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 K-theory 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 Dold-Thom theo* *rem. (5) Brown and Adams representability in the setting of triangulated ca* *tegories. (6) Operads and the May-Thomason 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 x3-x11. 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 1900-1960, Birkh"auser (19* *89); see also, Adams, Proc. Sympos. Pure Math. 22 (1971), 1-22 and Whitehead, Bull. Amer. Math. S* *oc. 8 (1983), 1-29. 0-1 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, 0-2 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 (A-MOD or MOD-A ), the category of groups, abelian groups, rings with unit (left or right A-modules, 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 0-3 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 f-1 . 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. 0-4 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. 0-5 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 one-to-one correspondence. [Note: If F and G are injective on objects, then C and D are isomorphic (ca* *tegorical "Schroeder-Bernstein").] 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. 0-6 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. 0-7 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 n-source_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 n-sink_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.] 0-8 (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 Stone-Cech 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 2-source 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 2-sink Y !fZ fOu-X 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_. 0-9 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 2-sink 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 2-source 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 2-source X * * P j!Y is called a pullback_of the 2-sink 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 2-source X f Z g!Y , a commutative diagram yf * * yj X --! * * P 0 j0 is said to be a pushout_square_if for any 2-sink 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 2-sink X !* * P j Y is called a pushout_of the 2-source 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 0-10 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 * * . .!. X2n-1 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 0-11 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. 0-12 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. 0-13 (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 ). 0-14 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 0-15 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 0-16 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- 0-17 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_n-simplex_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 n-skeleton_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](n-1), the frontier__[n] of [n]. Of* * course, 0-18 _[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 n-sim* *plexes indexed by X#n_then the simplicial maps x : [n] ! X (x 2 X#n) determine an arrow X#n.?_[n] --! X(n-1)? 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 0-23 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 P-localization 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). * * 0-24 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 * * 0-25 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) . * * 0-26 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 * * 0-27 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 * * 0-28 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 T-ALG is isomorphic a (b x) ! ab x * * x ! 1 x to A-MOD . 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 Eilenberg-Moore: Given a triple T = (T; m; ffl) in C , call FT the functor C ! T-ALG * * that sends X f!Y to (T X; mX ) Tf!(T Y; mY ), call GT the functor T-ALG ! 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 T-algebras 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 T-ALG = SET and the comparison functor TOP !* * SET is the forgetful functor. 0-29 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 ! * *T-ALG 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 Eilenberg-Moore 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 = m-1 = 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 0-30 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 one-to-one 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* * (S-1 C, LS), where S-1 C is a metacategory and LS : C ! S-1 C is a functor such that 8 * *s 2 S, LSs is an isomorphism, (S-1 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: S-1 C ! D such that F = F 0* *O LS. S-1 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 S-1 C = HTOP . __ [Note: If S is the class of all morphisms rendered invertible by LS (the sa* *turation_of __-1 S), then the arrow S-1 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 S-1C 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.] 0-31 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 2-source 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 2-source 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 S-1 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: S-1 C is a metacategory. To guarantee that S-1 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 n-equivalences, 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 0-32 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 S-1* * C is isomorphic to a category and LS : C ! S-1 C has a right adjoint that is full an* *d faithful, while F 0: S-1 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 S-1 C and T * *factors as C ! S-1 C ! D ! C , the arrow S-1 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 T-1 D 0 : S-1C ! T-1 D G0O F0* * id-1 phic to categories and F induce functors F such that * * S C , thus ae G G0: T-1 D ! S-1C F0O G0* * idT-1D S-1C are equivalent. In particular, when G is full and faithful, S-1C is equ* *ivalent to D (the saturation T-1 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 A-MOD * *, A (as a left A-module) is a separator and in MOD-A , A (as a right A-module) is a separator. Given a category C , a set U of objects in C is said to be a coseparating_* *set_if for 0-33 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 A-MOD or MOD-A , 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 0-34 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 A-MOD or MOD-A . 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_. 