Cellular Spaces
by
E. Dror Farjoun
0. Introduction and Main Results
The aim of the present work is to study, in a systematic way, the possibili*
*ty of con-
structing new spaces out of a given collection of persumably better understood *
*spaces.
Recently there were several developments in this direction: There has been an i*
*ntensive
study of the construction of the classifying space of a compcat Lie group -as a*
* homotopy
limit- out of a collection of much smaller subgroups [JMO?]. This was generaliz*
*ed by [D-
W?] and was used effectively by several authors to construct new realizations o*
*f interesting
algebras as cohomology algebras of spaces. In a different vein [S?H] introduced*
* 'thick class-
es'. These classes are closed under cofibration. In the stable category they ar*
*e in fact the
closure under cofibrations and desuspensions of a single space, say V(n), const*
*ructed by
[?] and [?] Thus each one of these classes is precisely the class of all spectr*
*a that can be
built form a single spectra by taking cofibres of arbitrary maps (and desuspens*
*ions). Now
unstably the question arrise naturally:
1. Question: What is the collection of all spaces that can be built by repeate*
*d cofibration
fron these generic spaces V (n).?
This question is closely related to a problem posed by F. Adams in 1970:
Classify all E*- acyclic spaces for a give generalized homology theory E*.
In paticular, under what condition a K-acyclic space can actually be built *
*by a possibly
infinite process of repeated cofibration from the 'elementary' four cells space*
*:e V (1) first
constructed stably by Adams but now [Mis?] [C-N?] considered as a finite dimen*
*sional
p-torsion space[?] [?]?.
Our approach is to start with an arbitrary space A and consider the class o*
*f all spaces
gotten from it by repeatedly taking all pointed homotopy colimits along arbitra*
*ry diagrams
starting with A itself. We get a full subcategory denoted here by C.(A). This c*
*lass comes
with a functor CWA [Bou?] that associates to to every space a member of the cla*
*ss in a
natural fashion. Much of the present work can be done in the category of genera*
*l topological
spaces. In that context the usual class of CW complexes introduced by Whitehead*
* is simply
the class of spaces built from the zero sphere, the class of 1-connected comple*
*xes is that
built out of the two-sphere etc. But one can just as well start with 'singular'*
* spaces such
as the Hawaian rings and get a full blown homotopy theory.
Thus our basic concept in the present study is C.(A) = the smallest full su*
*bcategory of
S* closed under arbitrarypointed hocolim and weak homotopy equivalences that co*
*ntains
A. The functor CWA starts with an arbitrary space A and associates to any space*
* X the
`best A-approximation,' the space in C.(A) which is the closest space to X. It *
*can be built
out of copies of A by gluing them together along base-point preserving maps in *
*a similar
fashion to the construction of the usual CW -approximation to a given topologic*
*al space
X. The latter is, in fact, equivalent to the space CWS0X, where A = S0 is the z*
*ero sphere.
The functor CWA , is very closely related to the localization (or periodiza*
*tion) functor
PA with respect to A ?Bou] ?EDF]. By definition whenever a space X is A-periodi*
*c i.e.
the pointed function complex from A to X is contractible the A-CW approximation*
* to X
is contractible. In [N] Nofech introduced a model category structure on simplic*
*ial sets or
topological spaces in which weak equivalences are maps that induces isomorphism*
*s on the
A- homotopy groups. In that context the A- periodic (or shall we say A- trivial*
* ) spaces
apear as the fibrant objects which are weakly equivalent to a point while the A*
*- CW or
A- cellular appear as the cofibrant objects. Thus there is a sort of duality b*
*etween A
periodic spaces and A-cellular spaces. In the case where A is the mod-p Moore s*
*paces this
is well established: under mild conditions the homotopy fibre of the rational l*
*ocalization
and more generally rationally acyclic spaces can be built from these Moore spac*
*e Building
Blocks, via cofibrations. But we shall see that whenever [A; X] ' * the space C*
*WA X is
homotopy equivalent to the homotopy fibre of X ! PA X (see 0.7 below). As an e*
*xample
of application we address the following
0.1 Problem: Given a h*-acyclic space X for a generalized homology theory h*,*
* when
is X the limit of its finite h*-acyclic complexes?
It turns out that for h* = K = Morava's K-theories, and X = GEM a genera*
*lized
Eilenberg-MacLane space positive answer is available under some dimensional res*
*triction.
2
More generally, using some recent work of Bousfield and Thompson one can show t*
*hat for
the universal theories Sn considered in ?? ] ? ] the following holds:
0.2 Theorem. For any of the theories S(n) there exist N n such that every
N-connected space X, with N X a S(n)-acyclic space, is V (n)-cellular space and*
* in
particular a limit of its finite S(n)-acyclic subcomplexes.
Totally unrelated is the following.
f
0.3 Theorem. Let A = A0 any suspension and let A ! X!- X [ CA be a
homotopy cofibration sequence. The homotopy fibre F of f is an A-cellular spac*
*e. In
particular if "h*(A) ' 0 for some generalized homology then "h*(F ) ' 0.
Here is an interesting generalization of a theorem of Serre which follows q*
*uite easily
from the present approach:
0.4 Theorem. Let A be any pointed, finite type connected space. Let X be an*
*y finite
A-cellular space, with H"*(X; Z=pZ) 6= 0 for some p. Then ssi(X; A) = [iA; X] 6*
*= 0 for
infinitely many dimensions i 0.
One immediate corollary is for X = A.
0.4.1 Corollary. Let A be any connected space with "H*(X; Z=pZ) 6 0 for som*
*e p.
There are infinitely many `'s for which [`A; A] 6= *.
For the case A = S1=the circle we get well known theorems of Serre about th*
*e higher
homotopy groups of finite simply connected complexes.
A proper context for the present approach is to define a map f: X ! Y to b*
*e weak
A-equivalence if map*(A; f) is a weak homotopy equivalence. It turns out that *
*for any
pointed simplicial set A or any pointed compact topological space A, the notion*
* of weak
A-equivalence can be embedded in a model category in the sense of Quillen ?? ]*
* on the
category of all topological spaces or pointed simplicial sets ?N]. This is disc*
*ussed briefly in
section 3.
3
For finite p-torsion spaces one can classify all the types of equivalences *
*that arise this
way between spaces.
As in the case of the periodization functor PA the functor CWA has pleasant*
* formal
properties that makes it accessible and useful. Let us summarize some of these *
*properties
as follows:
0.5 Theorem. Let the class C.(A) S* be the closure under arbitrary pointed
hocolims and weak equivalences of the singleton {A} containing the pointed spac*
*e A.
1. There is a homotopy idempotent functor CWA : S* ! S* which is augmented by
o: CWA X ! X with CWA o a homotopy equivalence, and therefore CWA is a poin*
*ted
simplicial functor taking values in C.(A).
2. The map CWA X ! X is initial among all maps f: Y ! X with
map *(A; f): map*(A; Y ) ! map *(A; X)
a weak equivalence and terminal among all maps of members of C.(A) into X.
3. Each of the universal properties (2) above determine CWA X ! X up to an equ*
*ivalence
which is itself unique up to homotopy.
4. Let X~be any I-diagram of pointed spaces. One has a natural homotopy equiva*
*lence
CWA ho limIX~' CWA ho limICWA X~:
'
5. There is a natural homotopy equivalence CWA (X x Y ) -! CWA (X) x CWA (Y ).
'
6. There is a natural homotopy equivalence CWA X -! CWA X, where A is the
suspension of A.
7. If Y = GEM then CWA Y is also a GEM. If, in addition, ssiY ' 0 for i r,*
* then
ssiCWA Y ' 0 for i 0.
8. CWkA CW`A X = CWn X where n = max (k; `).
9. PA CWA X ' CWA PA X ' *.
These properties of CWA follow in turn from certain closure properties of C*
*.(A).
4
0.5 Theorem. For any fibration F ! E ! B we have:
(1)If F; E 2 C.(A) then B 2 C.(A)
(2)If E; B 2 C.(A) then F 2 C.(A)
(3)If F; B 2 C.(A) then E 2 C.(A).
0.6 Theorem. For any pointed spaces A; X; Y the product X x Y is A-cellu*
*lar if
and only if both X and Y are A-cellular. A retract of any X in C.(A) is also in*
* C.(A).
Fibrations Associated to CWA
The above theorems 0.5-0.6 form the basis on which one can build a reasonab*
*le theory
of CWA very similar, but not identical, to the theory of PA , the A-localizatio*
*n or A-
periodization functor as developed in ?Bou] ?EDF] ?DF-S]. The main features of *
*the gen-
eral theory is to determine to what extent CWA preserves fibre sequences, relat*
*e CWkA
for various k , give practical criteria for a space to belong to C.(A) and dete*
*rmine for what
A and B one has C.(A)= C.(B) .
Here we give a review of some fibration theorems associated to CWA . In pa*
*rticular
it often preserves fibrations and commutes with loops `up to a GEM.' This close*
* analogy
with the results of ?DF-S] about PA X is perhaps explained by the following the*
*orem that
presents CWA X as the homotopy fibre of X ! PA X under certain circumstances:
0.7 Theorem. Whenever the natural composition
CWA X ! X ! PA X
is null homotopic, the sequence is a homotopy fibration sequence for any A; X 2*
* S*.
0.8 Corollary. In particular if PA X ' * then CWA X ! X is a homotopy
equivalence and so X 2 C.(A).
0.9 Corollary. If [A; X] ' * then the above sequence is a fibration sequen*
*ce.
0.10 Corollary. If the natural map CWA X ! CWA X or the natural map
PA X!- PA X is an equivalence then the above sequence (1.7) is a fibration.
5
The basic tool in analyzing preservation of fibration is the following theo*
*rem which is
dual to the implication PA X ' * ) P2A X is a GEM (= generalized Eilenberg Ma*
*cLane
space i.e. a possibly infinite product of K(G; h)'s for G abelian group) ?DF-S]:
Preservation of fibrations. As in [?] one of our main concerns is to show t*
*hat
CWA 'almost' preserves fibration. A moment's reflection on the example where A=*
* the
sphere shows that one cannot expect that it will preserve fibration without som*
*e "error
term". Thus our aim is to be able to show that often that error term is small a*
*nd man-
ageable. As was seen in [?[? a small error is in this context a space which is *
*a procuct of
(possibly infinite number ) of Eilenberg-MacLane spaces (GEM).
The simplest question in that direction concerns a situation where CWA kil*
*ls two
members in a fibre sequence -does it always kill the third?
