Parametrized Homotopy Theory J. P. May J. Sigurdsson Department of Mathematics, The University of Chicago, Chicago, IL 60637 Department of Mathematics, The University of Notre Dame, Notre Dame, IN, 46556-4618 E-mail address: may@math.uchicago.edu E-mail address: jsigurds@nd.edu 1991 Mathematics Subject Classification. Primary 19D99, 55N20, 55P42; Secondary 19L99, 55N22, 55T25 Key words and phrases. ex-space, ex-spectrum, model category, parametrized spectrum, parametrized homotopy theory, equivariant homotopy theory, parametrized stable homotopy theory, equivariant stable homotopy theory May was partially supported by the NSF. Abstract.Part I: We set the stage for our homotopical work with preliminary chapters on the point-set topology necessary to parametrized homotopy theo* *ry, the base change and other functors that appear in over and under categorie* *s, and generalizations of several classical results about equivariant bundles* * and fibrations to the context of proper actions of non-compact Lie groups. Part II: Despite its long history, the homotopy theory of ex-spaces re- quires further development before it can serve as the starting point for a* * rig- orous modern treatment of parametrized stable homotopy theory. We give a leisurely account that emphasizes several issues that are of independent i* *nterest in the theory and applications of topological model categories. The essent* *ial point is to resolve problems about the homotopy theory of ex-spaces that a* *re absent from the homotopy theory of spaces. In contrast to previously encou* *n- tered situations, model theoretic techniques are intrinsically insufficien* *t to a full development of the basic foundational properties of the homotopy cate* *gory of ex-spaces. Instead, a rather intricate blend of model theory and classi* *cal ho- motopy theory is required. However, considerable new material on the gener* *al theory of topologically enriched model categories is also required. Part III: We give a systematic treatment of the foundations of parametr* *ized stable homotopy theory, working equivariantly and with highly structured smash products and function spectra. The treatment is based on equivari- ant orthogonal spectra, which are simpler for the purpose than alternative kinds of spectra. Again, the parametrized context introduces many difficul* *ties that have no nonparametrized counterparts and cannot be dealt with using standard model theoretic techniques. The space level techniques of Part II only partially extend to the spectrum level, and many new twists are encou* *n- tered. Most of the difficulties are already present in the nonequivariant * *special case. Equivariantly, we show how change of universe, passage to fixed poin* *ts, and passage to orbits behave in the parametrized setting. Part IV: We give a fiberwise duality theorem that allows fiberwise reco* *g- nition of dualizable and invertible parametrized spectra. This allows dire* *ct application of the formal theory of duality in symmetric monoidal categori* *es to the construction and analysis of transfer maps. The relationship between transfer for general Hurewicz fibrations and for fiber bundles is illumina* *ted by the construction of fiberwise bundles of spectra, which are like bundle* *s of tangents along fibers, but with spectra replacing spaces as fibers. Using * *this construction, we obtain a simple conceptual proof of a generalized Wirthm"* *uller isomorphism theorem that calculates the right adjoint to base change along* * an equivariant bundle with manifold fibers in terms of a shift of the left ad* *joint. Due to the generality of our bundle theoretic context, the Adams isomorphi* *sm theorem relating orbit and fixed point spectra is a direct consequence. Contents Prologue 1 Part I. Point-set topology, change functors, and proper actions 7 Chapter 1. The point-set topology of parametrized spaces 8 Introduction 8 1.1. Convenient categories of topological spaces 8 1.2. Topologically bicomplete categories and ex-objects 11 1.3. Convenient categories of ex-spaces 14 1.4. Convenient categories of ex-G-spaces 17 1.5. Appendix: nonassociativity of smash products in T op* 19 Chapter 2. Change functors and compatibility relations 21 Introduction 21 2.1. The base change functors f!, f*, and f* 22 2.2. Compatibility relations 23 2.3. Change of group and restriction to fibers 25 2.4. Normal subgroups and quotient groups 28 2.5. The closed symmetric monoidal category of retracts 30 Chapter 3. Proper actions, equivariant bundles, and fibrations 33 Introduction 33 3.1. Proper actions of locally compact groups 33 3.2. Proper actions and equivariant bundles 37 3.3. Spaces of the homotopy types of G-CW complexes 37 3.4. Some classical theorems about fibrations 39 3.5. Quasifibrations 41 Part II. Model categories and parametrized spaces 43 Introduction 44 Chapter 4. Topologically bicomplete model categories 45 Introduction 45 4.1. Model theoretic philosophy: h, q, and m-model structures 46 4.2. Strong Hurewicz cofibrations and fibrations 47 4.3. Towards classical model structures in topological categories 50 4.4. Classical model structures in general and in K and U 53 4.5. Compactly generated q-type model structures 55 Chapter 5. Well-grounded topological model categories 59 iii iv CONTENTS Introduction 59 5.1. Over and under model structures 60 5.2. The specialization to over and under categories of spaces 63 5.3. Well-grounded topologically bicomplete categories 66 5.4. Well-grounded categories of weak equivalences 68 5.5. Well-grounded compactly generated model structures 72 5.6. Properties of well-grounded model categories 73 Chapter 6. The qf-model structure on KB 77 Introduction 77 6.1. Some of the dangers in the parametrized world 78 6.2. The qf model structure on the category K =B 79 6.3. Statements and proofs of the thickening lemmas 82 6.4. The compatibility condition for the qf-model structure 84 6.5. The quasifibration and right properness properties 86 Chapter 7. Equivariant qf-type model structures 88 Introduction 88 7.1. Families and non-compact Lie groups 89 7.2. The equivariant q and qf-model structures 90 7.3. External smash product and base change adjunctions 94 7.4. Change of group adjunctions 97 7.5. Fiber adjunctions and Brown representability 100 Chapter 8. Ex-fibrations and ex-quasifibrations 102 8.1. Ex-fibrations 102 8.2. Preservation properties of ex-fibrations 104 8.3. The ex-fibrant approximation functor 106 8.4. Preservation properties of ex-fibrant approximation 108 8.5. Quasifibrant ex-spaces and ex-quasifibrations 110 Chapter 9. The equivalence between Ho GKB and hGWB 112 Introduction 112 9.1. The equivalence of Ho GKB and hGWB 112 9.2. Derived functors on homotopy categories 114 9.3. The functors f* and FB on homotopy categories 115 9.4. Compatibility relations for smash products and base change 117 Part III. Parametrized equivariant stable homotopy theory 121 Introduction 122 Chapter 10. Enriched categories and G-categories 125 Introduction 125 10.1. Parametrized enriched categories 125 10.2. Equivariant parametrized enriched categories 127 10.3. G-topological model G-categories 129 Chapter 11. The category of orthogonal G-spectra over B 132 Introduction 132 11.1. The category of IG -spaces over B 132 CONTENTS v 11.2. The category of orthogonal G-spectra over B 136 11.3. Orthogonal G-spectra as diagram ex-G-spaces 139 11.4. The base change functors f*, f!, and f* 140 11.5. Change of groups and restriction to fibers 143 11.6. Some problems concerning non-compact Lie groups 145 Chapter 12. Model structures for parametrized G-spectra 148 Introduction 148 12.1. The level model structure on GSB 149 12.2. Some Quillen adjoint pairs relating level model structures 152 12.3. The stable model structure on GSB 153 12.4. The ss*-isomorphisms 156 12.5. Proofs of the model axioms 159 12.6. Some Quillen adjoint pairs relating stable model structures 163 Chapter 13. Adjunctions and compatibility relations 167 Introduction 167 13.1. Brown representability and the functors f* and FB 168 13.2. The category GEB of excellent prespectra over B 170 13.3. The level ex-fibrant approximation functor P on prespectra 172 13.4. The auxiliary approximation functors K and E 175 13.5. The equivalence between Ho GPB and hGEB 177 13.6. Derived functors on homotopy categories 178 13.7. Compatibility relations for smash products and base change 180 Chapter 14. Module categories, change of universe, and change of groups 184 Introduction 184 14.1. Parametrized module G-spectra 184 14.2. Change of universe 188 14.3. Restriction to subgroups 192 14.4. Normal subgroups and quotient groups 195 Part IV. Duality, transfer, and base change isomorphisms 197 Chapter 15. Fiberwise duality and transfer maps 198 Introduction 198 15.1. The fiberwise duality theorem 199 15.2. Duality and transfer maps 201 15.3. The bundle construction on parametrized spectra 205 15.4. -free parametrized -spectra 208 15.5. The fiberwise transfer for ( ; )-bundles 209 15.6. Sketch proofs of the compatible triangulation axioms 212 Chapter 16. The Wirthm"uller and Adams isomorphisms 214 Introduction 214 16.1. A natural comparison map f!-! f* 215 16.2. The Wirthm"uller isomorphism for manifolds 216 16.3. The fiberwise Wirthm"uller isomorphism 219 16.4. The Adams isomorphism 221 16.5. Proof of the Wirthm"uller isomorphism for manifolds 222 vi CONTENTS Bibliography 231 Prologue What is this book about and why is it so long? Parametrized homotopy theory concerns systems of spaces and spectra that are parametrized as fibers over poi* *nts of a given base space B. Parametrized spaces, or "ex-spaces", are just spaces over* * and under B, with a projection, often a fibration, and a section. Parametrized spec* *tra are analogous but considerably more sophisticated objects. They provide a world* * in which one can apply the methods of stable homotopy theory without losing track of fundamental groups and other unstable information. Parametrized homotopy theory is a natural and important part of homotopy theory that is implicitly ce* *ntral to all of bundle and fibration theory. Results that make essential use of it ar* *e widely scattered throughout the literature. For classical examples, the theory of tran* *sfer maps is intrinsically about parametrized homotopy theory, and Eilenberg-Moore type spectral sequences are parametrized K"unneth theorems. Several new and current directions, such as "twisted" cohomology theories and parametrized fixed point theory cry out for the rigorous foundations that we shall develop. On the foundational level, homotopy theory, and especially stable homotopy theory, has undergone a thorough reanalysis in recent years. Systematic use of Quillen's theory of model categories has illuminated the structure of the subje* *ct and has done so in a way that makes the general methodology widely applicable to other branches of mathematics. The discovery of categories of spectra with associative and commutative smash products has revolutionized stable homotopy theory. The systematic study and application of equivariant algebraic topology * *has greatly enriched the subject. There has not been a thorough and rigorous study of parametrized homotopy theory that takes these developments into account. It is the purpose of this mo* *no- graph to provide such a study, although we shall leave many interesting loose e* *nds. We shall also give some direct applications, especially to equivariant stable h* *omo- topy theory where the new theory is particularly essential. The reason this stu* *dy is so lengthy is that, rather unexpectedly, many seemingly trivial nonparametri* *zed results fail to generalize, and many of the conceptual and technical obstacles * *to a rigorous treatment have no nonparametrized counterparts. For this reason, the resulting theory is considerably more subtle than its nonparametrized precursor* *s. We indicate some of these problems here. The central conceptual subtlety in the theory enters when we try to prove th* *at structure enjoyed by the point-set level categories of parametrized spaces desc* *ends to their homotopy categories. Many of our basic functors occur in Quillen adjoi* *nt pairs, and such structure descends directly to homotopy categories. Recall that an adjoint pair of functors (T, U) between model categories is a Quillen adjoint pair, or a Quillen adjunction, if the left adjoint T preserves cofibrations and* * acyclic cofibrations or, equivalently, the right adjoint U preserves fibrations and acy* *clic 1 2 PROLOGUE fibrations. It is a Quillen equivalence if, further, the induced adjunction on* * ho- motopy categories is an adjoint equivalence. We originally hoped to find a model structure on parametrized spaces in which all of the relevant adjunctions are Q* *uillen adjunctions. It eventually became clear that there can be no such model structu* *re, for altogether trivial reasons. Therefore, it is intrinsically impossible to la* *y down the basic foundations of parametrized homotopy theory using only the standard methodology of model category theory. The force of parametrized theory largely comes from base change functors ass* *o- ciated to maps f :A -! B. The existing literature on fiberwise homotopy theory says surprisingly little about such functors. This is especially strange since* * they are the most important feature that makes parametrized homotopy theory useful for the study of ordinary homotopy theory: such functors are used to transport information from the parametrized context to the nonparametrized context. One of the goals of our work is to fill this gap. On the point-set level, there is a pullback functor f* from ex-spaces (or sp* *ectra) over B to ex-spaces (or spectra) over A. That functor has a left adjoint f! and* * a right adjoint f*. We would like both of these to be Quillen adjunctions, but th* *at is not possible unless the model structures lead to trivial homotopy categories* *. We mean literally trivial: one object and one morphism. We explain why. It will be clear that the explanation is generic and applies equally well to a variety of * *sheaf theoretic situations where one encounters analogous base change functors. Counterexample 0.0.1. Consider the following diagram. OE ; ______//_B OE|| i0|| fflffl| fflffl| B __i1_//B x I Here ; is the empty set and OE is the initial (empty) map into B. This diagram * *is a pullback since B x{0}\B x{1} = ;. The category of ex-spaces over ; is the trivi* *al category with one object, and it admits a unique model structure. Let *B denote the ex-space B over B, with section and projection the identity map. Both (OE!,* * OE*) and (OE*, OE*) are Quillen adjoint pairs for any model structure on the categor* *y of ex-spaces over B. Indeed, OE! and OE* preserve weak equivalences, fibrations, * *and cofibrations since both take *; to *B . We have (i0)* O (i1)! ~=OE!O OE* since* * both composites take any ex-space over B to *B . If (i1)!and (i0)* were both Quillen* * left adjoints, it would follow that this isomorphism descends to homotopy categories. If, further, the functors (i1)!and (i0)* on homotopy categories were equivalenc* *es of categories, this would imply that the homotopy category of ex-spaces over B with respect to the given model structure is equivalent to the trivial category. Information in ordinary homotopy theory is derived from results in parametri* *zed homotopy theory by use of the base change functor r!associated to the trivial m* *ap r :B -! *. For this and other reasons, we choose our basic model structure to be one such that (f!, f*) is a Quillen adjoint pair for every map f :A -! B and is a Quillen equivalence when f is a homotopy equivalence. Then (f*, f*) cannot be a Quillen adjoint pair in general. However, it is essential that we still h* *ave the adjunction (f*, f*) after passage to homotopy categories. For example, taki* *ng f to be the diagonal map on B, this adjunction is used to obtain the adjunction PROLOGUE 3 on homotopy categories that relates the fiberwise smash product functor ^B on ex-spaces over B to the function ex-space functor FB . To construct the homotopy category level right adjoints f*, we shall have to revert to more classical met* *hods, using Brown's representability theorem. However, it is not clear how to verify * *the hypotheses of Brown's theorem in the model theoretic framework. Counterexample 0.0.1 also illustrates the familiar fact that a commutative d* *ia- gram of functors on the point-set level need not induce a commutative diagram of functors on homotopy categories. When commuting left and right adjoints, this i* *s a problem even when all functors in sight are parts of Quillen adjunctions. There* *fore, proving that compatibility relations that hold on the point-set level descend t* *o the homotopy category level is far from automatic. In fact, proving such "compatibi* *l- ity relations" is often a highly non-trivial problem, but one which is essentia* *l to the applications. We do not know how to prove the most interesting compatibility relations working only model theoretically. Even in the part of the theory in which model theory works, it does not work* * as expected. There is an obvious naive model structure on ex-spaces over B in which the weak equivalences, fibrations, and cofibrations are the ex-maps whose maps * *of total spaces are weak equivalences, fibrations, and cofibrations of spaces in t* *he usual Quillen model structure. This "q-model structure" is the natural starting point* * for the theory, but it turns out to have severe drawbacks that limit its space level utility and bar it from serving as the starting point for the development of a * *useful spectrum level stable model structure. In fact, it has two opposite drawbacks. * *First, it has too many cofibrations. In particular, the model theoretic cofibrations n* *eed not be cofibrations in the intrinsic homotopical sense. That is, they fail to s* *atisfy the fiberwise homotopy extension property (HEP) defined in terms of parametrized mapping cylinders. This already fails for the sections of cofibrant objects and* * for the inclusions of cofibrant objects in their cones. Therefore the classical the* *ory of cofiber sequences fails to mesh with the model category structure. Second, it also has too many fibrations. The fibrant ex-spaces are Serre fib* *ra- tions, and Serre fibrations are not preserved by fiberwise colimits. Such coli* *mits are preserved by a more restrictive class of fibrations, namely the well-sectio* *ned Hurewicz fibrations, which we call ex-fibrations. Such preservation properties * *are crucial to resolving the problems with base change functors that we have indica* *ted. In model category theory, decreasing the number of cofibrations increases the number of fibrations, so that these two problems cannot admit a solution in com- mon. Rather, we require two different equivalent descriptions of our homotopy categories of ex-spaces. First, we have another model structure, the "qf-model structure", which has the same weak equivalences as the q-model structure but h* *as fewer cofibrations, all of which satisfy the fiberwise HEP. Second, we have a d* *escrip- tion in terms of the classical theory of ex-fibrations, which does not fit natu* *rally into a model theoretic framework. The former is vital to the development of the stable model structure on parametrized spectra. The latter is vital to the solu* *tion of the intrinsic problems with base change functors. Before getting to the issues just discussed, we shall have to resolve various others that also have no nonparametrized analogues. Even the point set topology requires care since function ex-spaces take us out of the category of compactly generated spaces. Equivariance raises further problems, although most of our new foundational work is already necessary nonequivariantly. Passage to the spectrum 4 PROLOGUE level raises more serious problems. The main source of difficulty is that the u* *nder- lying total space functor is too poorly behaved, especially with respect to sma* *sh products and fibrations, to give good control of homotopy groups as one passes from parametrized spaces to parametrized spectra. Moreover, the resolution of base change problems requires a different set of details on the spectrum level * *than on the space level. The end result may seem intricate, but it gives a very powerful framework in which to study homotopy theory. We illustrate by showing how fiberwise duality and transfer maps work out and by showing that the basic change of groups isomo* *r- phisms of equivariant stable homotopy theory, namely the generalized Wirthm"ull* *er and Adams isomorphisms, drop out directly from the foundations. Costenoble and Waner [28] have already given other applications in equivariant stable homotopy theory, using our foundations to study Poincar'e duality in ordinary RO(G)-grad* *ed cohomology. Further applications are work in progress. The theory here gives perhaps the first worked example in which a model theo- retic approach to derived homotopy categories is intrinsically insufficient and* * must be blended with a quite different approach even to establish the essential stru* *ctural features of the derived category. Such a blending of techniques seems essential* * in analogous sheaf theoretic contexts that have not yet received a modern model th* *e- oretic treatment. Even nonequivariantly, the basic results on base change, smash products, and function ex-spaces that we obtain do not appear in the literature. Such results are essential to serious work in parametrized homotopy theory. Much of our work should have applications beyond the new parametrized the- ory. The model theory of topological enriched categories has received much less attention in the literature than the model theory of simplicially enriched cate- gories. Despite the seemingly equivalent nature of these variants, the topologi* *cal situation is actually quite different from the simplicial one, as our applicati* *ons make clear. In particular, the interweaving of h-type and q-type model structures th* *at pervades our work seems to have no simplicial counterpart. This interweaving do* *es also appear in algebraic contexts of model categories enriched over chain compl* *exes, where foundations analogous to ours can be developed. One of our goals is to gi* *ve a thorough analysis and axiomatization of how this interweaving works in general in topologically enriched model categories. History. This project began with unpublished notes, dating from the summer of 2000, of the first author [64]. He put the project aside and returned to it* * in the fall of 2002, when he was joined by the second author. Some of Parts I and * *II was originally in a draft of the first author that was submitted and accepted f* *or publication, but was later withdrawn. That draft was correct, but it did not in* *clude the "qf-model structure", which comes from the second author's 2004 PhD thesis [88]. The first author's notes [64] claimed to construct the stable model struc* *ture on parametrized spectra starting from the q-model structure on ex-spaces. Following [64], the monograph [47] of Po Hu also takes that starting point and makes that claim. The second author realized that, with the obvious definitions, the axiom* *s for the stable model structure cannot be proven from that starting point and that a* *ny naive variant would be disconnected with cofiber sequences and other essential * *needs of a fully worked out theory. His qf-model structure is the crucial new ingredi* *ent that is used to solve this problem. Although implemented quite differently, the applications of Chapter 16 were inspired by Hu's work. PROLOGUE 5 Thanks. We thank the referee of the partial first version for several helpf* *ul suggestions, Gaunce Lewis and Peter Booth for help with the point set topology, Mike Cole for sharing his remarkable insights about model categories, and Mike Mandell for much technical help. We thank Kathleen Lewis for working out the counterexample in Theorem 1.1.1 and Victor Ginzburg for giving us the striking Counterexample 11.6.2. We are especially grateful to Kate Ponto for a meticulou* *sly careful reading that uncovered many obscurities and infelicities. Needless to s* *ay, she is not to blame for those that remain. Part I Point-set topology, change functors, and proper actions CHAPTER 1 The point-set topology of parametrized spaces Introduction We develop the basic point-set level properties of the category of ex-spaces over a fixed base space B in this chapter. In x1.1, we discuss convenient categ* *ories of topological spaces. The usual category of compactly generated spaces is not adequate for our study of ex-spaces, and we shall see later that the interplay * *between model structures and the relevant convenient categories is quite subtle. In x1.