Z_in_CONCW * containing Z0 as an embedded pointed subcomplex*
* and
an element z 2 [Z; Y ]which restricts to z0 such that the associated natural tr*
*ansformation
[K; Z] ! [K; Y ] is a bijection for all K. Specialize and take K = Sn (n 2 N ) *
*to see that
the inclusion j : Y ! Z is a pointed homotopy equivalence (realization theorem)*
* and then
compose the inclusion i : X ! Z with a homotopy inverse for j to get a pointed *
*continuous
function f0 : X ! Y whose prehomotopy class is sent to x0.]
*
* _____
FACT If Y is a pointed connected CW complex whose homotopy groups are coun*
*table, then [_ ; Y ]
satisfies the finite MayerVietoris condition.
[Note: Under this assumption on Y , it follows that for all X in CONCW *,*
* the natural map [X; Y ] !
lim[Xk; Y ] is surjective (and even bijective provided that the homotopy groups*
* of Y are finite (cf. p. 550
& p. 587)).]
PROPOSITION 30 Suppose that F : HFCONCW *! SET is a cofunctor which
converts finite coproducts into finite products and weak pushouts into weak pul*
*lbacks. As
__ *
* __
sume: F satisfies the finite MayerVietoris condition_then the cofunct*
*or F :
PREHCONCW * ! SET is representable.
[By Proposition 28, there is an X in CONCW * and a natural isomorphism *
* :
[_ ; X] ! F . Repeating the reasoning used in the proof of Proposition 29, one *
*finds that
__ __
the extension : [_ ; X]pre! F is a natural isomorphism as well.]
PROPOSITION 31 Suppose that F , F 0: HFCONCW * ! SET are cofunctors
which convert finite coproducts into finite products and weak pushouts into wea*
*k pullbacks.
__ __0
Assume: F and F satisfy the finite MayerVietoris condition. Fix natural isomor*
*phisms :
[_ ; X] ! F , 0: [_ ; X0] ! F 0, where X; X0are pointed connected CW complexes.*
* Let T :
F ! F 0be a natural transformation_then there is a pointed continuous function *
*f : X !
[K;?X] f*![K;?X0]
X0, unique up to prehomotopy, such that the diagram K y y0K commut*
*es
F K !T F 0K
K
for all K.
[Note: If F = F 0and T is the identity, then f : X ! X0 is a pointed homot*
*opy
equivalence.]
590
PROPOSITION 32 Any representing object in the Adams representability theor*
*em
is a group object in PREHCONCW *and all such have the same pointed homoto*
*py type.
FACT Let F : HFCONCW *! SET be a cofunctor which converts finite coprod*
*ucts into finite
products and weak pushouts into weak pullbacks. Assume: 8 K, #(FK) !_then F is*
* representable in
the large.
[Note: It is unknown whether the cardinality assumption can be dropped.]
ae
Given pointed connected CW complexes XY, a pointed continuous function f *
*: X !
Y is said to be a phantom_map__if it is prehomotopic to 0. Let Ph(X; Y ) be the*
* set of pointed
homotopy classes of phantom maps from X to Y _then there is an exact sequence
* ! Ph(X; Y ) ! [X; Y ] ! lim[Xk; Y ]
in SET *. Of course, [0] 2 Ph(X; Y ) but #(Ph (X; Y )) > 1 is perfectly possibl*
*e. Example:
Take X = K(Q ; 3); Y = K(Z ; 4) () [X; Y ] H4(Q ; 3) Ext(Q ; Z) R ), realize*
* X as
the pointed mapping telescope of the sequence S3 ! S3 ! . .,.the kth map having*
* degree
k, and note that up to homotopy, every OE : K ! X factors through S3() Ph (X; Y*
* ) =
[X; Y ]).
Is the arrow [X; Y ] ! lim[Xk; Y ] always surjective? While the answer is "*
*yes" under various assump
tions on X or Y , what happens in general has yet to be decided.
[Note: By contrast, there is a bijection Ph(X; Y ) ! lim1[Xk; Y ] of pointe*
*d sets (GrayMcGibbony).]
EXAMPLE Meierzhas shown that Ph(K(Z; n); Sn+1) Ext(Q ; Z) for all positiv*
*e even n. Special
case: Ph(P 1(C ); S3) Ext(Q ; Z).
[Note: Suppose that G is an abelian group which is countable and torsion fr*
*ee_then 9 X & Y :
Ph(X; Y ) Ext(G; Z) (Roitbergk).]
EXAMPLE (Universal_Phantom_Maps_) Let X be a pointed connected CW complex*
*. Assume:
X has a finite number of cells in each dimension_then it is clear that f : X ! *
*Y is a phantom map iff
8 n > 0, fX(n)is nullhomotopic. Denote by tel+X the pointed telescope of X whi*
*ch starts at X(1)rather
than X(0). Recall that the projection p : tel+X ! X is a pointed homotopy equiv*
*alence (cf. p. 312).
_________________________
yTopology 32 (1993), 371394.
zQuart. J. Math. 29 (1978), 469481.
kTopology Appl. 59 (1994), 261271.
591
W
Now collapse each integral joint of tel+X to a point, i.e., mod out by X(n).*
* The resulting quotient can
W W n>0
be identified with X(n)and the arrow : tel+X ! X(n)is a phantom map. It *
*is universal
n>0 n>0 __
in the sense that if f : X ! Y is a phantom map and if f = f O p, then there is*
* a pointed continuous
W __ W
function F : X(n)! Y such that f' F O . This is because the inclusion i : *
* X(n)! tel+X
n>0 W n>0
is a closed cofibration, hence Ci X(n)(cf. p. 324). Corollary: All phantom *
*maps out of X are
n>0
nullhomotopic iff is nullhomotopic. ae
[Note: Here is an application. Suppose that X are pointed connected CW co*
*mplexes with a finite
Y
number of cells in each dimension. Claim: If f : X ! Y and g : Y ! Z are phanto*
*m maps, then g O f :
W F p1 *
* W
X ! Z is nullhomotopic. To see this, observe that the composite X(n)! Y ! t*
*el+Y ! Y (n)
n>0 *
* n>0
is a phantom map. Accordingly, its restriction to each X(n)is nullhomotopic, so*
* actually Op1OF ' 0.
Therefore g O f ' (_gO p1) O (__fO p1) ' (G O O p1) O (F O O p1) ' G O ( *
*O p1 O F) O O p1 ' 0.]
61
x6. ABSOLUTE NEIGHBORHOOD RETRACTS
From the point of view of homotopy theory, the central result of this xis t*
*he CWANR
theorem which says that a topological space has the homotopy type of a CW compl*
*ex iff
it has the homotopy type of an ANR. But absolute neighborhood retracts also hav*
*e a life
of their own. For example, their theory is an essential component of infinite d*
*imensional
topology.
Consider a pair (X; A), i.e., a topological space X and a subspace A X. Le*
*t Y be
a topological space. Suppose given a continuous function f : A ! Y _then the ex*
*tension
question is: Does there exist a continuous function F : X ! Y such that F A = *
*f? While
this is a complex multifaceted issue, there is an evident connection with the t*
*heory of
retracts. For if we take Y = A, then the existence of a continuous extension r *
*: X ! A
of the identity map idA amounts to saying that A is a retract of X. Every retr*
*act of a
Hausdorff space X is necessarily closed in X. On the other hand, if A is closed*
* in X, then
with no assumptions on X, a continuous function f : A ! Y has a continuous exte*
*nsion
F : X ! Y iff Y is a retract of the adjunction space X tf Y . The opposite e*
*nd of the
spectrum is when A is dense in X. In this case, one can be quite specific and w*
*e shall start
with it.
Let (X; A) be a pair with A dense in X. Write oX and oA for the correspon*
*ding
______
topologies. Define a map Ex : oA ! oX by Ex(O) = X  A  O, the bar denoting cl*
*osure
S
inaX_theneEx (O) \ A = O and Ex (O) = {U : U 2 oX & U \ A = O}. Obviously,
Ex(;) = ;
Ex (A) = X and 8 O; P 2 oA : Ex(O \ P ) = Ex(O) \ Ex(P ). Put Ex (O) = {Ex (*
*O) :
O 2 O}(O oA ).
PROPOSITION 1 Let A be a dense subspace of a topological space X; let Y be *
*a regu
lar Hausdorff space_then a given f 2 C(A; Y ) admits a continuous extension F 2*
* C(X; Y )
iff X = [Ex (f1 (V)) for every open covering V of Y .
[The conditionaiseclearly necessary. As for the sufficiency, suppose that *
*X 6= ; and
#(Y ) > 1. Call oXo the topologies on X and Y .
Y
(F *) Define a map F *: oY ! oX by
[ ___
F *(V ) = {Ex (f1 (V 0)) : V 02 oY & V 0 V }:
ae *
Note that FF*(;)(=Y;) =aXnd 8 V1; V2 2 oY : F *(V1 \ V2) = F *(V1) \ F *(V2)*
*. Let
S S
{Vj} oY _then F *( Vj) F *(Vj) and in fact equality prevails. To see this*
*, write
j j
62
S __
Vj = [V, where V is the set of all V 2 oY : V Vj (9 j). Take a V 02 oY :
_j_ S ___ S ___ S
V 0 Vj. Since Y = (Y  V 0) ([V), X = Ex (f1 (Y  V 0)) ([Ex (f1 (V))).*
* But
j ___ S
; = Ex(f1 (V 0)) \ Ex(f1 (Y  V 0)) ) Ex(f1 (V 0)) [Ex (f1 (V)) F *(Vj)*
*, from
S S j
which it follows that F *( Vj) F *(Vj).
j j
(F*) Define a map F* : oX ! oY by
[
F*(U) = {V : V 2 oY & F *(V ) U}:
Note that 8 U 2 oX and 8 V 2 oY : V F*(U) , F *(V ) U. Indeed,_F *respec*
*ts
arbitrary unions._ We claim now that 8 x 2 X 9 y 2 Y : F*(X  {x}) =_Y__{y}.
Let F*(X  {x}) = Y  Bx. Case 1: Bx = ;. Here, X = F *(Y ) X  {x}, an
impossibility.aCasee2: #(Bx) > 1. Choose y1; y2 2 Bx : y1 6= y2. Choose V1, V2 *
*2 oY :
____ ____
V1 \ V2 = ; & y1y2 V1_then V1 \ V2 F*(X  {x}) ) F *(V1 \ V2) X  {x}, i.e.,
2 2 V2__ *
*____
F *(V1) \ F *(V2) X  {x}, thus either_F_*(V1)_or F *(V2) is contained in X  *
*{x} and so
either V1 or V2 is contained in F*(X  {x}) = Y  Bx, a contradiction. _*
*___
(F ) Define a map F : X ! Y byastipulatingethat F (x) = y iff F*(X  {*
*x}) =
1(V ) = F *(V )
Y  {y}. The definitions imply that F F1(V ) \ A = f1 (V()V 2 oY ), there*
*fore F 2
C(X; Y ) and F A = f.]
Retain the assumption that A is dense in X and Y is regular Hausdorff. As*
*sign
to each x 2 X the collection U(x) of all its neighborhoods_then a continuous fu*
*nction
f : A ! Y has a continuous extension F : X ! Y iff 8 x the filter base f(U(x)*
* \ A)
converges. The nontrivial part of this assertion is a simple consequence of the*
* preceding
result. For suppose that for some open covering V of Y : X 6= [Ex (f1 (V)). *
* Choose
x 2 X : x =2 [Ex (f1 (V)), so 8 U 2 U(x) and 8 V 2 V : U \ A 6 f1 (V ) or st*
*ill,
f(U \A) 6 V . But f(U(x)\A) converges to y 2 Y . Accordingly, there is (i) V0 2*
* V : y 2 V0
and (ii) U0 2 U(x) : f(U0 \ A) V0. Contradiction.
Here are two other applications.
(C) Suppose that Y is compact Hausdorff_then a continuous function f :*
* A !
Y has a continuous extension F : X ! Y iff for every finite open covering V of *
*Y there
exists a finite open covering U of X such that U \ A is a refinement of f1 (V).
In this statement, one can replace "compact" by "Lindel"of" if "finite" is *
*replaced by
"countable". More is true: It suffices to assume that Y is merely R compact (r*
*ecall that
every Lindel"of regular Hausdorff space is R compact).
63
(RC) Suppose that Y is R compact_then a continuous function f : A !*
* Y
has a continuous extension F : X ! Y iff for every countable open covering V of*
* Y there
exists a countable open covering U of X such that U \ A is a refinement of f1 *
*(V).
Q
[There is a closed embedding Y ! R . Postcompose f with a generic proje*
*ction
Q Q
R ! R and extend it to X. Form the associated diagonal map F : X ! R _then
Q
F is continuous and F A = f (viewed as a map A ! R ). Conclude by remarking *
*that
__ _____ __
F (X) = F (A ) F (A) Y = Y .]
[Note: The R compactness of Y is essential. Consider X = [0; ], A = Y = *
*[0; [,
and let f = idA (Y is not R compact, being countably compact but not compact).]
EXAMPLE The proposition can fail if the assumption "Y regular Hausdorff" is*
* weakened to "Y
Hausdorff". Let X be the set of nonnegative real numbers. Put D = {1=n : n = *
*1; 2; : :}:_then the
collection of all sets of the form U [ (V  D), where U and V are open in the u*
*sual topology on X, is also
a topology, call the resulting space Y . Observe that Y is Hausdorff but not re*
*gular. Let A = X  D and
define f 2 C(A; Y ) by f(x) = x. It is clear that there is no F 2 C(X; Y ) : F*
*A = f, yet for every open
covering V of Y , X = [Ex(f1(V)).
FACT Let A be a dense subspace of a topological space X; let Y be a regular*
* Hausdorff space_then
a given f 2 C(A; Y ) admits a continuous extension F 2 C(X; Y ) iff 8 x 2 X  A*
* 9 fx 2 C(A [ {x}; Y ):
fxA = f.
Let X and Y be topological spaces.
(EP) A subspace A X is said to have the extension_property_with_respe*
*ct_to_Y
(EP w.r.t. Y ) if 8 f 2 C(A; Y ) 9 F 2 C(X; Y ): F A = f.
(NEP) A subspace A X is said to have the neighborhood_extension_prope*
*rty_ae
with_respect_to_Y(NEP w.r.t. Y ) if 8 f 2 C(A; Y ) 9 F 2UC(AU; Y )(U open): F *
*A = f.
[Note: In this terminology, A is a retract (neighborhood retract) of X iff *
*A has the
EP (NEP) w.r.t. Y for every Y .]
Two related special cases of importance are when Y = R or Y = [0; 1]. If A *
*has the EP
w.r.t. R , then A has the EP w.r.t. [0; 1]. Reason: If f 2 C(A; [0; 1]) and if *
*F 2 C(X; R)
is a continuous extension of f, then min{1; max{0; F }} is a continuous extensi*
*on of f with
range a subset of [0; 1]. The converse is trivially false. Example: Let X be CR*
*H space_
then X, as a subspace of fiX, has the EP w.r.t. [0; 1] but X has the EP w.r.t. *
*R iff X is
pseudocompact (of course in general X, as a subspace of AEX, has the EP w.r.t. *
*R ). Bear
in mind that a CRH space is compact iff it is both R compact and pseudocompact.
64
[Note: Suppose that X is Hausdorff_then X is normal iff every closed subspa*
*ce has
the EP w.r.t. R (or, equivalently, [0; 1]).]
Suppose that X is a CRH space. Let A be a subspace of X.
(fi) If A has the EP w.r.t. [0; 1], then the closure of A in fiX is fi*
*A and conversely.
(AE) If A has the EP w.r.t. R , then the closure of A in AEX is AEA an*
*d conversely provided that
X is in addition normal.
[Note: The Niemytzki plane is a nonnormal hereditarily R compact space, s*
*o the unconditional
converse is false.]
Two subsets A and B ofaaetopological space X are said to be completely_sepa*
*rated_in
X if 9 OE 2 C(X; [0; 1]): OEAO=E0B.=F1or this to be the case, it is necessa*
*ry and sufficient
that A and B are contained in disjoint zero sets. Example: Suppose that X is a*
* CRH
space_then any two disjoint closed subsets of X, one of which is compact, are c*
*ompletely
separated in X (no compactness assumption being necessaryaifeX is in addition n*
*ormal).
[Note: It is enough to find a function f 2 C(X) : fAf0B .1 Reason: Ta*
*ke OE =
min {1; max{0; f}}. Moreover, 0 and 1 can be replaced by any real numbers r and*
* s with
r < s.]
PROPOSITION 2 Let A X_then A has the EP w.r.t. [0; 1] iff any two complete*
*ly
separated subsets of A are completely separated in X.
[Assume that A has the stated property. Fix an f 2 C(A; [0; 1]). To const*
*ruct an
extension F 2 C(X; [0; 1]) of f, we shall first define by recursion two sequenc*
*es {fn} and
{gn} subject to: fn 2 BC(A) & kfnk 3rn and gn 2 BC(X) &akgnke rn, where rn =
1P S = {x 2 A : f (x) *
*r }
(1=2)(2=3)n (so rn = 1). Set f1 = f. Given fn, let n+ n *
* n .
ae 1 Sn = {x 2 A : fn(x) *
* rn}

Since SnS+are completely separated in A, they are, by hypothesis, completely *
*separated
n ae
 = r
in X. Choose gn 2 BC(X) : gnSng +n & kgnk rn. Push the recursion forward
nSn = rn
1P
by setting fn+1 = fn  gnA. The series gn is uniformly convergent on X, thus*
* its sum
1
G is a continuous function on X : GA = f. Take F = max {0; G}.]
Application: Suppose that X is a CRH space_then any compact subset of X has*
* the
EP w.r.t. [0; 1] (cf. p. 214).
65
ae FACT Let A X; let f 2 BC(A)_then 9 F 2 BC(X) : FA = f iff 8 a; b 2 R : a *
*< b, the sets
f1(]  1; a])are completely separated in X.
f1([b; +1[)
PROPOSITION 3 Let A X_then A has the EP w.r.t. R iff A has the EP w.r.t.
[0; 1] and is completely separated from any zero set in X disjoint from it.
[Necessity: Let Z be a zero set in X disjoint from A : Z = Z(g), where g 2 *
*C(X; [0; 1]).
Put f = (1=g)A. Choose h 2 C(X) : hA = f. Consider gh.
Sufficiency: Fix an f 2 C(A). Because arctan Of 2 C(A; [ss=2; ss=2]), it h*
*as an exten
sion G 2 C(X; [ss=2; ss=2]). Let B = G1(ss=2)_thenaBeis a zero set in X disjo*
*int from
A, so there exists OE 2 C(X; [0; 1]) : OEAO=E1B.=P0ut F = tan(OEG): F 2 C(X*
*) & F A = f .]
Consequently, every zero set in X that has the EP w.r.t. [0; 1] actually ha*
*s the EP
w.r.t. R . On the other hand, a zero set in X need not have the EP w.r.t. [0; 1*
*]. Examples:
(1) Take for X the IsbellMrowka space (N )_then A = S is a zero set in X but S*
* does not
have the EP w.r.t. [0; 1]; (2) Take for X the Niemytzki plane_then A = {(x; y) *
*: y = 0}
is a zero set in X but A does not have the EP w.r.t. [0; 1].
EXAMPLE (Katetov_Space_) As a subspace of R , N has the EP w.r.t. [0; 1], *
*so the closure of
N in fiR is fiN . Let X = fiR  (fiN  N)_then fiX = fiR and X is a LCH spac*
*e which is actually
pseudocompact (an unbounded continuous function on X would be unbounded on a cl*
*osed subset of R
disjoint from N). However, X is not countably compact, thus is not normal (cf. *
*x1, Proposition 5). As a
subspace of X, N has the EP w.r.t. [0; 1] but does not have the EP w.r.t. R .
[Note: N is a closed Gffibut is not a zero set in X.]
A subspace A X is said to be Zembedded_in X if every zero set in A is the*
* intersection of A with
a zero set in X. Example: Any cozero set in X is Zembedded in X. If A has the *
*EP w.r.t. [0; 1], then
A is Zembedded in X (but not conversely), so, e.g., any retract of X is Zembe*
*dded in X. Examples:
Suppose that X is Hausdorff_then (1) Every subspace of a perfectly normal X is *
*Zembedded in X;
(2) Every Foesubspace of a normal X is Zembedded in X; (3) Every Lindel"of su*
*bspace of a completely
regular X is Zembedded in X.
FACT Let A X_then A has the EP w.r.t. R iff A is Zembedded in X and is c*
*ompletely
separated from any zero set in X disjoint from it.
[Note: It is a corollary that if A is a zero set in X, then A has the EP w.*
*r.t. R iff A is Zembedded
in X. Both the IsbellMrowka space and the Niemytzki plane contain zero sets th*
*at are not Zembedded.]
66
Application: Suppose that X is a Hausdorff space_then X is normal iff every*
* closed subset of X is
Zembedded in X.
PROPOSITION 4 Let A X_then A has the EP w.r.t. [0; 1] (R ) iff for every f*
*inite
(countable) numerable open covering O of A there exists a finite (countable) nu*
*merable
open covering U of X such that U \ A is a refinement of O.
[The proof of necessity is similar to but simpler than the proof of suffici*
*ency so we
shall deal just with it, assuming only that there exists a numerable open cover*
*ing U of X
such that U \ A is a refinementaofeO, thereby omitting the cardinality assumpti*
*onaoneU.
0 Z0
([0; 1]) Let SS00be two completely separated subsets of A; let Z00*
* be two
ae 0 0
disjoint zero sets in A : SS00Z Z00. Let O = {A  Z0; A  Z00}. Take U per O *
*and choose
a neighborhood finite cozeroasetecovering V of X such that V is aastarerefineme*
*nt of U (cf.