0.11 Theorem. For any pointed spaces A; X let f ! E ! B be a fibration seq*
*uence
if CWA F ' CWA E then CWA B is a GEM. Therefore if CW2A X ' * then CWA X
is a GEM.
0.12 Remark. In most cases it is not hard to determine the GEM that arise *
*in 0.11
since the A-cellular K(G; n)'s are not hard to understand in many cases see (5.*
*4) below.
Of course (0.11) is obvious for A = Sn for all it says is that if some n-connec*
*ted cover of
X is contractible then the (n - 1)-connected cover is an Eilenberg-MacLane spac*
*e.
With these techniques we deduce without much difficulties:
0.13 Theorem. Let A = A0; X 2 S*.
(1)The fibre of
CWA X ! CWA X
is a polyGEM.
(2)If F ! E ! B is a fibration then for any A 2 S* there exists a fibration
__
CWA F ! E ! CWA B
6
__
in which E 2 C.(A) is a `mixture' of A- and A-co-localization of E. If in add*
*ition
__
A = A0 is a suspension space, then the natural map CWA E ! E has a GEM as a
homotopy fibre, and therefore
CWA F ! CWA E ! CWA B
is a fibration, up to a polyGEM. Namely the homotopy fibre of the natural map
CWA F ! fibre(CWA E!- CWA B)
is a polyGEM (= a "generalized Postnikov n-stage").
Organization of the paper. In the first section we review some basic techn*
*ical
result about localization theory. We give a full proof of a crucial technical r*
*esults about
preservations of certain fibration by localization functors. In the second sect*
*ion we discuss
in some details general closure properties of "closed classes" such as C.(A). I*
*n particular
we show that they are closed under cartesian products, half-smash product and t*
*o some
extend under taking homotopy fibre. This gives us a useful generalization of an*
* important
lemma[Bou?] of Miller and Zabrodsky.
We prove 0.3 and 0.6 in that section. In the third section we discuss and p*
*rove (0.5)
(1,2,3,4,5) as well as formulate the appropriate Whitehead theorem for detectin*
*g homotopy
equivalences between A-cellular spaces. The more delicate fibration theorems a*
*nd (0.5)
(6-9) are discussed in section 4 and 5. where we prove the basic 0.7.
The last section is devoted to examples, discussion of E-acyclic spaces and*
* a proof of
0.2.
1. A Review of Homotopy Localization with Respect to a Map
In this paper we will use crucially several properties of Lf, the localizat*
*ion functor
with respect to a general map f: A!- D. Most of the time we will consider f: *
*A!- *
and in that case one denotes Lf by PA due to its close formal similarity to the*
* Postnikov
section functor Pn = Psn+1.
Basic Properties:
7
We recall from [ ] [ ] [ ] several basic properties of Lf and PA where A *
*is any space
and f : A ! D any map.
Given any spaces A; D; X and a map f : A ! D there exists co-augmented func*
*tors
X ! LfX and X ! PA X (where PA is a special notation for L(A!*) since its prope*
*rties
are very remenicent of those of the Postnikov section functor Pn). X ! LfX is *
*initial
among all maps X ! T to f-local spaces T (or f-periodic or f-divisible) i.e. sp*
*aces with
the function complex map:
(1.1) map (f; T ) : map (D; T ) ! map (A; T )
being a weak equivalence. The co-augmentation X ! LfX is also terminal among a*
*ll
maps X ! T which become equivalences upon taking their function complexes to any
f-local space.
The functor Lf can be defined in either the pointed or unpointed category o*
*f spaces
and its value for connected A, D, X does not depend, up to homotopy, on the cho*
*ice of
the category in which one works ({S} or{S*}).
Being defined on the unpointed category and being homotopy functor it has a*
*n asso-
ciated fibrewise localization functor that turns any fibration sequence F ! E !*
* B into a
fibration sequence LfF ! "E! B.
In addition Lf (or PA ) enjoys the following properties.
~=
1.2 There is a natural equivalence Lf(X x Y ) -! Lf(X) x Lf(Y ).
1.3 Every fibration sequence F ! E ! B with LfF ' * gives a homotopy equivalence
LfE -'! LfB.
1.4 There is a natural equivalence LfX ' Lf X.
__ __ __
1.5 If PA X is the homotopy fibre of X ! PA X then PA (P AX) ' *. Similarly LfL*
* f X '
*. (Notice f).
8
1.6 If F ! E ! B any fibre map and B is A-periodic (i.e. map *(A; B) ' *) or *
*more
generally PA B ' PA B then PA F ! PA E ! B is also a fibre sequence. Simi*
*larly if
F is A-periodic that F ! PA E ! PA B is a fibration sequence.
1.7 Lf hocolimIX~= Lf hocolimILfX~, and in particular LfX ' * implies LfkX ' **
* for
all k.
1.8 If LfX ~ * and LgY ' * (or PW Y ~ *) then Lf^g(X ^ Y ) ' * (or Lf^W X ^ Y *
*~ *).
1.9 If PA B ' * and PB C ' * then PA C ' *.
1.10 If for all ff 2 I X(ff) is f-local where X(ff) is a member of an I-diagram*
* X~ indexed
by a small category I, then so is holimIX~.
1.11 If Y is an n-connected GEM then so is LfY for any f : A ! B.
GEM-Properties
In addition to the above list the functor Lf the most fundamental property*
* of Lf is
the following
1.12 GEM Theorem. Let F!- E!- B be a fibration sequence of pointed conne*
*cted
spaces. Assume Lf B ' Lf E ' *. Then Lf F is a GEM while LfF ' *.
1.12.1 Corollry:. If Lf X ' * then L2f X ' GEM.
1.12.2 Corollry:. The homotopy fibre of P2A X ! PA X is a GEM for any X;*
* A.
Proof. The first corollary follows immediately using the adjunction (1.4) *
*and the stan-
dard loop space fibration over X. The second corollary follows from the first *
*using (1.5)
and noting that the map in question is in fact a localization map.
We now turn to the proof of (1.12)
9
We first show that Lf F is an 1-loop space using ?S] ?B-F]. We define a (no*
*n-special)
-space as follows.
_
F n = fibre of(E _ : :_:E!- B _ : :_:B): (n - copies ofE; B)
We observe that the functor that assigns to a finite pointed set S the wedge _s*
*X of copies
of X for any X 2 S*, gives a (non-special) -space: i.e. a functor from {finite*
* pointed
sets} to spaces. This functor assigns to every pointed set its smash product wi*
*th the given
space X-a construction that is clearly natural. We now consider this constructi*
*on for E
and B. ae oe ae oe
W W
The homotopy-fibre being a functor in S* and E !- B being a*
* map
n n0 n n0
_
of -spaces we conclude that F .above is a -space.
We claim: The natural map (see diagram below)
_
fn: Fn!- F x : :x:F
induces an equivalence on Lf (fn).
_
Since F 1= F this implies that
_ _
Lf F n = Lf (F x : :x:F )= (Lf F )n
= (F1)n
_ _
Thus Lf (F .) is a special -space and therefore Lf F 1 = Lf F is an 1-loop spac*
*e.
Consider the diagram that depicts the above constructions for n = 2: This *
*diagram is
built from the lower right square by taking homotopy fibres.
(B * B) ----! X ----! E * E ----! B * B
?? ? ?
y ?y ?y
_
F 2 ----! E _ E ----! B _ B
?? ? ?
y ?y ?y
F x F ----! E x E ----! B x B
10
By (1.3) in order to prove the claim it is sufficient to show that Lf X ' *. Fi*
*rst notice
Lf (E * E) ' Lf ((E ^ E). But Lf(E ^ E) ' * since LfE ' * (1.8). Thus
Lf E * E and also Lf (B * B) ' * (1.7). Now consider Lf (B * B). By (1.4)
Lf (B ^ B) = L2f (B ^ B):
But (1.8) Lf (B^B) ' * since (1.8) LfB ' * and Ps1B ' *, thus (1.7) L2f (B^
B) ' *
This proves our claim since it implies (1.3):
Lf X ' Lf (E * E) ' *:
Therefore Lf F is 1-loop space and in particular we can write: Lf F = Y .
claim: map*(2F; Y ) ' * i.e. map*(F; Y ) ' *:
This follows from universality (1.1): We have factorization: But
F ----! Y = Lf F
?? x
y ??9!
Lf F
in which: Claim: Lf F ~ *. Moreover: LfF ~ *. This is clear from (1.3) for*
* the
fibration:
B!- F!- E
and LfB ' Lf B ' * ' *. All the more so Lf kF ~ * and thus (kF!- Y )
is null. The claim being proven we can conclude from Bousfield's key lemma (1.1*
*3) below
that F!- Lf F factors through the universal GEM asociated with F namely the in*
*finite
symmetric product: SP 1F .
SP 1F
??
y
F ----! Y = Lf F
But SP 1F , the Dold-Thom functor on F is a GEM. Applying Lf to the factorizat*
*ion
we get that Lf F is a retract of a GEM since Lf = Lf Lf . But a retract of a G*
*EM is
a GEM. This concludes the proof.
11
1.13 Bousfield's Key Lemma. Let X be a connected, Y a simply-connected spac*
*es.
Assume map *(2X; Y ) ' *. Then map *(X; Y ) ~=map *(SP kX; Y ) for any k 1. ?5,
6.9].
1.14 Remark. A way to understand (1.13) is to interpret it as saying that t*
*he space
SP kX can be built by successively glueing together copies of `X for ` 1 with *
*precisely
one copy for ` = 1. Since the higher suspension 2+jX (j 0) will not contribute*
* anything
to map *( SP `X; Y ) we are left with map *(X; Y ).
More precisely, it can be easily seen by adjunction that (1.13) is equivale*
*nt to the
following:
For any space X the suspension of the Thom-Dold map t:X ! SP kX induces a
homotopy equivalence P2X (t) upon localization with respect to the double susp*
*ension of
X.
In fact the same holds for the James functor JkX and other cases.
This is a correct reformulation because by universality (1.1) a map t induc*
*es a ho-
motopy equivalence on the f-localization iff it becomes an equivalence upon tak*
*ing the
function complex of t into any f-local space. In this form (1.13) can be verifi*
*ed using (1.7)
and a homotopy colimit presentation of the Dold-Tho m functors in ?6,6.4],and u*
*sing the
fact discussed above that the inclusion (X _ X ! X x X) becomes an equivalence *
*after
localization with respect to the above double suspension.