0-35 __ * *__ [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 . .!.Xn-1 dn-1!Xn dn!Xn+1 ! . .i.s said to be exact_pr* *ovided that imdn-1 kerdn for all n. [Note: A short_exact_sequence_is an exact sequence of the form 0 ! X0 ! X !* * X00! 0.] (Ker-Coker 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. 0-36 (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.] 0-37 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 A-MOD and MOD-A 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. 0-38 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 A-MOD is isomorphic to [A; AB]+ and MOD-A is isomorphic to [AOP ; AB]+. [Note: A right A-moduleZX and a left A-module 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 0-39 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 S-1CC is isomorphic to a category. As such, it is abelian and LSC : C ! S-1CC 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 ! S-1CC has a right adjoint iff C is closed under copro* *ducts, in which case S-1CC 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 S-1CCis 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 one-t* *o-one correspondence between the Serre classes in C which are closed under coproducts* * and the Giraud subcategories of C . 0-40 [Note: The Gabriel-Popescu theorem says that every Grothendieck category wi* *th a separator is equivalent to a Giraud subcategory of A-MOD 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 V-category 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 .] 0-41 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) ffli|d | idff|l | | |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 V-category. 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. 0-42 A V -category is small_if its class of objects is a set; otherwise it is la* *rge_. V-CAT , 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 V-categories with M smal* *l_then the category V [M ; N] whose objects are the V-functorsZM ! 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 0-43 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 O-graph_is * *an object A and a pair of morphisms s; t : A ! O. O-GR is the category whose objects a* *re the O-graphs 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 O-graph. Therefore O-GR is a monoidal category: Take A A0 to be A xO A0 and * *let e be (O; idO; idO). A monoid M in O-GR is an internal category in C with objec* *t element O. 0-44 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; : :;:ssn-1), dn =* * (ss2; : :;:ssn), di = (ss1; : :;:c O (ssn-i; ssn-i+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 ! . .!.Xn-1 fn-1!Xn and 8 < X1 ! . .!.Xn (i = 0)f Of dif = X0 ! . .!.Xi-1 --------i--i-1----!Xi+1! . .!.Xn (0 < i < n); : X0 ! . .!.Xn-1 (i = n) idXi sif = X0 ! . .!.Xi---! Xi! . .!.Xn. The abstract definition thus reduces to the* *se formulas since f corresponds to the n-tuple (fn-1; : :;: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 0-45 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 G-set, 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 G-set_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) 0-46 in tranG to the morphism (h; g . h) in grdG induces an isomorphism nertranG ! n* *ergrdG of simplicial sets. For (g0; : :;:gn) ! (gn; gn-1gn; : :;:g0. .g.n) is the arrow nerntranG !* * nerngrdG, its inverse being (g0; : :;:gn) ! (gng-1n-1; gn-1g-1n-2; : :;: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, flflflflflflcom|aemute 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 , ffllflflflflco|mmute. 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 n-1G, di = F T i-1mT n-i-1G (0 < i < n), dn = F T n-1, and si = F T if* *flT n-iG. 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 O|G|||||ae______________________wG |||||||||||||a|eo ffNlG||||||||||||| | |||||||||||||a|eo NNQ ||||||||||||| | ||||||||||||| | _____T ||||||||||||| | |||||||||||||T|O T OTGO Gfl w ||||||||||||| | ||||||||| |AAADmG flflf||||||||||||||||fl 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 Tn-1 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 Tn-1 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 0-47 is defined by hi= si0(fflTn-i+1G)di0(0 i n), then d0O h0 = id, dn+1 O hn = sn* *0O fflG O O dn0, and 8 < hj-1O di(i < j) aeh O s (i j) diO hj = diO hi-1(i = j > 0); siO hj = j+1 i : : hjO si-1(i > j) hjO di-1(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.] 1-1 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'oe-compact ' ' 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, OE|A = 0 . (LCH 2) X is oe-compact 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 1-2 ` (LCH 3) X is paracompact iff X admits a representation X = Xi, wher* *e the i Xi are pairwise disjoint nonempty open oe-compact subspaces of X. (LCH 4) X is second countable iff X is oe-compact 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 < oe-point_finite_ : oe-neighborhood_finite_oe-discrete : _______ 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 oe-closure_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. 1-3 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 oe-discrete. 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 oe-closure 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 oe-discrete closed refinement. [Note: This definition is partially suggested by the fact that X is paraco* *mpact iff every open covering of X has a oe-discrete open refinement.] 1-4 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 oe-closure 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 oe-clo* *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 Foe-subspaces (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. 1-5 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 oe-discrete 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 (Isbell-Mrowka_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. 1-17), 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 Isbell-Mrowka 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 Isbell-Mrowka 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}): 1-6 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), 807-823; see also Topology Appl. 28 (1988), 113-* *125. zTopology Proc. 4 (1979), 393-405. 1-7 __ 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 Isbell-Mrowka space (N ) is perfect; however, it is not normal (cf. p. 1-12). 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) (Bennett-Lutzerz) 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 oe-discrete 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), 517-520. zGeneral Topology Appl. 2 (1972), 49-54. 