0= X  S{V 2 V : V \ Z0 = ;} W 0
x1, Proposition 13). Put WW 00= X  S{V 2 V : V \ Z00= ;}_then W 00are disj*
*oint
ae 0 0
zero sets in X : ZZ00W W 00. Therefore S0 and S00are completely separated in *
*X, thus,
by Proposition 2, A has the EP w.r.t. [0; 1].
(R ) Let Z be a zero set in X : A \ Z = ;, say Z = Z(f), where f 2 C(X*
*; [0; 1]).
The collection O = {f1 (]1=n; 1]) \ A} is a countable cozero set covering of A*
*, hence is
numerable (cf. p. 125). Take U per O and choose a neighborhood finite cozero s*
*et covering
V = {Vj : j 2 J} of X and a zero set covering Z = {Zj : j 2 J} of X such that V*
* is a
refinement of U with Zj Vj (8 j) (cf. p. 125). Given j, 9 nj : Zj\A f1 (]1=*
*nj; 1])\A.
S
Put W = Zj\ f1 ([1=nj; 1])_then W is a zero set in X containing A and disjoi*
*nt from
j
Z, so A and Z are completely separated in X. Since the first part of the proof*
* implies
that A necessarily has the EP w.r.t. [0; 1], it follows from Proposition 3 tha*
*t A has the
EP w.r.t. R .]
FACT Let A X_then A is Zembedded in X iff for every finite numerable open*
* covering O of A
there exists a cozero set U containing A and a finite numerable open covering U*
* of U such that U \ A is a
refinement of O.
LEMMA Let (X; d) be a metric space; let A be a nonempty closed proper subsp*
*ace
of X_then there exists a subset {ai: i 2 I} of A and a neighborhood finite open*
* covering
{Ui: i 2 I} of X  A such that 8 i : x 2 Ui) d(x; ai) 2d(x; A).
[Assign to each x 2 X  A the open ball Bx of radius d(x; A)=4. The collec*
*tion
{Bx : x 2 X  A} is an open covering of X  A, thus by paracompactness has a ne*
*ighbor
hood finite open refinement {Ui: i 2 I}. Each Uidetermines a point xi2 XA : Ui*
* Bxi,
67
from which a point ai 2 A : d(xi; ai) (5=4)d(xi; A). Obviously, 8 x 2 Ui : d(*
*x; ai)
(3=2)d(xi; A) and d(xi; A) (4=3)d(x; A).]
DUGUNDJI EXTENSION THEOREM Let (X; d) be a metric space; let A be a closedae
subspace of X. Let E be a locally convex topological vector space. Equip C(A;*
*CE)(X;wE)ith
the compact open topology_then there exists a linear embedding ext: C(A; E) ! C*
*(X; E)
such that 8 f 2 C(A; E), ext(f)A = f and the range of ext(f) is contained in t*
*he convex
hull of the range of f.
[Assume that A is nonempty, proper and, using the notation of the lemma, ch*
*oose a
partition of unity {i : i 2 I} on X  A subordinate to {Ui : i 2 I}. Given f 2 *
*C(A; E),
let (
P f(x) (x 2 A)
ext(f)(x) = i(x)f(ai) (x 2 X  A):
i
Then ext(f)A = f and it is clear that ext(f)(X) is contained in the convex hul*
*l of
f(A). The continuity of ext(f) is built in at the points of X  A. As for the*
* points of
A, fix a0 2 A and let N be a balanced convex neighborhood of zeroaineE. Choose*
* a
ffi > 0 : d(a; a0) ffi ) f(a)  f(a0) 2 N(a 2 A). Suppose that dx(2xX;aA *
* . If
0) *
*< ffi=3
i(x) > 0, then, from the lemma, d(x; ai) 2d(x; A), hence d(ai; a0) 3d(x; a0) *
*< ffi.
Consequently,
X X
ext(f)(x)  ext(f)(a0) = i(x)(f(ai)  f(a0)) 2 i(x)N N:
i i
Therefore ext(f) 2 C(X; E). By construction, ext is linear and onetoone, so *
*the only
remaining issue is its continuity. Take a nonempty compact subset K of X and l*
*et
O(K; N) = {F 2 C(X; E) : F (K) N}. Put KA = K \ A [ {ai 2 A : K \ Ui 6= ;}. Let
O(KA ; N) = {f 2 C(A; E) : f(KA ) N}. Plainly, f 2 O(KA ; N) ) ext(f) 2 O(K; N*
*).
Claim: KA is compact. To see this, let {xn} be a sequence in KA . Since K \A is*
* compact,
we can suppose that {xn} has no subsequence in K \ A, thus without loss of gene*
*rality,
xn = ain for some in : K \ Uin 6=.; Pick yn 2 K \ Uin and assume that yn ! y 2 *
*K.
Case 1: y 2 K \ A. Here, d(xn; y) = d(ain; y) 3d(yn; y) ! 0. Case 2: y 2 K \ (*
*X  A).
There is a neighborhood of y that meets finitely many of the Ui and once yn is *
*in this
neighborhood, the index in is constrained to a certain finite subset of I, whic*
*h means that
{xn} has a constant subsequence.]
[Note: Suppose that E is a normed linear space_then the image of extBC(A; *
*E) is
contained in BC(X; E) and, per the uniform topology, ext: BC(A; E) ! BC(X; E) i*
*s a
linear isometric embedding: 8 f 2 BC(A; E), kfk = k ext(f)k. ]
68
In passing, observe that if the ai are chosen from some given dense subset *
*A0 A,
then the range of ext(f) is contained in the union of f(A) and the convex hull *
*of f(A0).
The Dugundji extension theorem has many applications. To mention one, it is*
* a key ingredient in
the proof of a theorem of Milyutin to the effect that if X and Y are uncountabl*
*e metrizable compact
Hausdorff spaces, then C(X) and C(Y ) are linearly homeomorphic (Pelczynskiy). *
*Extensions to the case
of noncompact X and Y have been given by Etcheberryz.
[Note: The BanachStone theorem states that if X and Y are compact Hausdorf*
*f spaces, then X
and Y are homeomorphic provided that the Banach spaces C(X) and C(Y ) are isome*
*trically isomorphic
(Behrendsk).]
Is Dugundji's extension theorem true for an arbitrary topological vector sp*
*ace E? In
other words, can the "locally convex" supposition on E be dropped? The answer i*
*s "no",
even if E is a linear metric space (cf. p. 612).
[Note: A topological vector space E is said to be a linear_metric_space_if *
*it is metriz
able. Every linear metric space E admits a translation invariant metric (Kakuta*
*ni) but E
need not be normable.]
Let X be a CRH space; let A be a nonempty closed subspace of X. Let E be a*
* locally con
vex topological vector space (normed linear space)_then a linear operator T : C*
*(A; E) ! C(X; E)
(T : BC(A; E) ! BC(X; E)) continuous for the compact open topology (uniform top*
*ology) is said to
be a linear_extension_operator_if for all f in C(A; E) (BC(A; E)) : TfA = f. *
*Write LEO (X; A; E)
(LEO b(X; A; E)) for the set of linear extension operators associated with C(A;*
* E) (BC(A; E)). Assum
ing that X is metrizable, the Dugundji extension theorem asserts: 8 A; C(A; E) *
*(BC(A; E)) possesses a
linear extension operator (and even more in that the "same" operator works for *
*both). Question: What
conditions on X or A serve to ensure that LEO(X; A; E) (LEO b(X; A; E)) is not *
*empty?
EXAMPLE (The_Michael_Line_) Take the set R and topologize it by isolating t*
*he points of P,
leaving the points of Q with their usual neighborhoods. The resulting space X i*
*s Hausdorff and hereditarily
paracompact but not locally compact. And A = Q is a closed subspace of X which,*
* however, is not a
Gffiin X. Let E = C(P ), P in its usual topology_then E is a locally convex top*
*ological vector space
(compact open topology). Claim: LEO (X; A; E) is empty. For this, it suffices t*
*o exhibit an f 2 C(A; E)
_________________________
yDissertationes Math. 58 (1968), 192; see also Semadeni, Banach Spaces of C*
*ontinuous Functions,
PWN (1971), 379.
zStudia Math. 53 (1975), 103127; see also Hess, SLN 991 (1983), 103110.
kSLN 736 (1979), 138140.
69
thatacannotebe extended to an F 2 C(X; E). If P has its usual topology, then th*
*e continuous function
A x P! R has no continuous extension X x P ! R (thus X x P is not norma*
*l). Defining
(x; y) ! 1=(y  x)
f 2 C(A; E) by f(x)(y) = 1=(y  x), it follows that f has no extension F 2 C(X;*
* E).
A Hausdorff space X is said to be submetrizable_if its topology contains a *
*metrizable topology.
Examples: (1) The Michael line is submetrizable and normal but not perfect; (2)*
* The Niemytzki plane is
submetrizable and perfect but not normal.
FACT Let X be a submetrizable CRH space. Suppose that A is a nonempty close*
*d subspace of X
with a compact frontier_then 8 E; LEO(X; A; E) (LEO b(X; A; E)) is not empty.
[Note: In view of the preceding example, the hypothesis on A is not superfl*
*uous.]
When E = R, denote by LEO (X; A) (LEO b(X; A)) the set of linear extension *
*operators for C(A)
(BC(A)).
EXAMPLE LEO b(X; A) can be empty, even if X is a compact Hausdorff space. *
*For a case in
point, take X = fiN & A = fiN  N. Claim: LEO (X; A) (= LEO b(X; A)) is empty.*
* Suppose not and
let T : C(A) ! C(X) be a linear extension operator. Fix an uncountable collecti*
*on U = {Ui : i 2 I}
ofanonemptyepairwise disjoint open subsets of A. Pick an ai 2 Ui and choose an*
* fi 2 C(A; [0; 1]):
fi(ai) = 1. Let O = {x 2 X : Tf (x) > 1=2}. Since X is separable, there exi*
*sts an uncountable
fi(A  Ui) = 0 i i T
subset I0 of I and a point x0 2 X : x0 2 Oi. Let n be some integer > kTk. Se*
*lect distinct indices
i2I0
2nP
ik (k = 1; : :;:2n) in I0. Put f = fik, so kfk = 1. A contradiction then resu*
*lts by writing
1
2nP 1
n = nkfk kTfk Tf(x0) = Tfik(x0) > 2n . _=.n
1 2
[Note: Let X be a compact Hausdorff space; let A be a nonempty closed subs*
*pace of X. Set
ae(X; A) = inf{kTk : T 2 LEO(X; A)} (where ae(X; A) = 1 if LEO(X; A) is empty).*
* Of course, ae(X; A) 1
and Benyaminiy has shown that 8 r : 1 r < 1, there exists a pair (X; A) : ae(X*
*; A) = r.]
The space X figuring in the preceding example is not perfect (no point of f*
*iN  N is a Gffiin fiN ).
Can one get a positive result if perfection is assumed? The answer is "no". Ind*
*eed, van Douwenz has
constructed an example of a CRH space X that is simultaneously perfect and para*
*compact, yet contains
a nonempty closed subspace A for which LEOb(X; A) = ;.
_________________________
yIsrael J. Math. 16 (1973), 258262.
zGeneral Topology Appl. 5 (1975), 297319.
610
The assumption that LEO b(X; A) is not empty 8 A has implications for the t*
*opology of X. To
quantify the situation, given r : 1 r < 1, let brbe the condition: 8 A, {T 2 L*
*EOb(X; A): kTk r} 6= ;.
Claim: If bris in force, then for any discrete collection A = {Ai: i 2 I} of no*
*nempty closed subsets of X
there is a collection U = {Ui: i 2 I} of open subsets of X such that (1) Ai Ui *
*& i 6= j ) Ui\ Aj = ;
and (2) ord(U) [r]. Thus put A = [A, let Oi: A ! [0; 1] be the characteristic *
*function of Ai, choose
T 2 LEO b(X; A) : kTk r, and consider U = {Ui : i 2 I}, where Ui = {x 2 X : TO*
*i(x) > r=[r] + 1}.
Example: Suppose that X satisfies brfor some r < 2_then X is collectionwise nor*
*mal.
[Note: Let X be the Michael line_then one can show that X satisfies b1, yet*
* LEO (X; A) = ; if
A = Q.]
FACT Let X be a Moore space. Assume: X satisfies brfor some r_then X is nor*
*mal and meta
compact.
Let X be a nonempty topological space_then an equiconnecting_structure_onaX*
*eis a
continuous function : IX2 ! X such that 8 x; y 2 X and 8 t 2 [0; 1] : (x;(y;*
*x0);=yx;&1) = y
(x; x; t) = x. A subset A X for which (IA2) A is called convex_. In order th*
*at X
have an equiconnecting structure, it is necessary that X be both contractible a*
*nd locally
contractible but these conditions are not sufficient as can be seen by consider*
*ing Borsuk's
cone (cf. p. 615). Example: Suppose that X is a contractible topological gr*
*oup. Let
H : IX ! X be a homotopy contracting X to its unit element e_then the prescript*
*ion
(x; y; t) = H(e; t)1H(xy1 ; t)y defines an equiconnecting structure on X. In *
*particular,
if X is a topological vector space, then H(x; t) = (1  t)x will do.
[Note: Let E be an infinite dimensional Banach space. Consider GL (E), the*
* group
of invertible bounded linear transformations T : E ! E. Equip GL (E) with the t*
*opology
induced by the operator norm_then GL (E) is a topological group and, being an o*
*pen
subset of a Banach space, has the homotopy type of a CW complex (cf. x5, Propos*
*ition
6). If E is actually a Hilbert space, then GL (E) is contractible (Kuipery) but*
* this need not
be true in general (even if E is reflexive), although it is the case of certain*
* specific spaces,
e.g., C([0; 1]) or Lp([0; 1]) (1 p 1). See Mityaginz for proofs and other rem*
*arks.]
FACT A nonempty topological space X has an equiconnecting structure iff the*
* diagonal X is a
strong deformation retract of X x X.
[Necessity: Given , consider the homotopy H : IX2 ! X2 defined by H((x; y);*
* t) = ((x; y; t); y).
_________________________
yTopology 3 (1965), 1930.
zRussian Math. Surveys 25 (1970), 59103.
611
Sufficiency: Given H, consider the equiconnecting structure : IX2 ! X defi*
*ned by
ae
(x; y; t) = p1(H((x; y); 2t))(0 t 1=2);
p2(H((x; y); 2  2t))(1=2 t 1)
where p1 and p2 are the projections onto the first and second factors.]
FACT Suppose that X is a nonempty topological space for which the inclusio*
*n X ! X x X is a
cofibration_then X has an equiconnecting structure iff Xaisecontractible.
[Choose a homotopy H : IX ! X contracting X to x0 : H(x; 0) = xand then d*
*efine : IX2 !
H(x; 1) = x0
X2 by ((x; y); t) = (H(x; t); H(y; t)) to see that X is a weak deformation retr*
*act of X x X.]
A nonempty topological space X is said to be locally_convex_if it admits an*
* equiconnecting structure
such that every x 2 X has a neighborhood basis comprised of convex sets. The *
*convex subsets of a
locally convex topological vector space are therefore locally convex, where (x;*
* y; t) = (1  t)x + ty. On
the other hand, the long ray L+ is not locally convex.
EXAMPLE Let K = (V; ) be a vertex scheme. Suppose that K is full_, i.e., if*
* F V is finite
and nonempty, then F 2 . Claim: K is locally convex. Thus fix a point * 2 V .*
* Let OE 2 K_then
P P
OE = bv(OE)Ov + (1  bv(OE))O*. Here, Ov (O*) is the characteristic funct*
*ion of {v} ({*}). Define fi :
v6=* v6=* P P
KxK ! K by fi(OE; ) = fi(OE; )vOv+(1 fi(OE; )v)O*, where fi(OE;*
* )v = min{bv(OE); bv( )}.
v6=* v6=*
The assignment ae
(OE; ; t) = (1  2t)OE + 2tfi(OE;(0) t 1=2)
(2  2t)fi(OE; ) + (2t(11)=2 t 1)
is an equiconnecting structure on K relative to which K is locally convex.
FACT Let A X, where X is metrizable and A is closed_then A has the EP w.r.*
*t. any locally
convex topological space.
PLACEMENT LEMMA Every metric space (X; d) can be isometrically embedded as
a closed subspace of a normed linear space E, where wt E = ! wt X.
[Denote by the collection of all nonempty finite subsets of X.aGivee the d*
*iscrete
topology. Fix a point x0 2 X. Attach to each x 2 X a function fx : oe ! d(!xR*
*; oe)  d(x
ae *
* 0; oe)
_then fx 2 BC() and the assignment : X !xBC()! f is an isometric embedding.
x
Note that fx0 0. Let E be the linear span of (X) in BC(). To see that (X) is c*
*losed
nP
in E, take a OE 2 E  (X), say OE = rifxi (ri real), put oe = {x0; : :;:xn} a*
*nd choose ffi
0
612
positive and less than (1=2) minikOE  fxik. Claim: No element of (X) can be wi*
*thin ffi of
OE. Suppose not, so 9 x 2 X : kOE  fxk < ffi. Since is an isometry,
d(x; xi) = kfxi fxk kOE  fxik  kOE  fxk > 2ffi  ffi = ffi;
from which kOE  fxk OE(oe)  fx(oe) = d(x; oe) ffi, a contradiction. Ther*
*e remains
the assertion on the weights. For this, let D be a dense subset of (X) of card*
*inality
: fx0 2 D_then the linear span of D is dense in E and contains a dense subset*
* of
cardinality !.]
[Note: One can obviously arrange that E is complete provided this is the c*
*ase of
(X; d).]
FACT Every CRH space X can be embedded as a closed subspace of a locally co*
*nvex topological
vector space E.
Let Y be a nonempty metrizable space.
(AR) Y is said to be an absolute_retract_(AR) if under any closed emb*
*edding
Y ! Z into a metrizable space Z, the image of Y is a retract of Z.
(ANR) Y is said to be an absolute_neighborhood_retract_(ANR) if under*
* any
closed embedding Y ! Z into a metrizable space Z, the image of Y is a neighbo*
*rhood
retract of Z.
[Note: There is no map from a nonempty set to the empty set, thus ; cannot*
* be
an AR, but there is a map from the empty set to the empty set, so we shall exte*
*nd the
terminology and agree that ; is an ANR.]
PROPOSITION 5 Let Y be a nonempty metrizable space_then Y is an AR (ANR)
iff for every pair (X; A), where X is metrizable and A X is closed, A has the *
*EP (NEP)
w.r.t. Y .
[The indirect assertion is obvious. Turning to the direct assertion, in vi*
*ew of the
placement lemma, Y can be realized as a closed subspace of a normed linear spa*
*ce E.
Assuming that Y is an AR, fix a retraction r : E ! Y . If now f : A ! Y is a co*
*ntinuous
function, then by the Dugundji extension theorem, 9 F 2 C(X; E) : F A = f. Con*
*sider
r O F .]
EXAMPLE Cautyy has given an example of a linear metric space E which is no*
*t an absolute
retract. So, for this E, the Dugundji extension theorem must fail.
_________________________
yFund. Math. 146 (1994), 8599.
613
[Note: Therefore a metrizable space that has an equiconnecting structure ne*
*ed not be an AR.]
A countable product of nonempty metrizable spaces is an AR iff all the fact*
*ors are
ARs. Example: [0; 1]n; Rn; [0; 1]!, and R ! are absolute retracts. A countable *
*product of
nonempty metrizable spaces is an ANR iff all the factors are ANRs and all but f*
*initely
manyaofethe factors are ARs. Example: S n and T n are absolute neighborhood re*
*tracts
n x Sn x . . .
but STnx Tn x . . .(! factors) are not absolute neighborhood retracts.
Every retract (neighborhood retract) of an AR (ANR) is an AR (ANR). An open
subspace of an ANR is an ANR.
EXAMPLE Let E be a normed linear space_then every nonempty convex subset of*
* E is an AR
and every open subset of E is an ANR. Assume in addition that E is infinite dim*
*ensional. Let S be
the unit sphere in E_then S is an AR. To establish this, it need only be shown *
*that S is a retract of
D, the closed unit ball in E. Fix a proper dense linear subspace E0 E (the ker*
*nel of a discontinuous
linear functional on E will do). In the notation of the Dugundji extension theo*
*rem, work with the pair
(D; S), picking the points defining extin S \ E0, and let f = idS_then there ex*
*ists a continuous function
ext(f) : D ! E such that ext(f)S = idS, with ext(f)(D) contained in S [ (D \ E*
*0), a proper subset of
D. Choose a point p in the interior of D : p =2ext(f)(D), let r : D  {p} ! S b*
*e the corresponding radial
retraction and consider r O ext(f). Corollary: Not every continuous function D *
*! D has a fixed point.
[Note: There is another way to argue. Kleey has shown that if E is an infin*
*ite dimensional normed
linear space and if K E is compact, then E and E  K are homeomorphic. In part*
*icular, E  {0} is
homeomorphic to E, thus is an AR, and so S, being a retract of E  {0}, is an A*
*R. Matters are trivial if
E is an infinite dimensional Banach space, since then E is actually homeomorphi*
*c to S.]
EXAMPLE Let Y be any set lying between ]0; 1[n and [0; 1]n_then Y is an AR.*
* Thus let f
be a closed embedding Y ! Z of Y into a metrizable space Z. Call j the inclusi*
*on Y ! [0; 1]n, so
j O f1 2 C(f(Y ); [0; 1]n). Choose a g 2 C(Z; [0; 1]n) : gf(Y ) = j O f1. Fi*
*x a compatible metric d on
Z and define a continuous function h : Z ! [0; 1]n x [0; 1] by sending z to (g(*
*z); min{1; d(z; f(Y ))}). The
range of h is therefore a subset of i0Y [ [0; 1]nx]0; 1]. Let r : i0Y [ [0; 1]n*
*x]0; 1] ! i0Y be the retraction
determined by projecting from the point (1=2; : :;:1=2; 1) 2 Rn+1 and let p : *
*i0Y ! Y be the canonical
map. That f(Y ) is a retract of Z is then seen by considering the composite f O*
* p O r O h.