2. Closed Classes and A-Cellular Spaces
In this section we discussed certain full subcategories of S* called closed*
* classes. The
main example of such classes is C.(A) for a given pointed A, but also the class*
* of spaces
that map trivially to all finite dimensional spaces is closed.
2.1 Definition. A full subcategory of pointed spaces C. S* is called "clo*
*sed" if it
is closed under weak equivalences and arbitrary pointed homotopy colimits: Nam*
*ely for
any diagram of space in C. (i.e. a functor X~: I ! C.) the space hocolim X~is a*
*lso in C..
We prove several closure theorems for any closed class C., the most importa*
*nt ones
12
being:
1. C. is closed under finite product.
2. If X 2 C. and Y any (unpointed) space then X o Y = (X x Y )= * xY is in C..
3. If F ! E ! B a fibration sequence and F; E in C. then so is B.
i
4. If A ! X!- X [ CA is any cofibration sequences and A is in C. then so is t*
*he fibre
of i.
2.2 Examples of Closed Classes:
1. The class C.(A): This is the smallest closed class that contains a given p*
*ointed space
A. It can be built by a process of transfinite induction by starting with the f*
*ull subcategory
containing the single space A and closing it repeatedly under arbitrary pointed*
* hocolim. In
section (3) below we give a `cellular' description of spaces in C.(A). We refer*
* to members
of C.(A) as A-cellular spaces.
f
2. The class C.(A!- B) = C.(f) here we start with any map (or a class of maps*
*) f 2 S*
of pointed spaces and consider all spaces X such that the induced map on pointe*
*d function
complexes
map *(X; A) ! map *(X; B)
is a (weak) homotopy equivalence of simplicial sets. Since map*(hocolimIXff; A)
= holimIopmap*(Xff; A), it is immediate that C.(f) is a closed class. This cla*
*ss is often
empty.
3. The class of spaces that map trivially to all finite dimensional spaces. Th*
*is includes
by Miller's theorem K(ss; 1) for a finite group ss.
2.3 Pointed and Unpointed Homotopy Colimits
Let A be a pointed space. We have considered C.(A) the smallest class of po*
*inted spaces
closed under arbitrary pointed_hocolim, and homotopy equivalences, which contai*
*ns the
space A. Notice that if we consider classes closed under arbitrary non-pointed*
*_hocolim
we get only two classes the empty class and the class of all unpointed space. T*
*his is true
13
since a class closed under unpointed hocolim that contains a contractible space*
*, contains
all weak homotopy types, since evey space is the free hocolim of its simplices.
Notice also that if A is not empty then C.(A) contains the one-point space *
** ' P -
hocolim (A ! A ! A ! : :):where all the maps in this infinite telescope are the*
* trivial
maps into the base point * 2 X.
In general given a pointed I-diagram X~ we can consider its homotopy (inver*
*se) limit
in either the pointed or unpointed category. By definition, these two spaces ha*
*ve the same
(pointed or unpointed) homotopy type they have in fact the same underlying spac*
*e. On
the other hand, the homotopy colimits of X~will generally have a different homo*
*topy type
when taken in the pointed or unpointed category: If *~2 X~ is the I-diagram of*
* base
points in X~ then we have a cofibration_; with NI = the classifying space (or t*
*he nerve) of
the category I.
NI ! free-hocolimX~! pointed-hocolimX~:
Since by definition we have a cofibration:
(I\ - *~) ! (I\ - X~) ! (I\ - | X~)
where is the "tensor product" of an I-diagram with Iop-diagram and | is the `p*
*ointed
tensor product." ?B-K p. 327 & p. 333]
Corollary. If the classifying space of the indexing category I is contracti*
*ble then
for any I-diagram pointed diagram Y~we have a homotopy equivalence free-hocolim*
*IY~'
pointed-hocolimI"Y.
Remark. Thus over the usual pushout diagram .- .!- . and over infinite*
* tower
.!- .!- : ::::h:ocolim takes the same value in the pointed and unpointed cate*
*gories.
But not e.g. over a discrete group.
In the present paper unless explicitly expressed otherwise hocolim mean poi*
*nted ho-
colim over pointed diagram. Thus for any small category I we have hocolimI{*} *
*= {*}.
Otherwise we use the notation free-hocolim, thus free-hocolimI{*} = BI = NI the*
* nerve
of I for every I.
14
2.4 Half smashes and products in closed classes: We now show that a closed cla*
*ss C.
is closed in an appropriate sense under half smash with an arbitrary unpointed*
* space (i.e.
C. is an ideal in S* under the operation C.!- C.o Y ), and under internal fini*
*te Cartesian
products (see 2.1 above). But first
2.5 Generalities about half-smash: Recall the notation
X o Y = (X x Y )= * xY
where X is pointed and Y is unpointed space. This gives a bifunctor S* x S!- *
*S*. There
is another bifunctor S x S*!- S* given by gmap(Y; X) where Y is unpointed and*
* X pointed
and where gmap(Y; X) is the space of all maps equipped with the base point Y!-*
* *!- X.
Thus the underlying space of mgap(Y; X) is the same as that of the free maps w*
*hile the
underlying space of X o Y is different in general from that of the base point *
*free product
Y 0x Y .
There are obvious adjunction identities
(i) map*(A o Y; X) = map *(A; gmap*(Y; X))
(ii) map*(A o Y; X) = gmap(Y; map*(A; X))
The first identity (i) says that for each Y 2 S the functor -oY : S*!- S**
* is left adjoint
to gmap(Y; -). Whereas identity (ii) says that for each A 2 S* the functor A o*
* -: S!- S* is
left adjoint to map*(A; -), where the latter is the space of pointed maps as a*
*n unpointed
space, i.e. forgetting its base point.
In particular we conclude
2.6 Proposition. For each A 2 S* and Y 2 S the functors -oY and Ao- commute
with colimits and hocolimits.
Notice that to say that A o -: S!- S* commutes with hocolim involves comm*
*uting
pointed hocolim i.e. the hocolim in S* with unpointed hocolim in S.
Explicitly: For any base point free diagram of space Y~: I!- S we have an*
* equivalence:
A o (free- hocolimIY~) = pointed- hocolimI(A o Y~):
15
2.7 Lemma. If Y is any unpointed space then for any indexing diagram I the*
* functor
- o Y : S* ! S* commutes with hocolimI, and if X is any pointed space the funct*
*ors
X o -: S!- S* and - ^ X: S* ! S* commute with hocolimI.
Proof. We have just considered X o -. Similarly - ^ Y is left adjoint to ma*
*p*(Y; -)
and again commutes with colim and hocolim.
2.8 Theorem. If X is any closed class C. space then:
(1)For any (unpointed) space Y the half-smash X o Y is in C.
(2)For any (pointed) B cellular-space Y the smash X ^ Y is an (A ^ B) cellu*
*lar-space
and an A-cellular space.
Proof. To prove (1) we start with an example showing that X o S1 is an X-ce*
*llular
space. In fact it can be gotten directly as a pointed hocolim of the push-out d*
*iagram:
fold
X _ X ----! X
? ?
fold?y ?y
X ----! X o S1
This diagram is gotten simply by half-smashing X with the diagram that presents*
* S1 as
free-hocolim of discrete sets:
{0; 1}----! {0}
?? ?
y ?y
{1} ----! S1
By inducation we present Sn+1 as a pushout *- Sn!- * which gives by indu*
*ction
X o Sn+1 as a pushout along X- X o Sn!- X, that arise since (2.5) (X o -) co*
*mmutes
with free-hocolim on the right (smashed) side. Since the filtration of Y by ske*
*leton Y0
`
Y1C : :p:resents Yn+1 = Yn [ (C Sn) we get upon half smashing with X a presen*
*tation
of Y o X as a pointed-hocolim.
Remark. Here is a `global' formulation of the above proof using (2.6): Pre*
*sent the
space Y as free-hocolimDY{*} where DY is any small category whose nerve is equi*
*valent to
16
Y , and {*~} is the DY -diagram consisting of the one-point space for each obje*
*ct of DY .
Now by (2.6) above:
X o Y = X o free- hocolimDY{*~} = pointed- hocolimDYX o {*~}
Thus X o Y is directly presented as a pointed hocolim of a pointed diagram con*
*sisting
solely of many copies of the space X itself. Now to prove (2) one just notices *
*that X ^ Y =
(X o Y )=X x {pt} so X ^ Y is certainly an X-cellular space. Now since pointed-*
*hocolim
commutes with smash-product we get by induction on the presentation of Y as a B*
* space
that X ^ Y is a A ^ B space as needed.
2.9 Theorem. Let F ! E ! B be any fibration of pointed spaces. If F and E *
*are
members of some closed class C. then so is B.
corollary. We shall see later (4.9) that this implies that if the base and *
*total spaces
are A-cellular for any A then the fibre is A-cellular.
corollry. (This is a generalization of [millerZab?][Bo?]): let F ! E ! B be*
* any
fibration sequence. If both the fibre and total space have a trivial function c*
*omplex to a
given pointed space Y then so does the base space B.
The second corollary follows immediately by observing (2.2) (2) that the cl*
*ass of spaces
with a trivial function complex to a given space is closed.
Proof: We define a sequence of fibrations Fi ! Ei ! B by E0 = E; F0 = F; E*
*i+1 =
Ei [ CFi and Fi+1 is the homotopy fibre of obvious map Ei+1 ! B. All Ei; Fi a*
*re
naturally pointed spaces.
17
F ----! E ----! B
??
y
F1 ----! E [ CF = E1 ----! B
??
y
F2 ----! E1 [ CF1 = E2 ----! B
??
y
: : :
E1 = B
By Ganea's theorem ? ?] Fi+1 ' Fi* B ' (Fi^ B) and therefore connectivity of
Fi+1 is at least i, since F0 is (-1)-connected. Notice that by definition since*
* E0; F0 are in
C. spaces so are Ei; Fi for all i. But since conn Fi! 1, we deduce that hocolim*
* Ei= B.
Therefore B is also in C., as needed.
We now turn to the somewhat surprising closure property of closed classes (*
*2.1) (4.)
2.10 Theorem. For any map A!- X of pointed space the homotopy fibre F of
X!- X [ CA satisfies PA F ' *.
2.10.1 Corollary. If A = A0 then above fibre F is an A-cellular space.
Proof. The proof uses the following diagram:
A
?
i?y
c
F ----! X ----! X [ CA
?? ? ?
y ?y =?y
__ _c
PA F ----! X ----! X [ CA
Where the solid arrows are given by the fibrewise localization (p. 4) of th*
*e top row.