* *2, we give basic facts about based and unbased topologically bicomplete categories. T* *his gives the language that is needed to describe the good formal properties of the various categories in which we shall work. We discuss convenient categories of ex-spaces in x1.3, and we discuss convenient categories of ex-G-spaces in x1.4. As a matter of recovery of lost folklore, x1.5 is an appendix, the substance* * of which is due to Kathleen Lewis. It is only at her insistence that she is not na* *med as its author. It documents the nonassociativity of the smash product in the ordin* *ary category of based spaces, as opposed to the category of based k-spaces. When wr* *it- ing the historical paper [70], the first author came across several 1950's refe* *rences to this nonassociativity, including an explicit, but unproven, counterexample i* *n a 1958 paper of Puppe [82]. However, we know of no reference that gives details, * *and we feel that this nonassociativity should be documented in the modern literatur* *e. We are very grateful to Gaunce Lewis for an extended correspondence and many details about the material of this chapter, but he is not to be blamed for the point of view that we have taken. We are also much indebted to Peter Booth. He is the main pioneer of the theory of fibered mapping spaces (see [5, 6, 7]) * *and function ex-spaces, and he sent us several detailed proofs about them. 1.1. Convenient categories of topological spaces We recall the following by now standard definitions. Definition 1.1.1. Let B be a space and A a subset. Let f :K -! B run over all continuous maps from compact Hausdorff spaces K into B. (i)A is compactly closed if each f-1 (A) is closed. (ii)B is weak Hausdorff if each f(K) is closed. (iii)B is a k-space if each compactly closed subset is closed. (iv)B is compactly generated if it is a weak Hausdorff k-space. Let Top be the category of all topological spaces and let K , wH , and U = K \wH be its full subcategories of k-spaces, weak Hausdorff spaces, and compact* *ly generated spaces. The k-ification functor k :Top -! K assigns to a space X the same set with the finer topology that is obtained by requiring all compactly cl* *osed subsets to be closed. It is right adjoint to the inclusion K -! T op. The weak 8 1.1. CONVENIENT CATEGORIES OF TOPOLOGICAL SPACES 9 Hausdorffication functor w :T op -! wH assigns to a space X its maximal weak Hausdorff quotient. It is left adjoint to the inclusion wH - ! Top. From now on, we work in K , implicitly k-ifying any space that is not a k-sp* *ace to begin with. In particular, products and function spaces are understood to be k-ified. With this convention, B is weak Hausdorff if and only if the diagonal * *map embeds it as a closed subspace of B x B. Let A xcB denote the classical cartesi* *an product in T op and recall that B is Hausdorff if and only if the diagonal embe* *ds it as a closed subspace of B xcB. The following result is proven in [56, App.x2* *]. Proposition 1.1.2. Let A and B be k-spaces. If one of them is locally compact or if both of them are first countable, then A x B = A xcB. Therefore, if B is either locally compact or first countable, then B is Hausdor* *ff if and only if it is weak Hausdorff. We would have preferred to work in U rather than K since there are many counterexamples which reveal the pitfalls of working without a separation prope* *rty. However, as we will explain in x1.3, several inescapable facts about ex-spaces * *force us out of that convenient category. Like U , the category K is closed cartesi* *an monoidal. This means that it has function spaces Map(X, Y ) with homeomorphisms Map (X x Y, Z) ~=Map (X, Map(Y, Z)). This was proven by Vogt [94], who uses the term compactly generated for our k- spaces. See also [99]. An early unpublished preprint by Clark [20] also showed this, and an exposition of ex-spaces based on [20] was given by Booth [6]. Philosophically, we can justify a preference for K over U by remarking that the functor w is so poorly behaved that we prefer to minimize its use. In U , c* *olimits must be constructed by first constructing them in K and then applying the funct* *or w, which changes the underlying point set and loses homotopical control. Howeve* *r, this justification would be more persuasive were it not that colimits in K that* * are not colimits in U can already be quite badly behaved topologically. For example, w itself is a colimit construction in K . We describe a relevant situation in w* *hich colimits behave better in U than in K in Remark 1.1.4 below. More persuasively, w is a formal construction that only retains formal contr* *ol because both colimits and the functor w are left adjoints. We will encounter ri* *ght adjoints constructed in K that do not preserve the weak Hausdorff property when restricted to U , and in such situations we cannot apply w without losing the adjunction. In fact, when restricted to U , the relevant left adjoints do not c* *ommute with colimits and so cannot be left adjoints there. We shall encounter other re* *asons for working in K later. An obvious advantage of K is that U sits inside it, so * *that we can use K when it is needed, but can restrict to the better behaved category* * U whenever possible. Actually, the situation is more subtle than a simple dichoto* *my. In our study of ex-spaces, it is essential to combine use of the two categories, requiring base spaces to be in U but allowing total spaces to be in K . We have concomitant categories K* and U* of based spaces in K and in U . We generally write T for U* to mesh with a number of relevant earlier papers. Using duplicative notations, we write Map (X, Y ) for the space K (X, Y ) of ma* *ps X - ! Y and F (X, Y ) for the based space K*(X, Y ) of based maps X - ! Y between based spaces. Both K* and T are closed symmetric monoidal categories 10 1. THE POINT-SET TOPOLOGY OF PARAMETRIZED SPACES under ^ and F [56, 94, 99]. This means that the smash product is associative, commutative, and unital up to coherent natural isomorphism and that ^ and F are related by the usual adjunction homeomorphism F (X ^ Y, Z) ~=F (X, F (Y, Z)). The need for k-ification is illustrated by the nonassociativity of the smash pr* *oduct in Top*; see x1.5. We need a few observations about inclusions and colimits. Recall that a map is an inclusion if it is a homeomorphism onto its image. Of course, inclusions need not have closed image. As noted by Strom [91], the simplest example of a non-closed inclusion in K is the inclusion i: {a} {a, b}, where {a, b} has the indiscrete topology. Here i is both the inclusion of a retract and a Hurewi* *cz cofibration (satisfies the homotopy extension property, or HEP). As is well-kno* *wn, such pathology cannot occur in U . Lemma 1.1.3. Let i: A -! X be a map in K . (i)If there is a map r :X -! A such that r O i = id, then i is an inclusion. I* *f, further, X is in U , then i is a closed inclusion. (ii)If i is a Hurewicz cofibration, then i is an inclusion. If, further, X is * *in U , then i is a closed inclusion. Proof. Inclusions i: A -! X are characterized by the property that a func- tion j :Y -! A is continuous if and only if iOj is continuous. This implies the* * first statement in (i). Alternatively, one can note that a map in K is an inclusion i* *f and only if it is an equalizer in K , and a map in U is a closed inclusion if and o* *nly if it is an equalizer in U [56, 7.6]. Since i is the equalizer of iOr and the identit* *y map of X, this implies both statements in (i). For (ii), let Mi be the mapping cylinde* *r of i. The canonical map j :Mi -! X x I has a left inverse r and is thus an inclusion * *or closed inclusion in the respective cases. The evident closed inclusions i1: A -* *! Mi and i1: X -! X x I satisfy j O i1 = i1 O i, and the conclusions of (ii) follow. The following remark, which we learned from Mike Cole [22] and Gaunce Lewis, compares certain colimits in K and U . It illuminates the difference between th* *ese categories and will be needed in our later discussion of h-type model structure* *s. Remark 1.1.4. Suppose given a sequence of inclusions gn :Xn -! Xn+1 and maps fn :Xn -! Y in K such that fn+1gn = fn. Let X = colimXn and let f :X -! Y be obtained by passage to colimits. Fix a map p: Z -! Y . The maps Z xY Xn -! Z xY X induce a map ff: colim(Z xY Xn) -! Z xY X. Lewis has provided counterexamples showing that ff need not be a homeomorphism in general. However, if Y 2 U , then a result of his [56, App. 10.3] shows that* * ff is a homeomorphism for any p and any maps gn. In fact, as in Proposition 2.1.3 below, if Y 2 U , then the pullback functor p*: K =Y - ! K =Z is a left adjoint and therefore commutes with all colimits. To see what goes wrong when Y is not 1.2. TOPOLOGICALLY BICOMPLETE CATEGORIES AND EX-OBJECTS 11 in U , consider the diagram colim(Z xY Xn) __ff//_Z xY X '|| || fflffl| fflffl| colim(Z x Xn)______//Z x X. Products commute with colimits, so the bottom arrow is a homeomorphism, and the top arrow ff is a continuous bijection. The right vertical arrow is an incl* *usion by the construction of pullbacks. If the left vertical arrow ' is an inclusion,* * then the diagram implies that ff is a homeomorphism. The problem is that ' need not be an inclusion. One point is that the maps Z xY Xn -! Z x Xn are closed inclusions if Y is weak Hausdorff, but not in general otherwise. Now assume that all spaces in sight are in U . Since the gn are inclusions, the relevant colimits, when compu* *ted in K , are weak Hausdorff and thus give colimits in U . Therefore the commutation * *of p* with colimits (which is a result about colimits in K ) applies to these part* *icular colimits in U to show that ff is a homeomorphism. The following related observation will be needed for applications of Quillen* *'s small object argument to q-type model structures in x4.5 and elsewhere. Lemma 1.1.5. Let Xn -! Xn+1, n 0, be a sequence of inclusions in K with colimit X. Suppose that X=X0 is in U . Then, for a compact Hausdorff space C, the natural map colimK (C, Xn) -! K (C, X) is a bijection. Proof. The point is that X0 need not be in U . Let f :C -! X be a map. Then the composite of f with the quotient map X -! X=X0 takes image in some Xn=X0, hence f takes image in Xn. The conclusion follows. Scholium 1.1.6. One might expect the conclusion to hold for colimits of se- quences of closed inclusions Xn-1 -! Xn such that Xn - Xn-1 is a T1 space. This is stated as [49, 4.2], whose authors got the statement from May. However, Lewis has shown us a counterexample. 1.2.Topologically bicomplete categories and ex-objects We need some standard and some not quite so standard categorical language. All of our categories C will be topologically enriched, with the enrichment giv* *en by a topology on the underlying set of morphisms. We therefore agree to write C (X, Y ) for the space of morphisms X -! Y in C . Enriched category theory would have us distinguish notationally between morphism spaces and morphism sets, but we shall not do that. A topological category C is said to be topologi* *cally bicomplete if, in addition to being bicomplete in the usual sense of having all* * limits and colimits, it is bitensored in the sense that it is tensored and cotensored * *over K . We shall denote the tensors and cotensors by X x K and Map (K, X) for a space K and an object X of C . The defining adjunction homeomorphisms are (1.2.1) C (X x K, Y ) ~=K (K, C (X, Y )) ~=C (X, Map(K, Y )). By the Yoneda lemma, these have many standard implications. For example, (1.2.2) X x * ~=X and Map (*, Y ) ~=Y, 12 1. THE POINT-SET TOPOLOGY OF PARAMETRIZED SPACES (1.2.3)X x (K x L) ~=(X x K) x L and Map(K, Map(L, X)) ~=Map (K x L, X). We say that a bicomplete topological category C is based if the unique map from the initial object ; to the terminal object * is an isomorphism. In that c* *ase, C is enriched in the category K* of based k-spaces, the basepoint of C (X, Y ) * *being the unique map that factors through *. We then say that C is based topologically bicomplete if it is tensored and cotensored over K*. We denote the tensors and cotensors by X ^ K and F (K, X) for a based space K and an object X of C . The defining adjunction homeomorphisms are (1.2.4) C (X ^ K, Y ) ~=K*(K, C (X, Y )) ~=C (X, F (K, Y )). The based versions of (1.2.2) and (1.2.3) are (1.2.5) X ^ S0 ~=X and F (S0, Y ) ~=Y, (1.2.6) X ^ (K ^ L) ~=(X ^ K) ^ L and F (K, F (L, X)) ~=F (K ^ L, X). Although not essential to our work, a formal comparison between the based and unbased notions of bicompleteness is illuminating. The following result all* *ows us to interpret topologically bicomplete to mean based topologically bicomplete whenever C is based, a convention that we will follow throughout. Proposition 1.2.7. Let C be a based and bicomplete topological category. Then C is topologically bicomplete if and only if it is based topologically bicomple* *te. Proof. Suppose given tensors and cotensors for unbased spaces K and write them as X n K and Map (K, X)* as a reminder that they take values in a based category. We obtain tensors and cotensors X ^ K and F (K, X) for based spaces K as the pushouts and pullbacks displayed in the respective diagrams X n * _____//X n K and F (K, X)_____//Map(K, X)* | | | | | | | | fflffl| fflffl| fflffl| fflffl| * _______//X ^ K * ________//Map(*, X)*. Conversely, given tensors and cotensors X ^ K and F (K, X) for based spaces K, we obtain tensors and cotensors X n K and Map (K, X)* for unbased spaces K by setting X n K = X ^ K+ and Map (K, X)* = F (K+ , X), where K+ is the union of K and a disjoint basepoint. As usual, for any category C and object B in C , we let C =B denote the category of objects over B. An object X = (X, p) of C =B consists of a total ob* *ject X together with a projection map p: X -! B to the base object B. The morphisms of C =B are the maps of total objects that commute with the projections. Proposition 1.2.8. If C is a topologically bicomplete category, then so is C* * =B. Proof. The product of objects Yi over B, denoted xB Yi, is constructed by taking the pullback of the product of the projections Yi- ! B along the diagonal B -! xiB. Pullbacks and arbitrary colimits of objects over B are constructed by taking pullbacks and colimits on total objects and giving them the induced proj* *ec- tions. General limits are constructed as usual from products and pullbacks. If * *X is an object over B and K is a space, then the tensor X xB K is just X x K together 1.2. TOPOLOGICALLY BICOMPLETE CATEGORIES AND EX-OBJECTS 13 with the projection X x K -! B x * ~=B induced by the projection of X and the projection of K to a point. Note that this makes sense even though the tens* *or x need have nothing to do with cartesian products in general; see Remark 1.2.10 below. The cotensor Map B(K, X) is the pullback of the diagram B __'_//_Map(K, B)oo__Map (K, X) where ' is the adjoint of B x K -! B x * ~=B. The terminal object in C =B is (B, id). Let CB denote the category of based objects in C =B, that is, the category of objects under (B, id) in C =B. An obj* *ect X = (X, p, s) in CB , which we call an ex-object over B, consists of on object * *(X, p) over B together with a section s: B -! X. We can therefore think of the ex-obje* *cts as retract diagrams p B __s__//X____//_B. The terminal object in CB is (B, id, id), which we denote by *B ; it is also an* * initial object. The morphisms in CB are the maps of total objects X that commute with the projections and sections. Proposition 1.2.9. If C is a topologically bicomplete category, then the cat* *e- gory CB is based topologically bicomplete. Proof. The coproduct of objects Yi 2 CB , which we shall refer to as the "wedge over B" of the Yiand denote by _B Yi, is constructed by taking the pusho* *ut of the coproduct qB -! qYi of the sections along the codiagonal qiB -! B. Pushouts and arbitrary limits of objects in CB are constructed by taking pushou* *ts and limits on total objects and giving them the evident induced sections and pr* *o- jections. The tensor X ^B K of X = (X, p, s) and a based space K is the pushout of the diagram B oo___ (X x *) [B (B x K)____//X x K, where the right map is induced by the basepoint of K and the section of X. The cotensor FB (K, X) is the pullback of the diagram B _s__//_Xo"o_Map B(K, X), where " is evaluation at the basepoint of K, that is, the adjoint of the eviden* *t map X x K -! X over B. Remark 1.2.10. Notationally, it may be misleading to write X xK and X ^K for unbased and based tensors. It conjures up associations that are appropriate for the examples on hand but that are inappropriate in general. The tensors in a topologically bicomplete category C may bear very little relationship to cartes* *ian products or smash products. The standard uniform notation would be X K. However, we have too many relevant examples to want a uniform notation. In particular, we later use the notations X xB K and X ^B K in the parametrized context, where a notation such as X B K would conjure up its own misleading associations. 14 1. THE POINT-SET TOPOLOGY OF PARAMETRIZED SPACES 1.3.Convenient categories of ex-spaces We need a convenient topologically bicomplete category of ex-spaces1 over a space B, where "convenient" requires that we have smash product and function ex- space functors ^B and FB under which our category is closed symmetric monoidal. Denoting the unit B x S0 of ^B by S0B, a formal argument shows that we will then have isomorphisms (1.3.1) X ^B K ~=X ^B (S0B^B K) and FB (K, Y ) ~=FB (S0B^B K, Y ) relating tensors and cotensors to the smash product and function ex-space funct* *ors. In particular, S0B^B K is just the product ex-space B x K with section determin* *ed by the basepoint of K. The point-set topology leading to such a convenient category is delicate, and there are quite a few papers devoted to this subject. They do not give exactly what we need, but they come close enough that we shall content ourselves with a summary. It is based on the papers [5, 6, 7, 10, 11, 57] of Booth, Booth and Brown, and Lewis; see also James [51, 52]. We assume once and for all that our base spaces B are in U . We allow the total spaces X of spaces over B to be in K . We let K =B and U =B denote the categories of spaces over B with total spaces in K or U . Similarly, we let KB * *and UB denote the respective categories of ex-spaces over B. Remark 1.3.2. The section of an ex-space in UB is closed, by Lemma 1.1.3. Quite reasonably, references such as [29, 51] make the blanket assumption that sections of ex-spaces must be closed. We have not done so since we have not checked that all constructions in sight preserve this property. Both the separation property on B and the lack of a separation property on X are dictated by consideration of the function spaces Map B(X, Y ) over B that we shall define shortly. These are only known to exist when B is weak Hausdorff. However, even when B, X and Y are weak Hausdorff, Map B(X, Y ) is generally not weak Hausdorff unless the projection p: X -! B is an open map. Categori- cally, this means that the cartesian monoidal category U =B is not closed carte* *sian monoidal. Wishing to retain the separation property, Lewis [57] proposed the fo* *l- lowing as convenient categories of spaces and ex-spaces over a compactly genera* *ted space B. Definition 1.3.3. Let O(B) and O*(B) be the categories of those compactly generated spaces and ex-spaces over B whose projection maps are open. Remark 1.3.4. Bundle projections over B are open maps. Hurewicz fibrations over B are open maps if the diagonal B -! B x B is a Hurewicz cofibration [57, 2.3]; this holds, for example, if B is a CW complex. However, the categories O(B) and O*(B) are insufficient for our purposes. Working in these categories, we only have the base change adjunction (f*, f*) o* *f x2.1 below for open maps f :A -! B, which is unduly restrictive. For example, we need the adjunction ( *, *), where : B -! BxB is the diagonal map. Moreover, the generating cofibrations of our q-type model structures do not have open project* *ion ____________ 1Presumably the prefix "ex" stands for "cross", as in "cross section". The * *unlovely term "ex- space" has been replaced in some recent literature by "fiberwise pointed space"* *. Used repetitively, that is not much of an improvement. The term "retractive space" has also been u* *sed. 1.3. CONVENIENT CATEGORIES OF EX-SPACES 15 maps. This motivates us to drop the weak Hausdorff condition on total spaces and to focus on KB as our preferred convenient category of ex-spaces over B. The cofibrant ex-spaces in our q-type model structures are weak Hausdorff, hence th* *is separation property is recovered upon cofibrant approximation. Therefore, use of K can be viewed as scaffolding in the foundations that can be removed when doing homotopical work. We topologize the set of ex-maps X -! Y as a subspace of the space K (X, Y ) of maps of total spaces. It is based, with basepoint the unique map that factors through *B . Therefore the category KB is enriched over K*. It is based topo- logically bicomplete by Proposition 1.2.8. Recall that we write xB Yi and _B Yi for products and wedges over B. We also write Y=BX for quotients, which are understood to be pushouts of diagrams *B - X -! Y . We give a more concrete description of the tensors and cotensors in K =B and KB given by Proposition 1.* *2.8 and Proposition 1.2.9. For a space X over B, we let Xb denote the fiber p-1(b).* * If X is an ex-space, then Xb has the basepoint s(b). Definition 1.3.5. Let X be a space over B and K be a space. Define X xB K to be the space X x K with projection the product of the projections X -! B and K -! *. Define Map B(K, X) to be the subspace of Map (K, X) consisting of those maps f :K -! X that factor through some fiber Xb; the projection sends such a map f to b. Definition 1.3.6. Let X be an ex-space over B and K be a based space. Define X ^B K to be the quotient of X xB K obtained by taking fiberwise smash products, so that (X^B K)b = Xb^K; the basepoints of fibers prescribe the secti* *on. Define FB (K, X) to be the subspace of Map B(K, X) consisting of the based maps K -! Xb X for some b 2 B, so that FB (K, X)b = F (K, Xb); the section sends b to the constant map at s(b). Remark 1.3.7. As observed by Lewis [57, p. 85], if p is an open map, then so are the projections of X ^B K and FB (K, Y ). Therefore O*(B) is tensored and cotensored over T . The category K =B is closed cartesian monoidal under the fiberwise cartesian product X xB Y and the function space Map B(X, Y ) over B. The category KB is closed symmetric monoidal under the fiberwise smash product X ^B Y and the function ex-space FB (X, Y ). We recall the definitions. Definition 1.3.8. For spaces X and Y over B, X xB Y is the pullback of the projections p: X -! B and q :Y -! B, with the evident projection XxB Y -! B. When X and Y have sections s and t, their pushout X _B Y specifies the coproduc* *t, or wedge, of X and Y in KB , and s and t induce a map X _B Y -! X xB Y over B that sends x and y to (x, tp(x)) and (sq(y), y). Then X ^B Y is the pushout in K =B displayed in the diagram X _B Y _____//X xB Y | | | | fflffl| fflffl| *B _______//X ^B Y. This arranges that (X ^B Y )b = Xb^Yb, and the section and projection are evide* *nt. The following result is [11, 8.3]. 16 1. THE POINT-SET TOPOLOGY OF PARAMETRIZED SPACES Proposition 1.3.9. If X and Y are weak Hausdorff ex-spaces over B, then so is X ^B Y . That is, UB is closed under ^B . Function objects are considerably more subtle, and we need a preliminary def- inition in order to give the cleanest description. Definition 1.3.10. For a space Y 2 K , define the partial map classifier "Yto be the union of Y and a disjoint point !, with the topology whose closed subspa* *ces are "Yand the closed subspaces of Y . The point ! is not a closed subset, and "* *Yis not weak Hausdorff. The name "partial map classifier" comes from the observation that, for any space X, pairs (A, f) consisting of a closed subset A of X and a continuous map f :A -! Y are in bijective correspondence with continuous maps "f:X -! "Y. Given (A, f), "frestricts to f on A and sends X - A to !; given "f, (A, f) is "f-1(Y ) and the restriction of "f. Definition 1.3.11. Let p: X -! B and q :Y -! B be spaces over B. Define Map B(X, Y ) to be the pullback displayed in the diagram Map B(X, Y )____//Map(X, "Y) | Map(id,"q)| | | |fflffl fflffl| B ____~____//_Map(X, "B). Here ~ is the adjoint of the map X x B -! "Bthat corresponds to the composite of the inclusion Graph (p) X x B and the projection X x B - ! B to the second coordinate. The graph of p is the inverse image of the diagonal under p x id:X x B -! B x B, and the assumption that B is weak Hausdorff ensures that it is a closed subset of X x B, as is needed for the definition to make se* *nse. Explicitly, ~(b) sends Xb to b and sends X - Xb to the point ! 2 "B. This definition gives one reason that we require the base spaces of ex-space* *s to be weak Hausdorff. On fibers, Map B(X, Y )b = Map (Xb, Yb). The space of sectio* *ns of Map B(X, Y ) is K =B(X, Y ). We have (categorically equivalent) adjunctions (1.3.12) MapB (X xB Y, Z) ~=Map B(X, MapB (Y, Z)), (1.3.13) K =B (X xB Y, Z) ~=K =B (X, MapB (Y, Z)). These results are due to Booth [5, 6, 7]; see also [10, x7], [11, x8], [51, IIx* *9], [57]. Examples in [10, 5.3] and [57, 1.7] show that Map B(X, Y ) need not be weak Hausdorff even when X and Y are. The question of when Map B(X, Y ) is Hausdorff or weak Hausdorff was studied in [10, x5] and later in [51, 52], but the defini* *tive criterion was given by Lewis [57, 1.5]. Proposition 1.3.14. Consider a fixed map p: X - ! B and varying maps q :Y - ! B, where X and the Y are weak Hausdorff. The map p is open if and only if the space Map B(X, Y ) is weak Hausdorff for all q. Proposition 1.3.15. If p: X -! B and q :Y -! B are Hurewicz fibrations, then the projections XxB Y -! B and MapB (X, Y ) -! B are Hurewicz fibrations. The second statement is false with Hurewicz fibrations replaced by Serre fibrat* *ions. Proof. The statement about X xB Y is clear. The statements about Map B(X, Y ) are due to Booth [5, 6.1] or, in the present formulation [6, 3.4];* * see also [51, 23.17]. 1.4. CONVENIENT CATEGORIES OF EX-G-SPACES 17 Definition 1.3.16. For ex-spaces X and Y over B, define FB (X, Y ) to be the subspace of Map B(X, Y ) that consists of the points that restrict to based maps Xb -! Yb for each b 2 B; the section sends b to the constant map from Xb to the basepoint of Yb. Formally, FB (X, Y ) is the pullback displayed in the diagram FB (X, Y_)_____//MapB(X, Y ) | |Map (s,id) | | B fflffl| fflffl| B ____t___//Y ~=Map B(B, Y ), where s and t are the sections of X and Y . The space of maps S0B-! FB (X, Y ) is KB (X, Y ), and we have adjunctions (1.3.17) FB (X ^B Y, Z) ~=FB (X, FB (Y, Z)), (1.3.18) KB (X ^B Y, Z) ~=KB (X, FB (Y, Z)). Proposition 1.3.14 implies the following analogue of Proposition 1.3.9. Proposition 1.3.19. If X and Y are weak Hausdorff ex-spaces over B and X -! B is an open map, then FB (X, Y ) is weak Hausdorff. We record the following analogue of Proposition 1.3.15. The second part is again due to Booth, who sent us a detailed write-up. The argument is similar to his proofs in [5, 6.1(i)] or [6, 3.4], but a little more complicated, and a gen* *eral result of the same form is given by Morgan [78]. Proposition 1.3.20. If X and Y are ex-spaces over B whose sections are Hurewicz cofibrations and whose projections are Hurewicz fibrations, then the p* *ro- jections of X ^B Y and FB (X, Y ) are Hurewicz fibrations. 1.4. Convenient categories of ex-G-spaces The discussion just given generalizes readily to the equivariant context. L* *et G be a compactly generated topological group. Subgroups of G are understood to be closed. Let B be a compactly generated G-space (with G acting from the left). We consider G-spaces over B and ex-G-spaces (X, p, s). The total space X is a G-space in K , and the section and projection are G-maps. The fiber Xb is a bas* *ed Gb-space with Gb-fixed basepoint s(b), where Gb is the isotropy group of b. Recall from [61, IIx1] the distinction between the category KG of G-spaces a* *nd nonequivariant maps and the category GK of G-spaces and equivariant maps; the former is enriched over GK , the latter over K . We have a similar dichotomy on* * the ex-space level. Here we have a conflict of notation with our notation for categ* *ories of ex-spaces, and we agree to let KG,B denote the category whose objects are the ex-G-spaces over B and whose morphisms are the maps of underlying ex-spaces over B, that is, the maps f :X -! Y such that f O s = t and q O f = p. Henceforward, we call these maps "arrows" to distinguish them from G-maps, which we often abbreviate to maps. For g 2 G, gf is also an arrow of ex-spaces over B, so that KG,B(X, Y ) is a G-space. Moreover, composition is given by G-maps KG,B(Y, Z) x KG,B(X, Y ) -! KG,B(X, Z). We obtain the category GKB by restricting to G-maps f, and we may view it as the G-fixed point category of KG,B. Of course, GKB (X, Y ) is a space and not a 18 1. THE POINT-SET TOPOLOGY OF PARAMETRIZED SPACES G-space. The pair (KG,B, GKB ) is an example of a G-category, a structure that we shall recall formally in x10.2. Since *B is an initial and terminal object in both KG,B and GKB , their mor- phism spaces are based. Thus KG,B is enriched over the category GK* of based G-spaces and GKB is enriched over K*. As discussed in [61, II.1.3], if we were * *to think exclusively in enriched category terms, we would resolutely ignore the fa* *ct that the G-spaces KG,B(X, Y ) have elements (arrows), thinking of these G-spaces as enriched hom objects. From that point of view, GKB is the "underlying cate- gory" of our enriched G-category. While we prefer to think of KG,B as a categor* *y, it must be kept in mind that it is not a very well-behaved one. For example, becau* *se its arrows are not equivariant, it fails to have limits or colimits. In contrast, the category GKB is bicomplete. Its limits and colimits are con- structed in KB and then given induced G-actions. The category KG,B, although not bicomplete, is tensored and cotensored over KG,*. The tensors X ^B K and cotensors FB (K, X) are constructed in KB and then given induced G-actions. They satisfy the adjunctions (1.4.1) KG,B(X ^B K, Y ) ~=KG,*(K, KG,B(X, Y )) ~=KG,B(X, FB (K, Y )) and, by passage to fixed points, (1.4.2) GKB (X ^B K, Y ) ~=GK*(K, KG,B(X, Y )) ~=GKB (X, FB (K, Y )). It follows that GKB is tensored and cotensored over GK* and, in particular, is topologically bicomplete. The category KG,B is closed symmetric monoidal via the fiberwise smash prod- ucts X ^B Y and function objects FB (X, Y ). Again, these are defined in KB and then given induced G-actions. The unit is the ex-G-space S0B= B x S0. The cat- egory GKB inherits a structure of closed symmetric monoidal category. We have homeomorphisms of based G-spaces (1.4.3) KG,B(X ^B Y, Z) ~=KG,B(X, FB (Y, Z)) and, by passage to G-fixed points, homeomorphisms of based spaces (1.4.4) GKB (X ^B Y, Z) ~=GKB (X, FB (Y, Z)). The first of these implies an associated homeomorphism of ex-G-spaces (1.4.5) FB (X ^B Y, Z) ~=FB (X, FB (Y, Z)). Nonequivariantly, the functor that sends an ex-space X over B to the fiber Xb has a left adjoint, denoted (-)b. It sends a based space K to the wedge Kb = B _ K, where B is given the basepoint b; the section and projection are evident. Nonobviously, the same set B _ K admits a quite different topology under which * *it gives a right adjoint to the fiber functor X 7! Xb. We shall prove the equivari* *ant analogue conceptually in Example 2.3.12, but we describe the left adjoint to the fiber functor explicitly here. Construction 1.4.6. Let b 2 B. Then the functor GKB - ! GbK* that sends Y to Yb has a left adjoint. It sends a based Gb-space K to the ex-G-space Kb given by the pushout Kb = (G xGb K) [G B. 1.5. APPENDIX: NONASSOCIATIVITY OF SMASH PRODUCTS IN T op* 19 Here G xGb K is the (left) G-space (G x K)= ~, where (gh, k) ~ (g, hk) for g 2 * *G, h 2 Gb, and k 2 K. The pushout is defined with respect to the map G -! B that sends g to gb and the map G -! G xGb K that sends g to (g, k0), where k0 is the (Gb-fixed) basepoint of K. The section is given by the evident inclusion of B a* *nd the projection is obtained by passage to pushouts from the identity map of B and the G-map ssb:G xGb K -! B given by ssb(g, k) = gb. Thus we first extend the group action on K from Gb to G and then glue the orbit of the basepoint of K to the orbit of b. If K is an unbased Gb-space, then (K+ )b = (G xGb K) q B. Remark 1.4.7. There is an alternative parametrized view of equivariance that is important in torsor theory but that we shall not study. It focuses on "topol* *og- ical groups GB over B" and "GB -spaces E over B", where GB is a space over a nonequivariant space B with a product GB xB GB -! GB that restricts on fibers to the products of topological groups Gb and E is a space over B with an action GB xB E -! E that restricts on fibers to actions Gb x Eb -! Eb. That theory intersects ours in the special case GB = G x B for a topological group G. Since* *, at least implicitly, all of our homotopy theory is done fiberwise, our work adapts* * with- out essential difficulty to give a development of parametrized equivariant homo* *topy theory in that context. 