FACT Let Y be an AR; let B be a nonempty closed subspace of Y _then B is a*
*n AR iff B is a
strong deformation retract of Y .
_________________________
yProc. Amer. Math. Soc. 7 (1956), 673674.
614
[To see that the condition is necessary,8fix a retraction r : Y ! B and def*
*ine a continuous function
< y (y 2 Y; t = 0)
h : i0Y [ IB [ i1Y ! Y by h(y; t) = y (y 2 B; 0 t .1)Since i0Y [ IB [ i1Y is*
* a closed subspace of
:
r(y) (y 2 Y; t = 1)
IY and since B is an AR, it follows from Proposition 5 that h has a continuous *
*extension H : IY ! Y .]
Let Y be an AR_then Y is homeomorphic to its diagonal Y which is therefore *
*a strong deformation
retract of Y x Y and this means that Y has an equiconnecting structure (cf. p. *
*610).
[Note: A metrizable locally convex topological space is an AR (cf. p. 611 *
*and Proposition 5) but
not every AR is locally convex.]
FACT Let Y be an ANR; let B be a closed subspace of Y _then B is an ANR if*
*f the inclusion
B ! Y is a cofibration.
[If B is an ANR, then so is i0Y [ IB (cf. p. 643 (NES4)), thus there exist*
*s a neighborhood O of
i0Y [ IB in IY and a retractionare: O ! i0Y [ IB. Chooseaaeneighborhood V of B *
*in Y : IV O and
fix OE 2 C(Y; [0; 1]) : OEB = 1 . Consider the map IY ! i0Y [ IB.]
OEY  V = 0 (y; t) ! r(y; OE(y)t)
Let Y be an ANR_then Y is homeomorphic to its diagonal Y , hence the inclus*
*ion Y ! Y x Y
is a cofibration. Consequently, Y is uniformly locally contractible (cf. p. 31*
*4) and 8 y0 2 Y , (Y; y0) is
wellpointed (cf. p. 315).
[Note: It is unknown whether every metrizable uniformly locally contractibl*
*e space is an ANR. Any
counterexample would necessarily have infinite topological dimension (cf. infra*
*).]
Thanks to the placement lemma and the fact that a retract of a contractible*
* (locally
contractible) space is contractible (locally contractible), every AR (ANR) is c*
*ontractible
(locally contractible). Both the broom and the cone over the Cantor set are con*
*tractible
but, failing to be locally contractible, neither is an ANR.
LEMMA Suppose that Y is a contractible ANR_then Y is an AR.
A locally path connected topological space X is said to be locally_nconnec*
*ted_(n 1)
provided that for any x 2 X and any neighborhood U of x there exists a neighbor*
*hood
V U of x such that the arrow ssq(V; x) ! ssq(U; x) induced by the inclusion V*
* ! U
is the trivial map (1 q n). If X is locally nconnected for all n, then X is*
* called
locally_homotopically_trivial_. Example: A locally contractible space is locall*
*y homotopi
cally trivial.
EXAMPLE Working in `2, let pk = (rk(2k + 1); 0; : :):, where rk = 1=2k(k +*
* 1) (k = 1; 2; : :):,
and put p0 = limkpk (= (0; 0; : :):). Denote by Xk(n) the set consisting of tho*
*se points x = {xi} : xi= 0
615
1S
(i > n + 1) and whose distance from pk is rk. The union {p0} [ Xk(n + 1) is *
*locally nconnected but
1 k=1
S
not locally (n + 1)connected, while the union {p0} [ Xk(k) is locally homot*
*opically trivial but not
k=1
locally contractible.
Let Y be a nonempty metrizable space.
(LC n) Y is locally nconnected iff for every pair (X; A), where X is *
*metrizable and A X is
closed with dim(X  A) n + 1, A has the NEP w.r.t. Y .
(C n+ LCn) Y is nconnected and locally nconnected iff for every pair*
* (X; A), where X is
metrizable and A X is closed with dim(X  A) n + 1, A has the EP w.r.t. Y .
Let Y be a nonempty metrizable space of topological dimension n.
(LC n+ dim n) Y is locally nconnected iff Y is locally contractible i*
*ff Y is an ANR.
(C n+ LCn + dim n) Y is nconnected and locally nconnected iff Y is c*
*ontractible and
locally contractible iff Y is an AR.
The proofs of these results can be found in Dugundjiy.
[Note: It follows that a metrizable space of finite topological dimension i*
*s uniformly locally con
tractible iff it is an ANR and has an equiconnecting structure iff it is an AR.]
EXAMPLE (Borsuk's_Cone_) There exists a contractible, locally contractible *
*compact metrizable1
*
* Q
space that is not an ANR. Choose a sequence: 0 = t0 < t1 < . .<.1, limtn = 1. I*
*nside the product [0; 1],
1 *
* 0
Q *
* 1S
for n = 1; 2; : :,:form Yn = [tn1; tn]x[0; 1]nx0x. .,.put Y1 = 1x [0; 1], and*
* let Y = ( frYn)[Y1 _
1 *
* 1
then Y is a compact connected metrizable space which we claim is locally contra*
*ctible yet has nontrivial
singular homology in every dimension, thus is not an ANR (cf. p. 620). Local c*
*ontractibility at the points
of Y  Y1 being obvious, let y1 = (1; y1; : :):2 Y1 and fix a neighborhood U of*
* y1 . There is no loss of
generality in assuming that U is the intersection of Y with a set [a0; 1]x[a1; *
*b1]x. .x.[ak; bk]x[0; 1]x. ...
Consider a neighborhood V of y1 that is the intersection of Y with a set [a0;a1*
*]ex [a1; b1] x . .x.[ak; bk] x
[ak+1; bk+1] x [0; 1] x . .,.where bk+1  ak+1 < 1. There are two cases: 1 =2*
*[ak+1; bk+1]. As both
0 =2*
*[ak+1; bk+1]
are handled in a similar manner, suppose, e.g., that 1 =2[ak+1; bk+1] and defin*
*e a homotopy H : IV ! U
between the inclusion V ! U and the constant map V ! y1 by letting H(v; t) be c*
*onsecutively
8
< (v0; v1; : :;:vk; (1  3t)vk+1; vk+2; : :):
: (3t  1 + (2  3t)v0; v1; : :;:vk; 0; vk+2; : :):
(1; y1 3(1  t)(y1 v1); y2 3(1  t)(y2 v2); : :)::
_________________________
yCompositio Math. 13 (1958), 229246; see also Kodama, Proc. Japan Acad. Sci*
*. 33 (1957), 7983.
616
8
< 0 t 1=3
Here, v = (v0; v1; : :):2 V and 1=3 t 2=3. That Y is not an ANR is seen by *
*remarking that frYn
:
2=3 t 1
is a retract of Y , hence Hn(frYn) Z is isomorphic to a direct summand of Hn(Y*
* ). The cone Y of Y
is a contractible, locally contractible compact metrizable space. And Y , as a *
*closed subspace of Y , is a
neighborhood retract of Y . Therefore Y is not an ANR. Finally, Y is not unifor*
*mly locally contractible,
so Y does not have an equiconnecting structure.
[Note: Other, more subtle examples of this sort are known (DavermanWalshy)*
*.]
FACT Let Y Rn_then Y is a neighborhood retract of Rn iff Y is locally comp*
*act and locally
contractible.
Haverz has shown that if a locally contractible metrizable space Y can be w*
*ritten as a countable
union of compacta of finite topological dimension, then Y is an ANR. Example: E*
*very metrizable CW
complex X is an ANR. Indeed, for this one can assume that X is connected (cf. P*
*roposition 12). But
then X, being locally finite, is necessarily countable, hence can be written as*
* a countable union of finite
subcomplexes.
Certain function spaces or automorphism groups that arise "in nature" turn *
*out to
be ARs or, equivalently, contractible ANRs. Example: Let E be an infinite dimen*
*sional
Hilbert space_then GL (E) is contractible (cf. p. 610). However, GL (E) is a*
*n open
subset of a Banach space, thus is an ANR. Conclusion: GL (E) is an AR.
EXAMPLE (Measurable_Functions_) Let Y be a nonempty metrizable space. Denot*
*e by MY the
set of equivalence classes of Borel measurable functions f : [0; 1] ! Y equippe*
*d with the topology of
convergence in measure_then MY is metrizable, a compatible metric being given b*
*y the assignment
R1
(f; g) ! 0d(f(x); g(x))dx, where d is a compatible metric on Y bounded by 1. N*
*huk has shown that
MY is an ANR.aClaim:eMY is contractible. To see this, fix a point y0 2 Y and co*
*nsider the homotopy
H(f; t)(x) = f(x) (x > t). Therefore MY is an AR.
y0 (x t)
[Note: Take Y = R_then MR is a linear metric space. But its dual M*Ris triv*
*ial, hence MR is not
locally convex.]
EXAMPLE (Measurable_Transformations_) Let be the set of equivalence classe*
*s of measure
_________________________
yMichigan Math. J. 30 (1983), 1730.
zProc. Amer. Math. Soc. 40 (1973), 280284.
kFund. Math. 124 (1984), 243254.
617
preserving Borel measurable bijections fl : [0; 1] ! [0; 1], i.e., let be the *
*automorphism group of the
618
measure algebra A of the unit interval. Equip with the topology of pointwise c*
*onvergence on A_then a
subbasis for the neighborhoods at a fixed fl0 2 is the collection of all sets *
*of the form {fl : flAfl0A < ffl}
(A 2 A & ffl > 0), being symmetric difference. With respect to this topology, *
* is a first countable
Hausdorff topological group, so is metrizable. Nhuyhas shown that is an ANR. *
*Claim: is contractible.
To see this, let B be the complement of A in [0; 1] and assign to each pair (A;*
* fl) its return partition, viz. the
sequence {n}, where 0 = B, 1 = A \ fl1A, and for n 2, n = Aa\efl1B \ . .\.fl*
*(n1)B \ fln A.
Define flA 2 by flA(x) = fln(x) (x 2 n), check that the map A x ! is con*
*tinuous, and consider
(A; fl) ! flA
the homotopy H(t; fl) = fl[t;1]. Therefore is an AR.
[Note: Confining the discussion to the unit interval is not unduly restrict*
*ive since the Halmosvon
Neumann theorem says that every separable, non atomic, normalized measure algeb*
*ra is isomorphic to A.]
Let X be a second countable topological manifold of euclidean dimension n. *
*Denote by
H(X) the set of all homeomorphisms X ! X endowed with the compact open topology_
then H(X) is a topological group (cf. p. 26). Moreover, H(X) is metrizable a*
*nd one
can ask: Is H(X) an ANR? If X is not compact, then the answer is "no" since the*
*re are
examples where H(X) is not even locally contractible (EdwardsKirbyz). If X is *
*compact,
then H(X) is locally contractible (Cernavskiik) and there is some evidence to s*
*upport a
conjecture that H(X) might be an ANR.
[Note: If X is not compact but is homeomorphic to the interior of a compact*
* topo
logical manifold with boundary, then H(X) is locally contractible (Cernavskii(i*
*bid.)). Ex
ample: H(R n) is locally contractible.]
EXAMPLE Take X = [0; 1]_then H([0; 1]) is homeomorphic to R! x {0; 1} (thus*
* is an ANR).
In other words, the claim is that the identity component He([0; 1]) of H([0; 1]*
*) is homeomorphic to R! .
Q1 2nQ
Form the product ]0; 1[n;iand define a homeomorphism between it and He([0;*
* 1]) by assigning to
n=0i=1
a typical string (xn;i) an order preserving homeomorphism OE : [0; 1] ! [0; 1] *
*via the following procedure.
Suppose that n is given and that there have been defined two sets of points
ae
An = {0 = a(n; 0) < a(n; 1) < . .<.a(n; 2n) = 1}
Bn = {0 = b(n; 0) < b(n; 1) < . .<.b(n; 2n) = 1};
with OE(a(n;ai))e= b(n; i). To extend the definition of OE to an order preservi*
*ng bijection An+1 ! Bn+1,
where An+1 An and both have cardinality 2n+1 + 1, distinguish two cases. Ca*
*se 1: n is odd. Let ffi
Bn+1 Bn
_________________________
yProc. Amer. Math. Soc. 110 (1990), 515522.
zAnn. of Math. 93 (1971), 6388.
kMath. Sbornik 8 (1969), 287333; see also Rushing, Topological Embeddings, *
*Academic Press (1973),
270293.
619
be the midpoint of [a(n; i  1); a(n; i)] and set fii= OE(ffi) = xn;i(b(n; i) *
* b(n; i  1)) + b(n; i  1). Case 2:
n is even. Let fiibe theamidpointeof [b(n; i  1); b(n; i)] and set ffi= OE1(f*
*ii) = xn;i(a(n; i)  a(n; i  1))
n}
+a(n; i  1). Define An+1 = An [ {ffi: i = 1; :,:;:2so that in obvious notati*
*on
Bn+1 = Bn [ {fii: i = 1; : :;:2n}
ae
An+1 = {0 = a(n + 1; 0) < a(n + 1; 1) < . .<.a(n + 1; 2n+1) = 1}
Bn+1 = {0 = b(n + 1; 0) < b(n + 1; 1) < . .<.b(n + 1; 2n+1) = 1};
8 1S
>:B = S Bn B
1
preserving bijection, hence admits an extension to an order preserving homeomor*
*phism OE : [0; 1] ! [0; 1].
[Note: H([0; 1]) and H(]0; 1[) are homeomorphic. In fact, the arrow of rest*
*riction H([0; 1]) ! H(]0; 1[)
is continuous and has for its inverse the arrow of extension H(]0; 1[) ! H([0; *
*1]), which is also continuous.
Corollary: H(]0; 1[) is an ANR. Corollary: H(R ) is an ANR.]
EXAMPLE Take X = S1_then H(S1) is homeomorphic to R! xS 1x{0; 1} (thus is a*
*n ANR). To
see this, it suffices to observe that H(S1) is homeomorphic to G x S1, where G *
*is the subgroup of H(S1)
consisting of those OE which fix (1; 0).
Therefore, if X is a compact 1manifold, then H(X) is an ANR. The same conc*
*lusion obtains if X is
a compact 2manifold (LukeMasony) but if n > 2, then it is unknown whether H(X*
*) is an ANR.
EXAMPLE Take X = [0; 1]!, the Hilbert cube_then H(X) (compact open topolog*
*y) is metriz
able and Ferryz has shown that H(X) is an ANR.
LEMMA Let K = (V; ) be a vertex scheme_then Kb is an ANR.
[There are three steps to the proof.
(I) Fix a point * =2V and put V* = V [ {*}. Let * be the set of all no*
*nempty
finite subsets of V*. Call K* the associated vertex scheme. Claim: K*b is an *
*AR. Indeed,
the inclusion K*b ! `1(V*) is an isometric embedding with a convex range.
(II) Let * be the subspace of K*b consisting of O*, the characterist*
*ic function
of {*}, and those OE 6= O* : OE1(]0; 1]) \ V 2 . Claim: * is an AR. To establi*
*sh this, it
suffices to exhibit a retraction r : K*b ! *. Take a OE 2 K*b. Case 1: OE =*
* O*. There is
no choice here: r(O*) = O*. Case 2: OE 6= O*. Suppose that OE1(]0; 1]){*} = {*
*v0; : :;:vn}.
_________________________
yTrans. Amer. Math. Soc. 164 (1972), 275285.
zAnn. of Math. 106 (1977), 101119.
620
Order the vertexes vi so that OE(v0) . . . OE(vn). Denote by k the maximal in*
*dex:
{v0; : :;:vk} 2 and define r(OE) by the following formulas:
( P
r(OE)(*) = 1  r(OE)(v)
v2V
r(OE)(v) = 0 (v 2 V  {v0; : :;:vk})
and ae
k = n : r(OE)(vi) = OE(vi) (0 i k)
k < n : r(OE)(vi) = OE(vi)  OE(vk+1) (0 i k):
One can check that r is welldefined and continuous.
(III) Since *  {O*} is open in *, it is an ANR. Claim: Kb is a retr*
*act of
*  {O*}, hence is an ANR. To see this, consider the map OE ! OE__OE(*)O*_1.]*
*OE(*)
A topological space is said to be a (finite_, countable_) CW_space__if it h*
*as the homotopy
type of a (finite, countable) CW complex. The following theorems characterize t*
*hese classes
in terms of ANRs.
CWANR THEOREM Let X be a topological space_then X has the homotopy type
of a CW complex iff X has the homotopy type of an ANR.
[If X has the homotopy type of a CW complex, then there exists a vertex sch*
*eme K
such that X has the homotopy type of K (cf. x5, Proposition 2) or still, the *
*homotopy
type of Kb (cf. x5, Proposition 1) and, by the lemma, Kb is an ANR. Convers*
*ely, if X
has the homotopy type of an ANR Y , use the placement lemma to realize Y as a c*
*losed
subspace of a normed linear space E. Fix an open U E : U Y and a retraction
r : U ! Y . Since U has the homotopy type of a CW complex (cf. x5, Proposition *
*6), the
domination theorem implies that the same is true of Y .]
COUNTABLE CWANR THEOREM Let X be a topological space_then X has the
homotopy type of a countable CW complex iff X has the homotopy type of a second
countable ANR.
[If X has the homotopy type of a countable CW complex, then there exists a *
*count
able locally finite vertex scheme K such that X has the homotopy type of K (c*
*f. x5,
Proposition 3 and p. 514). Therefore K = Kb is Lindel"of, hence second co*
*untable,
and, by the lemma, Kb is an ANR. Conversely, if X has the homotopy type of a *
*second
countable ANR Y , then the "E" figuring in the preceding argument is second cou*
*ntable,
therefore the "U" has the homotopy type of a countable CW complex (cf. x5, Prop*
*osition
6) and the countable domination theorem can be applied.]
621
FINITE CWANR THEOREM Let X be a topological space_then X has the homo
topy type of a finite CW complex iff X has the homotopy type of a compact ANR.
[One direction is easy: If X has the homotopy type of a finite CW complex, *
*then there
exists a finite vertex scheme K such that X has the homotopy type of K = Kb*
* (cf. x5,
Proposition 3), which, by the lemma, is an ANR. The converse, however, is diffi*
*cult: Its
proof depends on an application of a number of theorems from infinite dimension*
*al topology
(Westy).]
Application: The singular homology groups of a compact ANR are finitely gen*
*erated
and vanish beyond a certain point and the fundamental group of a compact connec*
*ted
ANR is finitely presented.
According to the CWANR theorem, if Y is an ANR, then it and each of its op*
*en subsets has the
homotopy type of a CW complex. On the other hand, it can be shown that every me*
*trizable space with
this property is an ANR (Cautyz).
FACT Let Y be a nonempty metrizable space_then Y is an AR iff Y is a homoto*
*pically trivial
ANR.
[A connected CW complex is homotopically trivial iff it is contractible. Qu*
*ote the CWANR theorem.]
Let X andaYebe topological spaces. Let O = {O} be an open covering of Y _th*
*en two continuous
functions f : X ! Yare said to be Ocontiguous_if 8 x 2 X 9 O 2 O : {f(x); g(*
*x)} O.
g : X ! Y
LEMMA Suppose that Y is anaANR_thenethere exists an open covering O = {O} o*
*f Y such that
for any topological space X : f 2 C(X; YO)contiguous ) f ' g.
g 2 C(X; Y )
[Choose a normed linear space E containing Y as a closed subspace. Fix a ne*
*ighborhood U of Y in
E and a retraction r : U ! Y . Let C = {C} be a covering of U by convex open se*
*ts. Put O = C \ Y .
Take two Ocontiguous functions f and g. Define h : IX ! E by h(x; t) = (1  t)*
*f(x) + tg(x)_then
h(IX) U, so H = r O h is a homotopy IX ! Y between f and g.]
Let X be a topological space, U = {U} an open covering of X. Let K = (V; ) *
*be a vertex scheme_
then a function f : K(0) ! X is said to be confined_by U if 8 oe 2 9 U 2 U :*
* f(oe \ K(0)) U.
_________________________
yAnn. of Math. 106 (1977), 118.
zFund. Math. 144 (1994), 1122.
622
LEMMA Suppose that Y is an ANR. Let O = {O} be an open covering of Y _then *
*there exists
an open refinement P = {P} of O such that for every vertex scheme K = (V; ) and*
* every function
f : K(0) ! Y confined by P there exists a continuous function F : K ! Y suc*
*h that F K(0) = f and
8 oe 2 ; 8 P 2 P : f(oe \ K(0)) P ) 9 O 2 O : F(oe) [ P O.
[Choose a normed linear space E containing Y as a closed subspace. Fix a ne*
*ighborhood U of Y in
E and a retraction r : U ! Y . Let C = {C} be a refinement of r1(O) consisting*
* of convex open sets. Put
P = C \ Y _then P is an open refinement of O which we claim has the properties *
*in question. Thus let
K = (V; ) be a vertex scheme. Take a function f : K(0) ! Y confined by P. Giv*
*en oe 2 , write Coefor
the convex hull of f(oe\K(0)), itself a subset of some element C 2 C. Const*
*ruct by induction continuous
functions n : K(n) ! U subject to: 0 = f; n+1 K(n) = n, and 8 oe 2 ; n(oe*
* \ K(n)) Coe.
Here the point is that if n has been constructed and if oe is an (n + 1)simple*
*x, then oe  K(n),
therefore the restriction of n to oe  can be continuously extended to *
*oe, Coebeing an AR. This
done, define : K ! U by  K(n) = n. Since each n is continuous, so is . Co*
*nsider F = r O .]