1-8 submetacompact (cf. p. 1-12). On the other hand: (i) The ordinal space [0; ] x * *[0; [ is nonnormal and countably paracompact but not subparacompact; (ii) The Isbell-Mrowka space (N )* * is nonnormal and subparacompact but not countably paracompact (cf. p. 1-12). [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 oe-discrete 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- = {S-ff: ff < } following properties: (1) 8 ff : #(N - (S+ff[ S-ff)) = !. (2) 8 ff; 8 fi : fi < ff ) S+fi< S+ffand S-fi< S-ff. (3) 8 ff : #(S+ff\ S-ff) < !. (4) 8 ff; 8 n 2 N: #{fi : fi < ff & S+ff\ S-fi Fn} < ! (Fn = {1; : :;:* *n}). There is then no H N such that 8 ff : S+ff H and S-ff 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 S-0= ;. Given S+ffand S-ff, decompose N - (S+ff[ S-* *ff) into three infinite pairwise disjoint sets N+ff, N-ff, and Nff. Put ae + + + Sff+1= Sff[ Nff () N - (S+ [ S- ) N ): S-ff+1= S-ff[ N-ff 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), 179-186; see also Balogh, Proc. Amer. Math. Soc. 124* * (1996), 2555-2560. 1-9 S S S-ffj Fniand write T+ for (S+ffi- Fni). Note that 8 ff < : S+ff< T+ and 8 * *i : #(T+ \ S-ffi) < !. ji i T S If Ii= {ff : ffi ff < ffi+1& T+ S-ff 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 S-ff- * *{S-ffj: 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+ \ S-ff 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 \ S-ffiis 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 S-ff\ H are both finite for eve* *ry ff < . Choose an n 2 N : W = {ff : S-ff\ 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\ S-fi 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} [ ((S-ff- S-fi) -* * 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 f|K = (by "p* *ressing down", i.e., Fodor's lemma). Put ae H+ = (S+ [ (N \ U+ )) - S- H- = (S- [ (N \ U- )) - S+ : 1-10 Then H+ \ H- = ;. Let ff < be arbitrary. Using the cofinality of K and the rel* *ation f|K = , 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. 1-17* *). 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 Isbell-Mrowka sp* *ace (N ) is pseudocompact but not countably compact (cf. p. 1-12). 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), 505-516. zCanad. J. Math. 30 (1978), 243-249. 1-11 Since X is completely regular, there exists a continuous function fn : X ! [0; * *n] such that P fn(xn) = n, fn|X - 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. 1-12). 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 R-open * *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. 1-10). 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 Isbell-Mrowka 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 oe-discrete 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 : oei|U < 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 oe-discrete. 1-25 [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 oe-discrete 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 - __Pn-2(n 3) 1 n 1 2n-1 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 : cr-1(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 1-28 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| < 2-i}_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 f-1(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 : t-1(]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) 1-29 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. 1-30 [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. 5-4 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. 1-31 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. 0-21 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 1-32 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 Isbell-Mrowka * *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. 0-21), 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), 392-395. 1-33 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. 0-21). 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. 3-28) 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), 133-152. 1-34 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 OE-1(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), 309-316. zArch. Math. 22 (1971), 545-555; see also Wyler, General Topology Appl. 3 (1* *973), 225-242. 1-35 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 OE-1(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. 0-21). 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), 273-298; see also Hoffmann, Arch. Math.* * 32 (1979), 487-504. 1-36 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 OE-1(A) = OE-1(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), 181-195. 1-37 EXAMPLE (k-Spaces_) The monocoreflective hull in TOP of the category of c* *ompact spaces is the category of k-spaces. In other words, a topological space X is a k-space_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 k-space. 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 k-space (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. 1-29, 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), 205-212. 1-38 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), 314-320. zGeneral Topology Appl. 4 (1974), 51-72; see also Quaestiones Math. 2 (1977)* *, 355-377. 1-39 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)). 2-1 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. 0-33) 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. 2-2 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 (Hofman-Lawsony). 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)!!foY-1(Q)(Q 2 oZ ), where * *oY has the _________________________ yTrans. Amer. Math. Soc. 246 (1978), 285-310 (cf. 304-306). 2-3 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 f-1(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. 2-2). 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. 2-2). 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.] 2-4 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. 5-11)). 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 oe-compact. [Note: X countable at infinity ) X oe-compact. Example: P is not oe-compact* *, 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), 183-189. 2-5 EXAMPLE Q is oe-compact 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 ! f-1aisecontinuous.] 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 , f-1 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). 