1.5.Appendix: nonassociativity of smash products in T op* In a 1958 paper [82], Puppe asserted the following result, but he did not gi* *ve a proof. It was the subject of a series of e-mails among Mike Cole, Tony Elmendor* *f, Gaunce Lewis and the first author. Since we know of no published source that gi* *ves the details of this or any other counterexample to the associativity of the sma* *sh product in T op*, we include the following proof. It is due to Kathleen Lewis. Let Q and N be the rational numbers and the nonnegative integers, topologized as subspaces of R and given the basepoint zero. Consider smash products as quo- tient spaces, without applying the k-ification functor. Then we have the follow* *ing counterexample to associativity. Theorem 1.5.1. (Q ^ Q) ^ N is not homeomorphic to Q ^ (Q ^ N). Proof. Consider the following diagram. Q x Q xON idxp0ooooo || OOOOpxidOO oooo | OO wwooo | OOO'' Q x (Q ^ N) q| (Q ^ Q) x N | | | | s| | |r fflffl| t fflffl|~= fflffl| Q ^ (Q ^ N)oo___Q ^ Q ^ N_____//(Q ^ Q) ^ N Here Q ^ Q ^ N denotes the evident quotient space of Q x Q x N. The maps p, p0, q, r, and s are quotient maps. Since N is locally compact, p x idis also a quot* *ient map, hence so is r O (p x id). The universal property of quotient spaces then g* *ives the bottom right homeomorphism. Since Q is not locally compact, idx p0 need not be a quotient map, and in fact it is not. The map t is a continuous bijecti* *on given by the universal property of the quotient map q, and we claim that t is n* *ot a homeomorphism. To show this, we display an open subset of Q ^ Q ^ N whose image under t is not open. 20 1. THE POINT-SET TOPOLOGY OF PARAMETRIZED SPACES Let fi be an irrational number, 0 < fi < 1, and let fl = (1 - fi)=2. Define * *V 0(fi) to be the open subset of R x R that is the union of the following four sets. (1)The open ball of radius fi about the origin (2)The tubes [1, 1)x(-fl, fl), (-1, -1]x(-fl, fl), (-fl, fl)x[1, 1), and (-fl, * *fl)x (-1, -1] of width 2fl about the axes. (3)The open balls of radius fl about the four points ( 1, 0), (0, 1). (4)For each n 1, the open ballPof radius fl=2n about the four points ( fln, 0* *), (0, fln), where fln = 1 - k=n-1k=0fl=2k. To visualize this set, it is best to draw a picture. It is symmetric with re* *spect to 90 degree rotation. Consider the part lying along the positive x-axis. A tube of width 2fl covers the part of the x-axis to the right of (1, 0). A ball of ra* *dius fi centers at the origin. A ball of radius fl centers at (1, 0). Its vertical diag* *onal is the edge of the tube going off to the right. On the left, by the choice of fl, * *this ball reaches halfway from its center (1, 0) to the point (fi, 0) at the right edge o* *f the ball centered at the origin. The point (1 - fl, 0) at the left edge of the ball cent* *ered at (1, 0) is the center of another ball, which reaches half the distance from (1 -* * fl, 0) to (fi, 0). And so on: the point where the left edge of the nth ball crosses the x* *-axis is the center point of the (n + 1)st ball, which reaches half the distance from* * its center to the edge of the ball centered at the origin. Define V (fi) = V 0(fi)\(QxQ). Note that the only points of the coordinate a* *xes of R x R that are not in V 0(fi) are ( fi, 0) and (0, fi). Since fi is irratio* *nal, V (fi) contains the coordinate axes of QxQ. Because the radii of the balls in the sequ* *ence are decreasing, for each " > fi, there is no ffi > 0 such that ((-", ")x(-ffi, * *ffi))\(QxQ) is contained in V (fi). Now let ff be an irrational number, 0 < ff < 1. Let o be the basepoint of Q * *^ N and * be the basepoint of Q ^ Q ^ N. Let U be the union of {*} and the image under q of [n 1V (ff=n) x {n}. This is an open subspace of Q ^ Q ^ N since q-1(U) = Q x Q x {0} [ ([n 1V (ff=n) x {n}) is an open subset of Q x Q x N. We claim that t(U) is not open in Q ^ (Q ^ N). Assume that t(U) is open. Then s-1(t(U)) = (idx p0)(q-1(U)) is an open subset of Q x (Q ^ N), hence it contains an open neighborhood V of (0, o). Now V must contain ((-", ") \ Q) x W for some " > 0 and some open neighborhood W of o in Q ^ N. Since Q ^ N is homeomorphic to the wedge over n 1 of the spaces Q x {n}, W must contain the wedge over n 1 of subsets ((-ffin, ffin) \ Q) x {n}, where ffin > 0. By the definition of U, this implies* * that ((-", ") x (-ffin, ffin)) \ (Q x Q) V (ff=n). However, for n large enough that " > ff=n, there is no ffin for which this hold* *s. CHAPTER 2 Change functors and compatibility relations Introduction In the previous chapter, we developed the internal properties of the category GKB of ex-G-spaces over B. As B and G vary, these categories are related by various functors, such as base change functors, change of groups functors, orbi* *t and fixed point functors, external smash product and function space functors, and so forth. We define these "change functors" and discuss various compatibility rela* *tions among them in this chapter. We particularly emphasize base change functors. We give a general categori- cal discussion of such functors in x2.1, illustrating the general constructions* * with topological examples. In x2.2, we discuss various compatibility relations that * *relate these functors to smash products and function objects. In x2.3 and x2.4 we turn to equivariant phenomena and study restriction of group actions along homomorphisms. As usual, we break this into the study of restriction along inclusions and pullback along quotient homomorphisms. In x2.3, we discuss restrictions of group actions to subgroups, together with the associated induction and coinduction functors. We also consider their compa* *t- ibilities with base change functors. In particular, this gives us a convenient* * way of thinking about passage to fibers and allows us to reinterpret restriction to* * sub- groups in terms of base change and coinduction. That is the starting point of o* *ur generalization of the Wirthm"uller isomorphism in Part IV. In x2.4, we consider pullbacks of group actions from a quotient group G=N to G, together with the associated quotient and fixed point functors. Again, we al* *so consider compatibilities with base change functors. For an N-free base space E, we find a relation between the quotient functor (-)=N and the fixed point funct* *or (-)N that involves base change along the quotient map E -! E=N. The good properties of the bundle construction in Part IV can be traced back to this rel* *ation, and it is at the heart of the Adams isomorphism in equivariant stable homotopy theory. In x2.5, we describe a different categorical framework, one appropriate to e* *x- spaces with varying base spaces. We show that the relevant category of retracts* * over varying base spaces is closed symmetric monoidal under external smash product and function ex-space functors. The internal smash product and function ex-space functors are obtained from these by use of base change along diagonal maps. The external smash products are much better behaved homotopically than the internal ones, and homotopical analysis of base change functors will therefore play a ce* *ntral role in the homotopical analysis of smash products. 21 22 2. CHANGE FUNCTORS AND COMPATIBILITY RELATIONS In much of this chapter, we work in a general categorical framework. In some places where we restrict to spaces, more general categorical formulations are u* *n- doubtedly possible. When we talk about group actions, all groups are assumed to be compactly generated spaces but are otherwise unrestricted. 2.1. The base change functors f!, f*, and f* Let f :A -! B be a map in a bicomplete subcategory B of a bicomplete category C . We are thinking of U K or GU GK . We wish to define functors f!:CA -! CB , f* :CB -! CA , f*: CA -! CB , such that f!is left adjoint and f* is right adjoint to f*. The definitions of f* ** and f!are dual and require no further hypotheses. The definition of f* does not work in full generality, but it only requires the further hypothesis that C =B be ca* *rtesian closed. Thus we assume given internal hom objects Map B(Y, Z) in C =B that satisfy the usual adjunction, as in (1.3.13). One reason to work in this genera* *lity is to emphasize that no further point-set topology is needed to construct these base change functors in the context of ex-spaces. This point is not clear from * *the literature, where the functor f* is often given an apparently different, but na* *turally isomorphic, description. We work with generic ex-objects p t q A __s__//X_____//A and B ____//_Y____//B in this section. Definition 2.1.1. Define f!X and its structure maps q and t by means of the map of retracts in the following diagram on the left, where the top square is a pushout and the bottom square is defined by the universal property of pushouts and the requirement that q O t = id. Define f*Y and its structure maps p and s * *by means of the map of retracts in the following middle diagram, where the bottom square is a pullback and the top square is defined by the universal property of pullbacks and the requirement that p O s = id. f f ' A ______//B A _____//_B B ______//MapB(A, A) s || |t| s|| t|| t || |Map(id,s)| fflffl| fflffl| fflffl| fflffl| fflffl| fflffl| X _____//f!X f*Y ____//_Y f*X _____//MapB(A, X) p || |q| p|| q|| q | |Map(id,p) fflffl| fflffl| fflffl| fflffl| fflffl|| fflffl|| A __f___//B A __f__//_B B ______//Map(A, A) ' B Thinking of X and A as objects over B via f O p and f and observing that the adjoint of the identity map of A gives a map ': B -! Map B(A, A), define f*X and its structure maps q and t by means of the map of retracts in the above diagram on the right, where the bottom square is a pullback and the top square is defin* *ed by the universal property of pullbacks and the requirement that q O t = id. Proposition 2.1.2. (f!, f*) is an adjoint pair of functors: CB (f!X, Y ) ~=CA (X, f*Y ). 2.2. COMPATIBILITY RELATIONS 23 Proof. Maps in both hom sets are specified by maps k :X -! Y in C such that q O k = f O p and k O s = t O f. Proposition 2.1.3. (f*, f*) is an adjoint pair of functors: CA (f*Y, X) ~=CB (Y, f*X). Proof. A map k :f*Y = Y xB A -! X such that p O k = p and k O s = s has adjoint "k:Y -! Map B(A, X) such that Map (id, p) O "k= ' O q and "kO t = Map (id, s) O '. The conclusion follows directly. Remark 2.1.4. Writing these proofs diagrammatically, we see that the ad- junction isomorphisms are given by homeomorphisms in our context of topological categories. We specialize to ex-spaces (or ex-G-spaces), in the rest of the section. Obs* *erve that the fiber (f*X)b is the space of sections Ab -! Xb of p: Xb -! Ab. Remark 2.1.5. If f :A -! B is an open map and X is in U , then f*X is in U and UA (f*Y, X) ~=UB (Y, f*X) for Y 2 U , by [57, 1.5]. Example 2.1.6. Let f :A -! B be an inclusion. Then f*Y is the restriction of Y to A and f!X = B [A X. The ex-space f*X over B is analogous to the prolongation by zero of a sheaf over A. The fiber (f*X)b is Xa if a 2 A and a p* *oint {b} otherwise. To see this from the definition, recall that Map (;, K) is a poi* *nt for any space K and that Map B(A, X)b = Map (Ab, Xb). As a set, f*X ~=B [A X, but the topology is quite different. It is devised so that the map Y -! f*f*Y t* *hat restricts to the identity on Ya for a 2 A but sends Yb to {b} for b =2A is cont* *inuous. Example 2.1.7. Let r :B -! * be the unique map. For a based space X and an ex-space E = (E, p, s) over B, we have r*X = B x X, r!E = E=s(B), and r*E = Sec(B, E), where Sec(B, E) is the space of maps t: B - ! E such that p O t = id, with basepoint the section s. These elementary base change functors are the key to u* *sing parametrized homotopy theory to obtain information in ordinary homotopy theory. Let ": r!r* -! id and j :id-! r*r! be the counit and unit of the adjunction (r!, r*). Then r!r*X ~=B+ ^X and " is r+ ^id. Similarly, r!r*r!E ~=B+ ^E=B, and r!j :r!E -! r!r*r!E is the "Thom diagonal" E=B -! B+ ^ E=B. If p: E -! B is a spherical fibration with section, such as the fiberwise one-point compactific* *ation of a vector bundle, then r!E is the Thom complex of p. 2.2. Compatibility relations The term "compatibility relation" has been used in algebraic geometry in the context of Grothendieck's six functor formalism that relates base change functo* *rs to tensor product and internal hom functors in sheaf theory. We describe how the analogous, but simpler, formalism appears in our categories of ex-objects. We recall some language. We are especially interested in the behavior of base change functors with respect to closed symmetric monoidal structures that, in o* *ur topological context, are given by smash products and function objects. Relevant categorical observations are given in [40]. We say that a functor T :B -! A between closed symmetric monoidal categories is closed symmetric monoidal if T SB ~=SA , T (X ^B Y ) ~=T X ^A T Y, and T FB (X, Y ) ~=FA (T X, T Y ), 24 2. CHANGE FUNCTORS AND COMPATIBILITY RELATIONS where SB , ^B and FB denote the unit object, product, and internal hom of B, and similarly for A . These isomorphisms must satisfy appropriate coherence conditi* *ons. In the language of [40], the following result states that any map f of base spa* *ces gives rise to a "Wirthm"uller context", which means that the functor f* is clos* *ed symmetric monoidal and has both a left adjoint and a right adjoint. Proposition 2.2.1. If f :A -! B is a map of base G-spaces, then the functor f* :GKB - ! GKA is closed symmetric monoidal. Therefore, by definition and implication, f*S0B~=S0Aand there are natural isomorphisms (2.2.2) f*(Y ^B Z) ~=f*Y ^A f*Z, (2.2.3) FB (Y, f*X) ~=f*FA (f*Y, X), (2.2.4) f*FB (Y, Z) ~=FA (f*Y, f*Z), (2.2.5) f!(f*Y ^A X) ~=Y ^B f!X, (2.2.6) FB (f!X, Y ) ~=f*FA (X, f*Y ), where X is an ex-G-space over A and Y and Z are ex-G-spaces over B. Proof. The isomorphism f*S0B~=S0Ais evident since f*(B x K) ~=A x K for based G-spaces K. The isomorphism (2.2.2) is obtained by passage to quotients from the evident homeomorphism (Y xB A) xA (Z xB A) ~=(Y xB Z) xB A As explained in [40, xx2, 3], the isomorphism (2.2.2) is equivalent to the isom* *or- phism (2.2.3), and it determines natural maps from left to right in (2.2.4), (2* *.2.5), and (2.2.6) such that all three are isomorphisms if any one is. By a comparison* * of definitions, we see that the categorically defined map in (2.2.4), which is den* *oted ff in [40, 3.3], coincides in the present situation with the map, also denoted * *ff, on [11, p. 167]. As explained on [11, p. 178], in the point-set topological frame* *work that we have adopted, that map ff is a homeomorphism. Remark 2.2.7. Only the very last statement refers to topology. The categori- cally defined map ff should quite generally be an isomorphism in analogous cont* *exts, but we have not pursued this question in detail. An alternative self-contained * *proof of the previous proposition is given in Remark 2.5.6 below by using Proposition* * 2.2.9 to prove (2.2.5) instead of (2.2.4). In that argument, the only non-formal ingr* *edient is the fact that the functor D xB (-) commutes with pushouts. We shall later need a purely categorical coherence observation about the cat- egorically defined map ff of (2.2.4). In fact, it will play a key role in the p* *roof of the fiberwise duality theorem of x15.1. It is convenient to insert it here. Remark 2.2.8. Let T :B -! A be a symmetric monoidal functor. We are thinking of T as, for example, a base change functor f*. The map ff: T FB (X, Y ) -! FA (T X, T Y ) is defined to be the adjoint of T FB (X, Y ) ^A T X ~=T (FB (X, Y ) ^BTX)ev//_T Y. 2.3. CHANGE OF GROUP AND RESTRICTION TO FIBERS 25 The dual of X is DB X = FB (X, SB ), where SB is the unit of B. Taking Y = SB , the definition of ff implies that the top triangle commutes in the diagram ~= Tev T DB X ^A T X ________//_T (DB X ^B X)___//_T2SB2ffffff ffffff ff^Aid|| fffffevffffff ~=|| fflffl|ffffff fflffl| FA (T X, T SB ) ^A T_X~=//_DA f*X ^A f*X_ev__//_SA . The bottom triangle is a naturality diagram. The outer rectangle is [40, 3.7], * *but its commutativity in general was not observed there. However, it was observed in [40, 3.8] that its commutativity implies the commutativity of the diagram ~= T T DB X ^A T Y ____//_T (DB X ^B Y_)___//T FB (X, Y ) ff^ATY || |ff| fflffl| fflffl| DA T X ^A T Y _______________________//_FA (T X, T Y ), where :DB X ^B Y -! FB (X, Y ) is the adjoint of DB X ^B Y ^B X ~=DB X ^B X ^B Y ev^id//_SB ^B Y ~=Y. In other contexts, the analogue of (2.2.5) is called the "projection formula* *", and we shall also use that term. The following base change commutation relations with respect to pullbacks are also familiar from other contexts. We state the r* *esult for spaces but, apart from use of the fact that the functor D xB (-) commutes w* *ith pushouts, the proof is formal. Proposition 2.2.9. Suppose given a pullback diagram of base spaces g C _____//D i|| |j| fflffl|fflffl| A __f__//B. Then there are natural isomorphisms of functors (2.2.10) j*f!~=g!i*, f*j* ~=i*g*, f*j!~=i!g*, j*f* ~=g*i*. Proof. The first isomorphism is one of left adjoints, and the second is the corresponding "conjugate" isomorphism of right adjoints. Similarly for the thi* *rd and fourth isomorphisms. By symmetry, it suffices to prove the first isomorphis* *m. The functor j* = D xB (-) commutes with pushouts. For a space X over A regarded by composition with f as a space over B, C xA X ~=D xB X. This gives j*f!X = D xB (B [A X) ~=D [C (C xA X) = g!i*X. 2.3. Change of group and restriction to fibers This section begins the study of equivariant phenomena that have no non- equivariant counterparts. In particular, using a conceptual reinterpretation of* * the adjoints of the fiber functors (-)b, we relate restriction to subgroups to rest* *riction to fibers. Recall that subgroups of G are understood to be closed and fix an inclu* *sion ': H G throughout this section. Parametrized theory gives a convenient way of studying restriction along ' without changing the ambient group from G to H. 26 2. CHANGE FUNCTORS AND COMPATIBILITY RELATIONS Proposition 2.3.1. The category GKG=H of ex-G-spaces over G=H is equiv- alent to the category HK* of based H-spaces. Proof. The equivalence sends an ex-G-space (Y, p, s) over G=H to the H- space p-1(eH) with basepoint the H-fixed point s(eH). Its inverse sends a based H-space X to the induced G-space G xH X, with the evident structure maps. More formally, recall that there are "induction" and "coinduction" functors * *'! and '* from H-spaces to G-spaces that are left and right adjoint to the forgetf* *ul functor '* that sends a G-space Y to Y regarded as an H-space. Explicitly, for * *an H-space X, (2.3.2) '!X = G xH X and '*X = Map H(G, X). The latter is the space of maps of (left) H-spaces, with (left) action of G ind* *uced by the right action of G on itself. Similarly, when X is a based H-space, we ha* *ve the based analogues (2.3.3) '!X = G+ ^H X and '*X = FH (G, X). With this notation, some familiar natural isomorphisms take the forms (2.3.4) '!('*Y x X) ~=Y x '!X and '*Map ('*Y, X) ~=Map (Y, '*X) and, in the based case, (2.3.5) '!('*Y ^ X) ~=Y ^ '!X and '*F ('*Y, X) ~=F (Y, '*X). By the uniqueness of adjoints, or inspection of definitions, we see that these * *familiar change of groups functors are change of base functors along r :G=H -! *. Corollary 2.3.6. The change of group and change of base functors associated to ' and r agree under the equivalence of categories between HK* and GKG=H : '* ~=r*, '!~=r!, and '* ~=r*. We can generalize this equivalence of categories, using the following defini* *tions. We have a forgetful functor '*: GKB -! HK'*B. It doesn't have an obvious left or right adjoint, but we have obvious analogues of induction and coinduction th* *at involve changes of base spaces. The first will lead to a description of '* as a* * base change functor and thus as a functor with a left and right adjoint. Definition 2.3.7. Let A be an H-space and X be an H-space over A. Define '!:HKA -! GK'!Aby letting '!X be the G-space G xH X over '!A = G xH A. Define '*: HKA -! GK'*A by letting '*X be the G-space Map H(G, X) over '*A = Map H(G, A). For an H-space A and a G-space B, let (2.3.8) ~: G xH '*B = '!'*B -! B and :A -! '*'!A = '*(G xH A) be the counit and unit of the ('!, '*) adjunction. The following result says t* *hat ex-H-spaces over an H-space A are equivalent to ex-G-spaces over the G-space '!* *A. Proposition 2.3.9. The functor '!:HKA -! GK'!Ais a closed symmetric monoidal equivalence of categories with inverse the composite * * GK'!A-'! HK'*'!A-! HKA . 2.3. CHANGE OF GROUP AND RESTRICTION TO FIBERS 27 Applied to A = '*B, this equivalence leads to the promised description of '*: GKB -! HK'*B as a base change functor. Proposition 2.3.10. The functor '*: GKB -! HK'*B is the composite * GKB _~__//_GK'!'*B~=HK'*B Change of base and change of groups are related by various further consisten* *cy relations. The following result gives two of them. Proposition 2.3.11. Let f :A -! '*B be a map of H-spaces and "f:'!A -! B be its adjoint map of G-spaces. Then the following diagrams commute up to natur* *al isomorphism. "f! f"* GK'!AO_____//_GKBOOO GKB ______//GK'!A '!|| |~!O'!| '*|| ||*O'* | | fflffl| fflffl| HKA __f!_//_HK'*B HK'*B __f*_//_HKA Proof. Since "f= ~ O '!f, we have f"!O '!~=(~ O '!f)!O '!~=~!O ('!f)!O '!~=~!O '!O f!, where the last isomorphism holds because GxH (-) commutes with pushouts. Since f = '*f"O , we have f* O '* ~=('*f"O )* O '* ~= * O ('*f")* O '* ~= * O '* O "f*, where the last isomorphism holds because pulling the G action back to an H-acti* *on commutes with pullbacks. The reader may find it illuminating to work out these isomorphisms in the context of Proposition 2.3.1. That result leads to the promised conceptual rein* *ter- pretation of Construction 1.4.6. Example 2.3.12. For b 2 B, we also write b: * -! B for the map that sends * to b, and we write "b:G=Gb -! B for the induced inclusion of orbits. Thus b is a Gb-map and "bis a G-map. Under the equivalence GKG=Gb ~= GbK* of Proposition 2.3.1, "b*may be interpreted as the fiber functor GKB -! GbK* that sends X to Xb, "b!may be interpreted as the left adjoint of Construction 1.4.6 * *that sends K to Kb, and "b*specifies a right adjoint to the fiber functor, which we * *denote by bK. With these notations, the isomorphisms of Proposition 2.2.1 specialize to the following natural isomorphisms, where Y and Z are in GKB and K is in GbK*. (Y ^B Z)b ~=Yb^ Zb, FB (Y, bK) ~=bF (Yb, K), FB (Y, Z)b ~=F (Yb, Zb), (Yb^ K)b ~=Y ^B Kb, FB (Kb, Y ) ~=bF (K, Yb). 28 2. CHANGE FUNCTORS AND COMPATIBILITY RELATIONS Example 2.3.13. Several earlier results come together in the following situa- tion. Let f :A -! B be a G-map. For b 2 B, let b: {b} -! B and ib:Ab -! A denote the evident inclusions of Gb-spaces. We have the following compatible pu* *ll- back squares, the first of Gb-spaces and the second of G-spaces. fb GxGbfb Ab_____//{b} G xGb Ab _____//G=Gb ib|| |b| "-b|| "b|| fflffl| fflffl| |fflffl fflffl| A __f___//B A ____f____//_B Applying Proposition 2.2.9 to the right-hand square and interpreting the conclu* *sion in terms of fibers by Definition 2.3.7, we obtain canonical isomorphisms of Gb-* *spaces (f!X)b ~=fb!i*bX and (f*X)b ~=fb*i*bX, where X is an ex-G-space over A, regarded on the right-hand sides as an ex-Gb- space over A by pullback along ': Gb -! G. 2.4. Normal subgroups and quotient groups Observe that any homomorphism ` :G -! G0 factors as the composite of a quotient homomorphism ", an isomorphism, and an inclusion '. We studied change of groups along inclusions in the previous section. Here we consider a quotient homomorphism ffl: G -! J of G by a normal subgroup N. We still have a restricti* *on functor ffl*: JKA -! GKffl*A, and we also have the functors (-)=N :GKB -! JKB=N and (-)N :GKB -! JKBN obtained by passing to orbits over N and to N-fixed points. When B is a point, these last two functors are left and right adjoint to ffl*, but in general chan* *ge of base must enter in order to obtain such adjunctions. The following observation follo* *ws directly by inspection of the definitions. Proposition 2.4.1. Let j :BN - ! B be the inclusion and p: B -! B=N be the quotient map. Then the following factorization diagrams commute. (-)=N (-)N GKB _____//_JKB=N99 and GKB _____//_JKBN ss ttt99 p!|| ssss j*|| tttt N fflffl|(-)=Nsss fflffl|(-)tt GKB=N GKBN It follows that ((-)=N, p*ffl*) and (j!ffl*, (-)N ) are adjoint pairs. We have the following analogue of Proposition 2.3.11. 2.4. NORMAL SUBGROUPS AND QUOTIENT GROUPS 29 Proposition 2.4.2. Let f :A -! B be a map of G-spaces. Then the following diagrams commute up to natural isomorphisms. f! f* f! GKA _______//GKB GKB ______//GKA GKA _____//_GKB (-)=N || (-)=N|| (-)N|| |(-)N| (-)N|| |(-)N| fflffl| fflffl| fflffl| fflffl| fflffl| fflffl| JKA=N (f=N)//_JKB=N JKBN _____//JKAN JKAN _____//JKBN ! (fN )* (fN )! Proof. For ex-G-spaces X over A and Y over B, these isomorphisms are given by the homeomorphisms (X [A B)=N ~=X=N [A=N B=N, (Y xB A)N ~=Y N xBN AN , and (X [A B)N ~=XN [AN BN . As a quibble, the last requires A -! X to be a closed inclusion, but this will * *hold for the sections of compactly generated ex-G-spaces over A by Lemma 1.1.3(i). Specializing to N-free G-spaces, we obtain a factorization result that is an* *al- ogous to those in Proposition 2.4.1, but is less obvious. It is a precursor of* * the Adams isomorphism, which we will derive in x16.4. Proposition 2.4.3. Let E be an N-free G-space, let B = E=N, and let p: E -! B be the quotient map. Then the diagram (-)=N GKE _____//JKB;; vv p*|| vvvv fflffl|(-)Nvvv GKB commutes up to natural isomorphism. Therefore the left adjoint (-)=N of the functor p*"* is also its right adjoint. Proof. Let X be an ex-G-space over E with projection q. Comparing the pullbacks that are used to define the functors p* and Map Bin Definitions 2.1.1* * and 1.3.11, we find that p*X fits into a pullback diagram p*X _____//Map(E, "X) | | |"q | | fflffl| fflffl| B ______//Map(E, "E). Here (b), b = Ne, corresponds as in Definition 1.3.10 to the inclusion of the * *closed subset Ne in E. Passing to N-fixed points, we see that it suffices to prove tha* *t the following commutative diagram is a pullback. X=N ___~___//MapN(E, "X) q=N|| "q|| fflffl| fflffl| E=N = B _____//MapN(E, "E) 30 2. CHANGE FUNCTORS AND COMPATIBILITY RELATIONS Here ~ is induced from the adjoint of the map X x E -! "Xthat sends (x, e) to nx if e = nq(x) and sends (x, e) to ! otherwise. With this description, ~ is well-* *defined since E is N-free. It suffices to give a continuous inverse to the induced map OE: X=N -! Map N(E, "X) xMapN(E,E")E=N. If (f, Ne) is a point in the pullback, then f corresponds to a map Ne -! X, and OE-1(f, Ne) = Nf(e) in X=N. For continuity, note that OE-1 is obtained from the evaluation map Map (E, "X) x E -! "Xby passage to subquotient spaces. Remark 2.4.4. This leads to a useful alternative description of the functor '!:HKA - ! GK'!A, where A is an H-space and '!A = G xH A. We have the projection ss :G x A -! A of (G x H)-spaces, where the G x H actions on the source and target are given by (g, h)(g0, a) = (gg0h-1, ha) and (g, h)a = ha. Consider ex-H-spaces X over A as (G x H)-spaces with G acting trivially and let ffl: G x H -! H be the projection. We see from the definition that '!X = (ss*"*X)=H. Since GxA is an H-free (GxH)-space, we conclude from the previous result that '!X ~=(p*ss*"*X)H , where p: G x A -! G xH A = '!A is the quotient map. 2.5. The closed symmetric monoidal category of retracts Let B be a topologically bicomplete full subcategory of a topologically bico* *m- plete category C . We are thinking of U K or GU GK . We have the category of retracts CB . The objects of CB are the retractions B -s! X -p! B with B 2 B and X 2 C , abbreviated (X, p, s) or just X. The morphisms of CB are the evident commutative diagrams. When B = C , this is just a diagram category for the evident two object domain category. The importance of the category CB is apparent from its role in Definition 2.