These lemmas can be used to prove that if Y is an ANR of topological dimens*
*ion n, then Y is
dominated in homotopy by K, where K is a vertex scheme: dimK n, a result not*
* directly implied
by the CWANR theorem. In succession, let O be an open covering of Y per the fi*
*rst lemma, let P be
an open refinement of O per the second lemma, and let Q be a neighborhood finit*
*e star refinement of P
(cf. x1, Proposition 13)_then Q has a precise open refinement V of order n + 1*
* (cf. x19, Proposition
6). Obviously, dimN(V) n, N(V) the nerve of V. Fix a point yV in each V 2 N*
*(V)(0). Define
f : N(V)(0) ! Y by f(OV ) = yV . Claim: f is confined by P. For suppose that *
*oe = {V1; : :;:Vk} is
a simplex of N(V). Since V1 \ . .\.Vk 6= ; and since V is a star refinement of *
*P, there exists P 2 P :
V1 [ . .[.Vk P ) f(oe \ N(V)(0)) P. Now take F : N(V) ! Y as above and *
*choose a Vmap
G : Y ! N(V) (cf. p. 53). One can check that F O G and idYare Ocontiguous, *
*hence homotopic.
[Note: By analogous arguments, if Y is a compact (connected) ANR of topolog*
*ical dimension n,
then Y is dominated in homotopy by K, where K is a vertex scheme: dimK n and*
* K is compact
(connected).]
Application: Let Y be an ANR of topological dimension n_then the singular *
*homology groups of
Y vanish in all dimensions > n.
EXAMPLE Suppose that Y is a compact connected ANR: dimY = 1 & ss1(Y ) 6= 1_*
*then ss1(Y ) is
finitely generated and free. Consequently, Y has the homotopy type of a finite *
*wedge of 1spheres.
There are two variants of the CWANR theorem.
(Paired Version) A CW_pair_ is a pair (X; A), where X is a CW complex *
*and
A X is a subcomplex; an ANR_pair_is a pair (Y; B), where Y is an ANR and B Y *
*is
623
closed and an ANR. Working then in the category of pairs of topological spaces,*
* the result
is that an arbitrary object in this category has the homotopy type of a CW pair*
* iff it has
the homotopy type of an ANR pair.
(Pointed Version) A pointed_CW_complex_ is a pair (X; x0), where X is *
*a CW
complex and x0 2 X(0); a pointed_ANR_ is a pair (Y; y0), where Y is an ANR and *
*y0 2 Y .
Working then in the category of pointed topological spaces, the result is that *
*an arbitrary
object in this category has the homotopy type of a pointed CW complex iff it ha*
*s the
homotopy type of a pointed ANR.
[Note: There is also a CWANR theorem for the category of pointed pairs of *
*topolog
ical spaces.]
In HTOP 2, the relevant reduction is that if (X; A) is a CW pair, then the*
*re exists a vertex scheme
K and a subscheme L such that (X; A) (K; L), while in HTOP *, the relevan*
*t reduction is that if
(X; x0) is a pointed CW complex, then there exists a vertex scheme K and a vert*
*ex v0 2 V such that
(X; x0) (K; v0) (cf. p. 512).
Convention: The function spaces encountered below carry the compact open to*
*pology.
LEMMA Let X; Y , and Z be topological spaces.
(i) Let f 2 C(X; Y )_then the homotopy class of the precomposition ar*
*row
f* : C(Y; Z) ! C(X; Z) depends only on the homotopy class of f.
(ii) Let g 2 C(Y; Z)_then the homotopy class of the postcomposition ar*
*row
g* : C(X; Y ) ! C(X; Z) depends only on the homotopy class of g.
Application: The homotopy type of C(X; Y ) depends only on the homotopy typ*
*es of
X and Y .
[Note: By the same token, in TOP 2 the homotopy type of (C(X; A; Y; B); C(*
*X; B))
depends only on the homotopy types of (X; A) and (Y; B), whereas in TOP *the h*
*omotopy
type of C(X; x0; Y; y0) depends only on the homotopy types of (X; x0) and (Y; y*
*0).]
PROPOSITION 6 Let K be a nonempty compact metrizable space; let Y be a metr*
*iz
able space_then C(K; Y ) is an ANR iff Y is an ANR.
[Necessity: Assuming that Y is nonempty, embed Y in C(K; Y ) via the assign*
*ment
y ! j(y), where j(y) is the constant map K ! y. Fix a point k0 2 K and denote *
*by
e0 : C(K; Y ) ! Y the evaluation OE ! OE(k0). Because j O e0 is a retraction *
*of C(K; Y )
onto j(Y ), it follows that if C(K; Y ) is an ANR, then so is Y .
624
Sufficiency: Let (X; A) be a pair, where X is metrizable and A X is closed*
*. Let
f : A ! C(K; Y ) be a continuous function. Define a continuous function OE : A *
*x K ! Y
by setting OE(a; k) = f(a)(k). Since Y is an ANR, there is a neighborhood O of *
*A x K in
X x K and a continuous function : O ! Y with A x K = OE. Fix a neighborhood
U of A in X : U x K O. Define a continuous function F : U ! C(K; Y ) by setti*
*ng
F (u)(k) = (u; k). Obviously, F A = f, thus C(K; Y ) is an ANR (cf. Propositio*
*n 5).]
Keeping to the above notation, the compactness of K implies that ss0(C(K; Y*
* )) =
[K; Y ]. Assume in addition that Y is separable_then C(K; Y ) is separable. But*
* C(K; Y )
is also an ANR, hence its path components are open. Conclusion: #[K; Y ] !.
Here is another corollary. Suppose that X is a finite CW space_then, on the*
* basis of
the CWANR theorem, for any CW space Y , C(X; Y ) has the homotopy type of an A*
*NR,
hence is again a CW space.
[Note: Some assumption on X is necessary. Example: Give {0; 1} the discrete*
* topology
and consider {0; 1}!.]
EXAMPLE Let X be a topological space_then the free_loop_space_X of X is de*
*fined by the
X ! PX
? ?
pullback square y y , where is the Hurewicz fibration oe ! (oe(0); *
*oe(1)) and X ! X xX
X ! X x X
is the diagonal embedding. The arrow X ! X is a Hurewicz fibration and its fibe*
*r over x0 is (X; x0),
so if X is path connected, then the homotopy type of (X; x0) is independent of *
*the choice of x0. Since
X can be identified with C(S1; X) (compact open topology), the free loop space *
*of X is a CW space
when X is a CW space.
W1G *
* ! PX1G
? *
* ?
[Note: Given a topological group G, define W1Gby the pullback square y *
* y ,
X1Gx G*
* ! X1Gx X1G
where (x; g) = (x; x . g)_then W1G=G can be identified with B1Gand there is a w*
*eak homotopy equiv
alence B1G! (X1Gx G)=G (the action of G on itself being by conjugation).]
EXAMPLE Suppose that X and Y are path connected CW spaces for which there *
*exists an n
such that (i) X has the homotopy type of a locally finite CW complex with a fin*
*ite nskeleton and (ii)
ssq(Y ) = 0 (8 q > n)_then C(X; Y ) is a CW space.
[Take X to be a locally finite CW complex with a finite nskeleton X(n). On*
*e can assume that n is
> 0 because when n = 0, Y is contractible and the result is trivial. Consider t*
*he inclusion i : X(n)! X_
then the precomposition arrow i* : C(X; Y ) ! C(X(n); Y ) is a Hurewicz fibrati*
*on (cf. x4, Proposition 6)
625
and, in view of the assumption on Y , its fibers are either empty or contractib*
*le. But C(X(n); Y ) is a CW
space, thus so is C(X; Y ) (cf. Proposition 11).]
PROPOSITION 7 Let K be a nonempty compact metrizable space, L K a nonempty
closed subspace; let Y be a metrizable space, Z Y a closed subspace. Suppose t*
*hat Y is
an ANR_then C(K; L; Y; Z) is an ANR iff Z is an ANR.
[Assuming that Z is nonempty, one may proceed as in the proof of Propositio*
*n 6
and show that Z is homeomorphic to a retract of C(K; L; Y; Z), from which the n*
*ecessity.
Consider now a pair (X; A), where X is metrizable and A X is closed. Let f : *
*A !
C(K; L; Y; Z) be a continuous function. Define a continuous function OE : A x L*
* ! Z by
setting OE(a; `) = f(a)(`). Since Z is an ANR, there is a neighborhood O of A x*
* L in X x L
and a continuous function : O ! Z with A x L = OE. Fix a neighborhood U of A
__ __
inaXe: U x L O. Define a continuous function : A x K [ U x L ! Y by setting
(a; k) = f(a)(k) __
(u; `) = (u; `). Since Y is an ANR, there is a_neighborhood P of A x K [ U *
*x L in
XxK and a continuous function : P ! Y with AxK[U xL = . Fix a neighborhood
V of A in X : V x K P & V U. Define a continuous function F : V ! C(K; L; Y; *
*Z)
by setting F (v)(k) = (v; k). Obviously, F A = f, thus C(K; L; Y; Z) is an AN*
*R (cf.
Proposition 5).]
Take, e.g., (K; L) = (S n; sn) (sn = (1; 0; : :;:0) 2 R n+1; n 1) and let *
*y0 2 Y _
then ssn(Y; y0) = ss0(C(S n; sn; Y; y0)). Accordingly, if Y is separable, the*
*n ssn(Y; y0) is
countable. Example: The homotopy groups of a countable connected CW complex a*
*re
countable.
LOOP SPACE THEOREM Let (X; x0) be a pointed CW space_then the loop space
(X; x0) is a pointed CW space.
[Fix a pointed ANR (Y; y0) with the pointed homotopy type of (X; x0) (cf. *
*p. 6
22)_then (Y; y0) = C(S 1; s1; Y; y0) is a pointed ANR (cf. Proposition 7), so (*
*X; x0) =
C(S 1; s1; X; x0) is a pointed CW space.]
EXAMPLE Suppose that (X; x0) is path connected and numerably contractible.*
* Assume: X is
a CW space_then X is a CW space. Thus let f : K ! X be a pointed CW resolution.*
* Owing to the
loop space theorem, K is a CW space. But the arrow f : K ! X is a weak homotopy*
* equivalence
and since X is a CW space, it follows from the realization theorem that f is a *
*homotopy equivalence.
Therefore f is a homotopy equivalence (cf. p. 427).
[Note: Let X be the Warsaw circle_then X is not a CW space. On the other ha*
*nd, there exists a
continuous bijection OE : [0; 1[! X which is a regular Hurewicz fibration. As t*
*his implies that OE is a pointed
626
Hurewicz fibration (cf. p. 414), X has the same pointed homotopy type as [0; 1*
*[ (cf. p. 435), hence
is a CW space, so X is not numerably contractible.]
EXAMPLE (Classifying_Spaces_) Let G be a topological group_then B1Gis pat*
*h connected and
numerably contractible (inspect the Milnor construction). Moreover, according t*
*o x4, Proposition 36, G
and B1G have the same homotopy type. Taking into account the preceding example,*
* it follows that if
G is a CW space, then the same is true of B1G. Corollary: Any classifying space*
* for G is a CW space
provided that G itself is a CW space.
LEMMA Let X f!Z g Y be a 2sink. Assume: X; Y , and Z are ANRs_then Wf;g
is an ANR.
PROPOSITION 8 Let X f!Z g Y be a 2sink. Assume: X; Y , and Z are CW
spaces_then Wf;gisaaeCW space. ae ae
0 OE : X0 ! X f0 = f O OE
[Fix ANRs XY 0, homotopy equivalences : Y 0! Y , and put g0= g O _*
*then
0 g0
X0? f! Z  Y?0
there is a commutative diagram OyE k y , thus the arrow Wf0;g0! W*
*f;gis
X !f Z g Y
a homotopy equivalence (cf. p. 425). Choose a homotopy equivalence i : Z ! Z0,*
* where
Z0 is an ANR. There is an arrow Wf0;g0! WiOf0;iOg0and it too is a homotopy equi*
*valence.
But from the lemma, WiOf0;iOg0is an ANR.]
For a case in point, let X and Y be CW spaces_then 8 f 2 C(X; Y ), Wf is a *
*CW
space, and 8 f 2 C(X; x0; Y; y0), Ef is a CW space.
FACT Let p : X ! B be a regular Hurewicz fibration. Assume: 9 b0 2 B such *
*that (B; b0) and
Xb0are CW spaces_then 8 x0 2 Xb0, (X; x0) is a CW space.
[By regularity, there is a lifting function 0 : Wp ! PX with the property t*
*hat 0(x; o) 2 j(X)
whenever o 2 j(B). Define f : (B; b0) ! Xb0 by f(o) = 0(x0; o)(1), so f(j(b0)) *
*= x0. The mapping
fiber Ef of f has the same homotopy type as (X; x0).]
PROPOSITION 9 Suppose that p : X ! B is a Hurewicz fibration and let 0 2
C(B0; B). Assume: X; B, and B0 are CW spaces_then X0 = B0xB X is a CW space.
[In view of the preceding proposition, this follows from x4, Proposition 18*
*.]
Application: Let p : X ! B be a Hurewicz fibration, where X and B are CW
spaces_then 8 b 2 B, Xb is a CW space.
627
p0
[Note: Let X be a CW space. Relative to a base point, work first with P X !*
* X to
see that X is a CW space and then consider X p1!X to see that X is a CW space,
thereby obtaining an unpointed variant of the loop space theorem.]
PROPOSITION 10 Suppose that p : X ! B is a Hurewicz fibration and let O B.
Assume: X is an ANR, B is metrizable, and the inclusion O ! B is a closed cofib*
*ration_
then XO is an ANR.
[The inclusion XO ! X is a closed cofibration (cf. x4, Proposition 11), a c*
*ondition
which is characteristic (cf. p. 614).]
Application: Let p : X ! B be a Hurewicz fibration, where X and B are ANRs_*
*then
8 b 2 B, Xb is an ANR.
[Given b 2 B, the inclusion {b} ! B is a closed cofibration (cf. p. 614).]
EXAMPLE Let (Y; B; b0) be a pointed pair. Assume: Y and B are ANRs, with B*
* Y closed.
Let (Y; B) be the subspace of Y consisting of those o such that o(1) 2 B_then (*
*Y; B) is an ANR.
(Y; B) ! Y
? ?
In fact, Y is an ANR and there is a pullback square y yp1.
B ! Y
EXAMPLE Take Y = Sn x Snx . .(.! factors), y0 = (sn; sn; : :):_then Y is n*
*ot an ANR.
Nevertheless, for every pair (X; A), where X is metrizable and A X is closed, *
*A has the HEP w.r.t. Y
(cf. p. 641). Therefore Y is an AR. Still, Y is not an ANR. Indeed, none of th*
*e fibers of the Hurewicz
fibration p1 : Y ! Y is an ANR.
PROPOSITION 11 Suppose that p : X ! B is a Hurewicz fibration. Assume: B is
a CW space and 8 b 2 B; Xb is a CW space_then X is a CW space.
[Fix a CW resolution f : K ! X. Consider the Hurewicz fibration q : Wf ! X(*
*f =
q O s). Since s : K ! Wf is a homotopy equivalence, Wf is a CW space. Moreover,*
* q is
a weak homotopy equivalence and the composite p O q : Wf ! B is a Hurewicz fibr*
*ation.
The fibers (p O q)1(b) = q1 (Xb) are therefore CW spaces. Comparison of the h*
*omotopy
sequences of p O q and p shows that the arrow qb : q1 (Xb) ! Xb is a weak homo*
*topy
equivalence, hence a homotopy equivalence. Because B is numerably contractible *
*(being
a CW space), one can then apply x4, Proposition 20 to conclude that q : Wf ! X *
*is a
homotopy equivalence.]
[Note: If p : X ! B is a Hurewicz fibration and if X and the Xb are CW spac*
*es, then
it need not be true that B is a CW space (consider the Warsaw circle).]
628
Let p : X ! B be a Hurewicz fibration, where X is metrizable and B and the *
*Xbare ANRs. Question:
Is X an ANR? While the answer is unknown in general, the following lemma implie*
*s that the answer is
"yes" provided that the topological dimension of X is finite (cf. p. 615). Inf*
*inite dimensional results can
be found in Ferryy.
LEMMA Suppose that p : X ! B is a Hurewicz fibration. Assume: B is an ANR *
*and 8 b 2 B,
Xb is locally contractible_then X is locally contractible.
[Fix x0 2 X, put b0 = p(x0), and let U be any neighborhood of x0. Since p h*
*as the slicing structure
property (cf. p. 414), it is an open map.aeAccordingly, one can assume at the*
* outset that there is a
continuous function : p(U) ! PB such that (b)(0) = b& (b0)(t) = b0 (0 t 1)*
*. Using the local
(b)(1) = b0
contractibilityaofeXb0, choose a neighborhood O0 U\Xb0of x0in Xb0and a homotop*
*y OE : IO0 ! U\Xb0
satisfying OE(x; 0) =(xu0 2 U \ Xb0). Fix a neighborhood U0 of x0 : U0 U and*
* O0 = U0\ Xb0. Let
OE(x; 1) = u0
0 : Wp ! PX be a lifting function with the property that 0(x; o) 2 j(X) wheneve*
*r o 2 j(B). Define
F 2 C(U; PX) by F(x) = 0(x; (p(x))). Because F(x0) = j(x0) 2 {oe 2 PX : oe([0; *
*1]) U0}, there is
a neighborhood V U0aofex0 such that 8 x 2 V , F(x)(t)a2eU0 (0 t 1). If now H*
* : IV ! U is the
homotopy H(x; t) = F(x)(2t) (0 t 1=2), then H(x; 0) = x, i.e., the *
*inclusion V ! U
OE(F(x)(1); 2t (1)1=2 t 1) H(x; 1) = u0
is inessential.]
Let Y be a metrizable space. Suppose that Y admits a covering V by pairwise*
* disjoint
open sets V , each of which is an ANR_then Y is an ANR. To see this, assume th*
*at Y
is realized as a closed subspace of a metrizable space Z. Fix a compatible metr*
*ic d on Z.
Given a nonempty V 2 V, put OV = {z : d(z; V ) < d(z; Y  V )}_then OV is open *
*in Z
and OV \ Y = V . Moreover, the OV are pairwise disjoint. By hypothesis, there e*
*xists an
S
open subset UV of OV containing V and a retraction rV : UV ! V . Form U = UV *
*, a
V
neighborhood of Y in Z, and define a retraction r : U ! Y by rUV = rV .
What is less apparent is that the same assertion is still true if the V are*
* not pairwise
disjoint.
LEMMA Let Y be a metrizable space. Suppose that Y = Y1 [ Y2, where Y1 and Y2
are open and ANRs_then Y is an ANR.
[This is proved in a more general context on p. 643 (cf. NES3).]
PROPOSITION 12 Let Y be a metrizable space. Suppose that Y admits a covering
V by open sets V , each of which is an ANR_then Y is an ANR.
_________________________
yPacific J. Math. 75 (1978), 373382.
629
[Use the domino principle (cf. p. 124).]
Application: Every metrizable topological manifold is an ANR, hence by the*
* CW
ANR theorem has the homotopy type of a CW complex.
In particular, every compact topological manifold is an ANR, hence by the f*
*inite CW
ANR theorem has the homotopy type of a finite CW complex. If X and Y are finite*
* CW
complexes, then #[X; Y ] ! (cf. p. 623). Specializing to the attaching proc*
*ess (and
recalling that the inclusion Sn1 ! D n is a closed cofibration), it follows th*
*at the set of
homotopy types of compact topological manifolds is countable.
[Note: One can even prove that the set of homeomorphism types of compact to*
*polog
ical manifolds is countable (CheegerKistery).]
The use of the term "set" in the above is justified by remarking that the f*
*ull subcategory of TOP
whose objects are the compact topological manifolds has a small skeleton.
EXAMPLE Let p : X ! B be a covering projection. Suppose that X is metrizab*
*le and B is an
ANR_then X is an ANR.
[Note: The assumption that X is metrizable is superfluous.]
EXAMPLE Let p : X ! B be a Hurewicz fibration. Assume: X is an ANR and B i*
*s a path
connected, numerably contractible, paracompact Hausdorff space_then B is an ANR*
*. For let O be an
open subset of B with the property that the inclusion O ! B is inessential, say*
* homotopic to O ! b.
Since XO is fiber homotopy equivalent to O xXb (cf. p. 424), secO(XO ) is none*
*mpty (cf. x4, Proposition
1), so O is homeomorphic to a retract of XO , an ANR. Therefore B is locally an*
* ANR, hence an ANR
(recall that locally metrizable + paracompact ) metrizable; cf. p. 119).
EXAMPLE Let X be an aspherical compact topological manifold. Assume: O(X) *
*6= 0_then the
path component of the identity in C(X; X) is contractible.
[Since C(X; X) is an ANR (cf. Proposition 6), the path component of the ide*
*ntity in C(X; X) is a
K(Cen ss; 1) (cf. p. 530 ff.), where ss = ss1(X). On the other hand, the assum*
*ption O(X) 6= 0 implies that
Cen ss is trivial.]
Let X and Y be metrizable spaces. Let A be a closed subspace of X and let *
*f : A ! Y be a
continuous function_then Borgesz has shown that X tf Y is metrizable iff every *
*point of X tf Y belongs
_________________________
yTopology 9 (1970), 149151.
zProc. Amer. Math. Soc. 24 (1970), 446451.
630
to a compact subset of countable character, i.e., having a countable neighborho*
*od basis in X. In particular,
this condition is satisfied if X tf Y is first countable or if A is compact.
[Note: In any event, X tf Y is a perfectly normal paracompact Hausdorff spa*
*ce (AD 5(cf. p. 31)).]
LEMMA Let B be a closed subspace of a metrizable space Y such that the inc*
*lusion B ! Y is a
cofibration. Suppose that B and Y  B are ANRs_then Y is an ANR.
[Fix a Strom structure ( ; ) on (Y; B) and put V = 1([0; 1[). Show that V*
* is an ANR.]