Thus the fibre map _cis induced from the composition X [ CA ! B aut F!- B out *
*PA F .
Here as in [??? ] [ ] we use the fact that the fibration Y!- B out .Y!- B an*
*d Y , where
out Y (out .Y ) is the space of un-pointed (res. pointed) self equivalences of *
*Y , classifies all
18
fibrations with fibre Y . Taking F to be the usual path space we have a well de*
*fined map i0
into the homotopy fibre. Since map(A; PA F ) ' * by construction of PA F the co*
*mposition
__
A!- X!- X factorizing through PA F is null-homotopic, where the null homotopy*
* comes
from the cone A!- F!- F [ CA!- PA F that defines PA F . This null homotopy g*
*ives a
__
well defined map c0: X [ CA!- X rendering the diagram strictly commutative.
*
* __
Therefore the fibration _cis a split fibration having c0: as a section. Als*
*o since F!- X
factors through X [ CA it is null homotopic map. But the splitting of _cimplie*
*s from
__
the long exact sequence of the fibration that the map PA F!- X is injective o*
*n pointed
__
homotopy class [W; -]* for any W 2 S*. And since F!- PA F!- X is null homoto*
*pic we
conclude that F!- PA F is null. Now idempotency of PA implies PA F ~ * as need*
*ed.
Proof of Corollary. We know from (1.8) that PA0 F ' * implies that F is a A*
*0-
cellular space, this is proved in (4.5) below without of course using the prese*
*nt corollary.
Closure under Products
Many of the pleasant properties of CWA depend on its commutation with finit*
*e prod-
ucts. This commutation rest on the following basic closure properly of any clos*
*ed class.
2.11 Theorem. Any closed class C. is closed under any finite product: If *
*X; Y 2 C.
then so is X x Y .
Remark. It is well known that an infinite product of S1's does not have the*
* homotopy
type of a CW -complex i.e. the class of all CW -complexes in T op* is not close*
*d under
arbitrary products.
Remark. If A = A0 and B = B0 where A; B 2 C., then A x B is easily seen to *
*be
in C. via the cofibration
A0* B0! A0_ B0! A0x B0:
Since A0* B0~= A0^ B0 one uses (2.8).
19
We owe the proof to W. Dwyer. An independent proof can be extracted from [B*
*ou?
-1].
Proof. We filter Y by its usual skeleton filtration Yn+1 = Yn [ en+1 : :.:
We may assume X; Y are connected. For brevity of notation we add one pointe*
*d cell
at a time but the proof works verbatim for an arbitrary number of cells. Let P *
*(n) be the
subspace of X x Y given by
P (n) = {*} x Y [ X x Yn:
Clearly the tower P (n) ,! P (n + 1) is "cofibrant" and its colimit X x Y is eq*
*uivalent to
its homotopy colimit. Since C. is closed under hocolim it is sufficient to show*
*, by induction
that P (n) 2 C. for all n 0 . For n = 0, we have P (0) = X _ Y clearly in C.. *
*Now P (n)
is given as homotopy pushout diagram:
'
X x Sn-1 [ * x Dn ----! X o Sn-1 ----! {*} x Y [ X x Yn-1
?? ? ?
y ?y ?y
'
X x Dn ----! X o Dn ----! P (n)
coming from the presentation of Xn as a pushout over pointed diagram: Xn-1- S*
*n-1!-
Dn. Since the upper-left corner is the half smash X o Sn it is in C. by lemma 2*
*.5 above.
Notice that all the maps are pointed. Therefore P (n) is a homotopy pushout of *
*members
of C. as needed.
2.12 Corollary. For any two A-cellular spaces X; Y their product X x Y i*
*s an
A-cellular space.
Proof. Consider the class C.(A). By the theorem just proved it is closed un*
*der finite
product, therefore the product of any two A-cellular spaces is A-cellular.
3. A-Homotopy Theory and the Construction of CWA X
In this section we describe some initial elements of A-homotopy theory. Thi*
*s frame-
work replaces the usual sphere S0 in usual homotopy theory of CW -complexes or *
*simplicial
20
sets by an arbitrary space A. It can be considered in the framework of general *
*compactly
generated spaces where A can be chosen to be any such space. We will however re*
*strict
our discussion to A 2 S* a pointed space.
In particular there is a model category structure on S* denoted by SA*where*
* a weak
equivalence f: X ! Y is a map that induces a usual weak equivalence
map*(A; f): map*(A; X) ! map *(A; Y )
or function complexes, and A-fibre maps are defined similarly. Cofibrations ar*
*e then
determined by lifting property ?N].
The analog of a cofibrant object i.e. CW -complex is an A-cellular space. T*
*he natural
homotopy groups in this framework are A-homotopy groups
ssi(X; A)= [iA; X]*
= ssimap *(A; X; null)
= [A; iX]*:
The classical Whitehead theorem about CW -complexes takes here the form:
3.1 A-Whitehead theorem: A map f: X ! Y between two pointed connected A-cellular
spaces has a homotopy inverse (in the usual sense) if and only if it induces a *
*homotopy
equivalence on pointed function complexes
'
(*) map*(A; X) -! map *(A; Y ):
or equivalently, iff f induces an isomorphism on the pointed homotopy classes:
~=
(**) [A o Sn; X]* -! [A o Sn; Y ]*
for all n 0. If the two pointed function complexes are connected i.e. ss0(X;*
* A) '
ss0(Y; A) ' * or if A = A0 is a suspension then a necessary and sufficient cond*
*ition
is that it induces an isomorphism on A-homotopy groups:
~=
ss*(X; A) -! ss*(Y ; A):
21
Proof. (compare [D-?-Z]). It is sufficient to show that under (*) for every*
* W 2 C.(A) we
'
have map*(W; X) -! map (W; Y ) is a homotopy equivalence. This can be easily sh*
*own by a
transfinite induction on the presentation of W as a hocolim of spaces in C.(A).*
* Namely one
need only show that the class of spaces W for which map*(Y; f) is a homotopy eq*
*uivalence
is a closed class. But this is the content of (2.1). Since by assumption it c*
*ontains A, it
follows that it contains also C.(A) and therefore by our assumption it contains*
* both X and
Y . Thus we get a homotopy inverse to X ! Y by taking Y = W . This completes*
* the
proof.
3.2 An elementary construction of CWA X
Given A; X 2 S* we construct, in a natural way, a map CWA X ! X. It will be*
* clear
from the construction that CWA (-) is a functor S* -! S*. Compare [B?ou-1].
3.3 Half-suspensions "nX: A basic building block for CWA is the half-smash Sn o*
*A =
Sn x A [ Dn+1 x {*} with the base point {*} x {*}. We denote this space by "n A*
*, and
call them half n-suspensions.
Just as an homotopy class ff 2 ssnmap *(A; X; null) in the null component i*
*s respresent-
ed by a pointed map nA ! X so a does a map "ff: "nA ! X represents an element in
ssnmap *(A; X; f) of the f-component where f: A ! X is any map. The map f is go*
*tten
from "ffby restricting "ffto * x A "nA.
Notice that if A itself is a suspension A = B then "n A ~=nA _ A ?E.D.F] bu*
*t in
general such a decomposition does not hold. Thus for suspension A = B an elemen*
*t "ff
as above is given simply by a pair (ff _ f): nA _ A ! X. In that case of course*
* all the
components of map*(A; X) has the same homotopy type.
ff
3.4 Construction of CWA X. Let c0: C0X = fVf2I"iA -! X be the wedge of all t*
*he
pointed maps "iA ! X from all half-suspensions "iA to X. Clearly the map c0 ind*
*uces
a surjection on the homotopy classes ["iA; -] for every i 0. We now proceed t*
*o add
enough `A-cells' to C0, so as to get an isomorphism on these classes. We take *
*the first
(transfinite) limit ordinal = (A) bigger than the cardinality of A itself (= c*
*ardinality
of the simplicies or cells or points in A).
The ordinal = (A) clearly has the limit property: Given any transfinite t*
*ower of
22
spaces of length
Y0 ! Yi: :Y:n! Yw ! Yw+1 : :Y:ff! : :(:ff < )
every map "iA ! lim!Yfffactors through "iA ! Yfifor some ordinal fi < .
ff<
Proof. This is clear for every individual cell of "iA and since the number *
*of this cells
is strictly smaller than the cardinality of , it is true for "iA.
We proceed to construct a -tower of correction C0 = C0X!- C1X!- C2X!-
: :C:fiX : :t:o our original map C0!- X:
W W W
D0 = "iA "iA = D1 Dfi= "iA
K0? K1 Kfi
?yk1 ??y ??y
W
C0 = "iA ----! C1X ! C2X : : : CfiX ! : :(:fi )
I0? ? ?
?yc0 ?y ?ycfi
X = X = X = X = : :=:X : : :
Since C0 ! X is surjective on the A-homotopy of all components of map*(A; X) we
proceed to kill the kernel in a functorial fashion. In order to preserve funct*
*oriality we
kill it over and over again: First notice that any element "ff: "nA ! X repres*
*enting an
A-homotopy class in the component "ff|{*} x A = f: A ! X is null homotopic in t*
*hat
component iff "ffcan be extended along the map
(e) "nA = Sn x A [ Dn+1 x {*} ,! Dn+1 x A:
Now let k0: D0 ! C0 be the wedge of all maps g: "iA ! C0 with a given extension*
* as
(e) of c0 O g (the space D0 being a point if there are no such extensions). Thu*
*s D0 ! C0
captures every null homotopic map "iA ! C0 ! X many times. The map D0 ! C0 is
given by g. We define C1X as the push-out along the extension to Dn+1 x A:
W " W
i A ----! Dn+1 x A
K0? K0 ?
?y ?y
C0 ----! C1 = C1X
23
In this fashion we proceed by induction. The map C1X ! X is given by the null *
*ho-
motopies in the indexing set of D0 = Vk"iA. Taking limits at limits ordinal we *
*define a
0
functorial tower CfiX for fi . We now define CWA X = C X. This is the classi*
*cal
small object argument ??Q, p...] [Bou...].