2-6 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: f-1 2 . Deter* *mine __ __ relatively compact open O & P : f-1 (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 ) g-1 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, g-1 (L) O V , i.e., g-1 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, hi|Ui= idUi. Every stable homeomorphism of Rn is isotopic to the identity. [Take k = 1 and consider a homeomorphism h : Rn ! Rn forawhicheh|U = 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) 2-7 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 o-1* *(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. 6-17)).] An isotopy {ht: 0 t 1} is said to be invertible_if the collection {h-1t: * *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 one-to-one, 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, h-1t(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 __ht|X = 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 H-1 (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_. 2-8 [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), 171-178; see also van Douwen, Topology Ap* *pl. 39 (1991), 3-32. 2-9 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}. 2-10 [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 f-1 (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 f-1 (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 Bessaga-Pelczynski, Selecte* *d Topics in Infinite Dimensional Topology, PWN (1975), 63. zMath. Sbornik 8 (1969), 287-333. 2-11 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. 1-13)).] 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.] 2-12 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. 2-10). 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 2-13 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. 2-11). Examples (Macky): (1) Every countably compact space is a CB space; (2) Ever* *y CB space is count- ably paracompact. EXAMPLE The Isbell-Mrowka space (N ) is a pseudocompact LCH space which is* * not countably paracompact (cf. p. 1-12), 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. 2-8), 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), 467-472. 2-14 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__2n-1, 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) = sup|f|(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) = * *|c|pK (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) : F|K = 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).] 2-15 [Note: Recall that in general, X completely regular 6) kX completely regula* *r (cf. p. 1-36).] 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 f|K_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 {fi|K} 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. 2-14).] 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), 197-198. 2-16 Remark: It can be shown that Proposition 15 goes through if the hypothesis * *"completely metrizable" is weakened to "topologically complete" (McCoy-Ntantuy). EXAMPLE Let X be a LCH space. Assume: X is paracompact_then C(X) is Baire. ` [Using LCH3 (cf. p. 1-2), 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. STONE-WEIERSTRASS 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_2-fq_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), 157-166. 2-17 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 = sup|f|_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 = OEn|X0_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* *0|K0 = f0|K0 & 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. 2-18 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 Anderson-Kadec theorem* * says that E is homeomorphic to R! (for a proof, see Bessaga-Pelczynskiy). 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. 2-15). 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), 471-474. kProc. Japan Acad. Sci. 30 (1954), 294-298. 2-19 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 & fx|X - 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 * * xi|K -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 = f|K ) 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: 2-20 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. 2-16).* * 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 : sup|f| 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 . 2-21 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, L-O 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. 2-24). 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. 2-20), 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. 2-20)* *. 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. 2-20)). 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, 2-22 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 = supnn|fn| 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 R-compact (Gillman-Henrikseny). For such X, C(X) is th* *erefore barrelled but not bornological. Given a closed subset A of X, let IA = {f : f|A = 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 : fx|Ux 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 I|X - U is an ideal in C(X - U) (Tietze), hence by the sub* *lemma, equals C(X - U). Choose an f 2 I : f|X - 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 = sup|f - fOE| < ffl ) f 2 I.] X PROPOSITION 23 The closed subsets of X are in a one-to-one correspondence * *with the closed ideals of C(X) via A ! IA . _________________________ yTrans. Amer. Math. Soc. 77 (1954), 340-362 (cf. 360-362). 2-23 [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 ____ I|K is an ideal in C(K) (cf. p. 2-14), thus I|K = IA\K (cf. Proposition 22), a* *nd from __ this it follows that IA I= I.] Application: The points of X are in a one-to-one 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 one-to-one 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(|f|2) = 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 R-compact and if O : C(X) ! R is a character, then in the terminol* *ogy of p. 19-6 & p. 19-7, 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 2-n, apply O * *to get O(f) = i=1 2 _________________________ yThe Stone-Cech Compactification, Springer Verlag (1974), 18. 2-24 O(gn) 2-n, and let n ! 1. It therefore follows that \FO is nonempty, say x 2 \* *FO (cf. p. 19-7). 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 one-to-one correspondence X ! "C(X), viz. x ! Ox* *, where Ox(f) = f(x). If X is not R-compact, 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. 3-1 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 2-source 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 OE|A = 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 Isbell-Mrowka 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. 