* *1.1: focus on this category is natural when we consider base change functors. In our examples, B and C are enriched and topologically bicomplete over the appropriate category of spaces, U for B and K for C . For a space K 2 K , the tensors - x K and cotensors Map (K, -) applied to retractions give retractions, and we have t* *he adjunction homeomorphisms (2.5.1) CB (X x K, Y ) ~=K (K, CB (X, Y )) ~=CB (X, Map(K, Y )). The category GKGU is closed symmetric monoidal under an external smash product functor, denoted X ZY , and an external function ex-space functor, deno* *ted ~F(Y, Z). If X, Y , and Z are ex-spaces over A, B, and A x B, respectively, then X Z Y is an ex-space over A x B and ~F(Y, Z) is an ex-space over A. We have (2.5.2) GKAxB (X Z Y, Z) ~=GKA (X, ~F(Y, Z)), which gives the required adjunction in GKGU . It specializes to parts of (1.4.2* *) when A or B is a point. The ex-space X Z Y is the evident fiberwise smash product, w* *ith (X Z Y )(a,b)= Xa ^ Yb. The fiber ~F(Y, Z)a is FB (Y, Za), where Za is the ex-s* *pace over B whose fiber Za,bover b is the inverse image of (a, b) under the projecti* *on Z -! A x B. Rather than describe the topology of the ex-space ~F(Y, Z) directly, we give alternative descriptions of X Z Y and ~F(Y, Z) in terms of internal sma* *sh products and internal function ex-spaces. Let ssA and ssB be the projections of A x B on A and B and observe that ss*AX ~=X x B and ss*BY ~=A x Y . If one like* *s, 2.5. THE CLOSED SYMMETRIC MONOIDAL CATEGORY OF RETRACTS 31 the following results can be taken as a definition of the external operations a* *nd a characterization of the internal operations, or vice versa. Lemma 2.5.3. The external smash product and function ex-space functors are determined by the internal functors via natural isomorphisms X Z Y ~=ss*AX ^AxB ss*BY and ~F(Y, Z) ~=ssA*FAxB (ss*BY, Z), where X, Y , and Z are ex-spaces over A, B, and A x B, respectively. With these isomorphisms taken as definitions, the adjunction (2.5.2) follows from the adjunctions (ss*A, ssA*), (ss*B, ssB*), and (^AxB , FAxB ). Lemma 2.5.4. The internal smash product and function ex-space functors are determined by the external functors via natural isomorphisms X ^B Y ~= *(X Z Y ) and FB (X, Y ) ~=~F(X, *Y ), where X and Y are ex-spaces over B and : B -! B x B is the diagonal map. With these isomorphisms taken as definitions, the adjunction (^B , FB ) foll* *ows from the adjunctions ( *, *) and (2.5.2). Since * is symmetric monoidal and the composite of either projection ssi:B x B -! B with is the identity map of B, we see that, if we have constructed both internal and external smash product* *s, then they must be related by natural isomorphisms as in Lemmas 2.5.3 and 2.5.4. Remark 2.5.5. The first referee suggests that we point out another consisten* *cy check. The fiber ( *Y )(b,c)is a point if b 6= c and is Yb if b = c. Therefor* *e the fiber over b of the restriction ( *Y )b of *Y to {b} x B is Yb[ (B - {b}), sui* *tably topologized, and ~F(X, *Y )b = FB (X, ( *Y )b)b ~=F (Xb, Yb) = FB (X, Y )b. Remark 2.5.6. The description of the internal smash product in terms of the external smash product sheds light on the basic compatibility isomorphisms (2.2* *.2) and (2.2.5). For maps f :A -! B and g :A0- ! B0 and for ex-spaces X over B and Y over B0, it is easily checked that (2.5.7) f*Y Z g*Z ~=(f x g)*(Y Z Z). Similarly, for ex-spaces W over A and X over A0, (2.5.8) f!W Z g!X ~=(f x g)!(W Z X). Now take A = A0, B = B0 and f = g. For ex-spaces Y and Z over B, f*(Y ^B Z) ~=f* *B(Y Z Z) ~=( B O f)*(Y Z Z). On the other hand, using (2.5.7), f*Y ^A f*Z ~= *A(f x f)*(Y Z Z) ~=((f x f) O A )*(Y Z Z). The right sides are the same since B O f = (f x f) O A . Similarly, f!(f*Y ^A X) ~=f! *A(f x id)*(Y Z X) ~=f!((f x id) O A )*(Y Z X), while Y ^B f!X ~= *B(idx f)!(Y Z X). 32 2. CHANGE FUNCTORS AND COMPATIBILITY RELATIONS Since the diagram A fxid A _____//A x A____//_B x A f|| |idxf| fflffl| fflffl| B _________B______//B x B is a pullback, the right sides are isomorphic by Proposition 2.2.9. It is illuminating conceptually to go further and consider group actions from an external point of view. For groups H and G, an H-space A, and a G-space B, we have an evident external smash product (2.5.9) Z: HKA x GKB ! (H x G)KAxB . For an ex-H-space X over A and an ex-G-space Y over B, X ZY is just the internal smash product over the (H x G)-space A x B of ss*Hss*AX and ss*Gss*BY , where t* *he ss0s are the projections from H x G and A x B to their coordinates. It is easil* *y seen that this definition leads to another (Z, ~F) adjunction. When H = G, the diagonal : G -! G x G is a closed inclusion since G is compactly generated. We can pull back along , and then our earlier external smash product X Z Y over the G-space *(A x B) is given in terms of (2.5.9) as the pullback *(X Z Y ). Note that, by Proposition 2.3.10, * here can be viewed as a base change functor. CHAPTER 3 Proper actions, equivariant bundles, and fibrations Introduction Much of the work in equivariant homotopy theory has focused on compact Lie groups. However, as was already observed by Palais [81], many results can be generalized to arbitrary Lie groups provided that one restricts to proper actio* *ns. These are well-behaved actions whose isotropy groups are compact, and all actio* *ns by compact Lie groups are proper. The classical definition of a Lie group [17, * *p. 129] includes all discrete groups (even though they need not be second countabl* *e) and, for discrete groups, the proper actions are the properly discontinuous one* *s. In the parametrized world, the homotopy theory is captured on fibers. When we restrict to proper actions on base spaces, the fibers have actions by the co* *mpact isotropy groups of the base space. So even though our primary interest is still* * in compact Lie groups of equivariance, proper actions on the base space provide the right natural level of generality. We set the stage for such a theory in this c* *hapter by generalizing various classical results about equivariant bundles and fibrati* *ons to a setting focused on proper actions by Lie groups. The reader interested primar* *ily in the nonequivariant theory should skip this chapter since only some very stan* *dard material in it is relevant nonequivariantly. In x3.1, we recall some basic results about proper actions of locally compact groups. We use this discussion to generalize some results about equivariant bun* *dles in x3.2. We generalize Waner's equivariant versions of Milnor's results on spac* *es of the homotopy types of CW complexes in x3.3. In x3.4, we recall and generalize classical theorems of Dold and Stasheff about Hurewicz fibrations. We also reca* *ll an important but little known result of Steinberger and West that relates Serre* * and Hurewicz fibrations. We recall the definition of equivariant quasifibrations in* * x3.5. 3.1. Proper actions of locally compact groups We recall relevant definitions and basic results about proper actions in this section. For appropriate generality and technical convenience, we let G be a lo* *cally compact topological group whose underlying topological space is compactly gen- erated. Local compactness means that the identity element, hence any point, has a compact neighborhood. We see from Proposition 1.1.2 that G is Hausdorff and, since all compact subsets are closed, it follows that each neighborhood of any * *point contains a compact neighborhood. Remark 3.1.1. We comment on the assumptions we make for G. If G is any topological group whose underlying space is in K , then an action of G on X in K may not come from an action in T op. The point is that the product G x X in K is defined by applying the k-ification functor to the product G xc X in T op, and not every action G x X -! X need be continuous when viewed as a function 33 34 3. PROPER ACTIONS, EQUIVARIANT BUNDLES, AND FIBRATIONS G xcX -! X. However, when G is locally compact, G xcX is already in K by Proposition 1.1.2, and k-ification is not needed. There is then no ambiguity ab* *out what we mean by a G-space, and we need not worry about refining the topology on products with G. Another reason for restricting to locally compact groups is that many useful properties of proper actions only hold in that case. In the literature, such re* *sults are usually derived for actions on Hausdorff spaces, but we shall see that weak Hausdorff generally suffices. We begin with some standard equivariant terminology. Definition 3.1.2. Let X be a G-space and let H G. (i)An H-tube U in X is an open G-invariant subset of X together with a G-map ss :U -! G=H. If x 2 U and H = Gx, then U is a tube around x. A tube is contractible if ss is a G-homotopy equivalence. (ii)An H-slice S in X is an H-invariant subset such that the canonical G-map G xH S -! GS X is an embedding onto an open subset. Then GS is an H-tube with S = ss-1(eH). Conversely, if (U, ss) is an H-tube in X, then S = ss-1(eH) is an H-slice and U = GS. On isotropy subgroups, we then have Gy = Hy H for all y 2 S, but equality need not hold. If x 2 S and H = Gx, then S is a slice through x. (iii)We say that X has enough slices if every point x 2 X is contained in an H-slice for some compact subgroup H. This implies that every point x has compact isotropy group, but in general it does not imply that there must be a slice through every point x. (iv)A G-numerable cover of X is a cover {Uj} by tubes such that there exists a locally finite partition of unity by G-maps ~j:X -! [0, 1] with support Uj. The following is the equivariant generalization of [33, 6.7]. Proposition 3.1.3. Any G-CW complex admits a G-numerable cover by con- tractible tubes. Proof. The proof given by Dold [33] in the nonequivariant case goes through with only a minor change in the initial construction, which we sketch. From the* *re, the technical details are unchanged. Let Xn be the n-th skeletal filtration of* * a G-CW complex X. Let X`ndenote the subspace obtained by deleting the centers G=H x 0 of all n-cells in Xn and let rn :`Xn-! Xn-1 denote the obvious retract. Starting from the interior en = G=H x (Dn - Sn-1) of an n-cell cn, define Vnm inductivelySfor m n by setting Vnn= en and Vnm+1 = r-1m+1(Vnm). Then the union Vn1 = m n Vnm is a contractible tube, where the projection to G=H is induced by the projection of en to G=H x 0. We now give the definition of a proper group action in K . We shall see that the definition could equivalently be made in U . For further details, but in T * *op, see for example [13, 32]. Recall that a continuous map is proper if it is a clo* *sed map with compact fibers. Definition 3.1.4. A G-space X in GK is proper (or G-proper) if the map ` :G x X -! X x X specified by `(g, x) = (x, gx) is proper. 3.1. PROPER ACTIONS OF LOCALLY COMPACT GROUPS 35 We warn the reader that the definition is not quite the standard one. We are working in the category K , and the product X x X on the right hand side is the k-space obtained by k-ifying the standard product topology on X xc X. In T op there are various other notions of a proper group action; see [4] for a ca* *reful discussion. They all agree for actions of locally compact groups on completely regular spaces. If X is proper, then the isotropy groups Gx are compact since t* *hey are the fibers `-1(x, x). Moreover, since points are closed subsets of G, the d* *iagonal X = `({e} x X) must be a closed subset of X x X and thus X must be weak Hausdorff. This means that proper G-spaces must be in U . Since G is locally compact, we have the following useful characterizations. Proposition 3.1.5. For a G-space X in GK the following are equivalent. (i)The action of G on X is proper. (ii)The isotropy groups Gx are compact and for all (x, y) 2 X x X and all neig* *h- borhood U of `-1(x, y) in GxX, there is a neighborhood V of (x, y) in X xX such that `-1(V ) U. (iii)The isotropy groups Gx are compact and for all (x, y) 2 X x X and all neig* *h- borhoods U of {g | gx = y} in G, there is a neighborhood V of (x, y) such that {g 2 G | ga = b for some (a, b)}2 VU. (iv)The space X is weak Hausdorff and every point (x, y) 2 X x X has a neigh- borhood V such that {g 2 G | ga = b for some (a, b)}2 V has compact closure in G. Proof. This holds by essentially the same proof as [4, 1.6(b)]. One must on* *ly keep in mind that we are now working in K rather than in T op and adjust the argument accordingly. Corollary 3.1.6. If G is discrete, then a G-space X is proper if and only if any point (x, y) 2 X x X has a neighborhood V such that {g 2 G | ga = b for some (a, b)}2 V is finite. Corollary 3.1.7. If G is compact, then any G-space in GU is proper. Remark 3.1.8. There is an alternative description of the set displayed in Proposition 3.1.5 that may clarify the characterization. Define OE: G x X x X -! X x X by OE(g, x, y) = (gx, y). For V X x X, let OEV be the restriction of OE to G * *x V and let ss :G x V -! G be the projection, which is an open map since G x V has the product topology. Then the displayed set is ssOE-1V( X ). If X x X = X xcX, then the condition in Proposition 3.1.5 is equivalent to the more familiar one * *that any two points x and y in X have neighborhoods Vx and Vy such that {g 2 G | gVx \ Vy 6= ;} has compact closure in G. Proposition 3.1.9. Proper actions satisfy the following closure properties. 36 3. PROPER ACTIONS, EQUIVARIANT BUNDLES, AND FIBRATIONS (i)The restriction of a proper action to a closed subgroup is proper. (ii)An invariant subspace of a proper G-space is also proper. (iii)Products of proper G-spaces are proper. (iv)If X is a proper Hausdorff G-space in GK and C is a compact Hausdorff G-space, then the G-space Map (C, X) is proper. (v)An H-space S is H-proper if and only if G xH S is G-proper. Proof. The first three are standard and elementary; see for example [32, I.5.10]. The fifth is [4, 2.3]. We prove (iv). We must show that the map ` :G x Map(C, X) -! Map (C, X) x Map(C, X) is proper, which amounts to showing that it is closed and that the isotropy gro* *ups Gf are compact for f 2 Map (C, X). For the latter, let {gi} be a net in Gf and fix c 2 C. Note that f(gic) = gif(c). Since C is compact, we can assume by passing to a subnet that {gic} converges to some ~c2 C. Let V be a neighborhood of (f(c), f(~c)) such that B = {g 2 G | ga = b for some (a, b)}2 V has compact closure. Since C is compact, C xC xMap (C, X) has the usual product topology. Since the map C x C x Map(C, X) -! X x X that sends (c, d, f) to (f(c), f(d)) is continuous and the net {c, gic, f} conv* *erges to (c, ~c, f), the net {(f(c), f(gic))} = {(f(c), gif(c))} must converge to (f(c),* * f(~c)). It follows that a subnet of {gi} lies in B and therefore has a converging sub-subn* *et. To show that ` is closed, let A be a closed subset of G x Map (C, X) and let {(fi, gifi)} be a net in `(A) that converges to (f, F ). We must show that (f, * *F ) is in `(A). For c 2 C, the net {g-1ic} has a subnet that converges to some ~c, by * *the compactness of C, so we may as well assume that the original net converges to ~* *c. Let V be a neighborhood of (f(~c), F (c)) such that B0= {g 2 G | ga = b for some (a, b)}2 V has compact closure. By continuity and the compactness of C, there is a compact neighborhood K1xK2 of (~c, c) that (f, F ) maps into V . Since {(fi, gifi)} con* *verges to (f, F ), there is an h such that (fi, gifi)(K1 x K2) V for i h. It follo* *ws that there is a k h such that (fi(g-1ic), gifi(g-1ic)) 2 V for all i k. Then t* *he subnet {gi}i k is contained in B0 and therefore has a sub-subnet that converges to some g 2 G. We have now seen that our original net {(gi, fi)} in A has a subnet {(gij, fij)} that converges to (g, f), and (g, f) 2 A since A is closed.* * By the continuity of `, {`(gi, fi)} must converge to (f, F ) = `(g, f) 2 `(A). In this* * last statement, we are using the uniqueness of limits, which we ensure by requiring X and C to be Hausdorff. The following theorem of Palais [81], as generalized by Biller [4], is funda* *men- tal. Those sources work in T op, but the arguments work just as well in U . Theorem 3.1.10 (Palais). Let X be a G-space in GU . (i)If X has enough slices, then it is proper. (ii)Conversely, if X is completely regular and proper, then it has enough slic* *es. (iii)If G is a Lie group and X is completely regular and proper, then there is a slice through each point of X. 3.3. SPACES OF THE HOMOTOPY TYPES OF G-CW COMPLEXES 37 Proof. Part (i) is given by [4, 2.4]. Part (iii) is given by [81, 2.3.3]. P* *art (ii) is deduced from part (iii) in [4, 2.5]. 3.2. Proper actions and equivariant bundles We introduce here the equivariant bundles to which we will apply our basic foundational results in Part IV. As we explain, Theorem 3.1.10 allows us to gen* *er- alize some basic results about such bundles from actions of compact Lie groups * *to proper actions of Lie groups. Let be a normal subgroup of a Lie group such that = = G and let q : -! G be the quotient homomorphism. By a principal ( ; )-bundle we mean the quotient map p: P - ! P= where P is a -free -space such that acts properly on P . It follows that the induced G-action on B = P= is proper. If F is a -space, then we have the associated G-map E = P x F -! P x * ~=P= , which we say is a -bundle with structure group and fiber F . For compact Lie groups, bundles of this general form are studied in [55], which generalizes the* * study of the classical case = G x given in [54]. A summary and further references are given in [68, Chapter VII]. We recall an observation about such bundles. Lemma 3.2.1. For b 2 B, the action of on F induces an action of the isotro* *py group Gb on the fiber Eb through a homomorphism aeb:Gb -! such that q O aeb is the inclusion Gb -! G and Eb ~=ae*bF . Proof. Choose z 2 P such that ss(z) = b. The isotropy group z intersects in the trivial group, and q maps z isomorphically onto Gb. Let aeb be the comp* *osite of q-1 :Gb -! z and the inclusion z -! . Since the subspace {z}xF of P xF is z-invariant and maps homeomorphically onto Eb on passage to orbits over , the conclusion follows. Note that changing the choice of z changes aebby conjug* *ation by an element of and changes the identification of Eb with F correspondingly. Bundles should be locally trivial. When P is completely regular, local trivi* *ality is a consequence of Theorem 3.1.10(iii), just as in the case when is a compact Lie group [55, Lemma 3], and this justifies our bundle-theoretic terminology. N* *ote that if P is completely regular, then so is B = P= . Lemma 3.2.2. A completely regular principal ( ; )-bundle P is locally trivi* *al. That is, for each b 2 B, there is a slice Sb through b and a homeomorphism ~= x Sb _____//_p-1(GSb) qx1 || |p| fflffl|~= fflffl| G xGb Sb_______//GSb where only intersects in the identity element and is mapped isomorphica* *lly to Gb by q. The -action on Sb is given by pulling back the Gb-action along q. 3.3.Spaces of the homotopy types of G-CW complexes In this section, we recall and generalize the equivariant version of Milnor'* *s re- sults [76] about spaces of the homotopy types of CW complexes. For compact Lie groups, Waner formulated and proved such results in [95, x4]. With a few obser- vations, his proofs generalize to deal with proper actions by general Lie group* *s. 38 3. PROPER ACTIONS, EQUIVARIANT BUNDLES, AND FIBRATIONS We first note the following immediate consequence of Proposition 3.1.3 and Theo- rem 3.1.10. Theorem 3.3.1. For any locally compact group G, a G-CW complex is proper if and only if it is constructed from cells of the form G=KxDn, where K is comp* *act. We also note the following recent "triangulation theorem" of Illman [48, The- orem II]. It is this result that led us to try to generalize some of our result* *s from compact Lie groups to general Lie groups. Theorem 3.3.2 (Illman). If G is a Lie group that acts smoothly and properly on a smooth manifold M, then M has a G-CW structure. Many of our applications of this result are based on the following observati* *on. Lemma 3.3.3. If H and K are closed subgroups of a topological group G and K is compact, then the diagonal action of G on G=H x G=K is proper. Proof. The proof given in [32, I.5.16] that G acts properly on G=K generali* *zes directly. Set X = G=H x G. Let G act diagonally from the left and let K act on the second factor from the right. Note that these actions commute. It suffices * *to show that ` :GxX -! X xX is proper. Indeed, consider the commutative square G x X ____`____//X x X | | | | fflffl|~` |fflffl G x X=K ____//_X=K x X=K. The right vertical map is proper and the left vertical map is surjective. There* *fore, by [32, VI.2.13], the bottom horizontal map is proper if the top horizontal map* * is proper. Since X is a free G-space, ` is proper if and only if the image Im(`) i* *s a closed subspace of X x X and the map OE: Im(`) -! G specified by OE(x, gx) = g is continuous. The diagonal subspace of G=H x G=H is closed, and its preimage under the map i :X x X -! G=H x G=H specified by i((xH, y), (~xH, ~y)) = (~yy-1xH, ~xH) is precisely Im(`), which is therefore closed. The function OE is the restrict* *ion to Im(`) of the continuous map : X x X -! G specified by ((xH, y), (~xH, ~y)) = ~yy-1 and is therefore continuous. We shall also make essential use of the following corollary of Theorem 3.3.2. Corollary 3.3.4. If X is a proper G-CW complex, then, viewed as an H-space for any closed subgroup H of G, X has the structure of an H-cell complex. Proof. Each cell G=K x Dn has K compact. Since G acts smoothly and properly on the smooth manifold G=K, the closed subgroup H also acts smoothly and properly. We use the resulting H-CW structure on all of the cells to obtain an H-cell structure. It is homotopy equivalent to an H-CW complex obtained by "sliding down" cells that are attached to higher dimensional ones, but we shall* * not need to use that. 3.4. SOME CLASSICAL THEOREMS ABOUT FIBRATIONS 39 Theorem 3.3.5 (Milnor, Waner). Let G be a Lie group and (X; Xi) be an n-ad of closed sub-G-spaces of a proper G-space X. If (X; Xi) has the homotopy type * *of a G-CW n-ad and (C; Ci) is an n-ad of compact G-spaces, then (X; Xi)(C;Ci)has the homotopy type of a G-CW n-ad. Proof. We only remark how the proof of Waner for the case of actions by a compact Lie group generalizes to the case of proper actions by a Lie group. Def* *ine a G-simplicial complex to be a G-CW complex such that X=G with the induced cell structure is a simplicial complex. In [95, x5], Waner proves that any G-CW comp* *lex is G-homotopy equivalent to a colimit of finite dimensional G-simplicial comple* *xes and cellular inclusions and that a G-space dominated by a G-CW complex is G- homotopy equivalent to a G-CW complex. The arguments apply verbatim to any topological group G. The rest of the argument requires two key lemmas. In [95, 4.2], Waner defines the notion of a G-equilocally convex, or G-ELC, G-space. The first lemma says that every finite dimensional G-simplicial complex is G-ELC. The essential star* *ting point is that orbits are G-ELC, the proof of which uses the Lie group structure just as in [95, p.358] in the compact case. From there, Waner's proof [95, x6] * *goes through unchanged. The second says that any completely regular, G-paracompact, G-ELC, proper G-space is dominated by a G-CW complex. When G is compact Lie, this is proven in [95, x7]. However, the hypothesis on G is only used to guaran* *tee the existence of enough slices, hence the proof holds without change for proper actions of Lie groups, indeed of locally compact groups. The rest of the proof goes as in [76, Theorem 3]. One only needs to make two small additional observations. First, if a G-simplicial complex K has the homot* *opy type of a proper G-space X, then it is proper. This holds since if f :K -! X is* * a homotopy equivalence, then Gk Gf(k)is compact. Second, for an n-ad (K; Ki) of G-simplicial complexes and a compact n-ad (C; Ci), (X; Xi)(C;Ci)is proper si* *nce it is a subspace of the proper G-space XC ; see (i) and (iv) of Proposition 3.1* *.9. Since it is also completely regular, G-paracompact, and G-ELC, it is dominated * *by a G-CW complex, and the result follows from the steps above. 3.4.Some classical theorems about fibrations A basic principle of parametrized homotopy theory is that homotopical infor- mation is given on fibers. We recall two relevant classical theorems about Hure* *wicz fibrations and a comparison theorem relating Serre and Hurewicz fibrations. We begin with Dold's theorem [33, 6.3]. The nonequivariant proof in [64, 2.6] is g* *ener- alized to the equivariant case in Waner [96, 1.11]. Waner assumes throughout [9* *6] that G is a compact Lie group, but that assumption is not used in the cited pro* *of. Theorem 3.4.1 (Dold). Let G be any topological group and let B be a G-space that has a G-numerable cover by contractible tubes. Let X -! B and Y -! B be Hurewicz fibrations. Then a map X -! Y over B is a fiberwise G-homotopy equiv- alence if and only if each fiber restriction Xb -! Yb is a Gb-homotopy equivale* *nce. We next recall and generalize a classical result that relates the homotopy t* *ypes of fibers to the homotopy types of total spaces. Nonequivariantly, it is due to Stasheff [89] and, with a much simpler proof, Sch"on [84]. The generalization * *to the equivariant case, for compact Lie groups, is given by Waner [96, 6.1]. With Theorems 3.4.1, 3.3.5 and 3.3.2 in place, Sch"on's argument generalizes directl* *y to 40 3. PROPER ACTIONS, EQUIVARIANT BUNDLES, AND FIBRATIONS give the following version. Since the result plays an important role in our wor* *k and the argument is so pretty, we can't resist repeating it in full. Theorem 3.4.2 (Stasheff, Sch"on). Let G be a Lie group and B be a proper G-space that has the homotopy type of a G-CW complex. Let p: X -! B be a Hurewicz fibration. Then X has the homotopy type of a G-CW complex if and only if each fiber Xb has the homotopy type of a Gb-CW complex. Proof. First assume that X has the homotopy type of a G-CW complex. For b 2 B, let ': Gb -! G be the inclusion and consider the Gb-map '*p: '*X -! '*B of Gb-spaces. It is still a Hurewicz fibration, as we see by using the left ad* *joint G xGb (-) of '*. By Corollary 3.3.4, '*X and '*B have the homotopy types of Gb-CW complexes. Factor '*p through the inclusion into its mapping cylinder i: '*X -! M'*p. Since Gb is compact, it follows from Theorem 3.3.5 that the homotopy fiber Fbi = (M'*p; {b}, '*X)(I;0,1)has the homotopy type of a Gb-CW complex. Since Fbi is homotopy equivalent to Fb'*p, by the gluing lemma, and Fb'*p is homotopy equivalent to the fiber Xb, this proves the forward implicati* *on. For the converse, assume that each fiber Xb has the homotopy type of a Gb- CW complex. Let fl : X -! X be a G-CW approximation of X. The mapping path fibration of fl gives us a factorization of fl as the composite of a G-hom* *otopy equivalence : X -! Nfl and a Hurewicz fibration q :Nfl -! X. We may view q as a map of fibrations over B. Nfl______q_____//BX BBB """" pOqBBB__B""p"""" B The fibers of p O q have the homotopy types of Gb-CW complexes by the first part of the proof, since X is a G-CW complex, and the fibers of p have the homotopy types of Gb-CW complexes by hypothesis. Comparison of the long exact sequences associated to p O q and p gives that q restricts to a Gb-homotopy equivalence on each fiber. Noting that we can pull back a numerable cover by contractible tubes along a homotopy equivalence B -! B0, where B0 is a G-CW complex, it follows from Theorem 3.4.1 that q is a homotopy equivalence. Although it no longer plays a role in our theory, the following little known result played a central role in our thinking. It shows that the dichotomy betwe* *en Serre and Hurewicz fibrations diminishes greatly over CW base spaces. It is due* * to Steinberger and West [90], with a correction by Cauty [16]. Theorem 3.4.3 (Steinberger and West; Cauty). A Serre fibration whose base and total spaces are CW complexes is a Hurewicz fibration. We believe that this remains true equivariantly for compact Lie groups, and * *it certainly remains true for finite groups. Before we understood the limitations* * of the q-model structure, we planned to use this result to relate our model theore* *tic homotopy category of ex-spaces over a CW complex B to a classical homotopy category defined in terms of Hurewicz fibrations and thereby overcome the probl* *ems illustrated in Counterexample 0.0.1. Such a comparison is still central to our * *theory, and it is this result that convinced us that such a comparison must hold. 3.5. QUASIFIBRATIONS 41 3.5. Quasifibrations For later reference, we recall the definition of quasifibrations. Here G can* * be any topological group. Definition 3.5.1. A map p: E -! Y in K is a quasifibration if the map of pairs p: (E, Ey) -! (Y, y) is a weak equivalence for all y in Y . A map p: E -!