Since c0 induce surjection on A-homotopy sets ["iA; X] for i 0, on all com*
*ponents
we get immediately that so does cfifor all fi . The limit property of = (A) *
*now
easily implies that C X ! X is injective in ssi( ; A; f) for any f: A!- X. Sin*
*ce every null
homotopic composition iA ! C X ! X factors through iA ! CfiX ! X for some fi,
a composition that is also null homotopic by commutativity. Therefore this map*
* is null
homotopic in Cfi+1X and thus in C X as needed.
3.5 A smaller non-functorial A-cellular approximation can be built by choosing *
*represen-
tatives in the associated homotopy classes. But it is clear that in general eve*
*n if A; X are
W W
of finite complexes CWA X may not be of finite type since CWS2(S1 Sn) Sn s*
*ince
W 1
this construction is just the universal cover of S1 Sn.
3.6 Corollary. Let A be a finite complex. Then for any countable space X we*
* have
the following form:
_
CWA X = ( "iA) ['1 "C"i1A ['2 "C"i2A
: :[:'`"C"i`A [ : : :
Where the "characteristic maps" '` are defined over "i`A for 0 ` < 1, and ther*
*efore
CWA X is also countable cell complex.
3.7 Corollary. In case A a finite suspension space A = B of pointed B we ha*
*ve
"iA = iA_A and "CiA = CiA_A and therefore in order to kill the kernels of Cfi! X
it is sufficient to attach cones over the usual iA ! Cfi. Thus in this case the*
* A-cellular
approximation to X has the usual form
i_ j
CWA X = i1A ['1 Ci2A ['2 C : :i:2A : : :
Which is just the usual CW -complex for A = S1 = S0, and X any connected CW
complex.
24
As in usual homotopy theory any map X ! Y can be turned into a cofibratio*
*n X ,!
X0 ! Y where X ,! X0 is an A-cofibration i.e. X0 is gotten from X by adding *
*"A-
cells" and X0!- Y is a trivial fibration i.e. in particular it induces an i*
*somorphism on
A-homotopy groups. Thus if Y ' * we get X0 ' PA X since map*(A; PA X) ' map **
*(A; *)
and X -! PA X is an A-cofibration.
If, on the other hand we take X ' *, the factorization becomes * ! CWA Y *
*! Y where
CWA Y now appears as the A-cellular approximation to X with the same A-homoto*
*py in
all dimensions.
3.8 Universality properties We now show that r: CWA X!- X has two universal*
*ity
properties:
(U1) (Bou. 7.5) The map r is initial among all maps f: Y!- X with map*(A; f) *
*a homotopy
equivalence. Namely for any such map there is a factorization "f:
r
CWA X ----! X
?
"f?y
f
Y
and such "fwith f O "f~ r is unique up to homotopy.
(U2) The map r is terminal among all map !: W!- X of spaces W 2 C(A) into X. *
*Namely
for every ! there is a "!: W!- CWA X with r O "!~ ! unique up to homotop*
*y.
Proof. Both (U1) and (U2) are easy consequences of the functoriality of C*
*WA when
coupled with the A-Whitehead theorem. Thus to prove (U1) consider CWA (f): CW*
*A Y!-
CWA X. This map is an A-equivalence between two A-cellular spaces, therefore*
* it is a
homotopy equivalence. Uniqueness follows by a simple diagram chase using nat*
*urality
and idempotency of CWA . To prove U(2): One gets a map A!- CWA X by notici*
*ng
that CWA W ' W , so CW (!) gives the unique factorization. Furthermore, uniqu*
*eness of
factorization implies that each one of these universality properties determin*
*e CWA X up
to an equivalence which itself is unique up to homotopy. This proves 1.4 (1)-*
*(3).
3.13 Proposition. The following conditions or pointed spaces are equivale*
*nt:
25
(1) For any space X there is an equivalence CWA X ' CWB X.
(2) C.(A) ' C.(B)
(3) A map f: X ! Y is an A-equivalence if and only if it is a B-equivalence.
(4) A = CWB A and B = CWA B.
Proof. These equivalences follow easily from the universal properties of CW*
*A X ! X.
(1),(2) Since the members of C.(A) are precisely the space X for which CWA *
*X ' X
this is clear from universality.
(1),(3) Clearly map (B; f) is an equivalence CWB f is a homotopy equivalenc*
*e. But
since by (1),(2) CWB f ' CWA f we get (3).
(2),(4) One direction is immediate. If A = CWB A, then A 2 C.(B) and thus *
*by
theorem An 2 C.(B) and therefore C. C.(B). Thus we get (2).
3.9 Theorem (1.7). For any A; X; Y 2 S* there is a homotopy equivalence
: CWA (X x Y )!- CWA X x CWA Y:
Proof. There is an obvious map
g: CWA X x CWA Y!- X x Y
It is clear that g induces a homotopy equivalence map(A; g) and therefore the m*
*ap in
the theorem induce the same equivalence map(A; ). But by corollary (2.11) the r*
*ange of
is an A-cellular space. Thus by the A-Whitehead theorem is a homotopy equival*
*ence.
6.1 Lemma. If X ' CWA X and Y is a retract of X then Y ' CWA X.
Proof. The retraction r : X -! Y implies that the map CWA Y ! Y is a retra*
*ct of
the homotopy equivalence CWA X ! X. But a retract of an equivalence is an equiv*
*alence.
3.10 Finite A-Cellular Spaces Have Infinite A-Homotopy: It is well known
that 1-connected CW -complex have non-trivial homotopy groups in infinitely man*
*y di-
mensions. This has been generalized in many directions _ relaxing the assumpti*
*on of
26
finiteness. In this section we consider a different direction of generalizing. *
*Instead of con-
sidering [Sn; X] we will consider [nA; X] for any arbitrary space A: Instead o*
*f assuming
X is a finite simply connected CW -complex we assume X is a finite A-cellular s*
*pace for
any connected A: Namely a space gotten by finite number of steps starting with*
* a finite
wedge of copies of A and adding cones along maps from A to the earlier step.
_
X ' ( A) [ C`1A [ C`2A : :C:`kA: (`i 1)
3.11 Theorem. Let A be any pointed, finite type connected space. Let X be *
*any
finite A-cellular space, with "H*(X; Z=pZ) 6= 0 for some p. Then ssi(X; A) = [i*
*A; X] 6= 0
for infinitely many dimensions i 0.
One immediate corollary is for X = A.
3.12 Corollary. Let A be any connected space with "H*(X; Z=pZ) 6 0 for som*
*e p.
There are infinitely many `'s for which [`A; A] 6= *.
Proof. First we note that since we consider spaces built from A, by a finit*
*e number
of cofibration steps we get a space which is conic in the sense of ??HFLT] name*
*ly is
gotten from a single point by finite number of steps of taking mapping cones. N*
*ow since
H"*(X; Fp) 6= 0 for some p; X satisfies the hypothesis of ??HFLT] and so X does*
* not have
a finite generalized Postnikov decomposition, i.e. X cannot be a polyGEM, since*
* X is of
finite type.
On the other hand suppose ssi(X; A) ~= 0 for i N. Then map*(N A; X) ' *
since all the homotopy groups of this space vanish. In other words X is N A-pe*
*riodic
or PN A X ' X. We claim that PA X ' *. This is true since by assumption X is*
* an
A-cellular space. (Theorem P-5 above).
But from (??F-S]) we know that the homotopy fibre of PN A X ! PA X is a po*
*ly
GEM for any connected A; X. But we just saw that the homotopy fibre of that map*
* is X
itself which cannot be a polyGEM. This contradiction implies ssi(X; A) 6 0 for *
*infinitely
may i's as needed.
Remark. Notice that in order to prove the corollary we need not use the hea*
*vy result
of ?HFLT] since H*Y is a tensor algebra and is not nilpotent. Therefore A can*
*not
27
be a GEM if H"*(A; Z=pZ) 6= 0 by Moore-Smith ?M-5]. Hence there must be infinit*
*ely
many maps `A ! A for any such A.
Resolution of CWA X
Here we record two simplicial resolutions of any space by A-cellular simpli*
*cial space.
The diagonal of that simplicial space is not, in general CWA X, rather it is a *
*kind of dual
to totR.X = R1 X in ?B-K]. Its relation to CWA X is similar to the relation bet*
*ween
R1 X the Bousfield-K an R-completion functor and LHR , the Bousfield HR-homolog*
*ical
localization functor. Thus this diagonal may prove to be a useful approximation*
* to CWA X.
If [A; X] ' * and A is finite we show that the diagonal of these resolutions gi*
*ve CWA X.
Blanc-Stover resolution: For any space X?? ] gives a simplicial resolution of *
*X by
means of UnX : :U:2X!!U1X ! X where for all n 1 the space UnX is homotopy
W
equivalent to a large wedge of half-suspensions of A, namely "`iA. The reali*
*zation
i
kU.Xk = hocolimkUkX is homotopy equivalent to CWA X. Whenever A is finite and
[A; X] ' * ?B-T].
o
Dwyer's resolution:For any A; X we can associate a functorial map TA X!- X fro*
*m an
A-cellular space TA X to X as follows (see 2.5):
TA X = A o map *(A; X)
o(a; f)= f(a) 2 X:
Proposition. The functor TA enjoys the following properties:
1. If f: X ! Y is an A-equivalence then TA f is a weak homotopy equivalence.
2. The map o is surjective on A-homotopy classes ["k A; *_].
Proof. The first assertion is clear since TA f = A o map (A; f) which is an*
* equivalence
if f is an equivalence. For the second assertion consider map*(A; o): map*(A; *
*TA X) !
map *(A; X). This map clearly splits naturally: Take a pointed map g: A ! X to *
*the map
A ! TA X taking at A to the pair (a; g). A little computation shows that this c*
*anonically
splits map(A; o). The above natural splitting makes TA into a natural cotriple*
* with a
map: TA ! TA TA enjoy the properties of triple. Therefore we extend this tri*
*ple to a
simplical resolution of X by cellular A-space TA.X ! X.
28
Definition. Denote the diagonal of TA.X by TA1X.
Definition. If X = V "niA then TA1X ' X.
Proof. We first consider X = "n A. For n = 0 there is a canonical splittin*
*g of A o
map (A; A) ! A taking a to (a; id). This splits the simplicial resolution and y*
*ields TA1A '
A. Taking n 1 we have a splitting of
A o map (A; "nA) ! "nA:
Taking (t; a) to (a; ft) where ft is the t-level identity map ft: A ! "nA takin*
*g a ! (t; a);
here t varies over In = I x : :x:I.
Therefore again one gets
' n
T 1("n A) -! " A:
W
Now one can define a split for X = "niA component-wise, thereby proving the p*
*ropo-
i
sition.