3-2 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 3-3 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 : t-1(]0; 1[) ! X be the function [x; t] ! x; let p : t-1(]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. 3-22). 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. 1-27 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 2-source 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. 4-5). 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 3-4 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 OE|i0X &* * OE|IA 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) 3-5 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 n-1 ! D n is a closed cofibration. Proof: Define a retrac* *tion r : ID n ! i0D n[ IS n-1 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 n-1. Consequently, if f : Sn-1 ! 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 Sn-1 ! Sn of Sn-1 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 : Sn-1 ! 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 3-6 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.] 3-7 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 OE-1(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 = OE-1(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}). 3-8 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). 3-9 [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, OE-1([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 ) 3-10 & 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 H|I(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 = OE-1(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 = h|IA & F = HA O i0 = HB O * *i0 B : IX ! Y HB |IB = h|IB 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) 3-11 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 ss-1 (]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 ss-1 (1) and ss-1 (]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 ss-1 ([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 G|A : A ! P Y will then do the tr* *ick.] EXAMPLE (Telescope_Construction_) Let X0 X1 . .b.e an expanding sequenc* *e of topo- 3-12 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. 3-5). 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 = ss-1 (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).] 3-13 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. 1-28). [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 p1|U and p2|U relX , where p1 and p2 are the projections onto the first* * and second factors. Examples: (1) R n, D n, and Sn-1 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. 6-30) and in NES(stratifiab* *le) (cf. p. 6-43). Every metrizable topological manifold is stratifiable (cf. p. 6-29 ff.: metrizable ) * *stratifiable) and, being an ANR (cf. p. 6-28), is in NES(stratifiable) (cf. p. 6-44: 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). 3-14 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)} H-1 (U0). Since H-1 (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) H-1 (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 = OE-1([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. 3-10 to the triple (X x X; X ; A x A).] 3-15 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* * = f-1 (0) is a retract of f-1 ([0; 1[ )_then the inclusion A ! X is a closed cofibration. [Write r for the retraction f-1 ([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 OE-1f(0) = A. * *Put Hf(x; t) = H((x; r(x)); t) to obtain a homotopy Hf : IOE-1f([0; 1[ ) ! X of the* * inclusion OE-1f([0; 1[ ) ! X relA such that Hf O i1(OE-1f([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 3-16 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) * *: G0|A0= 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 . 3-17 ae [The inclusions X ! X x X are cofibrations (cf. p. 3-14), thus the same* * is true of the inclu- ae Y ! Y x Y sions X ! X xk X (cf. p. 3-8). 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. 3-15). [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. 3-8). 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)) 3-18 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 one-to- 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 . 3-19 [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 (!). 3-20 [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. 3-5). 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 n-1 = 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 3-21 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 kn-1 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] ! (fn-1 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.] 3-22 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 n-1= 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) ! eHq-1(X) ! eHq-1(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. 3-23 [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. 3-13)* *.] [Note: Y and Mf have the same homotopy type, so Y numerably contractible ) * *Mf numerably contractible (cf. p. 3-13).] Let X f Z g!Y be a 2-source_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 2-source 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 2-source 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. 3-24 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 2-sources 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 |; id|TOP 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 |; id|T* *OP is a 2-source X f-- Z -g* *-! Y ? ? * * ? X f Z g!Y and a morphism of 2-sources is a commutative diagram y y * * y . The X0 f--0 Z0 --* *!g0Y 0 double mapping cylinder is a functor |; id|TOP ! TOP . It has a right adjoint T* *OP ! |; id|TOP , viz. the functor that sends X to the 2-source 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 f--0 Z0 --!g0Y 0 are homotopy equivalences_then the arrow Mf;g! Mf0;g0is a homotopy equivalence. 3-25 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. SEGAL-STASHEFF 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(n-1) 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) : 3-26 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 (n-1) [i]x -- U[i]x _ --! BU I[n] I[n] ` |u ` |u |u V[i]x n -- V[i]x _n --! BV(n-1) I[n] I[n] By induction, F(n-1)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=Sn-1 = 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)Sn-1, 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 2-source in TOP * is a pushout in TOP *. Examples: (1) The pusho* *ut 3-27 *? --! (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), 175-190. zProc. Amer. Math. Soc. 109 (1990), 237-241; see also Morgan-Morrison, Proc.