* * Y in K =B or KB is a quasifibration if it is a quasifibration on total spaces. A * *G-map p: E -! Y is a quasifibration if each of its fixed point maps pH :EH - ! Y H is* * a nonequivariant quasifibration. The condition that p: (E, Ey) -! (Y, y) is a weak equivalence means that for all e 2 Ey the following two conditions hold. (i)p*: ssn(E, Ey, e) -! ssn(Y, y) is an isomorphism for all n 1. (ii)For any x 2 E, p(x) is in the path component of y precisely when the path component of x in E intersects Ey. In other words, the sequence ss0(Ey, e) -! ss0(E, e) -! ss0(Y, y) of pointed sets is exact. Warning 3.5.2. In contrast to the usual treatments in the literature, we do * *not require p to be surjective and therefore ss0(E, e) -! ss0(Y, y) need not be sur* *jective. Hurewicz and, more generally, Serre fibrations are examples of quasifibrations,* * and they are not always surjective, as the trivial example {0} -! {0, 1} illustrate* *s. Model categorically, one point is that the initial map ; -! Y is always a Serre fibration since the empty lifting problem always has a solution. The definition of a quasifibration is arranged so that the long exact sequen* *ce of homotopy groups associated to the triple (E, Ey, e) is isomorphic to a long * *exact sequence . .-.! ssn+1(Y, y) -! ssn(Ey, e) -! ssn(E, e) -! ssn(Y, y) -! . .-.! ss0(Y, y). Part II Model categories and parametrized spaces Introduction In Part III, we shall develop foundations for parametrized equivariant stable homotopy theory. In making that theory rigorous, it became apparent to us that substantial foundational work was already needed on the level of ex-spaces. That work is of considerable interest for its own sake, and it involves general poin* *ts about the use of model categories that should be of independent interest. Therefore, * *rather than rush through the space level theory as just a precursor of the spectrum le* *vel theory, we have separated it out in this more leisurely and discursive expositi* *on. In Chapter 4, which is entirely independent of our parametrized theory, we g* *ive general model theoretic background, philosophy, and results. In contrast to the simplicial world, we often have both a classical h-type and a derived q-type mo* *del structure in topologically enriched categories, with respective weak equivalenc* *es the homotopy equivalences and the weak homotopy equivalences. We describe what is involved in verifying the model axioms for these two types of model structures. In Chapter 5, we describe how the parametrized world fits into this general framework. There are several different h-type model structures on our categories of parametrized G-spaces, with different homotopy equivalences based on differe* *nt choices of cylinders. These mesh in unexpected ways. Understanding of this part* *ic- ular case leads us to a conceptual axiomatic description of how the classical h* *-type homotopy theory and the q-type model structure must be related in order to be able to do homotopy theory satisfactorily in a topologically enriched category. In Chapter 6, we work nonequivariantly and develop our preferred "q-type" model category structure, the "qf-model structure", on the categories K =B and KB . This chapter is taken directly from the second author's thesis [88]. In Chapter 7, we give the equivariant generalization of the qf-model struc- ture and begin the study of the resulting homotopy categories by discussing tho* *se adjunctions that are given by Quillen pairs. There is another new twist here in that we need to use many Quillen equivalent qf-type model structures. In fact, this is already needed nonequivariantly in the study of base change along bundl* *es f :A -! B. In Chapter 8, we discuss ex-fibrations and an ex-fibrant approximation func- tor that better serves our purposes than model theoretic fibrant approximation * *in studying those adjunctions that are not given by Quillen pairs. In Chapter 9, we describe our parametrized homotopy categories in terms of classical homotopy ca* *t- egories of ex-fibrations and use this description to resolve the issues concern* *ing base change functors and smash products that are discussed in the Prologue. 44 CHAPTER 4 Topologically bicomplete model categories Introduction In x4.1, we describe a general philosophy about the role of different model structures on a given category C . It is natural and important in many contexts, and it helps to clarify our thinking about topological categories of parametriz* *ed objects. In particular, we advertise a remarkable unpublished insight of Mike C* *ole. It is a pleasure to thank him for keeping us informed of his ideas. We describe* * how a classical "h-type" model structure and a suitably related Quillen "q-type" mo* *del structure, can be mixed together to give an "m-type" model structure such that the m-equivalences are the q-equivalences and the m-fibrations are the h-fibrat* *ions. This is a completely general phenomenon, not restricted to topological contexts. In xx4.2 and 4.3, we describe classical structure that is present in any top* *o- logically bicomplete category C . Here we follow up a very illuminating paper * *of Schw"anzl and Vogt [85]. There are two classes of (Hurewicz) h-fibrations and t* *wo classes of h-cofibrations, ordinary and strong. Taking weak equivalences to be * *ho- motopy equivalences, the ordinary h-fibrations pair with the strong h-cofibrati* *ons and the strong h-fibrations pair with the ordinary h-cofibrations to give two i* *n- terrelated model like structures. For each choice, all of the axioms for a pro* *per topological model category are satisfied except for the factorization axioms, w* *hich hold in a weakened form. To prove that C is a model category, it suffices to pr* *ove one of the factorization axioms since the other will follow. Again, the theory * *can easily be adapted to other contexts than our topological one. We signal an ambiguity of nomenclature. In the model category literature, the term "simplicial model structure" is clear and unambiguous, since there is only one model structure on simplicial sets in common use. In the topological contex* *t, we understand "topological model structures" to refer implicitly to the h-model structure on spaces for model structures of h-type and to the q-model structure* * on spaces for model structures of q-type. The meaning should always be clear from context. In x4.4, we give another insight of Cole, which gains power from the work of Schw"anzl and Vogt. Cole provides a simple hypothesis that implies the miss- ing factorization axioms for an h-model structure of either type on a topologic* *ally bicomplete category C . When we restrict to compactly generated spaces, the hy- pothesis applies to give an h-model structure on U . In K , this seems to fail,* * and we give a streamlined version of Strom's original proof [93], together with his* * proof that the strong h-cofibrations in K are just the closed ordinary h-cofibration* *s. This works in exactly the same way for the categories GK and GU , where G is any (compactly generated) topological group. 45 46 4. TOPOLOGICALLY BICOMPLETE MODEL CATEGORIES In x4.5, we describe how to construct compactly generated q-type model struc- tures, giving a slight variant of standard treatments. In particular, GK and GU have the usual q-model structures in which the q-equivalences are the weak equi* *va- lences and the q-fibrations are the Serre fibrations. Again, G can be any topol* *ogical group. However we only know that the model structure is G-topological when G is a compact Lie group. 4.1. Model theoretic philosophy: h, q, and m-model structures The point of model categories is to systematize "homotopy theory". The ho- motopy theory present in many categories of interest comes in two flavors. Ther* *e is a "classical" homotopy theory based on homotopy equivalences, and there is a mo* *re fundamental "derived" homotopy theory based on a weaker notion of equivalence than that of homotopy equivalence. This dichotomy pervades the applications, re- gardless of field. It is perhaps well understood that both homotopy theories ca* *n be expressed in terms of model structures on the underlying category, but this asp* *ect of the classical homotopy theory has usually been ignored in the model theoreti* *cal literature, a tradition that goes back to Quillen's original paper [83]. The "c* *lassi- cal" model structure on spaces was introduced by Strom [93], well after Quillen* *'s paper, and the "classical" model structure on chain complexes was only introduc* *ed explicitly quite recently, by Cole [21] and Schw"anzl and Vogt [85]. Perhaps for this historical reason, it may not be widely understood that the* *se two model structures can profitably be used in tandem, with the h-model structu* *re used as a tool for proving things about the q-model structure. This point of vi* *ew is implicit in [39, 61, 62], and a variant of this point of view will be essential* * to our work. In the cited papers, the terms "q-fibration" and "q-cofibration" were used for the fibrations and cofibrations in the Quillen model structures, and the te* *rm "h-cofibration" was used for the classical notion of a Hurewicz cofibration spe* *cified in terms of the homotopy extension property (HEP). The corresponding notion of an "h-fibration" defined in terms of the covering homotopy property (CHP) is fortuitously appropriate1. Just as the "q" is meant to suggest Quillen, the "h" is meant to suggest Hurewicz, as well as homotopy. It is logical to follow this idea further (as was not done in [39, 61, 62]) by writing q-fibrant, q-cofibran* *t, h-fibrant, and h-cofibrant for clarity. Following this still further, we shoul* *d also write "h-equivalence" for homotopy equivalence and "q-equivalence" for (Quillen) weak equivalence. The relations among these notions are as follows in all of t* *he relevant categories C : ________________________________ | h-equivalence=) q-equivalence | | h-cofibration(= q-cofibration | | h-cofibrant(= q-cofibrant | | h-fibration=) q-fibration | |_____h-fibrant=)__q-fibrant____ | Therefore, the identity functor is the right adjoint of a Quillen adjoint pa* *ir from C with its h-model structure to C with its q-model structure. It follows t* *hat we have an adjoint pair relating the classical homotopy category, hC say, to the ____________ 1However, the notation conflicts with the notation often used for Dold's no* *tion of a weak or "halb"-fibration. We shall make no use of that notion, despite its real importa* *nce in the theory of fibrations. We do not know whether or not it has a model theoretic role to p* *lay. 4.2. STRONG HUREWICZ COFIBRATIONS AND FIBRATIONS 47 derived homotopy category qC = Ho C . This formulation packages standard in- formation. For example, the Whitehead theorem that a weak equivalence between cell complexes is a homotopy equivalence, or its analogue that a quasi-isomorph* *ism between projective complexes is a homotopy equivalence, is a formal consequence of this adjunction between homotopy categories. Recently, Cole [23] discovered a profound new way of thinking about the di- chotomy between the kinds of model structures that we have been discussing. He proved the following formal model theoretic result. Theorem 4.1.1 (Cole). Let (Wh, Fibh, Cofh) and (Wq, Fibq, Cofq) be two mo- del structures on the same category C . Suppose that Wh Wq and Fibh Fibq. Then there is a mixed model structure (Wq, Fibh, Cofm ) on C . The mixed cofi- brations Cofm are the maps in Cofh that factor as the composite of a map in Wh and a map in Cofq. An object is m-cofibrant if and only it is h-cofibrant and o* *f the h-homotopy type of a q-cofibrant object. If the h and q-model structures are le* *ft or right proper, then so is the m-model structure. By duality, the analogue with the inclusion Fibh Fibq replaced by an inclu- sion Cofh Cofq also holds. In the category of spaces with the h and q-model structures discussed above, the theorem gives a mixed model structure whose m- cofibrant spaces are the spaces of the homotopy types of CW-complexes. This m-model structure combines weak equivalences with Hurewicz fibrations, and it might conceivably turn out to be as important and convenient as the Quillen mod* *el structure. It is startling that this model structure was not discovered earlier. The pragmatic point is two-fold. On the one-hand, there are many basic resul* *ts that apply to h-cofibrations and not just q-cofibrations. Use of h-cofibrations* * limits the need for q-cofibrant approximation and often clarifies proofs by focusing a* *tten- tion on what is relevant. Many examples appear in [39, 62, 61], where properties of h-cofibrations serve as scaffolding in the proof that q-model structures are* * in fact model structures. We shall formalize and generalize this idea in the next chapt* *er. On the other hand, there are many vital results that apply only to h-fibrati* *ons (Hurewicz fibrations), not to q-fibrations (Serre fibrations). For example, a l* *ocal Hurewicz fibration is a Hurewicz fibration, but that is not true for Serre fibr* *ations. The mixed model structure provides a natural framework in which to make use of Hurewicz fibrations in conjunction with weak equivalences. While we shall make * *no formal use of this model structure, it has provided a helpful guide to our thin* *king. The philosophy here applies in algebraic as well as topological contexts, but we shall focus on the latter. 4.2. Strong Hurewicz cofibrations and fibrations Fix a topologically bicomplete category C throughout this section and the ne* *xt. With no further hypotheses on C , we show that it satisfies most of the axioms * *for not one but two generally different proper topological h-type model structures.* * We alert the reader to the fact that we are here using the term "h-model structure" in a generic sense. When we restrict attention to parametrized spaces, we will use the term in a different specific sense derived from the h-model structure on underlying total spaces. The material of these sections follows and extends mat* *erial in Schw"anzl and Vogt [85]. We have cylinders X x I and cocylinders Map (I, X). When C is based, we focus on the based cylinders X ^ I+ and cocylinders F (I+ , X). In either case,* * these 48 4. TOPOLOGICALLY BICOMPLETE MODEL CATEGORIES define equivalent notions of homotopy, which we shall sometimes call h-homotopy. We will later use these and cognate notations, but, for the moment, it is conve* *nient to introduce the common notations Cyl(X) and Cocyl(X) for these objects. There are obvious classes of maps that one might hope would specify a model structure. Definition 4.2.1. Let f be a map in C . (i)f is an h-equivalence if it is a homotopy equivalence in C . (ii)f is a Hurewicz fibration, abbreviated h-fibration, if it satisfies the CH* *P in C , that is, if it has the right lifting property (RLP) with respect to the* * maps i0 : X -! Cyl(X) for X 2 C . (iii)f is a Hurewicz cofibration, abbreviated h-cofibration, if it satisfies th* *e HEP in C , that is, if it has the left lifting property (LLP) with respect to t* *he maps p0: Cocyl(X) -! X. These sometimes do give a model structure, but then the h-cofibrations must be exactly the maps that satisfy the LLP with respect to the h-acyclic h-fibrat* *ions, and dually. In general, that does not hold. We shall characterize the maps in C that do satisfy the LLP with respect to the h-acyclic h-fibrations and, dually,* * the maps that satify the RLP with respect to the h-acyclic h-fibrations. For this, * *we need the following relative version of the above notions. Definition 4.2.2. We define strong Hurewicz fibrations and cofibrations. (i)A map p: E -! Y is a strong Hurewicz fibration, abbreviated ~h-fibration, if it satisfies the relative CHP with respect to all h-cofibrations i : A -! X* *, in the sense that a lift exists in any diagram A ____i____//_X______//E44jjjjj;; jjjj ww i0|| jjjj|| ww |p| |fflffljjjjjfflffl|w fflffl| Cyl(A)_____//Cyl(X)____//Y. (ii)A map i : A ! X is a strong Hurewicz cofibration, abbreviated ~h-cofibrati* *on, if it satisfies the relative HEP with respect to all h-fibrations p : E ! Y* * , in the sense that a lift exists in any diagram A _____//Cocyl(E)____//Cocyl(Y:):44 u iiiiii i|| uuiii ||iiii |p0| fflffl|uuiifflffl|iiiii fflffl| X ________//E_____p_____//Y. We recall the standard criteria for maps to be h-fibrations or h-cofibration* *s. Define the mapping cylinder Mf and mapping path fibration Nf by the usual pushout and pullback diagrams f X _______//_Y and Nf _____//Cocyl(Y ) i0|| || || |p0| fflffl| fflffl| fflffl| fflffl| Cyl(X) ____//_Mf X ____f____//Y. Lemma 4.2.3. Let f be a map in C . 4.2. STRONG HUREWICZ COFIBRATIONS AND FIBRATIONS 49 (i)f is an h-fibration if and only if it has the RLP with respect to the map i0 : Nf -! Cyl(Nf). (ii)f is an h-cofibration if and only if it has the LLP with respect to the map p0 : Cocyl(Mf) -! Mf. The ~h-fibrations and ~h-cofibrations admit similar characterizations. These* * were taken as definitions in [85, 2.4]. Lemma 4.2.4. (i) A map p: E -! Y is an ~h-fibration if and only if it has the RLP with respect to the canonical map Mi -! Cyl(X) for any h-cofibration i : A ! X; this holds if and only if the canonical map Cocyl(E) ! Np has the RLP with respect to all h-cofibrations. (ii) A map i : A ! X is an ~h-cofibration if and only if it has the LLP with re* *spect to the canonical map Cocyl(E) -! Np for any h-fibration p : E ! Y ; this holds if and only if the canonical map Mi ! Cyl(X) has the LLP with respect to all h-fibrations. Observe that the map i0 : X -! Cyl(X) is an ~h-cofibration and the map p0 : Cocyl(X) -! X is an ~h-fibration. Since the cylinder objects associated to init* *ial objects are initial objects, ~h-fibrations are in particular h-fibrations. Simi* *larly, ~h- cofibrations are h-cofibrations. Observe too that every object is both ~h-cofib* *rant and ~h-fibrant, hence both h-cofibrant and h-fibrant. We shall see in x4.4 that these distinctions are necessary in K but disappe* *ar in U , where the h and ~hnotions coincide. Even there, however, the conceptual distinction sheds light on classical arguments. The results of this section and the next are quite formal. Amusingly, the ma* *in non-formal ingredient is just the use in the following proof of the standard fa* *ct that {0, 1} ! I has the LLP with respect to all h-acyclic h-fibrations. Lemma 4.2.5. Let i: A -! X and p: E -! B be maps in C . (i)If i is an h-acyclic h-cofibration, then i is the inclusion of a strong def* *ormation retraction r :X -! A. (ii)If i is the inclusion of a strong deformation retraction r : X ! A, then i* * is a retract of Mi ! Cyl(X). (iii)If p is an h-acyclic h-fibration, then p is a strong deformation retractio* *n. (iv)If p is a strong deformation retraction, then p is a retract of Cocyl(E) -!* * Np. Proof. The last two statements are dual to the first two. For (i), since the h-equivalence i is an h-cofibration, application of the HEP shows that i has a homotopy inverse r : X ! A such that ri = idA. Since {0, 1} -! I has the LLP with respect to h-acyclic h-fibrations, an adjunction argument shows that p(0,1* *)has the RLP with respect to h-cofibrations. Thus a lift exists in the diagram on the left, which means that r is a strong deformation retraction with inclusion i. A ____i____//_XGG Cocyl(i) | | GGrGG A __c__//Cocyl(A)___//_Cocyl(X)44i i0| i0|| GGG i i i1 fflffl|pr | G## i || i i i p(0,1)||A _____//Cyl(A)J_______________//A fflffl|iiii fi fflffl| | JJJJ || | X ________(iOr,idX)__//_X x X i| Cyl(i)JJJ | i| fflffl| J%%fflffl| fflffl| X _________i1_____//_Cyl(X)_fi//_X 50 4. TOPOLOGICALLY BICOMPLETE MODEL CATEGORIES For (ii), we are given fi in the diagram on the left displaying r as a strong d* *e- formation retraction with inclusion i. Then the diagram on the right commutes, where the composites displayed in the lower two rows are identity maps. Using t* *he universal property of Mi to factor the crossing arrows i0 and pr through Mi, we see that i is a retract of the canonical map Mi ! Cyl(X). 4.3. Towards classical model structures in topological categories We now have two candidates for a classical model structure on C based on the h-equivalences. We can either take the h-fibrations and the ~h-cofibrations* * or the h-cofibrations and the ~h-fibrations. The following result shows that all o* *f the axioms for a proper topological model category are satisfied except that, in ge* *neral, only a weakened form of the factorization axioms holds. Theorem 4.3.1. The following versions of the axioms for a proper topological model category hold. (i)The classes of h-cofibrations, ~h-cofibrations, h-fibrations and ~h-fibrati* *ons are closed under retracts. (ii)Let i be an h-cofibration and p be an h-fibration. The pair (i, p) has the* * lifting property if i is strong and p is h-acyclic or if p is strong and i is h-acy* *clic. (iii)Any map f : X ! Y factors as X __i_//_Mf__r_//_Y where i is an ~h-cofibration and r has a section that is an h-acyclic ~h-co* *fibration and as p X __s__//Nf____//_Y where p is an ~h-fibration and s has a retraction that is an h-acyclic ~h-f* *ibration. (iv)Let i : A ! X be an h-cofibration and p : E ! B be an h-fibration, where i or p is strong. Then the map C (i, p) : C (X, E) ! C (A, E) xC(A,B)C (X, B) induced by i and p is an h-fibration of spaces. It is h-acyclic if i or p i* *s acyclic and it is an ~h-fibration if both i and p are strong. (v)The h-equivalences are preserved under pushouts along h-cofibrations and pu* *ll- backs along h-fibrations. Proof. Part (i) is clear since all classes are defined in terms of lifting * *prop- erties. Part (ii) follows directly from Lemma 4.2.4 and Lemma 4.2.5. The factor- izations of part (iii) are the standard ones. We consider the first. The evid* *ent section j :Y - ! Mf is an h-acyclic ~h-cofibration since it is the pushout of o* *ne. Consider the lifting problem in the left diagram below, in which the middle ver* *tical composite is i. Here p is an h-acyclic h-fibration, and we choose a section s o* *f p. X FF FF i1|| FffFFF ___sOfiOjOfqff//_55 i0 fflffl|F##F~0 X q X 0 l l lE X ____//_Cyl(X)```//E55llll< | >> i0|| |jn>>> fflffl|~n fflff>>>qnl| Cyl(Nqn) ____//_VZn+1q >> VVVVV OO n+1>> VVVVVV OO >> VVVVV O >>O VVVVVOEOE>++V''O Y The map Cyl(Nqn) -! Y is the adjoint of the projection Nqn -! Cocyl(Y ) given by the definition of Nqn, and qn+1 is the induced map. The maps jn are h-acyclic ~h-cofibrations since they are pushouts of such maps. Let Z be the colimit of t* *he Zn and j and q be the colimits of the jn and qn. Certainly f = q O j and j is a* *n h- acyclic ~h-cofibration. By Hypothesis 4.4.1, Nq is the colimit of the Nqn. Sinc* *e the cylinder functor preserves colimits, we see by Lemma 4.2.3 that q is an h-fibra* *tion since the ~n give a lift Cyl(Nq) -! Z by passage to colimits. The dual version of Theorem 4.4.2 admits a dual proof. 54 4. TOPOLOGICALLY BICOMPLETE MODEL CATEGORIES Theorem 4.4.3 (Cole). If C is a topologically bicomplete category which sat- isfies the dual of Hypothesis 4.4.1, then the h-equivalences, ~h-fibrations, an* *d h- cofibrations specify a proper topological h-model structure on C . From now on, we break the symmetry by focusing on h-fibrations and ~h- cofibrations. These give model structures in K and U . Everything in the rest of the section works equally in GK and GU . The following theorem combines several results of Strom [91, 92, 93]. Theorem 4.4.4 (Strom). The following statements hold. (i)The h-equivalences, h-fibrations, and ~h-cofibrations give K a proper topo* *log- ical h-model structure. Moreover, a map in K is an ~h-cofibration if and on* *ly if it is a closed h-cofibration. (ii)The h-equivalences, h-fibrations, and ~h-cofibrations give U a proper topo* *logi- cal h-model structure. Moreover, a map in U is an ~h-cofibration if and only if it is an h-cofibration. Proof. Theorem 4.4.2 applies to prove the first statement in (ii), but it d* *oes not seem to apply to prove the first statement in (i). The reasons are explaine* *d in Remark 1.1.4. Taking Z = Y Iand p = p0 there, the comparison map ff specializes to the map colimNfn -! Nf of Hypothesis 4.4.1. It may be that ff is a homeo- morphism in this special case, but we do not have a proof. It is a homeomorphism when we work in U . The characterization of the ~h-cofibrations in U follows fr* *om Lemma 1.1.3 and their characterization in K . For (i), we give a streamlined version of Strom's original arguments that us* *es the material of the previous section to prove both statements together. We proc* *eed in four steps. The first step is Strom's key observation, the second and third * *steps give the second statement, and the fourth step proves the needed factorization axiom. Consider an inclusion i: A -! X. Step 1. By Strom's [91, Thm. 3], if i is the inclusion of a strong deformati* *on retract and there is a map _ :X -! I such that _-1(0) = A, then i has the LLP with respect to all h-fibrations. By Proposition 4.3.3(i), this means that i i* *s an h-acyclic ~h-cofibration. Step 2. If i is an h-cofibration, then the canonical map j :Mi -! X x I is an h-acyclic h-cofibration and therefore, by Lemma 4.2.5, the inclusion of a st* *rong deformation retract. If i is closed, then (X, A) is an NDR-pair and there exis* *ts OE: X -! I such that OE-1(0) = A. Define _ :X x I -! I by _(x, t) = tOE(x). Then _-1(0) = Mi. Applying Step 1, we conclude that j has the LLP with respect to all h-fibrations. By Lemma 4.2.4, this means that i is an ~h-cofibration. Step 3. We can factor any inclusion i as the composite A _i0_//_Ess_//X, where E is the subspace X x (0, 1] [ A x I of X x I and ss is the projection. N* *ote that A = _-1(0), where _ :E -! I is the projection on the second coordinate. By direct verification of the CHP [93, p. 436], ss is an h-fibration. If i is * *an ~h- cofibration, then it has the LLP with respect to ss, hence we can lift the iden* *tity map of X to a map ~: X -! E such that ~ O i = i0. It follows that i(A) is closed in X since i0(A) is closed in E. Step 4. Let f :X -! Y be a map. Use Theorem 4.3.1(ii) to factor f as p O s, where s: X -! Nf is the inclusion of a strong deformation retract and p is an 4.5. COMPACTLY GENERATED q-TYPE MODEL STRUCTURES 55 ~h-fibration. Use Step 3 to factor s as X _i0_//_Nf x (0, 1] [ X_xsIs//_Nf. Here i0 is the inclusion of a strong deformation retract and X = _-1(0), as in * *Step 3. By Step 1, i0 is an h-acyclic ~h-cofibration. By Step 3, p O ss is an h-fibr* *ation. There are several further results of Strom about h-cofibrations that deserve* * to be highlighted. In order, the following results are [92, Theorem 12], [93, Lemma 5], and [92, Corollary 5]. Proposition 4.4.5. If p: E -! Y is an h-fibration and the inclusion X Y is an ~h-cofibration, then the induced map p-1(X) -! E is an ~h-cofibration. Proposition 4.4.6. If i: A -! B and j :B -! X are maps in K such that j and j O i are h-cofibrations, then i is an h-cofibration. Proposition 4.4.7. If an inclusion A X is an h-cofibration, then so is the induced inclusion ~A X. In view of the characterization of ~h-cofibrations in Theorem 4.4.4, it is n* *atural to ask if there is an analogous characterization of ~h-fibrations. Only the fol* *lowing sufficient condition is known. It is stated without proof in [85, 4.1.1], and i* *t gives another reason for requiring the base spaces of ex-spaces to be in U . Proposition 4.4.8. An h-fibration p: E -! Y with Y 2 U is an ~h-fibration. __ Proof. Let k :A -! X be an h-acyclic h-cofibration and let j :A -! X be the induced_inclusion. By Propositions 4.4.7 and 4.4.6, j and the inclusion i: A A are h-cofibrations. By Lemma 4.2.5(i), k is the inclusion of a deforma* *tion retraction r :X -! A_and the deformation restricts to a homotopy from (i O r) O* * j to the identity on A. It follows that j and hence also i are h-acyclic. Since* * j is an h-acyclic ~h-cofibration, it has the LLP with respect to p, and we see by a * *little diagram chase that it suffices to verify that i has the LLP with respect to p. * *Factor p as the composite of s: E -! Np and q :Np: - ! Y , as usual. Since q is an ~h-fibration, (i, q) has the lifting property, and it suffices to show that (i,* * s) has the __ lifting property. Suppose given a lifting problem f :A -! E and g :A -! Np such that s O f = g O i. Note that s(e) = (e, cp(e)) for e 2 E, where cy denote* *s the constant path at y. Since Y is weak Hausdorff, the constant paths give a closed subset of Y Iand Np = Y IxY E is a closed subset of Y Ix E. Therefore s(E) is closed in Np. We conclude that __ ____ ______ ____ g(A ) g(A)= s(f(A) s(E)= s(E), __ which means that there is a lift A -! E. 4.5. Compactly generated q-type model structures We give a variant of the standard procedure for constructing q-type model structures. The exposition prepares the way for a new variant that we will expl* *ain in x5.4 and which is crucial to our work. Although our discussion is adapted to topological examples, C need not be topological until otherwise specified. We first recall the small object argument in settings where compactness allows use* * of sequential colimits. Definition 4.5.1. Let I be a set of maps in C . 