Theorem. For any finite space with [A; X] ' * TA1X ' CWA X.
Proof. We use a result of Blanc-Thompson according to which there is a func*
*tori-
al resolution UA.X ! X where UAnX ' V "niX for all n, and with the property that
kUA.X| ' CWA X. UA.X is essentially an A-version of the Blanc-Stover resolution*
* ?? ].
Now consider the bi-simplicial space TanUAkX. We claim that its diagonal D = di*
*agTAOUAOX
is CWA X. This is so because the diagonal is equivalent to the realization of t*
*he simplicial
space kTA1UAkXkk which by the above is equivalent to kUA XU ~=CWA X. But taki*
*ng
first the realization along the UAk-direction we get DnX ' TAnkUAkXk = TAACWA X*
* by
Thomson-Blanc's result. But clearly the map CWA X ! X induces an equivalence on*
* TAn
and TA1.
4. Commuting CWA with other functors (; ; PA )
In this section we consider the relations between CWA X; CWkA X; CWA X and
PA X. Technically speaking these are the most useful properties of CWA and in p*
*articular
29
these are crucial for understanding the preservation of fibration under it. We *
*get four main
results:
4.1 CWA X ~=CWA X (1.4)
4.2 If CWA X!- X!- CWA X is null homotopic then the sequence is a fibration.*
* (1.11)
4.3 If CW2A X ' * then CWA X is a GEM. (1.9)
4.4 The fibre of CW2A X!- CWA X is a polyGEM which is also A-cellular and 2A-
periodic.
We start with (2) since it exposes the close relationship between CWA X and*
* PA X and
in fact directly implies the rest under additional assumptions. In fact (2) mot*
*ivated our
interest in CWA .
4.5 Theorem (see 1.7 above). Consider the sequence
` r
CWA X!- X!- PA X
for arbitrary pointed spaces A; X. This sequence is a fibration sequence if (a*
*nd only if)
the composition r O ` is null homotopic. In particular, if [A; X] ' * or PA X *
*' * then this
is a fibration sequence.
Proof. With first prove the special case PA X ' *.
'
4.6 Proposition. For any CW -complex A; X in S*, if PA X ' * then CWA X -!*
* X
is a homotopy equivalence.
Proof. Consider the fibre sequence:
(*) X!- F!- CWA X!- X:
In this sequence we now show that F ' *.
In order to show that we show:
(1) map(A; F ) ' *
(2) PA F ' *
30
Clearly any space Y that satifies (1) i.e. is A-periodic does not change under*
* PA thus
(1) and (2) imply F ' *. The fibration (*) implies that map(A; F ) is the hom*
*otopy
fibre of map(A; CWA X)!- map(A; X) over the trivial component. But by defini*
*tion
of CWA the latter map is a homotopy equivalence thus its fibre is contractible *
*and (1)
holds. To prove (2) we use theorem P (4) in (0.3) above, with respect to the f*
*ibration
sequence X!- F!- CWA X. First notice that by P (4) PA X ' PA X which is by
~=
our assumption contractible. But now Theorem P (5) means that PA F -! PA CWA X *
*is a
homotopy equivalence. Theorem (1.5) above now implies PA F ' * as claimed in (2*
*). This
completes the proof of the proposition.
We now proceed with the proof of the theorem.
Let Y be the fibre of X!- PA X. By theorem P (3) we deduce that PA Y *
* ' *
'
and therefore by the proposition just proved we deduce CWA Y -! Y is a homoto*
*py
equivalence. The following claim now completes the proof:
Claim. CWA Y ' CWA X.
Proof. The map Y!- X gives us a map Y ' CWA Y!- CWA X. Since both spaces *
*are
A-cellular it suffices by the A-Whitehead theorem (3.1) to prove that we have a*
* homotopy
equivalence:
map *(A; CWA Y ) ' map *(A; CWA X):
Since for all W the map CWA W!- W is a natural A-equivalence it suffices t*
*o show
that map(A; Y )!- map (A; X) is a homotopy equivalence of function complexes. *
*Consider
first the set of components: By definition of Y as a fibre we have an exact s*
*equence of
pointed sets:
(*) [A; PA X]!- [A; Y ]!- [A; X]!- [A; PA X]:
First notice that by universal property of PA X the left-most group is zero. N*
*ow we claim
that it follows from the assumption r O ` ' * in our theorem, that the right-mo*
*st arrow
is null. This is because by the universal property of CWA (Theorem 1.4 (2), 3.*
*8 (U.2))
every mapA!- X factors (uniquely up to homotopy) through CWA X!- X, therefore
31
r
its composition with r: A!- X!- PA X must be null homotopic. Therefore the m*
*iddle
arrow in (*) is an isomorphism of sets. Now consider the pull-back sequence:
map *(A; Y )----! map *(A; X)
?? ?
y _r?y
* ----! map *(A; PA X; null) ' *
We just saw that _r= map *(A; r) carries the whole function complex to the null*
* component
of map(A; PA X). Therefore we can and do restrict the lower right corner of th*
*e square to
the null component. But we claim that the component of the null map in map(A; P*
*A X) is
contractible since its loop map (A; PA X; null) is by adjunction just map(A; P*
*A X) '
*, as needed. Now a pull back square with two lower corners contractible impli*
*es an
equivalence of the top arrow as needed.
4.7 Proof of 4.1. We now turn to the proof of basic adjunction (4.1). If [*
*A; X] ~ *,
then this equation was just proved since it follows from (2) and the correspond*
*ing equation
for PA namely the equivalence (1.4) above: PA X ' PA X. But here we prove it *
*in
general:
4.8 Theorem. For any A; X 2 S* we have a homotopy equivalence
r: CWA X -! CWA X: `:
Proof. The proof follows very closely the proof for the periodization funct*
*or X !
PA X. Let us recall in broad lines the proof. We use Segal's -space machine[1*
*][?] to
recognize CWA X as a loop space and the map CWA X!- X as a loop map. For
this we need only an augment functor S* ! S* that commutes with products: This*
* we
have in virtue of theorem above: Thus we can write X as a simplicial space Y. *
*with
Y1 = X; Y2 = X xX : :Y:n= Xn , and Y.is a simplicial space which is very specia*
*l namely
'
the natural map Yn -! Y1 x : :x:Y1(n-times) is a homotopy equivalence. Applying*
* CWA
to Y.and using CWA (Xn ) ' (CWA X)n we get immediately that X!- XCWA X is in
fact a loop map. We now use universality properties of CWA to get the desired e*
*quivalence.
First take
j : CWA X ! X
32
to be the loop of the structure map for CWA . We get a diagram
CWA X
rj
j
CWA X ----! X
It is easy to see that (j ) induce a homotopy equivalence on map(A; -). Therefo*
*re, by
universality of the map (j) we get the map r, which is unique up to homotopy. T*
*o get
___
the map ` we first construct a map "W `"
___
`0: CWA X ! W CWA X
___
Where W is the classifying functor otherwise denoted by B-. Here we use crucia*
*lly the
fact proven above that CWA X!- X is a loop map. One deloop this map to get a m*
*ap
___ ___ ___
W j: W CWA X!- X. We lift the structure map CWA X!- X to W CWA X. Again
___
this lift exists by universality of CWA X (3.8 U.1) since W (j) is easily seen*
* by adjunction
___
to induce homotopy equivalence on the mapping space from A: i.e. map(A; W j) is*
* a h.e.
Since these two maps were defined by universality it is easily checked as in ??*
*DF] that
these are mutual inverses up to homotopy.
4.9 Corollary (1.5). In the fibration (2.8) above B and E are in C.(A) the*
*n F
is in C.(A). If F and B are A cellular then E is A-cellular.
Proof. This is immediate from above by considering the fibrations E ! B ! F*
* and
B ! F ! E and the fact that CWA B ' CWA B so that the latter is an A-cellular,
and so is CWA E.
Deviation by GEM and PolyGEM. We now turn to consider the relations between
CWkA X for various k's. This will pave the way for considering the difference*
* between
CWA X and CWA X.
The slogan is "Whenever the function complex map*(A; X) is homotopically di*
*screte
(i.e. it is h.e. to a discrete space) the space CWA X is a GEM and thus the ab*
*ove set is
an abelian group" (compare ??Bou-4 6.7].
Recall that (1.19) if PA X ' * then P2A X is a GEM. Here we have a simila*
*r result
about CWA X (see .??..).
33
4.10 Theorem. Assume CW2A X ' *. Then CWA X is a GEM.
Proof. Consider the natural square of maps associated to any space:
j: CWA X ----! X
?? ?
y '?y
P2A j: P2A CWA X ----! P2A
Our assumption is equivalent to map*(2A; X) ' * i.e. X is 2A-periodic and *
*the
right vertical map is an equivalence. Now since PA CWA X ' * for any X; A by*
* 1.4
(11) above, we get from ??EDF-S] that P2A CWA X is a GEM. Therefore we conclu*
*de
that the canonical map CWA X ! X factors up to homotopy through a GEM. Since by
lemma 4.11 below CWA (GEM) is always a GEM we conclude that CWA X is a retract
of a GEM, thus a GEM.
4.11 Lemma. For any space M of the homotopy type of product of Eilenberg-Ma*
*cLane
spaces K(G; n) with abelian G (i.e. M is a GEM), we have CWA M is also a GEM.
Proof. This follows from the structure map K(Z; n) x M ! M for any GEM space
M, realizing K(Z; n) and M as strictly abelian groups which is always possible.*
* The
above action presents M as a module over K(Z; n) for any n 0. Since for any k*
* 2
K(Z; n) k . 0 = 0 for 0 2 M. Therefore for k 2 K(Z; n) we get a pointed map M!-*
* M and
thus we have an induced map CWA M!- CWA M. This consideration together with the
usual machinary [Bou] yield an action K(Z; n) x CWA M ! CWA M. Therefore CWA M
is presented as a K(Z; n)-module for any n 0, this it is a GEM.
4.12 Theorem. Let Y be a GEM with ssiY = 0 for i 0. Then ssiCWA Y ' 0 for
i n.
Proof. Assume sskCWA Y ~=G 6 0 for some k n. Since CWA Y = W is also a GEM
by theorem above, K(G; k) is a retract of W and therefore by proposition ... K(*
*G; k) is an
A-cellular space. Therefore the retraction CWA K(G; k)! CWA Y = W is equivale*
*nt to
the original retraction K(G; k)! W . For dimensional reasons since k n, the co*
*mposition
K(G; k) ! W ! Y is null. Applying CWA we get null map, a contradiction.