* * London Math. Soc. 53 (1986), 562-576. 3-28 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. 2-2).] (#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: Sm-1 * Sn-1= Sm+n-1 ) 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), 1265-1269. 3-29 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 OE-1(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. 3-30 [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]); f|A = 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 2-source_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 2-source * * ! *, 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. 3-24). [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 i0--iZ q Z --! X q Y 1 fqg 3-31 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 2-source 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. 3-6). 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. 3-19). 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 3-32 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 3-33 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. 3-21). The embedding j : X * *! X is a * * _ closed cofibration (cf. p. 3-21). It is not a pointed map but the retraction r * *: X ! X is 3-34 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. 3-22) 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 = ae-1({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 = H|IU. 3-35 Sufficiency: One can assume that U is closed (cf. p. 3-11). 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 it|X)(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. 3-8) 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. 3-17.] 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. 3-36 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. 3-28), 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.Xn-1 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 2-source (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 _ _ _ _ _ _ _ _ _ 2-source (X #Y )#Z X #Y #Z ! X #(Y #Z ) and, to be specific,aworkeon_the_left* *,acall-e ing the arrow OE. Define pointed continuous functions uv::XY!_X!bY_y (u|X)(* *x)(=vx|Y&)(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 it|X)(x) = x& (H O it|[0; 1])(T=)max 0; 2T_-_t_. H a* *nd G : IY ! Y (G O it|Y )(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 . 3-37 [Note: Nondegeneracy is not actually necessary for the truth of this conclu* *sion (cf. p. 3-33).] 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 3-38 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. 3-32, 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 -ff-fi 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 +(a-bffuse 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 3-39 ae 0 0 assumption, [(X#Y ); E] is abelian. Therefore the ff . [ff ; fi]m-u[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 R-modu* *le L = Ln together * * n0 with bilinear pairings [ ; ] : Ln x Lm ! Ln+m such that [x; y] = (-1)|x||y|+1[y* *; x] and (-1)|x||z|[[x; y]; z] + (-1)|y||x|[[y; z]; x] + (-1)|z||y|[[z; * *x]; y] = 0: L L is said to be connected_if L0 = 0. Example: Let A = An be a graded R-algeb* *ra. For x 2 An, n0 y 2 Am , put [x; y] = xy - (-1)|x||y|yx_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 R-module.] 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+m-1 (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 Z-module 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 Z-algebra (Pontryagin product), hence * *can be regarded as a graded Lie algebra over Z.] A pair (X; A) is said to be n-connected_(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; Sq-1) ! 3-40 (X; A) is homotopic relSq-1 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 n-connected; (2) (B n+1; Bn+1 - {0}) is n-connected. [Note: Take A = {x0}_then ssq(X; {x0}; x0) = ssq(X; x0), so X is n-connecte* *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 n-connected and Y is m-connected, 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 1-connected.] ae EXAMPLE Suppose that X are nondegenerate and X is n-connected 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 n-connected. 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 n-connected _then the arrow 2; X2 \ X1)m-connected 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 n-con* *nectedm-connected_ 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) n-connected * * __ __ __ m-connected , 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 3-41 * * __ of IA and IC [ i1B is a strong deformation retract of IB (cf. p. 3-6). 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 n-connected, (X; A) is m-connected_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 n-connnected_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. 3-22, 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)).] 3-42 The infinite cyclic group ssn(S n) is generated by [n], n the identity Sn !* * Sn. Form the Whitehead product [n; n] 2 ss2n-1(S n)_then the kernel of the suspension ho* *momor- phism ss2n-1(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 n-connected_then the arrow ssq(X1;* * X1 \ X2) ! (X2; X2\ X1) m-connected 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 n-connected_then the arrow * *ssq(K1; K1\ (K2; K2\ K1) m-connected 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) n-connected of K01\ K02. Therefore (K1 1 2 is and the open homotopy exci* *sion theorem is (K02; K02\ K01) m-connected 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), 1-317. 3-43 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 * * n-connected. (K2; L) * * m-connected 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. f|K1 = f1. f2 f|K2 = 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 f|U1 __ __f(U and __ & f|U1 \ U2 are weak homotopy equivalences. But by assu* *mption X = 2) X2 f|U2 * * ae intX1 [ intX2. Therefore __fis a weak homotopy equivalence (cf. p. 4-52). The i* *nclusions K1 ! K __ * * K2 ! K are closed cofibrations (cf. p. 3-13), 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. 