56 4. TOPOLOGICALLY BICOMPLETE MODEL CATEGORIES (i)A relative I-cell complex is a map Z0 -! Z, where Z is the colimit of a sequence of maps Zn -! Zn+1 such that Zn+1 is the pushout Y [X Zn of a coproduct X -! Y of maps in I along a map X -! Zn. (ii)I is compact if for every domain object X of a map in I and every relative I-complex Z0 -! Z, the map colimC (X, Zn) -! C (X, Z) is a bijection. (iii)An I-cofibration is a map that satisfies the LLP with respect to any map t* *hat satisfies the RLP with respect to I. Lemma 4.5.2 (Small object argument). Let I be a compact set of maps in C , where C is cocomplete. Then any map f : X -! Y in C factors functorially as a composite p X ___i_//W_____//Y such that p satisfies the RLP with respect to I and i is a relative I-cell comp* *lex and therefore an I-cofibration. Definition 4.5.3. A model structure on C is compactly generated if there are compact sets I and J of maps in C such that the following characterizations hol* *d. (i)The fibrations are the maps that satisfy the RLP with respect to J, or equi* *v- alently, with respect to retracts of relative J-cell complexes. (ii)The acyclic fibrations are the maps that satisfy the RLP with respect to I* *, or equivalently, with respect to retracts of relative I-cell complexes. (iii)The cofibrations are the retracts of relative I-cell complexes. (iv)The acyclic cofibrations are the retracts of relative J-cell complexes. The maps in I are called the generating cofibrations and the maps in J are call* *ed the generating acyclic cofibrations. We find it convenient to separate out properties of classes of maps in a mod* *el category, starting with the weak equivalences. Definition 4.5.4. A subcategory of C is a subcategory of weak equivalences if it satisfies the following closure properties. (i)All isomorphisms in C are weak equivalences. (ii)A retract of a weak equivalence is a weak equivalence. (iii)If two out of three maps f, g, g O f are weak equivalences, so is the thir* *d. Theorem 4.5.5. Let C be a bicomplete category with a subcategory of weak equivalences. Let I and J be compact sets of maps in C . Then C is a compactly generated model category with generating cofibrations I and generating acyclic * *cofi- brations J if the following two conditions hold: (i)(Acyclicity condition) Every relative J-cell complex is a weak equivalence. (ii)(Compatibility condition) A map has the RLP with respect to I if and only * *if it is a weak equivalence and has the RLP with respect to J. Proof. This is the formal part of Quillen's original proof of the q-model s* *truc- ture on topological spaces and is a variant of [44, 2.1.19] or [43, 11.3.1]. Th* *e fi- brations are defined to be the maps that satisfy the RLP with respect to J. The cofibrations are defined to be the I-cofibrations and turn out to be the retrac* *ts of relative I-cell complexes. The retract axioms clearly hold and, by (ii), the co* *fibra- tions are the maps that satisfy the LLP with respect to the acyclic fibrations,* * which gives one of the lifting axioms. The maps in J satisfy the LLP with respect to * *the 4.5. COMPACTLY GENERATED q-TYPE MODEL STRUCTURES 57 fibrations and are therefore cofibrations, which verifies something that is tak* *en as a hypothesis in the versions in the cited sources. Applying the small object ar* *gu- ment to I, we factor a map f as a composite of an I-cofibration followed by a m* *ap that satisfies the RLP with respect to I; by (ii), the latter is an acyclic fib* *ration. Applying the small object argument to J, we factor f as a composite of a relati* *ve J-cell complex that is a J-cofibration followed by a fibration. By (i), the fir* *st map is acyclic, and it is a cofibration because it satisfies the LLP with respect t* *o all fibrations, in particular the acyclic ones. Finally, for the second lifting axi* *om, if we are given a lifting problem with an acyclic cofibration f and a fibration p, th* *en a standard retract argument shows that f is a retract of an acyclic cofibration t* *hat satisfies the LLP with respect to all fibrations. Using the following companion to Definition 4.5.4, we codify the usual patte* *rn for verifying the acyclicity condition. Definition 4.5.6. A subcategory of a cocomplete category C is a subcategory of cofibrations if it satisfies the following closure properties. (i)All isomorphisms in C are cofibrations. (ii)All coproducts of cofibrations are cofibrations. (iii)If i: X -! Y is a cofibration and f :X -! Z is any map, then the pushout j :Y -! Y [X Z of f along i is a cofibration. (iv)If X is the colimit of a sequence of cofibrations in :Xn -! Xn+1, then the induced map i: X0 -! X is a cofibration. (v)A retract of a cofibration is a cofibration. In more general contexts, (iv) should be given a transfinite generalization,* * but we shall not have need of that. Note that if a subcategory of cofibrations is d* *efined in terms of a left lifting property, then all of the conditions hold automatica* *lly. Lemma 4.5.7. Let C be a cocomplete category together with a subcategory of cofibrations, denoted g-cofibrations, and a subcategory of weak equivalences, s* *atis- fying the following properties. (i)A coproduct of weak equivalences is a weak equivalence. (ii)If i: X -! Y is an acyclic g-cofibration and f :X -! Z is any map, then the pushout j :Y -! Y [X Z of f along i is a weak equivalence. (iii)If X is the colimit of a sequence of acyclic g-cofibrations in :Xn -! Xn+1, then the induced map i: X0 -! X is a weak equivalence. If every map in a set J is an acyclic g-cofibration, then every relative J-cell* * complex is a weak equivalence. We emphasize that the g-cofibrations are not the model category cofibrations and may or may not be the intrinsic h-cofibrations or ~h-cofibrations. They ser* *ve as a convenient scaffolding for proving the model axioms. Remark 4.5.8. The properties listed in Lemma 4.5.7 include some of the ax- ioms for a "cofibration category" given by Baues [1, pp 6, 182]. However, our purpose is to describe features of categories that are more richly structured t* *han model categories, often with several relevant subcategories of cofibrations, ra* *ther than to describe deductions from axiom systems for less richly structured categ* *ories, which is his focus. The g-cofibrations in Lemma 4.5.7 need not be the cofibrati* *ons of any cofibration category or model category. 58 4. TOPOLOGICALLY BICOMPLETE MODEL CATEGORIES The q-model structures on K and U are obtained by Theorem 4.5.5, taking the q-equivalences to be the weak equivalences, that is, the maps that induce i* *so- morphisms on all homotopy groups, and the q-fibrations to be the Serre fibratio* *ns. We also have the equivariant generalization, which applies to any topological g* *roup G. We introduce the following notations, which will be used throughout. Definition 4.5.9. Nonequivariantly, let I and J denote the set of inclusions i: Sn-1 -! Dn (where S-1 is empty) and the set of maps i0: Dn -! Dn x I. Equivariantly, let I and J denote the set of all maps of the form G=H x i, where H is a (closed) subgroup of G and i runs through the maps in the nonequivariant sets I and J. In the based categories K* and GK* we continue to write I and J for the sets obtained by adjoining disjoint base points to the specified maps. A map f :X -! Y of G-spaces is said to be a weak equivalence or Serre fibra- tion if all fixed point maps fH :XH - ! Y H are weak equivalences or Serre fibr* *a- tions. Just as nonequivariantly, we also call these q-equivalences and q-fibrat* *ions. Observe that q-equivalences are defined in terms of the equivariant homotopy gr* *oups ssHn(X, x) = ssn(XH , x) for H G and x 2 XH and that q-fibrations are defined* * in terms of the RLP with respect to the cells in J. If X0 -! X is a relative I or J-cell complex, then X=X0 is in GU and Lemma 1.1.5 gives all that is needed to verify the compactness hypothesis in De* *f- inition 4.5.1(ii). Taking the g-cofibrations to be the h-cofibrations, Lemma 4* *.5.7 applies to verify the acyclicity condition of Theorem 4.5.5. With considerable * *sim- plification, our verification of the compatibility condition for the qf-model s* *tructure in Chapter 6 specializes to verify it here. Nonequivariantly, the q-model struc* *ture is discussed in [37, x8] and, with somewhat different details, in [44, 2.4] (wh* *ere the details on transfinite sequences are unnecessary). Equivariantly, a detailed proof of the following result is given in [61, III* *x1]. The argument there is given for based G-spaces, in GT , but it works equally we* *ll for unbased G-spaces, in GK . Theorem 4.5.10. For any G, GK is a compactly generated proper model cate- gory whose q-equivalences, q-fibrations, and q-cofibrations are the weak equiva* *lences, the Serre fibrations, and the retracts of relative G-cell complexes. The sets I* * and J are the generating q-cofibrations and the generating acyclic q-cofibrations, * *and all q-cofibrations are ~h-cofibrations. If G is a compact Lie group, then the m* *odel structure is G-topological. The notion of a G-topological model category is defined in the same way as the notion of a simplicial or topological model category and is discussed forma* *lly in x10.3 below. The point of the last statement is that if H and K are subgroups o* *f a compact Lie group G, then G=H x G=K has the structure of a G-CW complex. By Theorem 3.3.2, this remains true when G is a Lie group and H and K are compact subgroups. We shall see how to use this fact model theoretically in Chapter 7. CHAPTER 5 Well-grounded topological model categories Introduction It is essential to our theory to understand the interrelationships among the various model structures that appear naturally in the parametrized context, both in topology and in general. This understanding leads us more generally to an axiomatization of the properties that are required of a good q-type model struc* *ture in order that it relate well to the classical homotopy theory on a topological * *category. The obvious q-model structure on ex-spaces over B does not satisfy the axioms, * *and in the next chapter we will introduce a new model structure, the qf-model struc* *ture, that does satisfy the axioms. As we recall in x5.1, any model structure on a category C induces a model structure on the category of objects over, under, or over and under a given obj* *ect B. When C is topologically bicomplete, so are these over and under categories. They then have their own intrinsic h-type model structures, which differ from t* *he one inherited from C . This leads to quite a few different model structures on the category CB of objects over and under B, each with its own advantages and disadvantages. Letting B vary, we also obtain a model structure on the category of retracts. We shall only be using most of these structures informally, but t* *he plethora of model structures is eye opening. In x5.2, we focus on spaces and compare the various classical notions of fib* *ra- tions and cofibrations that are present in our over and under categories. Altho* *ugh elementary, this material is subtle, and it is nowhere presented accurately in * *the literature. In particular, we discuss h-type, f-type and fp-type model structur* *es, where f and fp stand for "fiberwise" and "fiberwise pointed". For simplicity, we discuss this material nonequivariantly, but it applies verbatim equivariantly. The comparisons among the q, h, f, and fp classes of maps and model struc- tures guide our development of parametrized homotopy theory. We think of the f-notions as playing a transitional role, connecting the fp and h-notions. In t* *he rest of the chapter, we work in a general topologically bicomplete category C ,* * and we sort out this structure and its relationship to a desired q-type model struc* *ture axiomatically. Here we shift our point of view. We focus on three basic types of cofibratio* *ns that are in play in the general context, namely the Hurewicz cofibrations deter* *mined by the cylinders in C , the ground cofibrations that come in practice from a gi* *ven forgetful functor to underlying spaces, and the q-type model cofibrations. The * *first two are intrinsic, but we think of the q-type cofibrations as subject to negoti* *ation. In KB , the Hurewicz cofibrations are the fp-cofibrations and the ground cofibrati* *ons are the h-cofibrations, which is in notational conflict with the point of view * *taken in the previous chapter. 59 60 5. WELL-GROUNDED TOPOLOGICAL MODEL CATEGORIES In xx5.3 and 5.4, we ignore model theoretic considerations entirely. We desc* *ribe how the two intrinsic types of cofibrations relate to each other and to colimit* *s and tensors, and we explain how this structure relates to weak equivalences. We define the notion of a "well-grounded model structure" in x5.5. We believe that this notion captures exactly the right blend of classical and model catego* *rical homotopical structure in topological situations. It describes what is needed f* *or a q-type model structure in a topologically bicomplete category to be compatible with its intrinsic h-type model structure and its ground structure. Crucially, * *the q-type cofibrations should be "bicofibrations", meaning that they are both Hure* *wicz cofibrations and ground cofibrations. To illustrate the usefulness of the axiom* *ati- zation, and for later reference, we derive the long exact sequences associated * *to cofiber sequences and the lim1exact sequences associated to colimits in x5.6. A clear understanding of the desiderata for a good q-type model structure reveals that the obvious over and under q-model structure is essentially worthl* *ess for serious work in parametrized homotopy theory. This will lead us to introduce the new qf-model structure, with better behaved q-type cofibrations, in the next chapter. The formalization given in xx5.3-5.6 might seem overly pedantic were it only to serve as motivation for the definition of the qf-model structure. Howev* *er, we will encounter exactly the same structure in Part III when we construct the lev* *el and stable model structures on parametrized spectra. We hope that the formalization will help guide the reader through the rougher terrain there. We note parenthetically that there is still another interesting model struct* *ure on the category of ex-spaces over B, one based on local considerations. It is d* *ue to Michelle Intermont and Mark Johnson [49]. We shall not discuss their model structure here, but we are indebted to them for illuminating discussions. It is conceivable that their model structure could be used in an alternative developm* *ent of the stable theory, but that has not been worked out. Their structure suffers* * the defects that it is not known to be left proper and that, with their definition * *of weak equivalences, homotopy equivalences of base spaces need not induce equivalences* * of homotopy categories. We focus mainly on the nonequivariant context in this chapter, but G can be any topological group in all places where equivariance is considered. 5.1. Over and under model structures Recall from x1.2 that, for any category C and object B in C , we let C =B and CB denote the categories of objects over B and of ex-objects over B. We also ha* *ve the category B\C of objects under B. If C is bicomplete, then so are C =B, B\C and CB . We begin with some general observations about over and under model categories before returning to topological categories. We have forgetful functors U :C =B -! C and V :CB -! C =B. The first is left adjoint to the functor that sends an object Y to the object B x Y over B: (5.1.1) C (UX, Y ) ~=C =B (X, B x Y ). The second is right adjoint to the functor that sends an object X over B to the object X q B over and under B: (5.1.2) CB (X q B, Y ) ~=C =B (X, V Y ). As a composite of a left and a right adjoint, the total object functor UV :CB -* *! C does not enjoy good formal properties. This obvious fact plays a significant ro* *le in 5.1. OVER AND UNDER MODEL STRUCTURES 61 our work. For example, it limits the value of the model structures on CB that a* *re given by the following result. Proposition 5.1.3. Let C be a model category. Then C =B, B\C , and CB are model categories in which the weak equivalences, cofibrations, and fibratio* *ns are the maps over B, under B, or over and under B which are weak equivalences, fibrations, or cofibrations in C . If C is left or right proper, then so are C * *=B, B\C , and CB . Proof. As observed in [44, p. 5] and [37, 3.10], the statement about C =B is a direct verification from the definition of a model category. By the self-* *dual nature of the axioms, the statement about B\C is equivalent. The statement about CB follows since it is the category of objects under (B, id) in C =B. The* * last statement holds since pushouts and pullbacks in these over and under categories are constructed in C . When considering q-type model structures, we start with a compactly generated model category C . Using the adjunctions (5.1.1) and (5.1.2), we then obtain the following addendum to Proposition 5.1.3. Proposition 5.1.4. If C is a compactly generated model category, then C =B and CB are compactly generated. The generating (acyclic) cofibrations in C =B a* *re the maps i such that Ui is a generating (acyclic) cofibration in C . The genera* *ting (acyclic) cofibrations in CB are the maps i q B where i is a generating (acycli* *c) cofibration in C =B. We now return to the case when C is topologically bicomplete. Then it has the resulting "classical", or h-type, structure that was discussed in x4.3 and x4.4* *. If our philosophy in x4.1 applies to C , then it also has q and m-structures and the c* *ate- gories C =B and CB both inherit over and under model structures that are related as we discussed there. However, since C is topologically bicomplete, so is C =B* * by Proposition 1.2.8, and CB is based topologically bicomplete by Proposition 1.2.* *9. These categories therefore have classical h-type structures when they are regar* *ded in their own right as topologically bicomplete categories. To fix notation and * *avoid confusion we give an overview of all of these structures. We start with the h-classes of maps in C that are given in Definition 4.2.1 * *and Lemma 4.2.4. As in our discussion of spaces, we work assymmetrically, ignoring * *the ~h-fibrations and focusing on the candidates for h-type model structures given * *by the h-fibrations and ~h-cofibrations. We agree to use the letter h for the inhe* *rited classes of maps in C =B and CB , although that contradicts our previous use of h for the classical classes of maps in an arbitrary topologically bicomplete cate* *gory, such as C =B or CB . We shall resolve that ambiguity shortly by introducing new names for the classes of "classical" maps in those categories. Definition 5.1.5. A map g in C =B is an h-equivalence, h-fibration, h-co- fibration, or ~h-cofibration if Ug is such a map in C . A map g in CB is an h- equivalence, h-fibration, h-cofibration, or ~h-cofibration if V g is such a map* * in C =B or, equivalently, UV g is such a map in C . The ~h-cofibrations are h-cofibrations, but not conversely in general. Since* * the object *B = (B, id, id) is initial and terminal in CB , an object of CB is h-co* *fibrant (or ~h-cofibrant) if its section is an h-cofibration (or ~h-cofibration) in C .* * It is h- fibrant if its projection is an h-fibration in C . 62 5. WELL-GROUNDED TOPOLOGICAL MODEL CATEGORIES In C =B, we have the notion of a homotopy over B, defined in terms of X xB I or, equivalently, Map B(I, X). The adjective "fiberwise" is generally used in * *the literature to describe these homotopies. See, for example, the books [29, 51] on fiberwise homotopy theory. To distinguish from the h-model structure, we agree to write f rather than h for the fiberwise specializations of Definition 4.2.1 * *and Lemma 4.2.4. To avoid any possible confusion, we formalize this, making use of Proposition 4.3.3. Definition 5.1.6. Let g be a map in C =B. (i)g is an f-equivalence if it is a fiberwise homotopy equivalence. (ii)g is an f-fibration if it satisfies the fiberwise CHP, that is, if it has * *the RLP with respect to the maps i0: X -! X xB I for X 2 C =B. (iii)g is an f-cofibration if it satisfies the fiberwise HEP, that is, if it ha* *s the LLP with respect to the maps p0: MapB (I, X) -! X. (iv)g is an ~f-cofibration if it has the LLP with respect to the f-acyclic f-fi* *brations. A map g in CB is an f-equivalence, f-fibration, f-cofibration, or ~f-cofibratio* *n if V g is one in C =B. Again, f~-cofibrations are f-cofibrations, but not conversely in general. T* *he- orem 4.4.2 often applies to show that the f-fibrations and f~-cofibrations defi* *ne an f-model structure on C =B and therefore, by Proposition 5.1.3, on CB . As is always the case for an intrinsic classical model structure, every object of C =* *B is both f-cofibrant and ~f-cofibrant as well as f-fibrant. While this is obvious f* *rom the definitions, it may seem counterintuitive. It does not follow that every ob* *ject of CB is f-cofibrant since the two categories have different initial objects. In CB , we also have the notion of a homotopy over and under B, defined in terms of X ^B I+ or, equivalently, FB (I+ , X). The adjective "fiberwise pointe* *d" is used in [29, 51] to describe these homotopies. Again, for notational clarity, w* *e agree to write fp rather than h for the fiberwise pointed specializations of Definiti* *on 4.2.1 and Lemma 4.2.4, and we formalize this to avoid any possible confusion. Definition 5.1.7. Let g be a map in CB . (i)g is an fp-equivalence if it is a fiberwise pointed homotopy equivalence. (ii)g is an fp-fibration if it satisfies the fiberwise pointed CHP, that is, i* *f it has the RLP with respect to the maps i0: X -! X ^B I+ . (iii)g is a fp-cofibration if it satisfies the fiberwise pointed HEP, that is, * *if it has the LLP with_respect to the maps p0: FB (I+ , X) -! X. (iv)g is an fp-cofibration if it has the LLP with respect to the fp-acyclic fp- fibrations. ___ Again, fp-cofibrations are fp-cofibrations, but not conversely_in general, a* *nd Theorem 4.4.2 often applies to show that the fp-fibrations and fp-cofibrations * *de- fine an fp-model structure on CB . We summarize some general formal implications relating our classes of maps. Proposition 5.1.8. Let C , C =B and CB be topologically bicomplete categories with h, f, and fp-classes of maps defined as above. Then the following implicat* *ions hold for maps in CB . 5.2. THE SPECIALIZATION TO OVER AND UNDER CATEGORIES OF SPACES 63 ___________________________________________________ | fp-equivalence=) f-equivalence=) h-equivalence | | fp-cofibration(= f-cofibration=) h-cofibration | | ___ * * * | | fp-cofibration(= ~f-cofibration=) ~h-cofibration | |____fp-fibration=)__f-fibration_(=___h-fibration__ | Moreover, every object of CB is both fp-fibrant and fp-cofibrant. Proof. Trivial inspections of lifting diagrams show that an h-fibration is * *an__ f-fibration, an f-cofibration is an fp-cofibration, and an f~-cofibration is an* * fp- cofibration. Use of the adjunctions (5.1.1) and (5.1.2) shows that an f-cofibra* *tion is an h-cofibration, an ~f-cofibration is an ~h-cofibration, and an fp-fibratio* *n is an f- fibration. The last statement holds since fiberwise pointed homotopies with dom* *ain or target B are constant at the section or projection of the target or source. Remark 5.1.9. Assume that these classes of maps define model structures. Then the implications in Proposition 5.1.8 lead via Theorem 4.1.1 and its dual version to two new mixed model structures on CB , one with weak equivalences the f-equivalences and fibrations the fp-fibrations and one with weak equivalences * *the h-equivalences and cofibrations the ~f-cofibrations. The category CB of retracts introduced in x2.5 suggests an alternative model theoretic point of view. We give the basic definitions, but we shall not pursue* * this idea in any detail. Again, Theorem 4.4.2 often applies to verify the model cate* *gory axioms. Note that the intrinsic homotopies are given by homotopies of total obj* *ects over and under homotopies of base objects. Definition 5.1.10. Assume that CB is topologically bicomplete and let g be a map in CB . (i)g is an r-equivalence if it is a homotopy equivalence of retractions. (ii)g is an r-fibration if it satisfies the retraction CHP, that is, if it has* * the RLP with respect to the maps i0: X -! X x I for X 2 CB . (iii)g is an r-cofibration if it satisfies the retraction HEP, that is, if it h* *as the LLP with respect to the maps p0: Map(I, X) -! X. (iv)g is an ~r-cofibration if it has the LLP with respect to the r-acyclic r-fi* *brations. Remark 5.1.11. The initial and terminal object of CB are the identity retrac- tions of the initial and terminal objects of B and every object is both r-cofib* *rant and r-fibrant. It might be of interest to characterize the retractions for whi* *ch the map *B - ! (X, p, s) induced by s is an r-cofibration or for which the map (X, p, s) -! *B induced by p is an r-fibration. By specialization of the liftin* *g prop- erties, an ex-map over B that is an r-cofibration or r-fibration is an fp-cofib* *ration or fp-fibration in CB , but we have not pursued this question further. 5.2.The specialization to over and under categories of spaces Now we take C to be K or U . We discuss the relationships among our various classes of fibrations and cofibrations in this special case, and we consider wh* *en the f and fp classes of maps give model structures. Everything in this section appl* *ies equally well equivariantly. We first say a bit about based spaces, which are ex-spaces over B = {*}. Here the fact that * is a terminal object greatly simplifies matters. All of t* *he 64 5. WELL-GROUNDED TOPOLOGICAL MODEL CATEGORIES f-notions coincide with the corresponding h-notions, and our trichotomy reduces to the familiar dichotomy between free (or h) notions and based (or fp) notions. Recall that a based space is well-based, or nondegenerately based, if the inclu* *sion of the basepoint is an h-cofibration. Every based space is fp-cofibrant, and an fp-cofibration between well-based spaces is an h-cofibration [93, Prop. 9]. Ev* *ery based space is fp-fibrant, and an h-fibration of based spaces satisfies the bas* *ed CHP with respect to well-based source spaces. Of course, the over and under h-model structure differs from the intrinsic fp-model structure. None of the reverse implications in Proposition 5.1.8 holds in general. We g* *ave details of that result since it is easy to get confused and think that more is * *true than we stated. Scholium 5.2.1. On [29, p. 66], it is stated that a fiberwise pointed cofibr* *ation which is a closed inclusion is a fiberwise cofibration. That is false even when* * B is a point, since it would imply that every point of a T1-space is a nondegenerate basepoint. On [29, p. 69], it is stated that a fiberwise pointed map (= ex-map)* * is a fiberwise pointed fibration if and only if it is a fiberwise fibration. That * *is also false when B is a point, since the unbased CHP does not imply the based CHP. However, as for based spaces, the reverse implications in parts of Proposi- tion 5.1.8 often do hold under appropriate additional hypotheses. Proposition 5.2.2. The following implications hold for an arbitrary topologi- cally bicomplete category C . (i)A map in C =B between h-fibrant objects over B is an h-equivalence if and only if it is an f-equivalence. (ii)An ex-map between f-cofibrant ex-objects over B is an f-equivalence if and only if it is an fp-equivalence. Proof. The first part follows from Proposition 4.3.5(ii) since an f-equival* *ence in C =B is the same as an h-equivalence over B in C . The second part follows similarly from Proposition 4.3.5(i) since an fp-equivalence in CB is the same a* *s an f-equivalence under B in C =B. The following results hold for spaces. We are doubtful that they hold in gen* *eral. Proposition 5.2.3. The following implications hold in both GK and GU . (i)An ex-map between ~f-cofibrant ex-spaces is an f-cofibration if and only if* * it is an fp-cofibration. (ii)An ex-map whose source is ~f-cofibrant is an f-fibration if and only if it* * is an fp-fibration. Proof. Part (ii) is [29, 16.3]. Part (i) is stated on [93, p. 441] and the * *proof given there for based spaces generalizes using the following lemma. It is easy to detect f-cofibrations by means of the following result, whose * *proof is the same as that of the standard characterization of Hurewicz cofibrations (* *e.g. [71, p. 43], see also [91, Thm. 2], [92, Lem. 4] and [29, 4.3]). Lemma 5.2.4. An inclusion i: X -! Y in K =B is an f-cofibration if and only if (Y, X) is a fiberwise NDR-pair in the sense that there is a map u: Y -!