FACT Let X and Y be ANRs. Let A be a closed subspace of X and let f : A ! *
*Y be a continuous
function. Suppose that A is an ANR_then X tf Y is an ANR provided that it is me*
*trizable.
LEMMA Let B be a closed subspace of a metrizable space Y such that the inc*
*lusion B ! Y is a
cofibration. Suppose that B is an AR and Y B is an ANR_then Y is an AR if B is*
* a strong deformation
retract of Y .
[It follows from the previous lemma that Y is an ANR. But Y and B have the *
*same homotopy type
and B is contractible.]
FACT Let X and Y be ARs. Let A be a closed subspace of X and let f : A ! Y*
* be a continuous
function. Suppose that A is an AR_then X tf Y is an AR provided that it is metr*
*izable.
EXAMPLE Take X = [0; 1]2, A = [1=4; 3=4]x{1=2}, Y = [0; 1]3and let f : A !*
* Y be a continuous
surjective map_then X tf Y is a compact AR of topological dimension 3, yet it i*
*s not homeomorphic to
any CW complex.
Let (X; A) be a CW pair. Is it true that A has the EP w.r.t. any locally co*
*nvex topological vector
space? A priori, this is not clear since CW complexes are not metrizable in gen*
*eral. There is, however, a
class of topologically significant spaces, encompassing both the class of metri*
*zable spaces and the class of
CW complexes for which a satisfactory extension theory exists.
Let X be a Hausdorff space; let o be the topology on X_then X is said to be*
* stratifiable_if there
*
* _________
exists a function STX : N x o ! o, termed a stratification_, such that (a) 8 U *
*2 o, STX (n; U) U; (b)
S
8 U 2 o, STX (n; U) = U; (c) 8 U; V 2 o : U V ) STX(n; U) STX(n; V ). A str*
*atifiable space is
n
perfectly normal and every subspace of a stratifiable space is stratifiable. A *
*finite or countable product of
stratifiable spaces is stratifiable. A stratifiable space need not be compactly*
* generated and a compactly
generated space need not be stratifiable, even if it is regular and countable (*
*Fogedy). Example: Every
_________________________
yProc. Amer. Math. Soc. 81 (1981), 337338; see also Proc. Amer. Math. Soc. *
*92 (1984), 470472.
631
metrizable space is stratifiable. Example: The Sorgenfrey line, the Niemytzki p*
*lane, and the Michael line
are not stratifiable.
[Note: Junnilay has shown that every topological space is the open image of*
* a stratifiable space.]
FACT Let X be a topological space; let A = {Aj : j 2 J} be an absolute clos*
*ure preserving closed
covering of X. Suppose that each Aj is stratifiable_then X is stratifiable.
[X is necessarily a perfectly normal Hausdorff space (cf. p. 54). As for s*
*tratifiability, consider the
S
set P of all pairs (I; STI), where I J and STIis a stratification of XI = Ai*
*. Order P by stipulating
i
that (I0; STI0) (I00; STI00) iff I0 I00and for each_open_subset U of_XI00:____
STI00(n; U) \ XI0= STI0(n; U \ XI0) & STI00(n;\U)XI0= STI0(n;.U \ *
*XI0)
Every chain in P has an upper bound, so by Zorn, P has a maximal element (I0; S*
*TI0). Verify that
XI0= X.]
Application: Every CW complex is stratifiable.
[The collection of finite subcomplexes of a CW complex X is an absolute clo*
*sure preserving closed
covering of X.]
Application: Let E be a vector space over R. Equip E with the finite topolo*
*gy_then E is stratifiable.
[Fix a basis {ei: i 2 I} for E. Assign to each finite subset of I the span *
*of the corresponding ei. The
resulting collection of linear subspaces is an absolute closure preserving clos*
*ed covering of E.]
FACT Suppose that X and Y are stratifiable_then the coarse join X *cY is s*
*tratifiable.
Application: Let G be a stratifiable topological group_then 8 n; XnGis stra*
*tifiable.
1S
LEMMA Let X = Xn be a topological space, where Xn Xn+1 and Xn is strati*
*fiable and a
0
zero set in X, say Xn = OE1n(0) (OEn 2 C(X; [0; 1]). Suppose that there is a r*
*etraction rn : OE1n([0; 1[) ! Xn
such that 8 x 2 Xn  Xn1 (X1 = ;), the sets r1n(U) \ OE1n([0; t[) form a ne*
*ighborhood basis of x in X
(U a neighborhood of x in Xn and 0 < t 1)_then X is stratifiable.
[The assumptions imply that X is Hausdorff. To construct STX , fix a strati*
*fication STXn of Xn :
STXn (k; U) STXn(k + 1; U). Given an open subset U of X, denote by U(n; k) the*
* interior of
x 2 Xn : r1n(x) \ OE1n([0; 1=(k + 1)[) U
in Xn and for N = 1; 2; : :;:put[
STX (N; U) = r1n(STXn (N; U(n; k))) \ OE1n([0; 1=(k + 2)[*
*):]
n;kN
_________________________
yColloq. Math. Soc. Janos Bolyai 23 (1980), 689703; see also Harris, Pacifi*
*c J. Math. 91 (1980),
95104.
632
EXAMPLE (Classifying_Spaces_) Let G be a stratifiable topological group_*
*then X1G and B1G
are stratifiable.
[Since the XnGare stratifiable, the lemma can be used to establish the stra*
*tifiability of X1G. As for
B1G, in the notation of the Milnor construction, X1GOiis homeomorphic to OixG,*
* thus Oiis stratifiable
and so B1G admits a neighborhood finite closed covering by stratifiable subspac*
*es, hence is stratifiable.]
FACT Let X and Y be stratifiable. Let A be a closed subspace of X and let *
*f : A ! Y be a
continuous function_then X tf Y is stratifiable.
Application: Suppose that (X; A) is a relative CW complex. Assume: A is str*
*atifiable_then X is
stratifiable.
Let X be a topological space; let S and T be collections of subsets of X_th*
*en S is said to be cushioned_
__________
in T if there exists a function : S ! T such that 8 S0 S : [{S : S 2 S0} [{(*
*S) : S 2 S0}. For
example, if S is closure preserving, then S is cushioned in __S. A collection *
*S which is the union of a
countable number of subcollection Sn, each of which is cushioned in T , is said*
* to be oecushioned_in T .
Michaely has shown that a CRH space X is paracompact iff every open coverin*
*g of X has a oe
cushioned open refinement (cf. p. 13). This result can be used to prove that s*
*tratifiable spaces are para
compact. For suppose that U = {U} is an open covering of X. Put Un = {STX (n; U*
*) : U.2LU}et U0
__________ _________*
*_________
U_then 8 U 2 U0, STX(n; U) STX(n; [U0) STX (n; [U0) [U0, from which [{STX (n;*
* U) : U 2}U0
[U0, thus Un is cushioned in U and so U has a oecushioned open refinement. Th*
*erefore X is paracom
pact. Example: A nonmetrizable Moore space is not stratifiable (Bing (cf. p. 1*
*18)).
[Note: Another way to argue is to show that every stratifiable space is col*
*lectionwise normal and
subparacompact (cf. x1, Proposition 10 and the ensuing remark).]
Let X be a CRH space_then X is said to satisfy Arhangel'skii's_condition_if*
* there exists a sequence
*
* T
{Un} of collections of open subsets of fiX such that each Un covers X and 8 x 2*
* X : st(x; Un) X.
*
* n
Example: Every topologically1complete CRH space X satisfies Arhangel'skii's con*
*dition. In fact X is a Gffi
T
in fiX, thus X = Un (Un open in fiX) and so we can take Un = {Un}. Example: E*
*very Moore space
1
satisfies Arhangel'skii's condition.
FACT Let X be a CRH space. Suppose that X satisfies Arhangel'skii's condit*
*ion_then X is
compactly generated.
_________________________
yProc. Amer. Math. Soc. 10 (1959), 309314.
633
Let X be a CRH space_then Kullmany has shown that X is Moore iff X is subme*
*tacompact, has
a Gffidiagonal, and satisfies Arhangel'skii's condition. Since a stratifiable s*
*pace is paracompact and has a
perfect square, it follows that every stratifiable space satisfying Arhangel'sk*
*ii's condition is metrizable (Bing
(cf. p. 118)). Consequently, a nonmetrizable stratifiable space cannot be embe*
*dded in a topologically
complete stratifiable space. Example: Every stratifiable LCH space is metrizabl*
*e.
A Hausdorff space X is said to satisfy Ceder's_condition_if X has a oeclos*
*ure preserving basis. Ex
ample: Suppose that X is metrizable_then X satisfies Ceder's condition. Reason:*
* The NagataSmirnov
metrization theorem says that a regular Hausdorff space X is metrizable iff X h*
*as a oeneighborhood finite
basis. On the other hand, every CW complex satisfies Ceder's condition (cf. inf*
*ra) and a CW complex is
not in general metrizable.
FACT Let X be a Hausdorff space. Suppose that X is the closed image of a me*
*trizable space_then
X satisfies Ceder's condition.
S
Any X that satisfies Ceder's condition is stratifiable. Proof: Let O = On*
* be a oeclosure preserving
S __ n_
basis for X, attach to each closed A X : O(n; A) = X  {O : O 2 On & A \ O = *
*;} and then define
__________
STX : N x o ! o by setting STX(n; U) = X  O(n; X  U).
[Note: It is unknown whether the converse holds.]
EXAMPLE (M_complexes_) A topological space is said to be an M0_space_if it *
*is metrizable and,
recursively, a topological space is said to be an Mn+1_space_if it is homeomorp*
*hic to an adjunction X tfY ,
where X is an M0 space and Y is an Mn space. An M1_space_is a topological space*
* that is an Mn space
for some n.
A(topologicalSspace X is said to be an M_complex_if there exists a sequence*
* of closed M1 subspaces
X = Aj
Aj : j and the topology on X is the final topology determined by the i*
*nclusions Aj ! X.
Aj Aj+1
Example: Every CW complex is an M complex. Since an M complex is the quotient o*
*f a metrizable space,
an M complex is necessarily compactly generated. Therefore a subspace of an M c*
*omplex is an M complex
iff it is compactly generated. Every M complex satisfies Ceder's condition, hen*
*ce is stratifiable.
[Note: Not every CW complex is the closed image of a metrizable space.]
DUGUNDJI EXTENSION THEOREM Let X be a stratifiable space; let A be aaclosed*
*esub
space of X. Let E be a locally convex topological vector space. Equip C(A; *
*E)with the compact
C(X; *
*E)
_________________________
yProc. Amer. Math. Soc. 27 (1971), 154160.
634
open topology_then there exists a linear embedding ext: C(A; E) ! C(X; E) such *
*that 8 f 2 C(A; E),
ext(f)A = f and the rangeaofeext(f) is contained in the convex hull of the ran*
*ge of f.
[Normalize STX : STX(n; X) = X & STX (n; U) STX(n+1; U). Given x 2 U,*
* let n(x; U)
STX(1; X  {x}) = ; ____*
*______________
be the smallest integer n : x 2 STX (n; U). Put U(x) = STX (n(x; U); U)  STX *
*(n(x; U); X ,{x})a
neighborhoodaofex. Plainly, U(x) \ V (y) 6= ; & n(x; U) n(y; V ) ) y 2 U. On*
* the other hand,
n(x; X) = 1) {U : y 2 U(x)} 6= ;. Assuming that A is nonempty and proper, att*
*ach to each x 2
X(x) = X
X A : n(x) = max{n(a; O)(O 2 o) : a 2 A & x 2 O(a)}_then n(x) < n(x; X A). Si*
*nce every subspace
of X is stratifiable, X A is, in particular, paracompact. Thus the open coveri*
*ng {(X A)(x) : x 2 X A}
has a neighborhood finite open refinement {Ui : i 2 I}. Each Ui determines a po*
*int xi 2 X  A : Ui
(X  A)(xi), from which a point ai2 A and a neighborhood Oi of ai: xi2 Oi(ai) &*
* n(xi) = n(ai; Oi).
Choose a partition of unity {i: i 2 I} on X  A subordinate to {Ui: i 2 I}. Giv*
*en f 2 C(A; E), let
(
f(x) (x 2 A)
ext(f)(x) = P i(x)f(ai) (x 2 X  A):
i
Referring back to the proof of the Dugundji extension theorem in the metrizable*
* case and eschewing the
obvious, it is apparent that there are two nontrivial claims.
Claim 1: ext(f) is continuous at the points of A.
[Let a 2 A; let N be a convex neighborhood of f(a) in E. By the continuity *
*of f, there exists a
neighborhood O of a in X : f(A \ O) N. Assertion: ext(f)(O(a)(a)) N. Case 1: *
*x 2 A \ O(a)(a).
Here, x 2 A \ O and ext(f)(x) = f(x) 2 N. Case 2: x 2 (X  A) \ O(a)(a). Take a*
*ny index i : i(x) 6=
0() x 2 Ui)_then ; 6= Ui\O(a)(a) (X A)(xi)\O(a) ) xi2 O(a) ) n(a; O) n(xi) =*
* n(ai; Oi) )
ai2 O ) f(ai) 2 N ) ext(f)(x) 2 N.]
Claim 2: ext2 LEO(X; A; E). ae
[Define a function OE : X ! 2A by the rule OE(a) = {a} (a 2 A) , Ix*
* the set {i 2
OE(x) = {ai: i 2(Ix}xS2 X  A)
I : x 2 spti}. Given a nonempty compact subset K of X, put KA = OE(x). As*
*sertion: KA is
x2K
compact. Since the OE(x) are finite, hence compact, it will be enough to show t*
*hat for every x 2 X and
for every open subset V of A containing OE(x) there exists an open subset U of *
*X containing x such that
[OE(U) V . Case 1: x 2 X  A. Here one need only remark that there exists a ne*
*ighborhood U of x
in X  A : y 2 U ) OE(y) OE(x). Case 2: a 2 A. Let O be an open subset of X: O*
*E(a) = {a} O. If
x 2 A \ O(a)(a), then OE(x) = {x} O, while if x 2 (X  A) \ O(a)(a), then argu*
*ing as in the first claim,
8 i 2 Ix; ai2 O. Conclusion: [OE(O(a)(a)) A \ O.]]
[Note: Suppose that E is a normed linear space_then the image of extBC(A; *
*E) is contained in
BC(X; E) and, per the uniform topology, ext: BC(A; E) ! BC(X; E) is a linear is*
*ometric embedding:
8 f 2 BC(A; E), kfk = k ext(f)k.]
635
FACT Let A X, where X is stratifiable and A is closed_then A has the EP w.*
*r.t. any locally
convex topological space.
Is it true that if K is a compact Hausdorff space and X is stratifiable, th*
*en C(K; X) is stratifiable?
The answer is "no" even if K = [0; 1].
EXAMPLE Let X be the closed upper half plane in R 2. Topologize X as follo*
*ws: The basic
neighborhoods of (x; y) (y > 0) are as usual but the basic neighborhoods of (x;*
* 0) are the "butterflies"
Nffl(x) (ffl > 0), where Nffl(x) is the point (x; 0) together with all points i*
*n the open upper half plane having
distance < ffl from (x; 0) and lying beneath the union of the two rays emanatin*
*g from (x; 0) with slopes
ffl. Thus topologized, X is stratifiable (and satisfies Ceder's condition). Mor*
*eover, X is first countable
and separable. But X is not second countable, so X is not metrizable. Therefo*
*re X carries no CW
structure (since for a CW complex, metrizability is equivalent to first countab*
*ility). Claim: C([0; 1]; X) is
not stratifiable. To see this, assign to each r 2 R an element fr 2 C([0; 1]; X*
*) by putting fr(1=2) = (r; 0)
and then laying down [0; 1] symmetrically around the circle of radius 1 centere*
*d at (r; 1). The set {fr}
is a closed discrete subspace of C([0; 1]; X) of cardinality 2!. Construct a cl*
*osed separable subspace of
C([0; 1]; X) containing {fr} and finish by quoting Jones' lemma.
[Note: X is compactly generated (being first countable). However, C([0; 1]*
*; X) is not compactly
generated.]
Cautyy has shown that if X is a CW complex, then for any compact Hausdorff *
*space K, C(K; X) is
stratifiable, hence is perfectly normal and paracompact.
Let be an infinite cardinal. A Hausdorff space X is said to be collection*
*wise_normal_
if for every discrete collection {Ai: i 2 I} of closed subsets of X with #(I) *
* there exists
a pairwise disjoint collection {Ui: i 2 I} of open subsets of X such that 8 i 2*
* I : Ai Ui.
So: X is collectionwise normal iff X is collectionwise normal for every .
[Note: Recall that every paracompact Hausdorff space is collectionwise nor*
*mal (cf.
x1, Proposition 9).]
EXAMPLE If X is normal, then X is !collectionwise normal (cf. p. 114) an*
*d conversely.
Let be an infinite cardinal; let I be a set: #(I) = . Assuming that 0 =2*
*I, let
V = {0} [ I and put = {{0}; {i}(i 2 I)} [ {{0; i}(i 2 I)}_then K = (V; ) is a
vertex scheme. Equipping I with the discrete topology, one may view K as the*
* cone
_________________________
yArch. Math. (Basel) 27 (1976), 306311; see also Guo, Tsukuba J. Math. 18 (*
*1994), 505517.
636
I. Therefore K is contractible, hence so is Kb (cf. x5, Proposition 1), the*
* latter being
by definition the star_space_S() corresponding to . It is clear that S() is co*
*mpletely
metrizable of weight . The elements of S() are equivalence classes [i; t] of p*
*airs (i; t),
where (i0;at0)e~ (i00; t00) iff t0 = 0 = t00or i0 = i00& t0 = t00.aeThere is a *
*continuous map
ss : S()[!i[0;;1]t]a!ntd 8 i 2 I there is an embedding ei : [0;t1]!![S()i;*
*.t]The point
ei(0) is independent of i and will be denoted by 0 .
PROPOSITION 13 Let X be a Hausdorff space_then X is collectionwise normal
iff every closed subspace A of X has the EP w.r.t. S().
[Necessity: Fix an f 2 C(A; S()) and let : X ! [0; 1] be a continuous exte*
*nsion
of ss O f. Put Ai = f1 ({[i; t] : 0 < t 1}) : {Ai : i 2 I} is a discrete co*
*llection of
closed subsets of 1(]0; 1]). Since 1(]0; 1]) is an Foe, it too is collection*
*wise normal,
thus there exists a pairwise disjoint collection {Ui : i 2 I} of open subsets o*
*f X such
S
that 8 i 2 I : Ai Ui. Define a continuous function g : A [ (X  Ui) ! [0; 1]*
* by the
( i
gA =Sss O f
conditions gX  Ui= 0 and extend it to a continuous function G : X ! [0; *
*1]. Set
( i
eiO G(x) (x 2 Ui)S
F (x) = 0 (x 2 X  Ui)_then F 2 C(X; S()) and F A = f.
i
Sufficiency: Let {Ai: i 2 I} be a discrete collection of closed subsets of *
*X with #(I) =
S
. Put A = Ai_then A is a closed subspace of X. Define f 2 C(A; S()) piecewi*
*se:
i
fAi = [i; 1]. Extend f to F 2 C(X; S()) and consider the collection {Ui : i 2 *
*I}, where
Ui= F 1({[i; t] : 1=2 < t 1}).]
Application: The star space S() is an AR.
EXAMPLE Let be an infinite cardinal_then there exists a collectionwise no*
*rmal space X which
is not +collectionwise normal, + the cardinal successor of . For this, fix a s*
*et I+ of cardinality +
Q
and equip I+ with the discrete topology. There is an embedding I+ ! S(), the *
*terms of the product
*
* Q
being indexed by the elements of C(I+ ; S()). Let X be the result of retopologi*
*zing S() by isolating
Q
the points of S()  I+ .
Claim: X is collectionwise normal.
[Let {Ai: i 2 I} be a discrete collection of closed subsets of X with #(I) *
*= . Since XI+ is discrete,
there is no loss of generality in assuming that the Ai are contained in I+ . De*
*fine a continuous function
S
f : Ai! S() by fAi= [i; 1] and then, using Proposition 13, extend f to an el*
*ement F 2 C(I+ ; S()),
i Q
determining a projection pF : S() ! S() such that pF I+ = F. Consider the co*
*llection {Ui: i 2 I},
where Ui= p1F({[i; t] : 1=2 < t 1}).]
637
Claim: X is not +collectionwise normal.
[If X were +collectionwise normal, then it would be possible to separate t*
*he points of I+ by a
collection of nonempty pairwise disjoint open subsets of X of cardinality +. Ta*
*king into account how
Q
X is manufactured from S(), one arrives at a contradiction to an obvious coro*
*llary of the Hewitt
Pondiczery theorem.] 1
S
[Note: Give I+ x {0} [ (X  I+ ) x {1=n} the topology induced by the produ*
*ct X x [0; 1]_then
1
this space is perfectly normal and collectionwise normal but is not +collecti*
*onwise normal. And: It is
not a LCH space (cf. p. 115).]
KOWALSKY'S LEMMA Let be an infinite cardinal. Let Y be an AR of weight
_then every metrizable space X of weight can be embedded in Y !.
S *
* `
[Let U = Un be a oediscrete basis for X : Un = {Un(i) : i 2 In}, where I*
* = In
n S *
* n
and #(I) . Write [Un = Amn ; Amn closed in X. Fix distinct points a, b wh*
*ich
m
do not belong to I. Since wt Y = , there exists in Y a collection of nonempty p*
*airwise
disjoint open sets Vj (j 2 I [ {a; b}). Choose a point yj 2 Vj. Given n, define*
* a continuous
__ _____
function fn : [U n! Y by fnUn(i) = yi (i 2 In) and extend fn to a continuous f*
*unction
Fn :aXe! Y . Given mn, define a continuous function fmn : Amn [ (X  [Un) ! Y
by f fmn Amn = ya and extend fmn to a continuous function Fmn : X ! Y . *
* Let
mn X  [Un = yb
mn : X ! Y 2be the diagonal of Fn and Fmn . Let be the diagonal of the mn , *
*so
: X ! (Y 2)!2 Y !_then is an embedding.]