3-44 Given a p-dimensional cube C in Rq (q 1; 0 p q), denote by skdC its d-di* *mensional skeleton, i.e., the set of its d-dimensional faces. Put _C= [skp-1C_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 p-dimensional 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 skp-1C, f-1(A) \ D Kd(D) (Ld(D))_then there exists a continuous function g : C ! X wit* *h f ' g rel_Cand g-1(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 g-1(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 skq-1[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, Iq-10= Iq-1 x {0} (q > 1) & I0* *0= {0} (q = 1), Jq-1 = _Iq-1xI [Iq-1x{1} (q > 1) & J0 = {1} (q = 1), so _Iq= Iq-10[Jq-1 and _Iq* *-10= Iq-10\Jq-1_ then for any pointed pair (X; A; x0), ssq(X; A; x0) = [Iq; _Iq; Jq-1; X; A; x0]. [Note: A continuous function f : (Iq; _Iq; Jq-1) ! (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; Jq-1) ! (X1 [ X2; X2; x0). It will be shown b* *elow that 9 F 2 ff : pro(F-1 (X - X1)) \ pro(F-1 (X - X2)) = ;, pro: Iq ! Iq-1 the projection. Grant* *ed this, choose a continuous function OE : Iq-1 ! [0; 1] which is 1 on pro(F-1 (X -X1)) and 0 on * *_Iq-1[pro(F-1 (X -X2)). 3-45 Define : Iq ! Iq by (x1; : :;:xq) = (x1; : :;:xq-1; t + (1 - t)xq), where t = * *OE(x1; : :;:xq-1), and put g = F O _then g : (Iq; _Iq; Jq-1) ! (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 {f-1(X1); f-1(X2)} is an open * *covering of Iq, one can subdivide Iq into a collection C of q-dimensional 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 k-1 has been constructed. Claim: 9 a homotopy hk : IDk ! X2rel_Dksuch that hkO i0 = k-1|Dk and (hkO i* *1)-1(X2- X1\ X2) Kn+1(Dk). [Case 1: dim Dk = 0. Here, Kn+1(Dk) = ; and the point k-1(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 k-1(D_k) X1\ X2, hence k-1|Dk repr* *esents an element of ssdk(X2; X1\ X2) = 0 (dk = dimDk). Case 3: dimDk n + 1. Apply the compressi* *on lemma.] k-1S Extend hk to a homotopy Hk : Iq x I ! X of k-1rel[ {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(X1-X1\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 Jq-1. Definition: F = s () F 2 ff). If pro(F-1 (X - X1))a\epro(F-1 (X - X2)) were* * nonempty, then there would exist an x 2 Iq-1 and a cube D Iq-1 : x 2 Kn(D), an impossibilit* *y since q -1 < n+m. x 2 Lm (D) Turning to injectivity, let f; g : (Iq; _Iq; Jq-1) ! (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; Jq-1) 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(H-1(X - X1)) \ prox idI(H-1(X - X2)) = ;. It is this extra dimension that accounts for the res* *triction q < n + m. Choose a continuous function OE : Iq-1 x I ! [0; 1] which is 1 on prox idI(H-1(* *X - X1)) and 0 on (_Iq-1x I) [ (Iq-1 x _I) [ prox idI(H-1(X - X2)). Define : Iq x I ! Iq x I by * *(x1; : :;:xq; xq+1) = 3-46 (x1; : :;:xq-1; t+(1-t)xq; xq+1), where t = OE(x1; : :;:xq-1; xq+1)_then the co* *mposite HO is a homotopy between f and g : H O (_Iqx I) X1\ X2 & H O (Jq-1 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 = f-1(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 OE-1(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 = [ak-1; 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 ) 3-47 (D x Ik) Xi & H(I(D x Ik)) Yi; (2) (D x {ak-1}) Y1 \ Y2 ) (D x {ak-1}) X1 * *\ X2 & H(I(D x {ak-1})) 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 {ak-1}) 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 {ak-1}) 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 {ak-1}) * *! (Xi; X1 \ X2) & H : (D_x Ik [ D x {ak}; _Dx {ak-1}) 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.] 4-1 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 =* * p-1(O), which is therefore an object in TOP =O (with projection pO = p|XO ). The notat* *ion X|O is also used. In particular: Xb = p-1(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 = f|XO (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. 4-2 [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. 4-3 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. 4-26). 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 : s|A = 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|* *OE-1(]0; 1]), where OE 2 C(B; [0; 1])_then 8 ffl, 0 < ffl < 1, s|OE-1([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 n-connected_then B x T has the SEP. 4-4 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. 3-11). Since* * Y has the SEP, there exists a sectionateof Y : t|P = 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 : s|A = 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 = f-1 (]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 f-1 (1). Fix OE 2 C(B; [0; 1]) : O = OE-1(]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 f-1 (]1=2; 1]) \ OE-1(]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. 4-5 [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 X|-1S(]0; 1]) & s|A = sA : S is nonem* *pty (take S = ;, s = sO |ss-1 (]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 : ss-10(]0; 1]) ! [0; 1] by OE0(b) = min{1* *; 0(b)=ss0(b)}. Owing to the lemma, X|ss-10(]0; 1]) has the SEP (ss-10(]0; 1]) Oi0). On the ot* *her hand, OE-10(]0; 1]) is a halo of OE-10(1) in ss-10(]0; 1]) and s0|OE-10(1) admits an * *extension to OE-10(]0; 1]), viz. s0|OE-10(]0; 1]). Therefore s0|OE-10(1)acanebe extended to a section si0of* * X|ss-10(]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. 3-11 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 F|A = 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 F|U = H O i0 and H|I(A \ U) = h|I(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 = OE|U, 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.] 4-6 (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 = h|IV . To construct a homotopy H : IY ! X such that F = H O i0 and p O H = h, with H|IA =_HV |IA, take V closed (cf. HA 2, p. 3-11) 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.] 4-7 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* *. 4-8). 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. 3-26), 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 Sn-1, 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.] 4-8 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 n-manifolds 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: p-1(0) and p-1(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), 201-219; see also Husch, Proc. Amer. Ma* *th. Soc. 61 (1976), 155-156. zPacific J. Math. 30 (1969), 549-553. 