* * I such that X u-1(0) and a homotopy h: Y xB I -! Y over B such that h0 = id, ht = id on X for 0 t 1, and h1(y) 2 X if u(y) < 1. A closed inclusion 5.2. THE SPECIALIZATION TO OVER AND UNDER CATEGORIES OF SPACES 65 i : X -! Y in K =B is an f~-cofibration if and only if the map u above can be chosen so that X = u-1(0). We introduce the following names here, but we defer a full discussion to x8.* *1. Definition 5.2.5. An ex-space is said to be well-sectioned if it is ~f-cofib* *rant. An ex-space is said to be ex-fibrant or, synonomously, to be an ex-fibration if* * it is both ~f-cofibrant and h-fibrant. Thus an ex-fibration is a well-sectioned ex-s* *pace whose projection is an h-fibration. The term ex-fibrant is more logical than ex-fibration, since we are defining* * a type of object rather than a type of morphism of KB , but the term ex-fibration goes better with Serre and Hurewicz fibration and is standard in the literature* *. We have the following implication of Propositions 5.1.8 and 5.2.2. It helps explai* *n the usefulness of ex-fibrations. Corollary 5.2.6. Let g be an ex-map between ex-fibrations over B. (i)g is an h-equivalence if and only if g is an f-equivalence, and this hold i* *f and only if g is an fp-equivalence. (ii)g is an f-cofibration if and only if g is an fp-cofibration, and then g is* * an h-cofibration. (iii)g is an f-fibration if and only if g is an fp-fibration, and this holds if* * g is an h-fibration. Remark 5.2.7. The model theoretic significance of ex-fibrations over B is un- clear. They are fibrant and cofibrant objects in the mixed model structure on ex-spaces over B whose weak equivalences are the h-equivalences and whose cofi- brations are the ~f-cofibrations. However, the converse fails since there are * *well- sectioned f-fibrant ex-spaces that are f-equivalent to h-fibrant ex-spaces, hen* *ce are mixed fibrant, but are not themselves h-fibrant. The previous remark anticipated the following result on over and under model structures in the categories of spaces and ex-spaces over B. Note that Lemma 1.* *1.3 applies to K =B and KB as well as to K to show that both f-cofibrations and fp-cofibrations are inclusions which are closed when the total spaces are in U . Theorem 5.2.8. The following statements hold. (i)The f-equivalences, f-fibrations, and ~f-cofibrations give K =B a proper to* *po- logical model structure. Moreover, a map in K =B is an ~f-cofibration if and only if it is a closed f-cofibration. (ii)The f-equivalences, f-fibrations, and ~f-cofibrations give U =B a proper t* *opo- logical model structure. Moreover, a map in U =B is an ~f-cofibration if and only if it is an f-cofibration. ___ (iii)The fp-equivalences, fp-fibrations, and fp-cofibrations give UB an fp-model structure. (iv)The r-classes of maps give the category UU of retracts a proper topological r-model structure. Proof. Apart from the factorization axioms, the model structures follow from the discussion in 4.3. In particular, the lifting axioms, the properness, and * *the topological property of all of these model structures are given by Theorem 4.3.* *1. In (ii), (iii), and (iv), the factorization axioms follow from Theorem 4.4.2 si* *nce 66 5. WELL-GROUNDED TOPOLOGICAL MODEL CATEGORIES the argument in Remark 1.1.4 verifies Hypothesis 4.4.1. The rest of (i) can be proven by direct mimicry of the proof of Theorem 4.4.4, using Lemma 5.2.4, and the characterization of the ~f-cofibrations in (ii) follows. Remark 5.2.9._We do not know whether or not KB is an fp-model category or whether the fp-cofibrations in KB are characterized as the closed fp-cofibratio* *ns. We also do not know whether or not KU is an r-model category. The problem here is related to the fact that, while the sections of ex-spaces are always inclusi* *ons, they need not be closed inclusions unless the total spaces are in U . Steps 1 a* *nd 3 of the proof of Theorem 4.4.4 fail in KB , and we also do not see how to carry * *over Strom's original proofs in [92, 93]. Theorem 4.3.1 still applies, giving much o* *f the information carried by a model structure. Observe too that if i: A -! X is a map of well-sectioned ex-spaces over B, then i is an fp-cofibration if and only if * *it is an f-cofibration, by Proposition 5.2.2(iii). For ex-spaces that are not well-secti* *oned, we have little understanding_of fp-cofibrations, even_when_B is a point. We have little understanding of fp-cofibrations that are not f-cofibrations in any case. There is a certain tension between the fp and h-notions, with the f-notions serving as a bridge between the two. Fiberwise pointed homotopy is the intrinsi* *cally right notion of homotopy in KB , hence the fp-structure is the philosopically r* *ight classical h-type model structure on KB , or at least on UB . It is the one tha* *t is naturally related to fiber and cofiber sequences, the theory of which works for* *mally in any based topologically bicomplete category in exactly the same way as for b* *ased spaces, as we will recall in x5.6. A detailed exposition in the case of ex-spac* *es is given in [29, 51, 52]. However, with h replaced by fp, we do not have the implications that we em- phasized in the general philosophy of x4.1. In particular, with the over and un* *der q-model structure, q-cofibrations need not be fp-cofibrations and fp-fibrations* * need not be q-fibrations, let alone h-fibrations. The q-model structure is still rel* *ated to the h-model structure as in x4.1, but this does not serve to relate the q-model* * struc- ture to parametrized fiber and cofiber sequences in the way that we are familiar with in the nonparametrized context. This already suggests that the q-model str* *uc- ture might not be appropriate in parametrized homotopy theory. In the following four sections, we explore conceptually what is required of a q-type model struc* *ture to connect it up with the intrinsic homotopy theory in a topologically bicomple* *te category. 5.3.Well-grounded topologically bicomplete categories Let C be a topologically bicomplete category in either the based or the unba* *sed sense; we use the notations of the based context. In our work here, and in other topological contexts, C is topologically concrete in the sense that there is a * *faithful and continuous forgetful functor from C to spaces. In practice, appropriate "gr* *ound cofibrations" can then be specified in terms of underlying spaces. These cofibr* *ations should be thought of as helpful background structure in our category C . To avoid ambiguity, we use the term "Hurewicz cofibration", abbreviated no- tationally to cyl-cofibration, for the maps that satisfy the HEP with respect t* *o the cylinders in C . We also have_the notion of a strong Hurewicz cofibration, whic* *h we abbreviate notationally to cyl-cofibration. For example, the cyl-cofibrations i* *n K , K =B, and KB are the h-cofibrations, the f-cofibrations, and the fp-cofibration* *s, 5.3. WELL-GROUNDED TOPOLOGICALLY BICOMPLETE CATEGORIES 67 ___ respectively, and similarly for cyl-cofibrations. As we have seen, it often hap* *pens that cyl-cofibrations between suitably nice objects of C , which we shall call * *"well- grounded", are also ground cofibrations. We introduce language to describe this situation. The following definitions codify the behavior of the well-grounded o* *bjects with respect to the cyl-cofibrations, colimits, and tensors in C . It is conven* *ient to build in the appropriate equivariant generalizations of our notions, although we defer a formal discussion of G-topologically bicomplete G-categories to x10.2; * *see Definition 10.2.1. The examples in x1.4 give the idea. Definition 5.3.1. An unbased space is well-grounded if it is compactly gener- ated. A based space is well-grounded if it is compactly generated and well-base* *d. The same definitions apply to G-spaces for a topological group G. Let C be a topologically bicomplete category. Definition 5.3.2. A full subcategory of C is said to be a subcategory of wel* *l- grounded objects if the following properties hold. (i)The initial object of C is well-grounded. (ii)All coproducts of well-grounded objects are well-grounded. (iii)If i: X -! Y is a cyl-cofibration and f :X -! Z is any map, where X, Y , and Z are well-grounded, then the pushout Y [X Z is well-grounded. (iv)The colimit of a sequence of cyl-cofibrations between well-grounded objects* * is well-grounded. (v)A retract of a well-grounded object is well-grounded. (vi)If X is a well-grounded object and K is a well-grounded space, then X ^ K (X x K in the unbased context) is well-grounded. When C is G-topologically bicomplete, we replace spaces by G-spaces in (vi). Definition 5.3.3. A ground structure on C is a (full) subcategory of well- grounded objects together with a subcategory of cofibrations, called the ground cofibrations and denoted g-cofibrations, such that every cyl-cofibration between well-grounded objects is a g-cofibration. A map that is both a g-cofibration an* *d a cyl-cofibration is called a bicofibration. Thus a cyl-cofibration between well-grounded objects is a bicofibration. The need for focusing on bicofibrations and the force of the definition come from t* *he following fact. Warning 5.3.4. In practice, (iii) often fails if i is a g-cofibration betwee* *n well- grounded objects that is not a cyl-cofibration, as we shall illustrate in x6.1.* * In particular, in GKB with the canonical ground structure described below, it can already fail for an inclusion i of I-cell complexes, where I is the standard se* *t of generators for the q-cofibrations. In the next chapter, we will construct a q-type model structure for GKB with* * a set of generating cofibrations to which the following implication of Definition* *s 4.5.6 and 5.3.2 applies. Lemma 5.3.5. Let I be a set of cyl-cofibrations between well-grounded objects and let f :X -! Y be a retract of a relative I-cell complex W -! Z. Then f is a bicofibration. If W is well-grounded, then so are X, Y , and Z. Our categories of equivariant parametrized spaces have canonical ground stru* *c- tures. Recall that the classes of f and ~f-cofibrations in GU =B and GUB coinci* *de. 68 5. WELL-GROUNDED TOPOLOGICAL MODEL CATEGORIES Definition 5.3.6. A space over B is well-grounded if its total space is com- pactly generated. An ex-space over B is well-grounded if it is well-sectioned a* *nd its total space is compactly generated. In both GK =B and GKB , define the g- cofibrations to be the h-cofibrations. Note that the only distinction between well-sectioned and well-grounded ex- spaces is the condition on total spaces. The distinction is relevant when we co* *nsider relative I-cell complexes X0 -! X in GKB . If X0 is well-sectioned, then so is * *X, whereas X=X0 is an I-cell complex and is therefore well-grounded for any X0. Proposition 5.3.7. These definitions specify ground structures on GK =B and on GKB . Proof. For GK =B, the Hurewicz cofibrations are the f-cofibrations, and these are h-cofibrations. It is standard that GU =B has the closure properties specified in Definition 5.3.2. For GKB , the Hurewicz cofibrations are the fp- cofibrations. Between well-sectioned ex-spaces, these are f-cofibrations and th* *ere- fore h-cofibrations by Proposition 5.2.3(i). Parts (i)-(v) of Definition 5.3.2* * are clear since well-sectioned means ~f-cofibrant, which is a lifting property. Fin* *ally we consider part (vi). Recall that X ^B K can be constructed as the pushout of *B oo___X q (B x K) _____//X x K in the category of spaces over B. By the equivariant version of the NDR-pair characterization of f-cofibrations in Lemma 5.2.4, these spaces are f-cofibrant* * and the map on the right is an f-cofibration. This implies that X ^B K is f-cofibra* *nt. 5.4.Well-grounded categories of weak equivalences The following definition describes how the weak equivalences and the ground structure are related in practice. Definition 5.4.1. Let C be a topologically bicomplete category with a given ground structure. A subcategory of weak equivalences in C is well-grounded if t* *he following properties hold (where acyclicity refers to the weak equivalences). (i)A homotopy equivalence is a weak equivalence. (ii)A coproduct of weak equivalences between well-grounded objects is a weak equivalence. (iii)(Gluing lemma) Assume that the maps i and i0 are bicofibrations and the vertical arrows are weak equivalences in the following diagram. f Y ooi___X _____//_Z | | | | | | fflffl| fflffl||fflffl Y 0oi0oX0___f0_//Z0 Then the induced map of pushouts is a weak equivalence. In particular, pushouts of weak equivalences along bicofibrations are weak equivalences. (iv)(Colimit lemma) Let X and Y be the colimits of sequences of bicofibrations in :Xn -! Xn+1 and jn :Yn -! Yn+1 such that both X=X0 and Y=Y0 are well-grounded. If f :X -! Y is the colimit of a sequence of compatible weak 5.4. WELL-GROUNDED CATEGORIES OF WEAK EQUIVALENCES 69 equivalences fn :Xn -! Yn, then f is a weak equivalence. In particular, if each in is a weak equivalence, then the induced map i: X0 -! X is a weak equivalence. (v)For a map i: X -! Y of well-grounded objects in C and a map j :K -! L of well-grounded spaces, i j is a weak equivalence if i is a weak equivalen* *ce or j is a q-equivalence. Here, in the based context, i j is the evident induced map (X ^ L) [X^K (Y ^ K) -! Y ^ L. The gluing lemma implies that acyclic bicofibrations are preserved under push- outs, as of course holds for pushouts of acyclic cofibrations in model categori* *es. The special case mentioned in (iii) corresponds to the left proper axiom in mod* *el categories. As there, it can be used to prove the general case of the gluing le* *mma provided that we have suitable factorizations. Lemma 5.4.2. Assume the following hypotheses. (i)Weak equivalences are preserved under pushouts along bicofibrations. (ii)Every map factors as the composite of a bicofibration and a weak equivalen* *ce. Then the gluing lemma holds. Proof. We use the notations of Definition 5.4.1(iii) and proceed in three c* *ases. If f and f0are both weak equivalences, then, by (i), so are the horizontal a* *rrows in the commutative diagram Y ______//Y [X Z | | | | fflffl| fflffl| Y 0_____//Y 0[X0 Z0. Since Y -! Y 0is a weak equivalence, the right arrow is a weak equivalence by t* *he two out of three property of weak equivalences. If f and f0 are both bicofibrations, consider the commutative diagram pX _________i_________//Y9n fpppp | nnnnn | 99 pp | wwnnnn | 99 xxppp____|_____//_Y [ Z || 999 Z| | | ?X? | 99 | | | ??? | 99 | fflffl| | ?? fflffl| 9oo | X0 _______|_____???_//_Y [X_X0_//Y 0 | f0 qqq | ?? sss | qqqq | oooo??? sss fflffl|xxqqq fflffl|wwooOO yyss Z0 ______________//_Y [X Z0__//_Y 0[X0 Z0. The back, front, top, and two bottom squares are pushouts, and the middle com- posite X0 -! Y 0is i0. Since f and f0 are bicofibrations, so are the remaining three arrows from the back to the front. Similarly, i and its pushouts are bico* *fi- brations. Since X -! X0, Y -! Y 0, and Z -! Z0 are weak equivalences, (i) and the two out of three property imply that Y - ! Y [X X0, Y [X X0 -! Y 0, Y [X Z -! Y [X Z0, and Y [X Z0- ! Y 0[X0Z0are weak equivalences. Composing the last two, Y [X Z -! Y 0[X0 Z0 is a weak equivalence. 70 5. WELL-GROUNDED TOPOLOGICAL MODEL CATEGORIES To prove the general case, construct the following commutative diagram. f Y oo__i____X ________________//_LLZ99ss | | LLLL sss | | | LL%% sss~f | | | W | | | | | | | | |fflffli0 fflffl|f0| fflffl| Y 0oo______X0 ________|_______//_Z0 KKK | tt99 KKK | ttt K%%fflffl|~f0tt X0[X W Here we first factor f as the composite of a bicofibration and a weak equivalen* *ce ~fand then define a map ~f0by the universal property of pushouts. By hypothesis (i), W -! X0[X W is a weak equivalence, and by the two out of three property, so is ~f0. By the second case, Y [X W -! Y 0[X0 (X0[X W ) ~=Y 0[X W is a weak equivalence and by the first case, so is Y [X Z ~=(Y [X W ) [W Z -! (Y 0[X W ) [(X0[XW) Z0~=Y 0[X0 Z0. Remark 5.4.3. Clearly the previous result applies to any categories of weak equivalences and cofibrations that satisfy (i) and (ii). The essential point is* * that, in practice, we often need bicofibrations in order to verify (i). Similarly, but more simply, the following observation reduces the verificati* *on of Definition 5.4.1(v) to special cases. Here we assume that C is based. Lemma 5.4.4. Let i: X -! Y be a map in C and j :K -! L be a map of based spaces. Display i j in the diagram id^j X ^ K ___________________________//_X ^ L | k jjjjj | | jjjjj | | uuj | i^id| (X ^ L) [X^K (Y ^ K) i^id| | jj55j TTTT | | jjjjj TTTTT | fflffl|jjj i j T))T fflffl| Y ^ K ____________id^j___________//_Y ^ L. If the maps i ^ idand the pushout k of i ^ idalong id^ j are weak equivalences, then so is i j, and similarly with the roles of i and j reversed. Together with Lemma 5.3.5, the notion of a well-grounded category of weak equivalences encodes a variant of Lemma 4.5.7 that often applies when the latter does not. Lemma 5.4.5. If J is a set of acyclic cyl-cofibrations between well-grounded objects, then all relative J-cell complexes are weak equivalences. Proof. This follows from (ii), (iii), and (iv) of Definition 5.4.1, togethe* *r with the observation that if X0 -! X is a relative J-cell complex, then X=X0 is a J-* *cell complex and is therefore well-grounded, so that (iv) applies. There is an analogous reduction of the problem of determining when a functor preserves weak equivalences. 5.4. WELL-GROUNDED CATEGORIES OF WEAK EQUIVALENCES 71 Lemma 5.4.6. Let F :C -! D be a functor between topologically bicomplete categories that come equipped with subcategories of well-grounded weak equivale* *nces with respect to given ground structures. Let J be a set of acyclic cyl-cofibra* *tions between well-grounded objects in C . Assume that F has a continuous right adjoi* *nt and that F takes maps in J to weak equivalences between well-grounded objects. Then F takes a retract of a relative J-cell complex to an acyclic map in D. Proof. The functor F preserves cyl-cofibrations since it has a continuous r* *ight adjoint and hence F J consists of acyclic cyl-cofibrations between well-grounded objects. The conclusion follows from Lemma 5.4.5 and the fact that left adjoints commute with colimits and therefore the construction of cell complexes. Similarly, cell complexes are relevant to the verification of Definition 5.4* *.1(v). Recall that the cyl-cofibrations in K* are the fp-cofibrations, that is, the ba* *sed cofibrations. Lemma 5.4.7. Let I be a set of cyl-cofibrations between well-grounded objects of C and let J be a set of fp-cofibrations between well-based spaces. If i is a* * retract of a relative I-cell complex, j is a retract of a relative J-cell complex, and * *either I or J consists of weak equivalences, then i j is a weak equivalence. Proof. Assume that I consists of weak equivalences; the proof of the other case is symmetric. Since the functor - ^ K commutes with coproducts, pushouts, sequential colimits, and retracts, we can construct j ^ K by first applying - ^* * K to the generators, then construct the cell complex, and finally pass to retracts. * *Since -^K preserves cyl-cofibrations and well-grounded objects by Definition 5.3.2(vi* *), it takes maps in I to cyl-cofibrations between well-grounded objects. By Lemma 5.4* *.5, the resulting cell complex is acyclic and therefore so also is any retract of i* *t. Thus j ^ K is an acyclic bicofibration. Since such maps are preserved under pushouts, Lemma 5.4.4 applies to give the conclusion. The following classical example is implicit in the literature. Proposition 5.4.8. The q-equivalences in GK are well-grounded with respect to the ground structure whose well-grounded objects are the compactly generated spaces and whose g-cofibrations are the h-cofibrations. Proof. Parts (i), (ii), and, here in the unbased case, (v) of Definition 5.* *4.1 are clear, and (iv) follows easily from Lemma 1.1.5. The essential point is the gluing lemma of (iii). By passage to fixed point spaces, it suffices to prove * *this nonequivariantly. Using the gluing lemma for the proper h-model structure on K , we see that f and f0 can be replaced by their mapping cylinders. Then the induc* *ed map of pushouts is the map of double mapping cylinders induced by the original diagram. This map is equivalent to a map of excisive triads, and in that case t* *he result is [67, 1.3], whose proof is corrected in [98]. Proposition 5.4.9. The q-equivalences in GK =B and GKB are well-grounded with respect to the ground structures of Proposition 5.3.7. In these cases, one* * need only assume that the relevant maps in the gluing and colimit lemmas are ground cofibrations (= h-cofibrations), not both ground and Hurewicz cofibrations. Proof. We verify this for GKB . Part (i) of Definition 5.4.1 holds since an* *y fp- equivalence is a q-equivalence and part (iii) follows directly from the gluing * *lemma 72 5. WELL-GROUNDED TOPOLOGICAL MODEL CATEGORIES in GK . For part (ii), the total space of _B Xi is the pushout in GK of *B oo___q*B _____//qXi. Since the Xi are well-grounded, the map on the right is an h-cofibration, hence (ii) also follows from the gluing lemma in GK . In part (iv), the relevant quot* *ient in GKB is given by the pushout, X=BX0, of the diagram *B - X0 -! X. Since X=BX0 is well-grounded, the quotient total space is in U and one can apply Lemma 1.1.5 just as on the space level. Finally consider (v). As in the proof of Proposition 5.3.7(vi), X ^B K can be constructed as the pushout of the following diagram of f-cofibrant spaces over B. *B oo___X q (B x K) _____//X x K The map on the right is an f-cofibration. By the gluing lemma in GK , it suffic* *es to observe that X x K preserves q-equivalences in both variables since homotopy groups commute with products. 5.5. Well-grounded compactly generated model structures Let C be a topologically bicomplete category or, equivariantly, a G-topologi* *cally bicomplete G-category. In the notion of a "well-grounded model structure", we f* *or- mulate the properties that a compactly generated model structure on C should have in order to mesh well with the intrinsic h-structure on C described in x4.* *3. When C has such a model structure, and when the classical h-structure actually is a model structure, the identity functor on C is a Quillen left adjoint from * *the well-grounded model structure to the h-model structure. Thus this notion gives a precise axiomatization for the implementaton of the philosophy that we advertis* *ed in x4.1. We begin with a variant of Theorem 4.5.5. Theorem 5.5.1. Let C be a topologically bicomplete category with a ground structure, a subcategory of well-grounded weak equivalences, and compact sets I and J of maps that satisfy the following conditions. (i)(Acyclicity condition) Every map in J is a weak equivalence. (ii)(Compatibility condition) A map has the RLP with respect to I if and only * *if it is a weak equivalence and_has_the RLP with respect to J. (iii)Every map in I and J is a cyl-cofibration between well-grounded objects. Then C is a compactly generated model category with generating sets I and J of cofibrations and acyclic cofibrations. Every cofibration is a bicofibration and* * every cofibrant object is well-grounded. A pushout of a weak equivalence along a bico* *fi- bration is a weak equivalence and, in particular, the model structure is left p* *roper. The model structure is topological or, equivariantly, G-topological if the foll* *owing condition holds. (iv)i j is an I-cell complex if i: X -! Y is a map in I and j :K -! L is a map of spaces (or G-spaces) in I. Proof. By Lemma 5.4.5, Theorem 4.5.5 applies to verify the model axioms. Condition (iii) implies the statements about cofibrations and cofibrant objects* * by Lemma 5.3.5, and the gluing lemma implies the statement about pushouts of weak equivalences. In the last statement, the set I of generating cofibrations in t* *he relevant category of (based or unbased) spaces is as specified in Definition 4.* *5.9. By passage to coproducts, pushouts, sequential colimits, and retracts, (iv) imp* *lies 5.6. PROPERTIES OF WELL-GROUNDED MODEL CATEGORIES 73 that i j is a cofibration if i: X -! Y is a cofibration in C and j :K -! L is a q-cofibration of spaces (or G-spaces). Together with Lemma 5.4.7, this implies * *that the model structure is topological. We emphasize the difference between the acyclicity conditions stated in Theo- rem 4.5.5 and in Theorem 5.5.1. In the applications of the former, it is the ve* *rifi- cation of the acyclicity of J-cell complexes that is problemmatic, but in the l* *atter our axiomatization has built in that verification. Similarly, our axiomatizatio* *n has built in the verification of the acyclicity condition required for the model st* *ructure to be topological. Definition 5.5.2. A compactly generated model structure on C is said to be well-grounded if it is right proper and satisfies all of the hypotheses of the * *preceding theorem. It is therefore proper and topological or, equivariantly, G-topologica* *l. 5.6. Properties of well-grounded model categories Assume that C is a well-grounded model category. To derive properties of its homotopy category HoC , we must sort out the relationship between homotopies defined in terms of cylinders and homotopies in the model theoretic sense, which we call "model homotopies". Of course, the cylinder objects Cyl(X) in C have maps i0, i1: X -! Cyl(X) and p: Cyl(X) -! X, and i0 (or i1) and p are inverse homotopy equivalences since tensors with spaces preserve homotopies in the space variable. Definition 5.4.1(i) ensures that p is therefore a weak equivalence. * * This means that Cyl(X) is a model theoretic cylinder object in C , provided that we adopt the non-standard definition of [37]. With the language there, it need not* * be a good cylinder object since i0 q i1: X q X -! Cyl(X) need not be a cofibration. As pointed out in [37, p. 90], this already fails for spaces, where the inclus* *ion X q X -! X x I is not a q-cofibration unless X is q-cofibrant. With the standard definition given in [43, 44, 83], cylinder objects are required to have this co* *fibration property. Under that interpretation, the cylinder objects Cyl(X) would not qual* *ify as model theoretic cylinder objects in general. (We note parenthetically that "* *good cylinders" are defined in [85] in such a way as to include all standard cylinde* *rs in the category of spaces). We record the following observations. Lemma 5.6.1. Consider maps f, g :X -! Y in C . (i)If f is homotopic to g, then f is left model homotopic to g. (ii)If X is cofibrant, then Cyl(X) is a good cylinder object. (iii)If X is cofibrant and Y is fibrant, then f is homotopic to g if and only i* *f f is left and right model homotopic to g. Proof. Part (i) is [37, 4.6], part (ii) follows from Definition 5.3.2(iii),* * and part (iii) follows from [37, 4.23]. Let [X, Y ] denote the set of morphisms X - ! Y in Ho C and let ss(X, Y ) denote the set of homotopy classes of maps X -! Y . We shall only use the latter notation when homotopy and model homotopy coincide. Lemma 5.6.2 (Cofiber sequence lemma). Assume that C is based. Consider the cofiber sequence X -! Y -! Cf -! X -! Y -! Cf -! 