[Note: Suppose that Y is not compact_then every completely metrizable spac*
*e X
of weight can be embedded in Y ! as a closed subspace. For X, as a subspace *
*of
Y !, is a Gffi(being completely metrizable), thus on elementary grounds is home*
*omorphic
to a closed subspace of Y ! x R !: Take a compatible metric d on Y !, represen*
*t the
S
complement Y ! X as a countable union Bj of closed subsets Bj, let dj : Y !!*
* R be
j
the function y ! d(y; Bj), and consider the graph of the diagonal of the dj. Cl*
*aim: There
is a closed embedding8R ! Y !. To see this, fix a closed discrete subset {yn *
*: n 2 Z }
>>< S = 1S [2n; 2n + 1] ae
in Y . Let > 1S1 and define continuous functions f : S ! *
*Y by
>:T = [2n + 1; 2n + 2] g : T ! Y
ae 1 ae ae
f[2n; 2n + 1] = yn f F : R ! Y
g[2n + 1; 2n + 2] = yn. Extend g to a continuous function G : R ! Y a*
*nd let
H : R ! Y 2be the diagonal of F and G. If : R ! Y ! is any embedding, then *
*the
diagonal of and H is a closed embedding R ! Y !x Y 2 Y !.]
638
Application: Every metrizable space X of weight can be embedded in S()!.
Let be an infinite cardinal. Let X be a topological space_then a subspace *
*A X
is said to have the extension_property_with_respect_to_B()_(EP w.r.t. B()) if i*
*t has the
EP w.r.t. every Banach space of weight . Since every completely metrizable AR *
*can
be realized as a closed subspace of a Banach space (cf. p. 612), it is clear t*
*hat A has the
EP w.r.t. B() iff it has the EP w.r.t. every completely metrizable AR of weight*
* .
PROPOSITION 14 Fix a pair (X; A). Suppose that for some noncompact AR Y of
weight , A has the EP w.r.t. Y _then A has the EP w.r.t. B().
[Let E be a Banach space of weight . Owing to Kowalsky's lemma, E can be
realized as a closed subspace of Y !. Let f 2 C(A; E). By hypothesis, f has a c*
*ontinuous
extension F 2 C(X; Y !). Consider r O F , where r : Y !! E is a retraction.]
One conclusion that can be drawn from this is that A has the EP w.r.t. R *
*iff A
has the EP w.r.t B(!). So: If X is a Hausdorff space, then X is normal iff ever*
*y closed
subspace A of X has the EP w.r.t. every separable Banach space.
Another conclusion is that A has the EP w.r.t. S() iff A has the EP w.r.t.*
* B().
Consequently, if X is a Hausdorff space, then X is collectionwise normal iff e*
*very closed
subspace A of X has the EP with respect to B() (cf. Proposition 13). Corollar*
*y: A
Hausdorff space X is collectionwise normal iff every closed subspace A of X has*
* the EP
w.r.t. every Banach space.
FACT Let A X_then A has the EP w.r.t. R iff IA IX has the EP w.r.t. [0; 1*
*].
Let X be a topological space. Let {Un} be a sequence of open coverings of X*
*_then
{Un} is said to be a star_sequence_if 8 n, Un+1 is a star refinement of Un. By*
* means
of a standard construction from metrization theory, one can associate with a gi*
*ven star
*
* 1T
sequence {Un} a continuous pseudometric ffi on X such that ffi(x; y) = 0 iff y *
*2 st(x; Un),
*
* 1
a subset U X being open in the topology generated by ffi iff 8 x 2 U 9 n : st(*
*x; Un) U.
Let Xffibe the metric space obtained from X by identifying points at zero dista*
*nce from
one another and write p : X ! Xffifor the projection.
PROPOSITION 15 Let A X_then A has the EP w.r.t. B() iff for every numerable
open covering O of A of cardinality there exists a numerable open covering U *
*of X of
cardinality such that U \ A is a refinement of O.
639
[Necessity: Let O = {Oi : i 2 I} be a numerable open covering of A with #(I*
*) .
Choose a partition of unity {i : i 2 I} on A subordinate to O. FormatheeBanach *
*space
P A ! `1(I)
`1(I) : r = (ri) 2 `1(I) iff krk = ri < 1. The assignment d*
*efines a
i a ! (i(a))
continuous function f whose range is contained in S+ = {r : krk = 1} \ {r : 8 i*
*; ri 0}, a
closed convex subsetaofe`1(I). Therefore f has a continuous extension F : X ! S*
*+ . Let pi
1(I) ! R P
be the projection `r ! r ; let oei= piOF _then oeiA = iand oei(x) = 1 (*
*8 x 2 X).
i i
Put Ui= oe1i(]0; 1]) and apply NU (cf. p. 123) to see that the collection U =*
* {Ui: i 2 I}
is a numerable open covering of X of cardinality . And by construction, U \ A*
* is a
refinement of O.
Sufficiency: Let E be a Banach space of weight . Fix a dense subset E0 i*
*n E
of cardinality and let En be the open covering of E consisting of the open ba*
*lls of
radius 1=3n centered at the points of E0. Suppose that f : A ! E is continuous*
*_then
8 n; f1 (En) is a numerable open covering of A of cardinality , so there exis*
*ts a star
sequence {Un} of open coverings of X of cardinality such that 8 n, Un \ A is a
refinement of f1 (En). Viewed as a map from A endowed with the topology induce*
*d by
the pseudometric ffi associated with {Un}, f is continuous, thus passes to the *
*quotient to
give a continuous function fffi: Affi! E, where Affi=_p(A). Because fffiis actu*
*ally uniformly
__ *
* __
continuous, there exists a continuous extension fffi: Affi! E of fffito the clo*
*sure Affiof Affi
__ __
in Xffi. Choose Fffi2 C(Xffi; E) : FffiA ffi= fffiand consider F = FffiO p.]
Examples: Let X be a CRH space_then 8 (1) Every compact subspace of X has
the EP w.r.t. B(); (2) Every pseudocompact subspace of X which has the EP w.r.t*
*. [0; 1]
has the EP w.r.t. B(); (3) Every Lindel"of subspace of X which has the EP w.r.t*
*. R has
the EP w.r.t. B().
Suppose that X is collectionwise normal. Let A be a closed subspace of X; *
*let
O = {Oi: i 2 I} be a neighborhood finite open covering of A_then Proposition 15*
* im
plies that there exists a neighborhood finite open covering U = {Ui : i 2 I} of*
* X such
that 8 i 2 I, Ui\ A Oi. Question: Is it possible to arrange matters so that 8*
* i 2 I,
Ui \ A = Oi? The answer is "no" since Rudin's Dowker space fails to admit this*
* im
provement (PrzymusinskiWagey) but "yes" if X is in addition countably paracomp*
*act
(Katetovz).
_________________________
yFund. Math. 109 (1980), 175187.
zColloq. Math. 6 (1958), 145151.
640
Let (X; ffi) be a pseudometric space; let A be a closed subspace of X_then *
*A has the EP w.r.t.
every AR Y . Proof: Let Xffibe the metric space obtained from X by identifying *
*points at zero distance
from one another, write p for the projection X ! Xffi, and put Affi= p(A), a cl*
*osed subspace of Xffi.
Each f 2 C(A; Y ) passes to the quotient to give an fffi2 C(Affi; Y ) for which*
* there exists an extension
Fffi2 C(Xffi; Y ). Consider F = FffiO p.
The weight of a pseudometric is the weight of its associated topology.
LEMMA Let A X_then A has the EP w.r.t. B() iff every continuous pseudometr*
*ic on A of
weight can be extended to a continuous pseudometric on X.
[Necessity: Let ffi be a continuous pseudometric on A of weight . Let Affi*
*be the metric space
obtained from A by identifying points at zero distance from one another. Embed *
*Affiisometrically into a
Banach space E of weightae_then the projection A ! Affi E has a continuous exte*
*nsion : X ! E
and the assignment : X x X ! R is a continuous extension of ffi.
(x0; x00) ! k(x0)  (x00)k
Sufficiency: Let E be a Banach space of weight ; let f 2 C(A; E). Define a*
* pseudometric ffi on A
by ffi(a0; a00) = kf(a0)  f(a00)k_then ffi is continuous of weight , hence ad*
*mits a continuous extension
. Call X() the set X equipped with the topology determined by . Let A() be the *
*closure of A in
X(). Extend f continuously to a function f() : A() ! E and note that A() X() h*
*as the EP
w.r.t. E.]
FACT Let A be a zero set in X. Suppose that A has the EP w.r.t. B()_then A *
*has the EP w.r.t.
every AR Y of weight .
[Choose a OE 2 C(X; [0; 1]) : A = OE1(0). Fix a compatible metric d on Y .*
* Given f 2 C(A; Y ), define
a pseudometric ffi on A by ffi(a0; a00) = d(f(a0); f(a00)). Let be a continuou*
*s extension of ffi to X and
consider the sum of (x0; x00) and OE(x0)  OE(x00).]
Let X be a CRH space. Suppose that X is perfectly normal and collectionwis*
*e normal_then it
follows that every closed subspace A of X has the EP w.r.t. every AR.
FACT Let X be a submetrizable CRH space. Suppose that A X has the EP with *
*respect to every
normed linear space_then A is a zero set in X.
[Note: Take for X the Michael line and let A = Q_then X is a paracompact Ha*
*usdorff space, so
A has the EP w.r.t. every Banach space. On the other hand, X is submetrizable*
* but A is not a Gffi.
Therefore A does not have the EP w.r.t. every normed linear space.]
LEMMA Fix a pair (X; A). Suppose that A has the EP w.r.t. B()_then every
continuous function OE : i0X [ IA ! S() has a continuous extension : IX ! S() *
* .
641
[The restriction of OE to IA determines a continuous function A ! C([0; 1*
*]; S()).
But C([0; 1]; S()) is a completely metrizable AR (cf. the proof of Proposition*
* 6), the
weight of which is , so our assumption on A guarantees that this function has a
continuous extension X ! C([0; 1]; S()), leading thereby to a continuous functi*
*on :
IX ! S() whose restriction to IA is . Chooseaanef 2 C(X; [0; 1]) : f1 (0) = *
*{x :
OE(x; 0) = (x; 0)}. Let F be the function x X!!(S()x; f(x)). Because S() is c*
*ontractible,
ae
there is a homotopy H : IX ! S() such that H(x;H0)(=xOE(x;;0)1).=CFo(x)nsider*
* the function
ae
: IX ! S() defined by (x; t) = H((x;xt); t=f(t(xf(x))))(t.<]f(x))
PROPOSITION 16 Let A X_then A has the EP w.r.t. B() iff i0X [ IA, as a
subspace of IX, has the EP w.r.t. every completely metrizable ANR Y of weight .
[Necessity: Let f : i0X [ IA ! Y be continuous. Using Kowalsky's lemma, r*
*ealize
Y as a closed subspace of S()! and let r : O ! Y be a retraction (O open in S*
*()!).
Given a projection p : S()! ! S(), let OEp = p O f_then by what has been said a*
*bove,
OEp has a continuous extension p : IX ! S(). Therefore f has a continuous exten*
*sion
: IX ! S()!. Set P = 1(O). Since P is a cozero set in IX containing IA and
since the projection IX ! X takes zero sets to zero sets, there is a cozero set*
* U in X
such that A U and IU P . On the other hand, Aahasethe EP w.r.t. R , so it fol*
*lows
from Proposition 3 that 9 OE 2 C(X; [0; 1]) : OEOEA=X1 U.= 0Define F 2 C(I*
*X; Y ) by
F (x; t) = r((x; OE(x)t)) : F is a continuous extension of f.
Sufficiency: Let O = {Oi : i 2 I} be a neighborhood finite cozero set cover*
*ing of A
with #(I) . Put
P = {Oix]1=3; 1] : i 2 I} [ {i0X [ A x [0; 2=3[}:
Then P is a neighborhood finite cozero set covering of i0X [ IA of cardinality *
* , thus
Proposition 15 implies that there exists a numerable open covering V of IX of c*
*ardinality
such that V \ (i0X [ IA) is a refinement of P. Let U = V \ (i1X) : Uis a nume*
*rable
open covering of i1X such that U \ (i1A) is a refinement of P \ (i1A) = i1O. Fi*
*nish by
quoting Proposition 15.]
EXAMPLE Suppose that the inclusion A ! X is a cofibration_then i0X [ IA is*
* a retract of IX
(cf. x3, Proposition 1), so Proposition 16 implies that A has the EP w.r.t. eve*
*ry Banach space.
[Note: This applies in particular to a relative CW complex (X; A).]
Let X and Y be topological spaces.
642
(HEP) A subspace A X is said to have the homotopy_extension_property_*
*with_ae
respect_to_Y_(HEP w.r.t. Y ) if given continuous functions Fh::XI!AY! Ysuch t*
*hat F A =
h O i0, there is a continuous function H : IX ! Y such that F = H O i0 and HIA*
* = h.
[Note: In this terminology, the inclusion A ! X is a cofibration iff A has *
*the HEP
w.r.t. Y for every Y .] ae
Suppose that A has the HEP w.r.t. Y . Let fg22C(A;CY()A;bYe)homotopic. As*
*sume: f
has a continuous extension F 2 C(X; Y )_then g has a continuous extension G 2 C*
*(X; Y )
and F ' G. Therefore, under these circumstances, the extension question for con*
*tinuous
functions A ! Y is a problem in the homotopy category.
If A X is closed and if i0X [ IA, as a subspace of IX, has the EP w.r.t. Y*
* , then
it is clear that A has the HEP w.r.t. Y . Conditions ensuring that this is so a*
*re provided
by Proposition 16. Here are two illustrations.
(1) Every closed subspace A of a normal Hausdorff space X has the HEP *
*w.r.t.
every second countable completely metrizable ANR Y .
(2) Every closed subspace A of a collectionwise normal Hausdorff space*
* X has
the HEP w.r.t. every completely metrizable ANR Y .
[Note: Historically, these results were obtained by imposing in addition a *
*countable
paracompactness assumption on X. Reason: If X is a normal Hausdorff space, then*
* the
product IX is normal iff X is countably paracompact.]
If A X and if A has the EP w.r.t. B(), then A has the HEP w.r.t. everyacom*
*pletelye
metrizable ANR Y of weight . Proof: Take a pair of continuous functions Fh::*
*XI!AY! Y
ae
such that F A = h O i0 and define OE : i0X [ IA ! Y by OE(x;O0)E=(Fa(x);.t)*
*I=nh(a;vt)iew of
Proposition 16, the only issue is the continuity of OE. To see this, embed Y in*
* a Banach
space E of weight . Since IA, as a subspace of IX, has the EPaw.r.t.eB(),_h ha*
*s a
__ __ __ __ OE(x; 0) = F (x)
continuous extension h : IA ! E. Define OE: i0X [ IA ! E by __OE(__a; t) =_*
*__h(__a;tt)hen
__
OEis a welldefined continuous function which agrees with OE on i0X [ IA.
EXAMPLE The product Y = Snx Snx . .(.! factors) is not an ANR. But if X is *
*normal and
A X is closed, then A has the HEP w.r.t. Y .
FACT Suppose that X is Hausdorff. Let A be a zero set in X.
(1) If X is normal, then A has the HEP w.r.t. every second countable A*
*NR Y .
(2) If X is collectionwise normal, then A has the HEP w.r.t. every ANR*
* Y .
643
FACT Let Y be a nonempty metrizable space. Suppose that Y is locally contra*
*ctible_then Y is an
ANR iff for every pair (X; A), where X is metrizable and A X is closed, A has *
*the HEP w.r.t. Y .
Let X be a homeomorphism invariant class of normal Hausdorff spaces that is*
* closed
hereditary, i.e., if X 2 X and if A X is closed, then A 2 X .
Let X be the class consisting of the Hausdorff spaces satisfying Ceder's co*
*ndition_then it is unknown
whether X is closed hereditary.
A nonempty topological space Y is said to be an extension_space_for X if ev*
*ery closed
subspace of every element of X has the EP w.r.t. Y . Denote by ES(X ) the class*
* of extension
spaces for X . Obviously, if X 0 X 00, then ES (X 00) ES (X 0), so 8 X : ES (*
*normal )
ES (X ).
(ES 1) The class ES(X ) is closed under the formation of products.
(ES 2) Any retract of an extension space for X is in ES(X ). ae
(ES 3) Suppose that Y = Y1 [ Y2, where Y1 and Y2 are open and Y1Y 2
2
ES (X ) & Y1 \ Y2 2 ES(X )_then Y 2 ES(X ).
(ES 4) Assume: The elements of X areahereditarilyenormal. Suppose t*
*hat
Y = Y1 [ Y2, where Y1 and Y2 are closed and Y1Y2 ES (X ) & Y1 \ Y2 2 ES (X )_*
*then
2
Y 2 ES(X ).
(ES 5) Suppose that Yae= Y1 [ Y2, where Y1 and Y2 are closed_then Y 2
ES (X ) & Y1 \ Y2 2 ES(X ) ) Y1Y2 ES(X ).
2
EXAMPLE A nonempty topological space Y is an extension space for the class *
*of metrizable spaces
iff it is an extension space for the class of M complexes.
A nonempty topological space Y is said to be a neighborhood_extension_spac*
*e_for
X if every closed subspace of every element of X has the NEP w.r.t. Y . Den*
*ote by
NES (X ) the class of neighborhood extension spaces for X . Obviously, if X 0 X*
* 00, then
NES (X 00) NES (X 0), so 8 X : NES (normal ) NES (X ). Of course, ES (X ) NE*
*S (X ).
In the other direction, every contractible element of NES (X ) is in ES(X ).
[Note: It is convenient to agree that ; 2 NES (X ). So, if Y 2 NES (X ) and*
* if V Y
is open, then V 2 NES (X ).]
(NES1) The class NES (X ) is closed under the formation of finite prod*
*ucts.
(NES2) Any neighborhood retract of a neighborhood extension space for *
*X is in
NES (X ).
644
ae
(NES3) Suppose that Y = Y1 [ Y2, where Y1 and Y2 are open and Y1Y 2
2
NES (X )_then Y 2 NES (X ).
(NES4) Assume: The elements of X areahereditarilyenormal. Suppose t*
*hat
Y = Y1 [ Y2, where Y1 and Y2 are closed and Y1Y2 NES (X ) & Y1 \ Y2 2 NES (X *
*)_then
2
Y 2 NES (X ).
(NES5) Suppose that Y =aY1e[ Y2, where Y1 and Y2 are closed_then Y 2
NES (X ) & Y1 \ Y2 2 NES (X ) ) Y1Y2 NES (X ).
2
[Note: There is a slight difference between the formulation of ES3, and NES*
*3. Reason:
An empty intersection is permitted in NES3 but not in ES3 (consider X = [0; 1],*
* A = Y =
{0; 1}).]
EXAMPLE (CW_Complexes_) Metrizable CW complexes are ANRs (cf. p. 616).
(1) Every finite CW complex is in NES(normal).
(2) Every CW complex is in NES(compact) (but it is not true that every*
* CW complex is in
NES(paracompact)).
(3) Every CW complex is in NES(stratifiable).
[First, if K is a full vertex scheme, then K is a locally convex topologi*
*cal space (cf. p. 611), so
K 2 ES(stratifiable) (cf. p. 634). Second, if K is a vertex scheme and if L *
*is a subscheme, then L is
a neighborhood retract of K. Third, if X is a CW complex, then X is the retra*
*ct of a polyhedron (cf.
p. 512).]
FACT Every CW complex has the homotopy type of an ANR which is in NES(parac*
*ompact).
EXAMPLE Suppose that X = Y [aZeis metrizable. LetaKeand L be finite CW co*
*mplexes.
Assume: Every closed subspace of Y has the EP w.r.t. K _then every closed s*
*ubspace of X has the
Z L
EP w.r.t. K * L.
The "ES" arguments are similar to but simpler than the "NES" arguments. Of
the latter, the most difficult is the one for NES3, which runs as follows. Tak*
*e an X
in X and let A X be closed_then the claim is that 8 f 2 C(A; Y ) there exists *
*an
openaUe A and an F 2 C(U; Y ) : F A = f. Since X is covered by theaopenesets
f1 (Y1) [ (X  A) X1 X
f1 (Y2) [ (XaeA) and since X is normal, there exist closed sets X2 X *
*which
1 (Y ) [ (X  A) aeA = X \ A
cover X with X1X f 1 1 . Put 1 1 . There are now two c*
*ases,
2 f (Y2) [ (X  A) A2 = X2 \ A
depending on whether Y1\Y2 is empty or not. The second possibility is more invo*
*lved than
645
the first so we shall look only at it. Because Y1 \ Y2 2 NES (X ), the restrict*
*ion fA1 \ A2
has an extension f12 2 C(O; Y1 \ Y2), where O is some open subset of X1 \ X2 co*
*ntaining
__
A1 \ A2. Choose an open subset P of X1 \ X2 : A1 \ A2 P aPe O. Observing
__ __ f(x) (x 2 A)
that A \ P = A1 \ A2, define g 2 C(A [ P ; Y ) by g(x) = f __. B*
*e
ae ae __ 12(x) (xa2eP)
cause Y1Y2 NES (X ), the restriction gA1 [ P_ has an extension G1 2 C(O1*
*; Y1),
ae 2 2 NES (X ) ae gA2 [ P ae __ G2 2 C(O2*
*; Y2)
where O1O is some open subset of X1 containing A1 [ P_. Choose an open s*
*ubset
ae 2 ae __ __ X2 A2 [ P
P1 ofX1 A1 [ P_ P1 P1_ O1
P2 ofX2 : A2 [ P P2aeP2 O2______and_an_open_subset V X : A Vae& (X1 \
__ B1 = (P1  X2 \ V) [ P B1 O1
X2  P ) \ V = ;. Let B ________ __ __. It is clear that *
*, with
2 = (P2  X1 \aV)e[ P B2 O2
__ G1(x)(x 2 B1)
B1 \ B2 = P , so the prescription G(x) = G is a continuous extens*
*ion of
2(x)(x 2 B2)
f to B1 [ B2 A. The set (P1  X2) [ (P2  X1) [ P is open in X. Denote by U *
*its
intersection with V and let F = GU. ae
1 (Y )
[Note: To reduce NES4 to NES3, put instead A1A= f 1 1 . Sin*
*ce
ae________ 2 = f (Y2)
A1  A2 \ (A2__A1)_= ;
(A1  A2) \ A2  A1 = ; and since X is hereditarilyanormal,ethere_exists an o*
*pen set
__ X1 = U0 [ (A1 \ A2)
U0 X : A1  A2 U0 U 0 X  (A2  A1). Setting X ,
2 = (X  U0) [ (A1 \ A2)
the argument then proceeds as before.]