4-9 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 ) 4-10 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 = ss-1j(]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 +a1e-1ik(]0; 1]) (k = 1; : :;* *:r) and (2) A neighborhood V of y such that IV meets but a finite number of the ae-1i(]0; 1])* *. Therefore 1S {Yj : j 2 J} = {Yj : j 2 Ir} is a oe-neighborhood finite cozero set covering * *of Y , hence is 1 numerable.] LOCAL-GLOBAL 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 = h-1(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 - tk-1 < fflj, and select i accordingly: h(Yj x [tk-1; 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 = h|IV . With ss a haloing funct* *ion of V , put Ak = ss-1 ([tk; 1]) (k = 1; : :;:n) : Ak is a halo of Ak+1 in Yj and V * * is a halo 4-11 of A1 in Yj. Owing to Proposition 2, there exist homotopies Hk : Yj x [tk-1; t* *k] ! X having the following properties: p O Hk = h|Yj x [tk-1; tk], Hk(y; tk-1) = Hk-1* *(y; tk-1) (k > 1), H1(y; 0) = F (y), Hk|Ak x [tk-1; tk] = HV |Ak x [tk-1; tk]. The Hk thu* *s combine to determine a homotopy H : IYj ! X such that F |Yj = H O i0, p O H = h|IYj, and H|IA = 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. 4-12 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 4-13 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 p-1(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., h|I{y0} cons* *tant ) H|I{y0} constant. [Note: The local-global principle is valid in the regular situation (work w* *ith a suitable subspace of W to factor in the stationary condition).] 4-14 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) = OE-1(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. 3-14). 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. 5-12).] [Note: If X x B is compactly generated, then p is a Hurewicz fibration.] Cofibrations are embeddings (cf. p. 3-3). By analogy, one might expect that* * surjective Hurewicz fibrations are quotient maps. However, this is not true in general. * *Example: 4-15 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)}).] 4-16 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 = p|A_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 = OE-1(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. 4-10).] 4-17 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 a--e . . . 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. 4-18 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 = OE-1(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 = p|A 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. 4-16). Therefore th* *e inclusion i0Wp [ IWpA ! IWp is a closed cofibration (cf. p. 3-6 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 4-19 a Hurewicz fibration. [Choose a liftingafunctione0 : Wp0 ! P A0.aeUse the lifting principle to se* *cure lifting functions 1 : Wp1 ! P A1 such that 1|Wp0 = 0 . Define a lifting * *function 2 :aWp2e! P A2 2|Wp0 = 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 (Mayer-Vietoris_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 X1|B0 have the same fiber homotopy type_then there exists a Hurewicz fibratio* *n X ! B such that aeX2|B0 X1 & X|B1 have the same fiber homotopy type. X2 & X|B2 X0 p0--!B0 q0--Y0 ? ? ? 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 p-1(B0) ! X is a closed cofibration (cf. Proposition 11). Si* *nce X0 is contained in p-1(B0) and since the inclusion X0 ! X is a closed cofibration, the inclusion X* *0 ! p-1(B0) is a closed cofibration (cf. x3, Proposition 9). Proposition 13 then implies that the arrow* * i0p-1(B0) [ IX0 ! B0 is a Hurewicz fibration. Consequently (cf. Proposition 12), the commutative diagram i0p-1(B0) [ IX0-id-!i0p-1(B0) [ IX0 ?y ?y Ip-1(B0) --! B0 has a filler r : Ip-1(B0) ! i0p-1(B0) [ IX0. Therefore the inclusion X0 xB0 Y0 * *! p-1(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 p-1(B0) x* *B Y0 ! X xB Y is a closed cofibration (cf. Proposition 11).] Application: Consider the 2-sink 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. 4-20 p q Let X ! B Y be a 2-sink_then the fiber_join_X *B Y is the double mapping * *cylin- der of the 2-source 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. 3-23). * *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 2-sink, where p is a Hurewicz fibration. There is a com* *mutative diagram 4-21 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. 4-25). Consideration of k ?y * * k then leads X -- X xB Wq --!* * Y to a homotopy equivalence X *B Y ! X *B Wq (cf. p. 3-24). 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.] 4-22 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. 4-12), 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 2-sink, 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(t-11=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* *.] 4-23 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 2-sink_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. 4-24 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 2-sink_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 * * aeW-g [Let Wf . The projections f , W-g= {(y; o) : g(y) = o(1); o 2 C([1=2; 1]; Z)} (x; o) ! o(1=* *2) (y; o) Wf;g --! W-g ! Z are Hurewicz fibrations and the commutative diagram ?y ?y is * *a pullback square.] ! o(1=2) W+f --! Z P? -j-! Y? Every 2-sink 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* * 2-sink X f!Z g Y 4-25 f g X? --! Z? -- Y? and a morphism of 2-sinks 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 2-sink 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 2-sink X p!B q Y , let X __|_|BY beatheedouble mapping cylinder of * *the 2-source 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. 4-20 ff.).] 4-26 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. 4-24), so Pro* *position 19 is applicable.] 4-27 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 (f-1 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. 4-25). 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. 3-35). 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. 4-28 x6, Proposition 6). In particular: (HE(K); idK) is wellpointed (cf. p. 6-14) an* *d numerably contractible (cf. p. 3-13). 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 2-source 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. 3-24). 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 = X|B1 B2 B * * X2 = X|B2 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 4-29 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. 4-12), 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. 3-1* *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. 3-12) and telB i* *s a double mapping cylinder calculated on the B(n)(cf. p. 3-23). 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 Segal-Stasheff 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 .] 4-30 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 local-global 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. 4-15)). EXAMPLE Define f : [-1; 1] ! [-1; 1] by f(x) = 2|x| - 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.] 4-31 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 EXAMP