34
Problem: Is it true that for any A and a polyGEM X the space CWA X is also a po*
*lyGEM?
(A polyGEM is "generalized n-Postnikov stage").
Finally we turn to (4) which may not be the best possible result:
4.13 Theorem. For any A; X 2 S* the homotopy fibre F of
j: CW2A X!- CWA X
is a polyGEM.
Proof. This follows from (1.12): Notice that PA kills both the domain an*
*d range
of j above. Therefore PA F is a GEM. But map*(2A; j) is a homotopy equivalenc*
*e.
Therefore map*(2A; F ) ' * and F is 2A-periodic. Therefore by (1.12.2) the fib*
*re of
the map form F to PA F is also a GEM and we are done.
4.14 Commuting CWA with taking homotopy fibres.
We will now address the question of preservation of filbration by CWA . Lo*
*oking at
A = Sn we see immediately that X = CWSn+1 being the n-connected cover of X d*
*oes
not preserve fibration in general. However we shall see that when A is a suspen*
*sion the
functor CWA "almost" preserves fibrations, the error term being under control.
In order to measure the extent to which CWA preserves fibration we will now*
* compare
the fibre of the CW -approximation with the CW -approximation of the fibre via *
*a natural
map.
4.15 : CWA F!- Fib(CWA E!- CWA B).
associated to any fibration sequence F!- E!pB over a connected B. For E ' *
** we
get as a special case a map CWA B!- CWA X for any space B.
In order to construct one notices that the fibre of the map CWA (p) denote*
*d here by
FibCWA (p) maps naturally to F . This map induces an equivalence on function co*
*mplexes
map*(A; Fib(CWA p))!- map *(A; F ) since map*(A; -) commutes with taking homot*
*opy
fibres. Therefore, by the universal property (3.8) U1.there is a factorization*
* CWA F!-
fib(CWA (p)) unique up to homotopy.
Now in general one shows:
35
4.16 Proposition. Whenever A = 2A0 is a double the homotopy fibre of the
above natural is an extension of two GEM spaces: (2-GEM).
(GEM )2!- !- (GEM )1:
Moreover is an A-periodic, Ai-cellular space.
Proof. First we notice that by a straightforward argument one shows that ma*
*p*(A; ) '
*, i.e. is A-periodic. This is because the map map*(A; ) is a homotopy equival*
*ence.
Since the fibre J of P2A0 !- PA0 is a GEM (by 1.5 PA0 J ' * and J is 2A0-local
so use 1.12). Since P2A0 ' it is sufficient to show PA0 is a GEM. For this w*
*e use
(1.12). By (4.9) above both domain and range of are A0-cellular and thus (1.7)*
* both
are killed by PA0 . Therefore the condition of 1.12 is satisfied and PA0 is a *
*GEM. This
completes the proof.
4.17 Corollary. For A = 2A0the fibre of CWA X!- CWA X is a 2-poly GEM.
Proof. Apply the above to X!- *!- X.
Remark. Using the adjunction (4.8) (4.17) is a special case of 4.13 albeit *
*with more
control on the fibre.
4.18 Proof of 0.11. This follows directly from (4.10) since by (4.8) B sati*
*sfies the
condition: The triviality of the A-CW approximation is equivalent to the trivia*
*lity of the
pointed function complex from A and this follows directly form the condition in*
* 0.11.
5. A Fibration Theorem
5.1 Theorem F. Given any fibration of pointed space F ! E ! B one can map t*
*he
following fibration into it:
__
CWA F ----! E ----! CWA B
? ? ?
f?y g?y h?y
F ----! E ----! B
36
__
where E is A-cellular and g a weak A-equivalence.
__
5.2 Remark. Thus although E is A-cellular it is slightly removed from being*
* CWA E
since it is only A-equivalent to E not A-equivalent to it. We shall soon see th*
*at the fibre
E
of a canonical map E!- CWA E -! ! CWA E associated to such fibration is a GEM *
*for
A = A0, a suspension space.
Proof. This is similar to ?F] and ?Bou], and uses crucially the equivalence*
* CWA X '
CWA X and CWA (X x Y ) ' CWA X x CWA Y (3.9) and (4.8) We consider first the
associated principal fibration:
B ! F ! E
where we consider E as the `quotient space' up to homotopy of F under the actio*
*n B x
F ! F . This action gives rise to a map
CWA (B) x CWA F ! CWA F;
or CWA B x CWA F ! CWA F . We now use the equivalence (4.8) and the usual
arguments ?Bou] show that the original action B x F ! F gives rise to an action*
* of
CWA B on CWA F . Therefore we get a principal fibration
__
CWA B ! CWA F ! E
associated to that action. Classifiying this fibration gives us the desired seq*
*uence. Since in
__
that last fibration both fibre and total spaces are A-cellular so is E by (2.9)*
*. Further since
in the map of fibrations h is an A-equivalence and f is an A-equivalence we get*
* by the
usual long exact sequence for A-homotopy groups [`A; -]*, that g is an A-equiva*
*lence.
This completes the proof.
p
5.3 Corollary. Let F!- E!- B be any fibration sequence in S* and let 2A be
any double suspension in S*. If B is 2A-cellular and [A; p] ~ * then
CWA F!- CWA F!- B = CWA B
37
is also a fibration sequence.
5.4 Example. Take A = S1. Then the theorem states the easily checked fact t*
*hat over
a 2-connected space B one can define fibrewise universal covering space.
Proof. We consider the diagram:
Y = Y
?? ?
y ?y
__
CWA F ----! E ----! CW2A B = CWA B
? ?
f?y g?y k
F ----! E ----! B
This diagram is constructed using the theorem above. We claim that there is a *
*natural
__ __
equivalence: _e: E!- CWA E. First notice that by theorem 2.8 E is A-cellular.*
* There-
fore there is a unique natural factorization _e. To check that _eis a homotopy *
*equivalence
we use the A-Whitehead theorem and check that the map
__
map*(A; _e): map*(A; E)!- map (A; CWA E)
is a homotopy equivalence. Notice that map*(A; Y ) ' * since Y is the fibre o*
*f f and
map*(A; f) is a homotopy equivalence. We know from theorem F above(6.2) that G
is a 2A- equivalence. But m ap*(A; Y ) ' * since Y is the fibre of a A- equiv*
*alence
f. Therefore it is enough to check the function complex on the level of compone*
*nts,since
all the components are equivalent to each other. Our assumption guarantees tha*
*t it is
surjective on componenets and injectivity follows from the above mentioned prop*
*ery of Y .
6. Examples and E*-acyclic spaces.
Many well-known constructions in algebraic topology leads to A cellular spa*
*ces.
6.1 James functor JX. For any X the space JX ' X is an X cellular space:
CWX JX ' JX:
38
Proof. We have a filtration
JnX JX
with homotopy pushouts of pointed spaces:
JnX _ X ----! JnX x X
?? ?
y ?y
JnX ----! Jn+1X:
This gives an inductive definition of JnX. Since by theorem 2.9 a product of tw*
*o X-spaces
is an X-space we get by induction that Jn+1X is an X-space. Therefore JX = hoco*
*lim
JnX is also an X space.
Hilton-Milnor-James decomposition: This theorem provides a decomposition of
X for an arbitrary pointed X in as a wedge of smash-powers of X. Thus it gives
an explicit description of X as an X-cellular space: (Any smash-power of W is
W -cellular).
Using the adjunction relation (4.8 ) yields immediately that X is in fact a*
* X-
cellular without however saying anything about the nature of the decomposition.*
* One
computes: ___
CWX X = W CWX X
___
= W X = X
Here we used CWY Y = CWY Y = Y . We get that (X) is X-cellular
which is X-cellular.
Notice, however, that for non suspension CWY Y 6= Y . In fact Y is not a
Y -cellular space, rather the other way around as we saw X is X-cellular.
For example K(Z; 3) = CP 1 is not K(Z; 3)-cellular since any K(Z; 3)-cellul*
*ar
space must have vanishing reduced complex K-theory and CP 1 is not K-acyclic.
Similarly nSnX is also an X-cellular space.
6.2 Theorem. For any X the Dold-Thom functor SP 1X is an X-space.
This follows from a much more general observation about arbitrary "converge*
*nt func-
tors" of ?B-F] [Bou], or -spaces of ??Segal]. Let Oc Sets* be the full subcate*
*gory of
39
the objects n+ = {0; : :;:n} with base point 0 2 n+ , for n 0. A -space is a f*
*unctor
[: ! S*. It is special if [(n+ ) = U(1+ )x: :x:[(1+ ). Each -space determines *
*a functor
[S* ! S* with [X = diag([X.). where ([Xk).is the space associate by the -space *
*[ to
the set of k-simplices Xk of X. Thus every very special -space h: ! S* determi*
*nes a
reduced homology theory ss+ hX h*X.
6.3 Proposition. For any -space U and any X 2 S* the space UX is an X
cellular-space.
Proof. Almost by definition U can be written as the "tensor product of Op-s*
*paces
with -space: ??Bou-4 6.4]
UX ' X..U(.)
Where .denotes the "coend" coequalizer ??Mac] over . Notice that X.: Op ! space*
* is
+
Xn = X x : :x:X; (n + 1) - times:
+ +
Xn = map (n ; X) this gives a functor
Op ! spaces:
Now since by definition Xn+ is an X space and by lemma 2.3 above Xn+ ^Y is an X*
*-space
for any Y we get that UX is a hocolim of X-spaces and therefore a -space.
In order to deduce theorem above it is enough to show that SP 1X is equivalent *
*to UX for
some -space U: ! S*. But ??Bou 6.2] shows that choosing the discrete -space "Z*
*to be
"Z(n+ ) = Z : : :Z n-times and regarding "Zas a discrete -space, gives "ZX ' S*
*P 1X.
Therefore SP 1X is an X-space.
By the same token 1 S1 X is also an X-space since 1 S1 X can be realized as*
* a
-space.
Further examples of A cellular-spaces can be gotten from the following.
6.4 Proposition. Let n 3 then for any Morava K-theory K, there exists *
*an
integer k(n) such that h > h(n) ) K(G; n + k) = CWV (n)K(G; n + k). In particu*
*lar
K(G; n + k) are homotopy colimits of K-acyclic subcomplexes.
40
Proof. See ??EDF]. The point is that one can show that PV (n)K(G; n + k) ' *
** for
large k.