Why work with classes of normal Hausdorff spaces? Answer: If the class X co*
*ntains a space that is
not normal, then every nonempty Hausdorff Y 2 NES(X) is necessarily a singleton.
FACT Suppose that Y is an AR (ANR).
(1) Let X be the class of perfectly normal paracompact Hausdorff space*
*s_then Y 2 ES(X)
(NES (X)).
(2) Let X be the class of perfectly normal Hausdorff spaces_then Y 2 E*
*S(X) (NES (X)) iff Y
is second countable.
[For the necessity, remark that every collection of nonempty pairwise disjo*
*int open subsets of Y
is countable. Reason: The construction on p. 635 ff. furnishes a perfectly no*
*rmal Hausdorff space X
containing an uncountable closed discrete subspace A, the points of which canno*
*t be separated by a
collection of nonempty pairwise disjoint open subsets of X.]
(3) Let X be the class of paracompact Hausdorff spaces_then Y 2 ES(X) *
*(NES (X)) iff Y is
completely metrizable.
646
[To establish the necessity, assume, e.g., that Y is an AR. Let X be the re*
*sult of retopologizing fiY
by isolating the points of fiY Y . Every open covering of X has a oediscrete *
*open refinement, hence X is a
paracompact Hausdorff space. Since Y sits inside X as a closed subspace, there *
*is a retraction r : X ! Y .
On the other hand, Y is metrizable, thus is Moore, so Y satisfies Arhangel'skii*
*'s condition. Fix a sequence
*
* T
{Vn} of collections of open subsets of fiY such that each Vn covers Y and 8 y 2*
* Y : st(y; Vn) Y .
*
* n
Assign to a given V 2 Vn the open subset PV V determined by intersecting V wit*
*h the interior in fiY
S T
of r1(V \ Y ). Put Pn = {PV : V 2 Vn} : Pn Y & Y = Pn, therefore Y is topo*
*logically complete
n
or still, is completely metrizable.]
(4) Let X be the class of normal Hausdorff spaces_then Y 2 ES(X) (NES *
*(X)) iff Y is second
countable and completely metrizable.
FACT Let X be the class consisting of the Hausdorff spaces that can be real*
*ized as a closed subspace
of a product of a compact Hausdorff space and a metrizable space (the elements *
*of X are precisely those
paracompact Hausdorff spaces satisfying Arhangel'skii's condition)_then every A*
*R (ANR) is in ES(X)
(NES (X)).
[Suppose that X 2 X is closed in K x Z, where K is compact Hausdorff and Z *
*is metrizable. The
projection K x Z ! Z is closed and has compact fibers, thus the same is true of*
* its restriction p to
X. Fix a closed subspace A X. Take an AR Y of weight and let f 2 C(A; Y ).*
* Embed Y in
S()! and apply Proposition 13 to produce a continuous extension OE : X ! S()! o*
*f f. Write for
the diagonal of OE and p_then (A) is closed in S()! x p(X). Therefore the rest*
*riction to (A) of
the projection : S()! x p(X) ! S()! has a continuous extension : S()! x p(X)*
* ! Y . Put
F = O : F 2 C(X; Y ) & FA = f.]
Application: If K is a compact Hausdorff space and if Y is an ANR, then C(K*
*; Y ) is an ANR (so
for any CW complex X, C(K; X) is a CW space).
[Inspect the proof of Proposition 6, keeping in mind the preceding result.]
Suppose that G is a stratifiable topological group_then X1Gand B1Gare strat*
*ifiable (cf. p. 631) and
Cautyy has shown that if G is also in NES(stratifiable), then the same holds fo*
*r X1G and B1G. Example:
If G is an ANR, then X1G and B1G are ANRs (cf. p. 465).
LEMMA Let Y be a topological space. Suppose that Y admits a covering V by
pairwise disjoint open sets V , each of which is in NES(collectionwise normal)_*
*then Y is
in NES(collectionwise normal).
_________________________
yArch. Math. (Basel) 28 (1977), 623631.
647
[Let X be collectionwise normal, A X closed, and let f 2 C(A; Y ). Put AV*
* =
f1 (V ), fV = fAV _then there exists a neighborhood OV of AV in X and an F*
*V 2
C(OV ; V ) : FV AV = fV . Since {AV } is a discrete collection of closed subse*
*ts of X, there
exists a pairwise disjoint collection {UV } of open subsets of X such that 8 V *
*: AV UV .
S
Set U = (OV \ UV ) and define F : U ! Y by F OV \ UV = FV OV \ UV to get*
* a
V
continuous extension of f to U.]
Let Y be a topological space. Suppose that Y admits a numerable covering*
* V by
open sets V , each of which is in NES(collectionwise normal)_then, from the pro*
*of of
Proposition 12, it follows that Y is in NES(collectionwise normal).
FACT Let Y be a topological space. Suppose that Y admits a covering V by op*
*en sets V , each of
which is in NES(paracompact)_then Y is in NES(paracompact).
Application: Every topological manifold is in NES(paracompact).
[Note: This applies in particular to the Pr"ufer manifold, which is not met*
*rizable and contains a closed
submanifold that is not a neighborhood retract.]
Assume: IX X . Let Y 2 NES (X )_then for every pair (X; A), where X 2 X and
A X is closed, A has the HEP w.r.t. Y . Proof: i0X [ IA, as a closed subspace *
*of IX,
has the EP w.r.t. Y .
EXAMPLE (CW_Complexes_) If X is stratifiable and A X is closed, then A has*
* the HEP w.r.t.
any CW complex.
PROPOSITION 17 Assume: IX X . Let Y 2 NES (X ) and suppose that Y is
homotopy equivalent to a Z 2 ES(X )_then Y 2 ES(X ).
[Choose continuous functions OE : Y ! Z, : Z ! Y such that OOE ' idY, OE*
*O ' idZ.
Take an X in X and let A X be closed. Given f 2 C(A; Y ), 9 F 2 C(X; Z) : F Oi*
* = OEOf,
where i : A ! X is the inclusion. But A has the HEP w.r.t. Y and O F O i ' f,*
* so f
admits a continuous extension to X.]
FACT Suppose that X is an ANR. Let Y be a topological space such that ever*
*y closed subset
A X has the EP w.r.t. Y . Fix a weak homotopy equivalence K ! Y , where K is a*
* CW complex_then
every closed subset A X has the EP w.r.t. K.
[Owing to the CWANR theorem, the induced map [X; K] ! [X; Y ] is bijective*
* (cf. p. 515). On
the other hand, every closed subset A X has the HEP w.r.t. K (metrizable ) str*
*atifiable).]
71
x7. CTHEORY
A classical technique in algebraic topology is to work modulo a Serre class*
* of abelian
groups. I shall review these matters here, supplying proofs of the less familia*
*r facts.
Let C Ob AB be a nonempty class of abelian groups_then C is said to be a
Serre_class_providedathatefor any short exact sequence 0 ! G0 ! G ! G00! 0 in
0
AB , G 2 C iff GG002 C or, equivalently, for any exact sequence G0 ! G ! G00*
*in AB ,
ae
G0
G00 2 C ) G 2 C.
[Note: To show that a nonempty class C ObAB is a Serre class, it is usual*
*ly simplest
to check that C is closed under subgroups, homomorphic images, and extensions.]
Example: For any Serre class C, the subclass Ctorof torsion groups in C is *
*a Serre
class.
[Note: A Serre class C is said to be torsion_if C = Ctor.]
EXAMPLE (pPrimary_Abelian_Groups_) An abelian pgroup G is said to be p*
*primary_. The
rank_r(G) of a pprimary G is the cardinality of a maximal independent system i*
*n G. If G[p] = {g : pg = 0},
then G[p] is a vector space over Fp and dimG[p] = r(G). The final_rank_rf(G) of*
* a pprimary G is the
infimum of the r(pnG) (n 2 N). Every pprimary G can be written as G = G0 G00, *
*where G0is bounded
and r(G00) = rf(G00) (Fuchsy). Fix now a symbol 1, considered to be larger than*
* all cardinals. Given a
Serre class C of pprimary abelian groups, let (C) be the smallest cardinal num*
*ber > r(G) 8 G 2 C if such
a number exists, otherwise put (C) = 1, and let (C) be the smallest cardinalanu*
*mbere> rf(G) 8 G 2 C
if such a number exists, otherwise put (C) = 1. Obviously, (C) (C), (C) = 1 *
*or (C) ! .
(C) = 1 *
*or (C) !
And: C is preciselyatheeclass of pprimary G for which r(G)a !.
[Any torsion free abelian group G of infinite rank contains a free abelian *
*group of rank
= #(G).]
EXAMPLE Fix a cardinal number > !. Let T be the class of torsion abelia*
*n groups of
cardinality < ; let F be the class of torsion free abelian groups of cardinali*
*ty < . Take any Serre
class T of torsion abelian groups: T T _then the class C consisting of all abe*
*lian groups G which are
extensions of a group in T by a group in F is a Serre class such that Ctor= T *
*and tf(C) = F .
A characteristic_is a sequence O =a{Ope: p 2 }, where each Op is a nonneg*
*ative
0
integer or 1. Given characteristics OO00, write O0 ~ O00iff #{p : O0p6= O00p*
*} < ! and
O0p= 1 , O00p= 1_then ~ is an equivalence relation on the set ofacharacteristic*
*s,ean
0
equivalence class tbeing called a type_. The sum t0+t00of types tt00is the ty*
*pe containing
the characteristic {O0p+ O00p: p 2 } and t0 t00provided that O0p O00pfor almos*
*t all p,
t00 t0being the largest type t such that t+ t0 t00.
(Rational Groups) A nonzero abelian group G is said to be rational_if*
* it is
isomorphic to a subgroup of Q or still, is torsion free of rank 1. Such group*
*s can be
classified. For assume that G is rational, say G Q . Take g 2 G : g 6= 0. G*
*iven
p 2 , consider the set Sp(g) of nonnegative integers n such that the equation*
* pnx = g
has a solution in G. Put Op(g) = sup Sp(g), the pheight_of g_then O(g) = {Op(*
*g) :
p 2 } is a characteristic. Moreover, distinct nonzero elements of G determine *
*equivalent
characteristics. Definition: The type_t(G) of G is the type of the characteri*
*stic of any
nonzero element of G.aeEvery type t can be realized by a rational group, i.e., *
*t = t(G)
0
(9 G) and rational GG00are isomorphic iff t(G0) = t(G00) (in general, G0is is*
*omorphic to
a subgroup of G00iff t(G0) t(G00)).
L
Example: Suppose that Z G Q _then G=Z Z=pOpZ , {Op : p 2 } the
p
characteristic of 1, and Hom (G; G) is isomorphic to the subring of Q generated*
* by 1 and
the p1 : pG = G.
FACT If G and K are rational, then G K is rational and t(G K) = t(G) + t*
*(K).
73
FACT If G and K are rational, then Hom(G; K) = 0 if t(G) 6 t(K), but is ra*
*tional if t(G) t(K)
with t(Hom (G; K)) = t(K)  t(G).
Notation: T is a nonempty set of types such that (i) t02 T & t t0) t2 T a*
*nd
(ii) t0; t002 T ) t0+ t002 T ; T(AB ) being the class of abelian groups G whi*
*ch admit
Ln
a monomorphism G ! Gi, where the Gi are rational (n depending on G) and the
1
t(Gi) 2 T.
FACT A torsion free abelian group G of finite rank is in T(AB ) iff for ea*
*ch nonzero homomorphism
OE : G ! Q, t(OE(G)) 2 T.
PROPOSITON 2 Let C be a Serre class. Assume: tf(C) contains only groups of*
* finite
rank and at least one group of positive rank_then tf(C) = T(AB ) for some T .
[Let T be the set of types t such that a rational group of type t is in tf*
*(C). If
nL
G1; : :;:Gn are rational and if t(G1); : :;:t(Gn) belong to T, then Gi2 tf(C)*
* and every
n 1
L
subgroup of Gi is in tf(C). On the other hand, for any G 6= 0 in tf(C), there*
* are rational
1 n
L
G1; : :;:Gn and a monomorphism G ! Gi. Upon restricting to homomorphic image*
*s,
1
one can arrange that the Gi2 tf(C), so the t(Gi) 2 T. Since Caiseclosed under s*
*ubgroups,ae
0 *
* G0
T satisfies condition (i) above. As for condition (ii), let tt002 T . Choose*
* G00 : Z
ae ae ae
G0 t0= t(G0) O0
G00 Q & t00= t(G00)is representedabyethe characteristic O00 correspondi*
*ng to
0
1. Suppose first that 8 p, OpO00is finite. Let Z G Q : O(1) = O0+ O00. *
* Fix an
p
isomorphism OE : G0=Z ! G=G00and let K be the subgroup of G0 G composed of the
(g0; g) : OE(g0+ Z) = g + G00_then there is a short exact sequence 0 ! G00! K !*
* G0! 0,
hence K 2 C. But there is also an epimorphismaKe! G, thus Ga2eC and t0+ t002 T*
* .
0= O0 + O0 O0
Passing to the general case, write OO00=fO00 0;100, where f00take finite v*
*alues and
ae f + O0;1 Of ae
O00;1 0 00 G00;1
aeO000;1have values 0 or 1. Let Z Gf Q : Of(1) = Of + Of; letaZe G000;1 Q :
O00;1(1) = O00;1 G00;1
O000;1(1) =aO000;1.eFrom the foregoing, Gf 2 C; in addition, G000;1is isomo*
*rphic to a
0
subgroup of GG002 C. Therefore Gf G00;1 G000;12 C and Gf + G00;1+ G000;1 Q h*
*as
type t0+ t00.]
EXAMPLE Given T, let T be a Serre class of torsion abelian groups with the*
* property that the
74
L
type determined by a characteristic O belongs to T iff Z =pOpZ 2 T _then the *
*class C consisting of all
p
abelian groups G which are extensions of a group in T by a group in T(AB ) is a*
* Serre class such that
Ctor= T and tf(C) = T(AB ).
Every torsion abelian group G contains a basic_subgroup_B,ai.e.,eB is a dir*
*ect sum of
0 *
* aeB0 G0
cyclic groups, B is pure in G, and G=B is divisible. If GG00are torsion and i*
*f B00 G00
are basic, then G0 G00 B0 B00. Corollary: The tensor product of two torsion abe*
*lian
groups is a direct sum of cyclic groups.
LEMMA Let 0 ! G0! G ! G00! 0 be a short exact sequence of abelian groups.
Suppose that the image of G0 in G is pure_then for any K, the sequence 0 ! G0 K*
* !
G K ! G00 K ! 0 is exact and the image of G0 K in G K is pure.
[Note: Under the same assumptions, the sequence 0 ! Tor(G0; K) ! Tor(G; K) !
Tor(G00; K) ! 0 is exact and the image of Tor(G0; K) in Tor(G; K) is pure.]
A Serre class C is said to be a ring_if G; K 2 C ) G K 2 C, Tor(G; K) 2 C.
[Note: C is a ring provided that 8 G 2 C : G G 2 C, Tor(G; G) 2 C. This is*
* because
G; K 2 C ) G K (G K) (G K), Tor(G; K) Tor(G K; G K).]
EXAMPLE Let C be a ring. Fix a group G_then G=[G; G] 2 C iff 8 i, i(G)=i+1*
*(G) 2 C.
[The iterated commutator map i+1(G=[G; G]) ! i(G)=i+1(G) is surjective.]
EXAMPLE Let C be a ring. Fix a group G such that 8 n > 0, Hn(G) 2 C. Let M*
* 2 C be a
nilpotent Gmodule_then 8 n 0, Hn(G; M) 2 C.
[Since the (I[G])i. M=(I[G])i+1. M 2 C, it suffices to look at the case whe*
*n the action of G on M is
trivial.]
FACT Let C be a Serre class. Suppose that G 2 C_then for any finitely gene*
*rated K, G K and
Tor(G; K) belong to C.
PROPOSITION 3 Let C be a Serre class_then C is a ring iff Ctoris a ring.
[Setting aside the trivial case when C is the class of all abelian groups, *
*let us assume
that Ctor6= C is a ring. Fix G 2 C  Ctor: Tor(G; G) Tor(Gtor; Gtor) 2 Ctor; *
*Gtorthe
torsion subgroup of G. To deal with G G, put tf(G) = G=Gtorand consider the ex*
*act
sequences 8
< 0 ! Gtor G ! G G ! tf(G) G ! 0
: 00!!GtortGtor!fGtor(GG!)Gtor tf(G)G! 0 :
tor! tf(G) G ! tf(G) tf(*
*G) ! 0
75
Because Gtor Gtor2 Ctor, it will be enough to prove that Gtor tf(G) and tf(G) *
*tf(G)
are in C.
(I) Suppose that tf(C) contains a group of infinite rank. Choose > !*
* as in
Proposition 1 (so C contains all abelian groups of cardinality < ) : #(tf(G)) <*
* )
#(tf(G) tf(G)) < ) tf(G) tf(G) 2 C. There is a free group F in C and an epi
morphism F ! tf(G) ! 0, where rankF < . Let B be a basic subgroup of Gtorand fo*
*rm
the exact sequence 0 ! B F ! Gtor F ! Gtor=B F ! 0. Using the fact that B is
a direct sum of cyclic groups, B F B B : #(B ) < ) B F 2 C. Analogously,
by an application of the structure theorem for divisible abelian groups, Gtor=B*
* F 2 C.
Conclusion: Gtor F 2 C ) Gtor tf(G) 2 C.
(II) Suppose that tf(C) = T (AB ) (cf. Proposition 2). Let F be t*
*he free
abelian group generated by a maximal independent system in tf(G)_then there is *
*an exact
sequence 0 ! F ! tf(G) ! tf(G)=F ! 0 and tf(G)=F 2 Ctor. Tensor this sequence w*
*ith
Gtorto get another exact sequence F Gtor! tf(G) Gtor! tf(G)=F Gtor. Of cours*
*e,
tf(G)=F Gtor2 Ctor; moreover, F Gtor2 C, which implies that tf(G) Gtoritself*
* is in
C. Finally, the sequence 0 ! F tf(G) ! tf(G) tf(G) ! tf(G)=F tf(G) ! 0 is ex*
*act.
Obviously, F tf(G) 2 C and, repeating the preceding argument, tf(G)=F tf(G) 2*
* C,
hence tf(G) tf(G) 2 C.]
In what follows, ff and fl are functions having cardinal numbers as values,*
* the domain
of ff being x N and the domain of fl being .
Examples: (1) Let G be a torsion abelian group. Assume: G is a direct su*
*m of
L L
cyclic groups_then G ff(p; n) . (Z =pnZ ); (2) Let G be a torsion abelian*
* group.
p n L
Assume: G is divisible_then G fl(p).(Z =p1 Z); (3) Let G be a torsion abelia*
*n group.
p
Assume: G is pprimary and satisfies the descending chain condition on subgroup*
*s_then
L P
G ff(p; n) . (Z =pnZ ) fl(p) . (Z =p1 Z), where ff(p; n) < ! and fl(p) is*
* finite.
n n L
[Note: For use below, recall that Z =p1 Z is a homomorphic image of Z=pn*
*Z (in
L n
fact, every countable pprimary G is a homomorphic image of Z=pnZ ).]
n L L
Notation: Given a torsion Serre class C, ffffff(C) = {ff : ff(p; n) . *
*(Z =pnZ ) 2 C} and
L p n
flflfl(C) = {fl : fl(p) . (Z =p1 Z) 2 C}.
p
Observations: (i) fl0 2 flflfl(C) & fl fl0 ) fl 2 flflfl(C) and (ii) fl0, *
*fl002 flflfl(C) ) fl0+ fl002
flflfl(C).
L
Suppose that C is a torsion Serre class. Let G 2 C_then G G(p), G(p) t*
*he
p
pprimary component of G. Denote by C0 the subclass of C comprised of those G s*
*uch that
76
each G(p) is bounded, so 8 p, 9 M(p) : pM(p)G(p) = 0, and put ffffff0(C) = ffff*
*ff(C0) (meaningful,
C0 being Serre).
CARDINAL LEMMA Let C be a torsion Serre class_then 8 ff 2 ffffff(C), 9 ff0*
* 2 ffffff0(C)
& fl 2 flflfl(C) such that ff(p; n) ff0(p; n) + fl(p), where fl(p) ! or fl(p)*
* = 0.
1X
[Set oe(p; n) = ff(p; m) and choose M(p) such that oe(p; n) = oe(p; n +*
* 1) = . . .
m=n ae
(n M(p)). Define ff0 by ff0(p; n) = ff(p;0n)(n( !. Put N1 = {n : ff(*
*p; n) > !} :
n L L
#(N1 ) = ! and there are epimorphisms ff(p; n).(Z =pnZ ) ! nff(p; n).(Z *
*=pnZ ) !
L n L n2N1
ff(p; n) . (Z =pZ . . .Z=pnZ ) ! fl(p) . ( Z=pnZ ) ! fl(p) . (Z =p1 Z).]
n2N1 n
Given a torsion Serre class C, let C* be the subclass of those G such that *
*each G(p)
satisfies the descending chain condition on subgroups. Note that C* is Serre.
PROPOSITION 4 Let C be a torsion Serre class_then C is a ring iff C* is a *
*ring.
[Straightforward computations establish the necessity. As for the sufficien*
*cy, fix G 2 C
and let B be a basic subgroup of G. Applying the cardinal lemma, one finds that*
* BB 2 C.
But G G B B, thus G G 2 C. The verification that Tor(G; G) 2 C hinges on a
preliminary remark.
Claim: Suppose that C* is a ring_then 8 fl 2 flflfl(C), fl2 2 flflfl(C).
[Write fl = fl0 + fl00, where 8 p, fl0(p) is finite and fl00(p) ! or fl00(*
*p) = 0, so
fl2 = (fl0)2 + fl00. Since C* is a ring, (fl0)2 2 flflfl(C), hence fl2 2 flflfl*
*(C).]
Consider the exact sequences
8
< 0 ! Tor(B; G) ! Tor(G; G) ! Tor(G=B; G) ! 0
: 00!!Tor(B;TB)o!rTor(G;(B)B!;Tor(G=B;GB)=!B0) ! Tor(G;:G=B) ! Tor*
*(G=B; G=B) ! 0
L
Owing to the claim, Tor(G=B; G=B) 2 C. Proof: G=B fl(p) . (Z =p1 Z) ) Tor(G=*
*B;
L p
G=B) fl2(p) . (Z =p1 Z). In addition, Tor(B; B) B B 2 C. Therefore everyt*
*hing
p L
comes down to showing that Tor(B; G=B) 2 C or still, that fl(p) . B(p) 2 C. *
* Using
p
77
L
the cardinal lemma, represent B by B0 B1 with B0(p) = ff0(p; n) . (Z =pnZ *
*) and
L n
B1 (p) = ff1 (p; n) . (Z =pnZ ), subject to (ff0) 8 p, 9 M(p) : n M(p) ) ff*
*0(p; n) = 0
n
and (ff1 ) 9 fl1 2 flflfl(C) : 8 p; 8 n; ff1 (p; n) fl1 (p), where fl1 (p) !*
* or fl1 (p) = 0.
L L
From the definitions, fl(p) . B0(p) B0 ( fl(p) . (Z =pM(p)Z )) 2 C. Turn*
*ing to B1 ,
p p
for each p, there is a monomorphism fl(p) . B1 (p) ! (fl(p) + fl1 (p)) . (Z =p1*
* Z). Because
L
fl + fl1 2 flflfl(C), it follows that fl(p) . B1 (p) 2 C.]
p
Application: Let C be a Serre class. Assume: tf(C) contains a free group of*
* infinite
rank_then C is a ring.
EXAMPLE Not every Serre class is a ring. For instance, let C be the class *
*of all torsion abelian
L *
* P
groups G such that 8 p, G(p) is finite, so G(p) ff(p; n) . (Z=pnZ ), where r*
*(G(p)) = ff(p; n) < !
n *
* n
(cf. p. 71). Enumerate : p1 < p2 < . ._.then the subclass of C consisting of*
* those G for which the
*
* L
sequence {r(G(pk))=k} is bounded is a Serre class but it is not a ring (conside*
*r G = k . (Z=pkZ)).
*
* k
[Note: C is a Serre class and it is a ring.]
A Serre class C is said to be acyclic_if 8 G 2 C, Hn(G) 2 C (n > 0).
FACT Let C be a Serre class. Suppose that G 2 C is finitely generated_then*
* Hn(G) 2 C (n > 0).
L
If G is a torsion abelian group and if G G(p) is its primary decompositi*
*on, then
p L
8 n > 0, the Hn(G) are torsion and 8 p, Hn(G)(p) Hn(G(p)) () Hn(G) Hn(G(p))*
*).
p
[Note: 8 n > 0, G(p) bounded ) Hn(G(p)) bounded (in fact, pM(p)G(p) = 0 )
pM(p)Hn(G(p)) = 0).]
L L
Example: Q =Z Z=p1 Z ) Hn(Q =Z ) Hn(Z =p1 Z), where for n > 0,
p ae p
1 Z (n odd)
Hn(Z =p1 Z) = colimHn(Z =pkZ ) = Z=p0 (n even).
FACT Fix a prime p. For k = 1; 2; : :,:let Gk be a direct sum of k copies *
*ofnZ=pZ_then by the
P
K"unneth formula, 8 n > 0, Hn(Gk) = Gd(n;k), where d(1; k) = k and d(n; k+1) = *
* d(i; k)+(1(1)n)=2
*
*i=1
(hence d(n; k) kn).
FACT Fix a prime p. For k = 1; 2; : :,:let Gk be a direct sum of k copies *
*of Z=p1 Z_thennbythe1!
*
* k + _____
K"unneth formula, 8 n > 0, Hn(Gk) = Gd(n;k), where d(n; k) = 0 (n even) and d(n*
*; k) = n + 12
*
* _____
(n odd) (hence d(n; k) kn). *
* 2
78
LEMMA Suppose that C is a Serre class. Let 0 ! K ! G ! G=K ! 0 be a short
exact sequence in C_then for n > 0, Hn(G) 2 C provided that the Hp(G=K; Hq(K)) *
*2 C
(p + q > 0).
[Apply the LHS spectral sequence.]
[Note: By the universal coefficient theorem, Hp(G=K; Hq(K)) Hp(G=K)Hq(K)
Tor(Hp1(G=K); Hq(K)).]
THEOREM OF BALCERZYK Let C be a Serre class_then C is acyclic iff C is a
ring.
[Suppose that C is acyclic. Since G torsion ) Hn(G) torsion (n > 0), Ctoris*
* acyclic,
thus one can assume that C is torsion (cf. Proposition 3) and then, taking into*
* account
L
Proposition 4, work with C* (which is acyclic). So let G 2 C* : G G(p) ) G *
* G
L p
G(p) G(p) and #(G(p) G(p)) < ! ) G(p) G(p) H2(G(p)) H2(G(p)) G(p) )
p L
G G (H2(G(p)) H2(G(p)) G(p)) H2(G) H2(G) G 2 C*. To check that
p L
Tor(G; G) 2 C*, it is obviously enough to look at the case when G fl(p) . (*
*Z =p1 Z),
p
L L fl(p) + 1
where 8 p, fl(p) < !. Thus: H3(G) H3(fl(p) . (Z =p1 Z)) . (Z *
*=p1 Z)
p p 2
(cf. supra) and 2 fl(p)2+ 1 fl(p)2 ) fl2 2 flflfl(C) ) Tor(G; G) 2 C*.
Suppose that C is a ring. Let G 2 C_then there is a short exact sequence 0 *
*! Gtor!
G ! G=Gtor! 0. Accordingly, in view of the lemma, to prove that Hn(G) 2 C (n > *
*0), it
suffices to prove that Hp(G=Gtor; Hq(Gtor)) 2 C (p + q > 0). But Hp(G=Gtor; Hq(*
*Gtor))
Hp(G=Gtor) Hq(Gtor) Tor(Hp1(G=Gtor); Hq(Gtor)) and the verification that Hn(*
*G) 2
C (n > 0) reduces to when (i) G is torsion free or (ii) G is torsion.
(Torsion Free) If tf(C) is the class of all torsion free abelian grou*
*ps of cardinality
< ( > !) (cf. Proposition 1), then G 2 tf(C) ) #(Hn(G)) < ) Hn(G) 2 C (n > 0).
The other possibility is that tf(C) = T(AB ) for some T (cf. Proposition 2). U*
*nder these
circumstances, a given G 2 tf(C) contains a free subgroup F r . Z of finite ra*
*nk such
Lr
that the sequence 0 ! F ! G ! G=F ! 0 is exact. Here, G=F Ti is torsion
1
and the pprimaryacomponentseof each Ti are isomorphic to Z=pniZ or Z=p1 Z. The*
*refore
Hn(Ti) Ti0 (n(odd)n even)2 C (n > 0) ) Hn(T ) 2 C (n > 0) (K"unneth). On the*
* other
8
< r . Z (n r)
hand, Hn(F ) : n 2 C (n > 0). The lemma now implies that Hn(G) *
*2 C
0 (n > r)
(n > 0).
79
(Torsion) Let G 2 Ctor. Choose a basic subgroup B of G : 0 ! B ! G !
G=B ! 0_then, thanks to the lemma, one need only consider Hn(B) and Hn(G=B)
L
(n > 0). Using the cardinal lemma, represent B by B0B! B1 with B0(p) = ff0(p;*
* n).
L L n
(Z =pnZ ), B!(p) = ff!(p; n) . (Z =pnZ ), and B1 (p) = ff1 (p; n) . (Z =pnZ*
* ), subject to
P n n
(ff0) 8 p, ff0(p; n) < !, (ff!) 8 p, 9 M(p) : n M(p) ) ff!(p; n) = 0 & 8 n :
n
ff!(p; n) ! or ff!(p; n) = 0, and (ff1 ) 9 fl1 2 flflfl(C) : 8 p; 8 n, ff1 (p*
*; n) fl1 (p),
where fl1 (p) ! or fl1 (p) = 0. That Hn(B) 2 C (n > 0) results from the follo*
*wing
observations (modulo K"unneth): (O 0) 8 p, #(B0(p)) < !, hence there is a monom*
*orphism
Hn(B0(p)) ! nB0(p); (O !) 8 p, 8 ff !, Hn(ff . (Z =pkZ )) ff . (Z =pkZ ); (O *
*1 ) 8 p,
#(B1 (p)) fl1 (p), hence there is a monomorphism Hn(B1 (p)) ! fl1 (p) . (Z =p1*
* Z).
L
Finally, write G=B fl(p) . (Z =p1 Z) and fix n > 0. Case 1: n even ) Hn(G=*
*B) =
p
0. Case 2: n odd. If fl(p) !, then Hn(fl(p)0. (Z =p11Z)) fl(p) . (Z =p1 Z)*
*, while if
fl(p) + n__1_
fl(p) < !, then Hn(fl(p) . (Z =p1 Z)) B@ n + 1 2 CA. (Z =p1 Z) (cf. supra).*
* However,
_____
0 n  11 2
_____
B@fl(p) + 2 C (fl(p))n and fln 2 flflfl(C).]
n_+_1_ A
2
EXAMPLE Let C be a ring. Fix a nilpotent group G such that G=[G; G] 2 C_th*
*en 8 n > 0,
Hn(G) 2 C.
FACT Let C be a ring. Suppose that X is simply connected_then Hq(X) 2 C 8*
* q > 0 iff
Hq(X) 2 C 8 q > 0.
Application: Let C be a ring. Fix ss 2 C_then the Hq(ss; n) 2 C (q > 0).
If C is a Serre class, then a homomorphism f : G ! K of abelian groups is s*
*aid to be
Cinjective_(Csurjective_) if the kernel (cokernel) of f is in C, f being Cbi*
*jective_provided
that it is both Cinjective and Csurjective.
MOD C HUREWICZ THEOREM Let C be a Serre class. Assume: C is a ring.
Suppose that X is abelian_then if n 2, the condition ssq(X) 2 C (1 q < n) is
equivalent to the condition Hq(X) 2 C (1 q < n) and either implies that the Hu*
*rewicz
homomorphism ssn(X) ! Hn(X) is Cbijective.
EXAMPLE Let C be a ring. Suppose that X is a pointed connected CW space wh*
*ich is nilpotent.
Agreeing to write ss1(X) 2 C if ss1(X)=[ss1(X); ss1(X)] 2 C, fix n 2_then the *
*following conditions are
710
equivalent: (i) ssq(X) 2 C (1 q < n); (ii) Hq(X) 2 C (1 q < n); (iii) ss1(X) *
*2 C & Hq(eX) 2 C
(1 q < n). Furthermore, under (i), (ii), or (iii), the Hurewicz homomorphism s*
*sn(X) ! Hn(X) induces
a Cbijection ssn(X)ss1(X)! Hn(X).
[To illustrate the line of argument, assume (iii) and consider the spectral*
* sequence E2p;q Hp(ss1(X);
Hq(eX)) ) Hp+q(X) of the covering projection eX! X (cf. p. 562). Since ss1(X) *
*2 C is nilpotent,
E2p;02 C (p > 0) (cf. p. 79). In addition, the Hq(eX) (q > 0) are nilpotent ss*
*1(X)modules (cf. x5,
Proposition 17), thus E2p;q2 C (p 0; 1 q < n) (cf. p. 74) ) Hq(X) 2 C (1 q *
*< n) and there
is a Cbijection Hn(eX)ss1(X)! Hn(X). Owing to the mod C Hurewicz theorem, ssq(*
*X) ssq(eX) 2 C
(2 q < n) and the Hurewicz homomorphism ssn(eX) ! Hn(eX) is Cbijective. But *
*then the arrow
ssn(eX)ss1(X)! Hn(eX)ss1(X)is also Cbijective, ssn(eX) and Hn(eX) being nilpot*
*ent ss1(X)modules.]
A Serre class C is said to be an ideal_if G 2 C ) G K 2 C, Tor(G; K) 2 C f*
*or all K
in AB .
L *
* L
LEMMA Let C be a Serre class_then C is an ideal iff 8 G 2 C, Gi2 C, whe*
*re
i *
* i
is taken over any index set and 8 i; Gi G.
Example: Let C be an ideal. Suppose that G 2 [(sinX)OP ; AB ] is a coeffici*
*ent system
on X such that 8 oe, Goe 2 C_then 8 n 0, Hn(X; G) 2 C.
EXAMPLE The conglomerate of torsion Serre classes which are ideals is coda*
*ble by a set. For in
the set of subsets of F(N ; Z0 [{1}), write S ~ T iff each sequence in S is a *
*finite sum of sequences in T
and each sequence in T is a finite sum of sequences in S. Let E be the resulti*
*ng set of equivalence classes.
Claim: The conglomerate of torsion ideals is in a onetoone correspondence wit*
*h E. Thus given a torsion
ideal C, assign to G 2 C the sequence {xn(G)} 2 F(N ; Z0 [ {1}) by letting xn(G*
*) be the least upper
bound of the exponents of the elements in G(pn), where 8 n, pn < pn+1. Put SC =*
* {{xn(G)} : G 2 C}
and call [SC] 2 E the equivalence class corresponding to SC. To go the other wa*
*y, take an S and let CS
be the class of torsion abelian groups G with the property that there exists a *
*finite number of sequences
in S such that the nthterm of their sum is an upper bound on the exponents of t*
*he elements in G(pn)_
then CS is an ideal and S ~ T ) CS = CT, so C[S]makes sense. One has C ! [SC] !*
* C[SC]= C and
[S] ! C[S]! [SC[S]] = [S].
[Note: It is sufficient to consider torsion ideals since any ideal containi*
*ng a nonzero torsion free group
is necessarily the class of all abelian groups.]
MOD C WHITEHEAD THEOREM Let C be a Serre class. Assume: C is an ideal.
Suppose that X and Y are abelian and f : X ! Y is a continuous function_then *
*if
n 2, the condition f* : ssq(X) ! ssq(Y ) is Cbijective for 1 q < n and Csur*
*jective for
711
q = n is equivalent to the condition f* : Hq(X) ! Hq(Y ) is Cbijective for 1 *
*q < n and
Csurjective for q = n.
ae ae *
* ae
EXAMPLE Let X be abelian. Assume: 8 q, Hq(X) is finitely generated () *
*8 q, ssq(X)
Y Hq(Y ) *
* ssq(Y )
is finitely generated).
(char k = 0) Let f : X ! Y be a continuous function. Fix a field k o*
*f characteristic 0
and denote by F the class of finite abelian groups, T the class of torsion abel*
*ian groups_then if n 2,
the following conditions are equivalent: (1) f* : Hq(X) ! Hq(Y ) is Fbijective*
* for 1 q < n and F
surjective for q = n; (2) f* : Hq(X) ! Hq(Y ) is T bijective for 1 q < n and *
*T surjective for q = n; (3)
f* : Hq(X; k) ! Hq(Y ; k) is bijective for 1 q < n and surjective for q = n; (*
*4) f* : Hq(Y ; k) ! Hq(X; k)
is bijective for 1 q < n and injective for q = n.
(char k = p) Let f : X ! Y be a continuous function. Fix a field k o*
*f characteristic p
and denote by Fp the class of finite abelian groups with order prime to p, Tp t*
*he class of torsion abelian
groups with trivial pprimary component_then if n 2, the following conditions *
*are equivalent: (1)
f* : Hq(X) ! Hq(Y ) is Fpbijective for 1 q < n and Fpsurjective for q = n; (*
*2) f* : Hq(X) ! Hq(Y )
is Tpbijective for 1 q < n and Tpsurjective for q = n; (3) f* : Hq(X; k) ! H*
*q(Y ; k) is bijective for
1 q < n and surjective for q = n; (4) f* : Hq(Y ; k) ! Hq(X; k) is bijective f*
*or 1 q < n and injective
for q = n.
Example: If 8 n, f* induces an isomorphism Hn(X; Fp) ! Hn(Y ; Fp), then 8 n*
*, f* induces an
isomorphism ssn(X)(p) ! ssn(Y )(p) of pprimary components.
FACT Let X be a CW complex. Assume: X is finite and nconnected_then the H*
*urewicz homo
morphism ssq(X) ! Hq(X) is Cbijective for q < 2n + 1, where C is the class of *
*finite abelian groups.
81
x8. LOCALIZATION OF GROUPS
The algebra of this section is the point of departure for the developments *
*in the next
x. While the primary focus is on the "abeliannilpotent" theory, part of the m*
*aterial is
presented in a more general setting. I have also included some topological app*
*lications
that will be of use in the sequel.
The Serre classes in AB that are closed under the formation of coproducts *
*(and hence
colimits) are in a onetoone correspondence with the Giraud subcategories of A*
*B . Under
this correspondence, the class of all abelian groups corresponds to the class o*
*f trivial groups.
The remaining classes are necessarily torsion ideals and their determination is*
* embedded
in abelian localization theory.
[Note: Not every torsion ideal is closed under the formation of coproducts *
*(consider,
e.g., the class of bounded abelian groups).]
__
Notation: P is a set of primes, P its complement in the set of all primes.
Given P , put SP = {1}[{n > 1 : p 2 P ) p6 n}_then ZP = S1PZ is the local*
*ization
of Z at P and the inclusion Z ! ZP is an epimorphism in RG . Z P is a princip*
*al ideal
domain. Moreover, ZP is a subring ofaQeand every subring of Q is a ZP for a sui*
*table P .
L
The characteristic of 1 in Z P is 01 (p(2pP2)__P)) ZP =Z __Z=p1 Z. Examp*
*les: (1)
p2P
Take P = ; : ZP = Q ; (2) Take P = : ZP = Z; (3) Take P =  {p} : ZP = Z 1*
*_p; (4)
Take P =  {2; 5} : ZP = all rationals whose decimal expansion is finite.
[Note: Write Zp in place of Z{p} : Zp is a local ring and its residue field*
* is isomorphic
to Fp.]
EXAMPLE Suppose that P 6= ;_then as vector spaces over Q, Ext(Q ; ZP) R.
Equip SP with the structure of a directed set by stipulating that n0 n00if*
*f n0n00.
View (SP ; ) as a filtered category SP and let P : S P ! AB be the diagram *
*that
00
sends an object n to Z and a morphism n0 ! n00to the multiplication n__n0: Z ! *
*Z _
thenatheehomomorphism_colimP ! ZP is an isomorphism. Example: Z P Z=pnZ =
0 (p 2 P)
Z =pnZ (p 2 P ).
EXAMPLE Fix P 6= _then there is a short exact sequence 0 ! lim1H1(Z; Q[*
*ZP ]) !
H2(ZP ; Q[ZP ]) ! limH2(Z; Q[ZP ]) ! 0. Here, H2(ZP ; Q[ZP ]) 6= 0 (in fact, i*
*s uncountable (cf. p.
547)).
82
LEMMA Let P 0and P 00be two sets of primes_then (i) ZP0+Z P00= ZP0\P00and *
*(ii)
Z P0\Z P00= ZP0[P00,atheesum and intersection being as subgroups of Q . Further*
*more, the
biadditive function ZP0x(ZP00!zZP0\P000;dz00)e!fz0z00ines an isomorphism of r*
*ings: Z P0Z P00
Z P0\P00() ZP ZP ZP ).
00
Z P0[P00 i! ZP00
? ?
FACT There is a commutative diagram iy0 y j00and a short exact*
* sequence
ZP0 !j0ZP0\P00
0 ! ZP0[P00!Z P0 ZP00!Z P0\P00! 0 ((z) = (i0(z); i00(z)) & (z0; z00) = j0(z0) *
* j00(z00)), thus the
square is simultaneously a pullback and a pushout in AB .
An abelian group G is said to be SP_torsion_if 8 g 2 G, 9 n 2 SP : ng = 0.*
* Denote by
CP the class of SP torsion abelian groups_then CP is a Serre class which is cl*
*osed under
the formation of coproducts and every torsion Serre class with this property is*
* a CP for
some P . Examples: (1) Take P = ; : CP is the class of torsion abelian groups; *
*(2) Take
P = : CP is the class of trivial groups; (3) Take P = {p} : CP is the class o*
*f torsion
abelian groups with trivial pprimary component; (4) Take P =  {p} : CP is t*
*he class
of abelian pgroups.
__
[Note: G is SP torsion iff G is P primary or still, iff ZP G = 0.]
Let f : G ! K be a homomorphism of abelian groups_then