6.4 Classifying Spaces: It is not hard to see directly that Milnor's classif*
*ying space
construction leads to a description of BG, for any group-space G, as G-cellular*
* space i.e.
BG 2 C.(G). But this fact is a direct corollary of (??? ) and (?? ) above.*
* In fact to
___ ___ ___
check PG BG ' * one use ( ) to get PG BG = W PG BG ' W PG G ' W {*} = {*}.
Therefore BG = CWG (BG) as needed. In particular K(G; n + k) is a K(G; n)-space*
* for
any k 1. Moreover BG is always a G-space since (using 4.1)
CWG BG ' BCWG BG = BCWG G = BG:
>From this observation we get also
6.5 Corollary. Any connected space X in S*, is in C.(X).
6.6 Proposition. For all n; k 0 K(G; n + k) is a K(Z; n) cellular-space.
Proof. For n = 1 it is clear since K(Z; 1) = S1 and K(G; n + k) is a connec*
*ted CW -
complex. For n 2 K(G; n + k) is an abelion Eilenberg-Maclane complex. If G is *
*abelion
then G = dir lim Gffwhere G is the system of finitely generated abelion subgrou*
*ps. So
BG = P -hocolim BGff. Therefore it is sufficient to prove the proposition for G*
* = finitely
generated abelion group. In that case we have a homotopy fibre sequence
K(F; n) ! K(F 0; n) ! K(G; n):
which corresponds to the representation of G ' F 0=F as a quadrant of two finit*
*ely gen-
erated free abelion groups. Now K(F; n) and K(F 0; n) are finite products of KZ*
*; n) with
itself and therefore by (3.7) K(Z; n) cellular-space. Now by theorem (2.9) abov*
*e it follows
that K(G; n) is K(Z; n)-space.
6.7 Example. CWK(Z=pZ;1)K(Z=p2Z; 1) = K(Z=pZ; 1). Proof: Consider the map
g: Z=pZ ! Z=p2Z of abelian groups 1 ! p. This is the generator of Hom(Z=pZ; Z=p*
*2Z) '
Z=pZ, g induces Bg: K(Z=pZ; 1) ! K(Z=p2Z; 1). Since the source is clearly a K(Z*
*=pZ; 1)-
41
CW -space it is sufficient to show that Bg induces a homotopy equivalence on th*
*e pointed
function complex map(K(Z=pZ; 1); Bg). But the pointed function complex is homot*
*opi-
cally discrete with
map *(K(Z=pZ; 1); K(G; 1)) = Hom (Z=pZ; G):
Therefore the above map Bg gives us the correct CWA -approximation for A = K(Z=*
*pZ; 1).
6.8 Corollary. If A = K(Z=pkZ; n) and X = K(Z=p`Z; n) then
aeA if k `
CWA X =
X if k `:
Proof. This is clear using the above together with the fibration theorem:
Thus the fibration
xp
K(Z=p2Z; n) --! K(Z=p2Z; n) ! K(Z=pZ; n) x K(Z=pZn + 1)
by ( ) and ( ) above K(Z=pZ; n) as a K(Z=p2Z; n) - CW -space.
6.9 Example. Let X = Mn+1 (p`) and A = Mn+1 (p) be two Moore spaces, with
Hn(Mn+1 (p`); Z) = Z=p`Z. Then CWA X is a fibre in:
F ! X ! K(Z=p`-1Z; n):
while X = CWA X.
Proof. To compute the fibre of the composition X ! K(ssnX; n) ! K(Z=p`Z; n)*
* as
CWA X we consider the pointed function complex of Mn+1 (p) into the fibration. *
* Since
map(Mn+1 (p); K(Z=p`; Z)) ' Z=pZ the homotopically discrete, by cohomological c*
*ompu-
tation we first notice that the fibre has the correct function complex from Mn+*
*1 (p). We
then must show that the fibre is a Mn+1 (p) cellular-space. But the fibre is a*
*-p-torsion
space so it has an Hilton-Eckman cell decomposition Mn+1 (p) [ CMn+2 (Hn+1(F );*
* n +
1) [ : :w:here all the attacking maps can be taken to be pointed maps. This giv*
*es direct
representation of F as Mn+1 (p)-cellular since clearly
42
6.10 Lemma. For any p-group Gn the Moore space Mn+j (G; n + j)j 2 is an
Mn+1 (p)-space.
Proof. Use theorem 4.5
PMn+1(p) Mn+j (G; n + j) ' *
since the localization is an n-connected p-torsion space with all maps from*
* Mn+1 (p)
being null, thus this localization is contractible.
E*-acyclic spaces: The fibration (? ) relating PA X and CWA X can be used *
*to show
that certain E*-acyclic spaces are V (n)-cellular, where V (n) are the spaces i*
*ntroduced by
Smith ??Rav] [H-5].
6.11 Lemma. Let X ' CWA X where A is a finite complex with "E*A ~=0. Then X
is the limit of its finite E*-acyclic subcomplexes.
Proof. Recall from above the construction of CWA X. For a finite A the limi*
*t ordinal
(A) is the first infinite ordinal w. Therefore in that case
CWA X = limi<1(X1 ,! X2 ,! Xi,!)
where Xi are all subcomplexes of X. But now by induction we can show that each *
*Xi is
the limit of finite E*-acyclic subcomplex. Notice that if A is any E*-acyclic s*
*pace then so
is the half-suspension "n A = Sn o A = Sn x A=Sn x {*} by a Mayer-Vietoris argu*
*ment.
Now if by induction Xj = lim!A2(i) where A(i) are finite E*-acyclic then since *
*Xj+1 is a
ff
push-out along a collection of maps from "nA, Xj+1 is again lim!Afi(j + 1) wher*
*e A(j + 1)
fi
are all finite. This completes the proof.
Our principal tool to detect when is an E*-acyclic complex X the limit of i*
*ts finite
E*-acyclic subcomplexes is the following.
6.12 Proposition. Let A be an E*-acyclic finite complex. Then X is the limi*
*t of its
finite acyclic subcomplexes if PA X ' *.
Proof. This is immediate from theorem (0.8) and the lemma above.
43
We apply the proposition to the spaces to spaces V (n) of type n + 1. Thus*
* V (0) is
S0[pe2, and for every prime p and n 0 there exists a finite p-torsion space X(*
*n) of type
n. This means K"*X(n) = 0 for all m < n and K"(m)*X(n) 6= 0 for all m n, wh*
*ere
K denotes the n-th Morava K-theory (compare discussion in ?Rav] ?Thomp] ?H-S]
?Bou].)
We apply this proposition in three interesting cases using ?6, 9.14 and 13.*
*6].
6.13 Theorem. In the following cases every E*-acyclic space is in C.(V (n))*
* for an
appropriate n 0.
(1) For all n there exist m n with K(G; m + j) 2 C.V (n) for all j, and all p-*
*torsion
groups G.
(2) If "KC2X ' 0 then X 2 C.(V (n)) for any p-torsion 2-connected X.
(3) For every n 1 there exist N n so that if X is N-connected, p-torsion and
"S(n)*N X = 0 then X 2 C.(V (n)).
REFERENCES
[1] J.F. Adams, infinite loop spaces, Princeton U press..
[2] D. Anderson and L. Hodgkin, The K-theory of Eilenberg-MacLane complexes, T*
*opology, 7 (1968),
pp. 317-329.
[3] A. K. Bousfield and D. M. Kan, Homotopy Limit, Completions, and Localizati*
*ons, Springer
Lecture Notes in Math, 304 (1972), pp..
[4] A. K. Bousfield, Factorization systems in categories, J. of Pure and Appl.*
* Alg., 9 (1977), pp.
207-220.
[5] ______________, Localization and periodicity in unstable homotopy theory, *
*preprint (1992).
[6] A. K. Bousfield and E. M. Friedlander, Homotopy theory of -spaces, spectra*
*, and bisimplical
sets, Springer Lecture Notes in Math, 658 (1978), pp. 80-130.
[7] C. Casacuberta, G. Peshke, and M. Pfenniger, On orthogonal pairs in catago*
*ries and localiza-
tion,, (), pp..
[8] E. Dror Farjoun, The localization with respect to a map and v1-periodicity*
*, Proceeding Barcelona
Conference 1990 Springer Lecture Note, No .
[9] ______________, Localizations, fibrations and conic structures, (submitted*
*, to appear).
no10 E. Dror Farjoun and A. Zabrodsky, homotopy equvalences in diagrams of sp*
*aces, Lec Note in
Math 133??4 (?198?), pp..
[11] E. Dror Farjoun and J. H. Smith, Homotopy localizations nearly preserves f*
*ibrations, preprint
1992, (submitted to appear), pp..
[12] E. Devinatz, M. J. Hopkins, and J. H. Smith, Nilpotence and stable homotop*
*y, Ann. of Math,
128 (1988), pp. 207-242.
44
[13] T. Ganea, A generalization of homology and homotopy suspension,*
* Commetarii Math. Helv, 39
(1965), pp. 295-322.
[14] S Halperin, Y. Felix, J.-M.Lemaire and J.C. Thomas, Mod-p loop *
*space theory, invention. Math.
[15] M. J. Hopkins and J. H. Smith, Nilpotence and stable homotopy t*
*heory II,, (), pp..
[?] M. Mahowald and R. Thompson, K-theory and unstable Homotopy gro*
*ups, Contemp. Math., 96
(1989), pp. 273-279.
[?] G. Mislin, On localization with respect to K-theory, J. of Pure*
* & App Algebra (1975).
[?] S. A. Mitchell, Finite complexes with A(n)-free chomology, Topo*
*logy, 24 (1985), pp. 227-246.
[? by S. Mitchell]Finite complexes with A(n)-free cohomology, Topology, 24 (198*
*5), pp. 227-246.
[?] J.Moore and L. Smith, Generalized Eilenberg MacLane spaces, ?, *
*? (?), pp. ?.
[?] A. Nofech, Thesis,Hebrew University of Jerusalem.
no? G. Segal, Categories and Cohomology Theories, Topology, 13 (1*
*974), pp. 293-312.
[?] J. H. Smith, Finite complxes with vanishing line of small slope*
*.,, preprint (1992).
[?] D. G. Quillen, Homotopical Algebra, Springer Lecture Notes in M*
*ath, 43 (1967), pp..
*
* May 1992,
He*
*brew University,
*
* Jerusalem
Pu*
*rdue University
*
* W. Lafayette
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