2X -! . . . 74 5. WELL-GROUNDED TOPOLOGICAL MODEL CATEGORIES of a well-grounded map f :X -! Y . For any object Z, the induced sequence . .-.! [ n+1X, Z] -! [ nCf, Z] -! [ nY, Z] -! [ nX, Z] -! . .-.! [X, Z] of pointed sets (groups left of [ X, Z], Abelian groups left of [ 2X, Z]) is ex* *act. Proof. As usual, giving I the basepoint 1, we define CX = X ^ I, X = X ^ S1, and Cf = Y [f CX. If X is cofibrant, then X is well-grounded and X -! CX is a cofibration and therefore a bicofibration. If X and Y are cofibrant, then so is Cf, as one sees* * by solving the relevant lifting problem by first using that Y is cofibrant, then u* *sing that X -! CX is a cofibration, and finally using that Cf is a pushout. Thus, taking Z to be fibrant, the conclusion follows in this case from the sequence of homotopy classes of maps . .-.! ss( X, Z) -! ss(Cf, Z) -! ss(Y, Z) -! ss(X, Z), which is proven to be exact in the same way as on the space level. If X and Y are not cofibrant, let Qf :QX - ! QY be a cofibrant approximation to f. The gluing lemma applies to give that the canonical map CQf -! Cf is a weak equivalence. Therefore the conclusion follows in general from the special case* * of cofibrant objects. Warning 5.6.3. While the proof just given is very simple, it hides substanti* *al subtleties. It is crucial that cofibrant objects X be well-grounded, so that t* *he cyl-cofibration X -! CX is a bicofibration and the gluing lemma applies. Of course, the group structures are defined just as classically. The pinch m* *aps S1 ~=I={0, 1} -! I={0, 1_2, 1} ~=S1 _ S1 and I -! I={1_2, 1} ~=I _ S1 induce pinch maps X -! X _ X and Cf -! Cf _ X that give X the structure of a cogroup object in Ho C and Cf a coaction by X; 2X is an abelian cogroup object for the same reason that higher homotopy groups are abelian. Therefore [ X, Z] is a group, [Cf, Z] is a [ X, Z]-set, and [ X, Z] -! [Cf, Z] is a [ X, Z]-map. Lemma 5.6.4 (Wedge lemma). For any Xi and Y in C , [qXi, Y ] ~= [Xi, Y ]. Proof. This is standard, using that a coproduct of cofibrant approximations is a cofibrant approximation. Lemma 5.6.5 (Lim 1lemma). Assume that C is based. Let X be the colimit of a sequence of well-grounded cyl-cofibrations in :Xn -! Xn+1. Then, for any object Y , there is a lim1exact sequence of pointed sets * -! lim1[ Xn, Y ] -! [X, Y ] -! lim[Xn, Y ] -! *. Proof. The telescope TelXn is defined to be colimTn, where the Tn and a ladder of weak equivalences jn :Xn -! Tn and rn :Tn -! Xn are constructed 5.6. PROPERTIES OF WELL-GROUNDED MODEL CATEGORIES 75 inductively by setting T0 = X0 and letting jn+1 and rn+1 be the maps of pushouts induced by the following diagram. Xn __________Xn ____in___//_Xn+1 i1|| |2| ||2 fflffl|i(0,1)fflffl|jnqin fflffl| CylXn oo___Xn q Xn _____//Tn q Xn+1 p|| |||| |rnqid| fflffl| || fflffl| Xn oo_r___Xn q Xn idqin//_Xn q Xn+1 Since jn+1 is a pushout of the bicofibration i1: Xn -! Cyl(Xn), the gluing lemma and colimit lemma specified in Definition 5.4.1(iii) and (iv) apply to show tha* *t the induced maps TelXn -! colimXn = X are weak equivalences. As in the cofiber sequence lemma, we can use cofibrant approximation to redu* *ce to a question about ss(-, -). Then the telescope admits an alternative descript* *ion from which the lim1exact sequence is immediate. It would take us too far afield to go into full details of what should be a standard argument, but we give a sk* *etch since we cannot find our preferred argument in the literature. Recall that the classical homotopy pushout, or double mapping cylinder, of f f0 Y oo___X ____//_Y 0 is the ordinary pushout M(f, f0) of i0,1 fqf0 CylX oo___X q X _____//Y q Y 0. It fits into a cofiber sequence Y q Y 0-! M(f, g) -! X. There results a surjection from ss(M(f, g), Z) to the evident pullback, the ker* *nel of which is the set of orbits of the right action of ss( Y, Z) x ss( Y 0, Z) on ss* *( X, Z) given by x(y, y0) = ( f)*(y)-1x( f0)*(y0). The classical homotopy coequalizer C(f, g) of parallel maps f, g :X -! Y is the homotopy pushout of the coproduct f qg :X qX -! Y qY and the codiagonal r: X qX -! X. Using a little algebra, we see that ss(C(f, g), Z) maps surjectiv* *ely to the equalizer of f* and g* with kernel isomorphic to the set of orbits of ss* *( X, Z) under the right action of ss( Y, Z) specified by xy = ( f)*(y)-1x( g)*(y). In this language, TelXn is the classical homotopy coequalizer of the identity and the coproduct of the in, both being self maps of the coproduct of the Xn. By algebraic inspection, the lim1exact sequence follows directly. A quicker, l* *ess conceptual, argument is possible, as in [71, p. 146] for example. Remark 5.6.6. Let C be an arbitrary pointed model category with (for sim- plicity) a functorial cylinder construction Cyl. If X is cofibrant, let X deno* *te the quotient Cyl(X)=(X _ X). Quillen [83] constructed a natural cogroup structure on X in Ho C . For a cofibration X -! Y between cofibrant objects, he also constructed a natural coaction of X on the quotient Y=X. One can then define cofiber sequences in HoC just as in the homotopy category of a topological model category, and one can define fiber sequences dually. 76 5. WELL-GROUNDED TOPOLOGICAL MODEL CATEGORIES The cofiber sequences and fiber sequences each give HoC a suitably weakened form of the notion of a triangulation, called a "pretriangulation" [44, 83], an* *d they are suitably compatible. If HoC is closed symmetric monoidal one can take this a step further and formulate what it means for the pretriangulation to be compati* *ble with that structure, as was done in [74] for triangulated categories. However, proving the compatibility axioms from this general point of view would at best * *be exceedingly laborious, if it could be done at all. These purely model theoretic constructions of the suspension and looping fun* *c- tors and are more closely related to the familiar topological constructions* * than might appear. The homotopy category of any model category is enriched and biten- sored over the homotopy category of spaces (obtained from the q-model structure) [36, 44], and the suspension and loop functors are given by the (derived) tensor and cotensor with the unit circle. That is, X ' X ^ S1 and X ' F (S1, X). This general point of view is not one that we wish to emphasize. For topolog* *ical model categories, the structure described in this section is far easier to defi* *ne and work with directly, as in classical homotopy theory, and we have axiomatized wh* *at is required of a model structure in order to allow the use of such standard and elementary classical methods. In our topological context, the homotopy category HoC is automatically enriched over HoK* and ( , ) is a Quillen adjoint pair th* *at descends to an adjoint pair on homotopy categories that agrees with the purely model theoretic adjoint pair just described. The crucial point for our stable work is that a large part of this structure* * exists before one constructs the desired model structure. It can therefore be used as* * a tool for carrying out that construction. This is in fact how stable model categ* *ories were constructed in [39, 61, 62], but there the compatibility between q-type and h-type structures was too evident to require much comment. The key step in our construction of the stable model structure on parametrized spectra in Chapter 12 is to show that cofiber sequences induce long exact sequences on stable homotopy groups. That will allow us to verify that the stable equivalences are suitably * *well- grounded, and from there the model axioms follow as in the earlier work just ci* *ted. CHAPTER 6 The qf -model structure on KB Introduction In this chapter, we introduce and develop our preferred q-type model structu* *re, namely the qf-model structure. It is a Quillen equivalent variant of the q-model structure that has fewer, and better structured, cofibrations. For clarity of e* *xposi- tion, we work nonequivariantly in this chapter, which is taken from [88]. We begin by comparing the homotopy theory of spaces and the homotopy theory of ex-spaces over B, starting with a comparison of the q-model structures that we have on both. In the category K of spaces, we have the familiar situati* *on described in x4.1. The homotopy category HoK that we care about is defined in terms of q-equivalences, the intrinsic notion of homotopy is given by the class* *ical cylinders, and, since all spaces are q-fibrant, the category HoK is equivalent* * to the classical homotopy category hKc of q-cofibrant spaces (or CW complexes). Since the q-cofibrations are h-cofibrations, the q-model structure and the h-mo* *del structure on K mesh smoothly. Indeed, the classical and model theoretic homotopy theory have been used in tandem for so long that this meshing of structures goes without notice. In particular, although cofiber and fiber sequences are defined* * in terms of the h-model structure while the homotopy category is defined in terms * *of the q-model structure, the compatibility seems automatic. Now consider the category KB . The homotopy category HoKB that we care about is defined in terms of q-equivalences of total spaces, but we need some j* *ustifi- cation for making that statement. A map of q-fibrant ex-spaces is a q-equivalen* *ce of total spaces if and only if all of its maps on fibers are q-equivalences. This * *reformula- tion captures the idea that the homotopical information in parametrized homotopy theory should be encoded on the fibers, and it is such fiberwise q-equivalences* * that we really care about. It is only for q-fibrant ex-spaces, or ex-spaces whose pr* *ojec- tions are at least quasifibrations, that the homotopy groups of total spaces gi* *ve the "right answer". There are three notions of homotopy in sight, h, f, and fp. The last of these is the intrinsic one defined in terms of the relevant cylinders i* *n KB , and HoKB is equivalent to the classical homotopy category hKB cf of q-cofibrant and q-fibrant objects, defined with respect to fp-homotopy. It is still true t* *hat q-cofibrations are h-cofibrations. However, it is not true that q-cofibrations * *are fp- cofibrations, and it is the latter that are intrinsic to cofiber sequences. The* * classical and model theoretic homotopy theory no longer mesh. Succinctly, the problem is that the q-model structure is not an example of a well-grounded compactly generated model category. The task that lies before us is to find a model structure which does satisfy the axioms that we set out in x* *5.5 and therefore can be used in tandem with the fp-structure to do parametrized homotopy theory. Before embarking on this, we point out the limitations of the 77 78 6. THE qf-MODEL STRUCTURE ON KB q-model structure more explicitly in x6.1. There are two kinds of problems, tho* *se that we are focusing on in our development of the model category theory, and the more intrinsic ones that account for Counterexample 0.0.1 and which cannot be overcome model theoretically. Ideally, to define the qf-model structure, we would like to take the qf-cofi* *bra- tions to be those q-cofibrations that are also f-cofibrations. However, with t* *hat choice, we would not know how to prove the model category axioms. We get closer if we try to take as generating sets of cofibrations and acyclic cofibrations t* *hose generators in the q-model structure that are f-cofibrations, but with that choi* *ce we still would not be able to prove the compatibility condition Theorem 5.5.1(i* *i). However, using this generating set of cofibrations and a subtler choice of a ge* *nerat- ing set of acyclic cofibrations, we obtain a precise enough homotopical relatio* *nship to the q-equivalences that we can prove the cited compatibility. The constructi* *on of the qf-model structure is given in x6.2, but all proofs are deferred to the fol* *lowing three sections. 6.1. Some of the dangers in the parametrized world We introduce notation for the generating (acyclic) cofibrations for the q-mo* *del structures on K =B and KB . These maps are identified in Proposition 5.1.4, sta* *rt- ing from the sets I and J in K specified in Definition 4.5.9. We then make some comments about these maps that help explain the structure of our theory. Definition 6.1.1. For maps i: C -! D and d: D -! B of (unbased) spaces, we have the restriction dOi: C -! B and may view i as a map over B. We agree to write i(d) for either the map i viewed as a map over B or the map iqid:C qB -! DqB of ex-spaces over B that is obtained by taking the coproduct with B to adjo* *in a section. In either K =B or KB , define IB to be the set of all such maps i(d)* * with i 2 I, and define JB to be the set of all such maps j(d) with j 2 J. Observe th* *at in KB , each map in JB is the inclusion of a deformation retract of spaces unde* *r, but not over, B. Warning 6.1.2. We cannot restrict the maps d to be open here. That is one of the reasons we chose KB over O*(B) in x1.3. Warning 6.1.3. The maps in IB and JB are clearly not f-cofibrations, only h-cofibrations. Looking at the NDR-pair characterization of f-cofibrations give* *n in Lemma 5.2.4, we see that, with our arbitrary projections d, there is in general* * no way to carry out the required deformation over B. Since the maps in IB and JB are maps between well-sectioned spaces, they cannot be fp-cofibrations in gener* *al, by Proposition 5.2.3(i). Remark 6.1.4. Observe that the maps i in IB or JB are closed inclusions in U , so that those maps in IB or JB which are f-cofibrations are necessarily ~f-cofibrations and therefore both f~p-cofibrations and ~h-cofibrations, by Pro* *posi- tion 5.1.8 and Theorem 5.2.8. Warning 6.1.3 shows that the q-model structure is not well-grounded since its generating (acyclic) cofibrations are not fp-cofibrations. This may sound l* *ike a minor technicality, but that is far from the case. We record an elementary exam* *ple. Counterexample 6.1.5. Let B = I and define an ex-map i: X -! Y over I by letting X = {0} q I, Y = I q I, and i be the inclusion. The second copies of* * I 6.2. THE qf MODEL STRUCTURE ON THE CATEGORY K =B 79 give the sections, and the projections are given by the identity map on each co* *py of I. This is a typical generating acyclic q-cofibration, and it is not an fp-cofi* *bration. Let Z be the pushout of i and p: X -! I, where the latter is viewed as a map of ex-spaces over I. Then Z is the one-point union I _ I obtained by identifying the points 0. The section I -! Z is not an f-cofibration, so that Z is not well- sectioned. The same is true if we replace Y by Y 0= {1=(n + 1) | n 2 N} q I and obtain Z0. The map Z0- ! CIZ0 of Z0 into its cone over I is not an h-cofibration (and therefore not a q-cofibration). Thus we cannot apply the classical gluing lemma to develop cofiber sequences, as we did in x5.6. This and related problems prevent use of the q-model structu* *re in a rigorous development of parametrized stable homotopy theory. For example, consider q-fibrant approximation. If we have a map f :X -! Y with q-fibrant approximation Rf :RX -! RY , there is no reason to believe that CB Rf is q- equivalent to RCB f. We are about to overcome model-theoretically the problems pointed out in the warnings above. Turning to the intrinsic problems that must hold in any q-type model structure, we explain why the base change functor f* and the internal sma* *sh product cannot be Quillen left adjoints. Warning 6.1.6. If f :A -! B is a map and d: D -! B is a disk over B, we have no homotopical control over the pullback A xB D -! A in general. Warning 6.1.7. In sharp contrast to the nonparametrized case, the generating sets do not behave well with respect to internal smash products, although they * *do behave well with respect to external smash products. We have (D q A) Z (E q B) ~=(D x E) q (A x B). If the projections of D and E are d and e, then the projection of D x E is d x * *e. However, if A = B, then (D q B) ^B (E q B) ~=(d x e)-1( B) q (A x B). We have no homotopical control over the space (d x e)-1( B) in general. This has the unfortunate consequence that, when we go on to parametrized spectra in Part III, we will not be able to develop a homotopically well-behaved theory of point-set level parametrized ring spectra. However, we will be able * *to develop a satisfactory point-set level theory of parametrized module spectra ov* *er nonparametrized ring spectra. 6.2. The qf model structure on the category K =B Rather than start with a model structure on K to obtain model structures on K =B and KB , we can start with a model structure on K =B and then apply Proposition 5.1.3 to obtain a model structure on KB . This gives us the opportu* *nity to restrict the classes of generating (acyclic) cofibrations present in the q-m* *odel structure on K =B to ones that are f-cofibrations, retaining enough of them that we do not lose homotopical information. This has the effect that the generating (acyclic) cofibrations are f-cofibrations between well-grounded spaces over B, * *as is required of a well-grounded model structure. Such maps have closed images, hence are ~f-cofibrations, and therefore all of the cofibrations in the resulti* *ng model structure on K =B are ~f-cofibrations. 80 6. THE qf-MODEL STRUCTURE ON KB We call the resulting model structure the "qf-model structure", where f refe* *rs to the fiberwise cofibrations that are used and q refers to the weak equivalenc* *es. The latter are the same as in the q-model structure, namely the weak equivalenc* *es on total spaces, or q-equivalences. This model structure restores us to a situ* *a- tion in which the philosophy advertised in x4.1 applies, with the q and h-model structures on spaces replaced by the qf and f-model structures on spaces over B. Since f-cofibrations in KB are fp-cofibrations, by Proposition 5.1.8, the philo* *so- phy also applies to the qf and fp-model structures on KB , or at least on UB (s* *ee Theorem 5.2.8 and Remark 5.2.9). We need some notations and recollections in order to describe the generating (acyclic) qf-cofibrations and the qf-fibrations. Notation 6.2.1. For each n 1, embed Rn-1 in Rn = Rn-1 x R by sending x to (x, 0). Let en = (0, 1) 2 Rn. For n 0, define the following subspaces of* * Rn. Rn+ = {(x, t) 2 Rn | t 0} Rn-= {(x, t) 2 Rn | t 0} Dn = {(x, t) 2 Rn | |x|2 + t2 S1}n-1= {(x, t) 2 Rn | |x|2 + t2 = 1} Sn-1+= Sn-1 \ Rn+ Sn-1-= Sn-1 \ Rn- Here R0 = {0} and S-1 = ;. We think of Sn Rn+1 as having equator Sn-1, upper hemisphere Sn+with north pole en+1 and lower hemisphere Sn-. We recall a characterization of Serre fibrations. Proposition 6.2.2. The following conditions on a map p: E -! Y in K are equivalent; p is called a Serre fibration, or q-fibration, if they are satisfie* *d. (i)The map p satisfies the covering homotopy property with respect to disks Dn; that is, there is a lift in the diagram Dn ___ff__//E;;w | w | | ww |p |fflfflw fflffl| Dn x I __h__//Y. (ii)If h is a homotopy relative to the boundary Sn-1 in the diagram above, then there is a lift that is a homotopy relative to the boundary. (iii)The map p has the RLP with respect to the inclusion Sn+- ! Dn+1 of the upper hemisphere into the boundary Sn of Dn+1; that is, there is a lift in * *the diagram Sn+___ff_//E<< z | zz |p | z | |fflfflz |fflffl Dn+1 __~h_//Y. Proof. Serre fibrations p: E - ! Y are usually characterized by the first condition. Since the pairs (Dn x I, Dn) and (Dn x I, Dn [ (Sn-1 x I)) are home- omorphic, one easily obtains that the first condition implies the second. Simil* *arly a homeomorphism of the pairs (Dn+1, Sn+) and (Dn x I, Dn) gives that the first and third conditions are equivalent. A homotopy h: Dn x I -! Y relative to the boundary Sn-1 factors through the quotient map Dn x I -! Dn+1 that sends 6.2. THE qf MODEL STRUCTURE ON THE CATEGORY K =B 81 p _______ (x, t) to (x, (2t - 1) 1 - |x|2). Conversely, any map ~h:Dn+1 -! Y gives rise * *to a homotopy h: Dn x I -! Y relative to the boundary Sn-1. It follows that the second condition implies the third. Property (ii) states that Serre fibrations are the maps that satisfy the "di* *sk lifting property" and that is the way we shall think about the qf-fibrations. I* *n view of property (iii), we sometimes abuse language by calling a map h: Dn+1 -! Y a disk homotopy. The restriction to the upper hemisphere Sn+gives the "initial di* *sk" and the restriction to the lower hemisphere Sn-gives the "terminal disk". Definition 6.2.3. A disk d: Dn -! B in K =B is said to be an f-disk if i(d): Sn-1 -! Dn is an f-cofibration. An f-disk d: Dn+1 -! B is said to be a relative f-disk if the lower hemisphere Sn-is also an f-disk, so that the restr* *iction i(d): Sn-1 -! Sn-is an f-cofibration; the upper hemisphere i(d): Sn-1 -! Sn+ need not be an f-cofibration. Definition 6.2.4. Define IfBto be the set of inclusions i(d): Sn-1 -! Dn in K =B, where d: Dn -! B is an f-disk. Define JfBto be the set of inclusions i(d): Sn+-! Dn+1 of the upper hemisphere into a relative f-disk d: Dn+1 -! B; note that these initial disks are not assumed to be f-disks. A map in K =B is s* *aid to be (i)a qf-fibration if it has the RLP with respect to JfBand (ii)a qf-cofibration if it has the LLP with respect to all q-acyclic qf-fibrat* *ions, that is, with respect to those qf-fibrations that are q-equivalences. Note that JfBconsists of relative IfB-cell complexes and that a map is a qf-fib* *ration if and only if it has the "relative f-disk lifting property." With these definitions in place, we have the following theorem. Recall the definition of a well-grounded model category from Definition 5.5.2 and recall f* *rom Propositions 5.3.7 and 5.4.9 that we have ground structures on K =B and KB with respect to which the q-equivalences are well-grounded. Also recall the definiti* *on of a quasifibration from Definition 3.5.1. Theorem 6.2.5. The category K =B of spaces over B is a well-grounded model category with respect to the q-equivalences, qf-fibrations and qf-cofibrations.* * The sets IfBand JfBare the generating qf-cofibrations and the generating acyclic qf- cofibrations. All qf-cofibrations are also f~-cofibrations and all qf-fibratio* *ns are quasifibrations. Using Proposition 5.1.3 and Proposition 5.1.4, we obtain the qf-model struct* *ure on KB . We define a qf-fibration in KB to be a map which is a qf-fibration when regarded as a map in K =B, and similarly for qf-cofibrations. Theorem 6.2.6. The category KB of ex-spaces over B is a well-grounded model category with respect to the q-equivalences, qf-fibrations, and qf-cofibrations* *. The sets IfBand JfBof generating qf-cofibrations and generating acyclic qf-cofibrat* *ions are obtained by adjoining disjoint sections to the corresponding sets of maps in K =B. All qf-cofibrations are f~-cofibrations and all qf-fibrations are quasif* *ibra- tions. Since the qf-model structures are well-grounded, they are in particular prop* *er and topological. Furthermore, the qf-cofibrant spaces over B are well-grounded 82 6. THE qf-MODEL STRUCTURE ON KB and the qf-fibrant spaces over B are quasifibrant. Since qf-cofibrations are q- cofibrations, we have an obvious comparison. Theorem 6.2.7. The identity functor is a left Quillen equivalence from K =B with the qf-model structure to K =B with the q-model structure, and similarly f* *or the identity functor on KB . We state and prove two technical lemmas in x6.3, prove that K =B is a com- pactly generated model category in x6.4, and prove that the qf-fibrations are q* *uasi- fibrations and the model structure is right proper in x6.5. The -product condi* *tion of Theorem 5.5.1(iv) follows as usual by inspection of what happens on generati* *ng (acyclic) cofibrations and, as in the case A = * of Warning 6.1.7, the projecti* *ons cause no problems here. 6.3.Statements and proofs of the thickening lemmas We need two technical "thickening lemmas". They encapsulate the idea that no information about homotopy groups is lost if we restrict from the general di* *sks and cells used in the q-model structure to the f-disks and f-cells that we use * *in the qf-model structure. Lemma 6.3.1. Let (Sm , q) be a sphere over B. Then there is an h-equivalence ~: (Sm , ~q) -! (Sm , q) in K =B such that (Sm , ~q) is an IfB-cell complex wit* *h two cells in each dimension. Lemma 6.3.2. Let (Dn, q) be a disk over B. Then there is an h-equivalence :(Dn, ~q) -! (Dn, q) relative to the upper hemisphere Sn-1+such that (Dn, ~q)* * is a relative f-disk. The rest of the section is devoted to the proofs of these lemmas. The reader may prefer to skip ahead to x6.4 to see how they are used to prove Theorem 6.2.* *5. Proof of Lemma 6.3.1. To define the map ~: (Sm , ~q) -! (Sm , q), we begin by defining some auxiliary maps for each natural number n m. They will in fact be continuous families of maps, defined for each s 2 [1_2, 1]. The parameter s * *will show that ~ is an h-equivalence. First we define the map OEn+:Dn \ Rn+-! As [ s . Sn-1+ from the upper half of the disk Dn to the union of the equatorial annulus _______________ As = Dn-1 - s . Dn-1= {(x, 0) 2 Rn :s |x| 1} and the upper hemisphere s . Sn-1+= {(x, t) 2 Rn :t 0 and |(x, t)|}= s to be the projection from the south pole -en. Similarly, we define OEn-:Dn \ Rn--! As [ s . Sn-1- to be the projection from the north pole en. The map OEn+is drawn schematically in the following picture. Each point in the upper half of the larger disk lies * *on a 6.3. STATEMENTS AND PROOFS OF THE THICKENING LEMMAS 83 unique ray from -en. The map OEn+sends it to the intersection of that ray with As [ s . Sn-1; two such points of intersection are marked with dots in the pict* *ure. ________________________________________________* *_____________________________________________________________________________* *%% _________________________________________________* *_________________________________________________________________________% __________________________________________________* *________________________________________________________________________% 7___________________________________________________* *_____________________________________________________________________________* *_____________________________________________________________%%o7 ___________________________________________________* *_____________________________________________________________________________* *_______%7 ___________________________________________________* *_____________________________________________________________________________* *________________________________%%77os.Dn ___________________________________________________* *_____________________________________________________________________________* *_______%77 ___________________________________________________* *_____________________________________________________________________________* *_____________________________________________________________%%77 __________________________________________________* *________________________________________________________________________%77 _________________________________________________* *_________________________________________________________________________%%7 ______________________________________________* *_____________________________________________________________________________* *__7-eDn * * * * n Next we use the maps OEn to define a continuous family of maps fns:Dn -! Dn for s 2 [1_2, 1] by induction on n. We let f0s:D0 -! D0 be the unique map and we define f1s:D1 -! D1 by 8 >1 if t ,s :-1 if t -s; it maps [-s, s] homeomorphically to [-1, 1]. We define fns:Dn -! Dn by 8 >>>s-1 . (x, t) if |(x, t)|, s >>>-1 n n fn-1s(OEn+(x, t))if |(x, t)| s, t 0 and |OEn+(x,,t)| s >>> -1 n n >>:s . OE- (x,it)f |(x, t)| s, t 0 and |OE-,(x, t)| = s fn-1s(OEn-(x, t))if |(x, t)| s, t 0 and |OEn-(x,.t)| s The map fnsis drawn in the following picture. The smaller ball s . Dn is mapped homeomorphically to Dn by radial expansion from the origin. Next comes the region in the upper half of the larger ball that is inside the cone and outside* * the smaller ball. Each segment of a ray from the south pole -en that lies in that r* *egion is mapped to a point which is determined by where we mapped the intersection of that ray-segment with the smaller ball (which was radially from the origin to t* *he boundary of Dn). Third is the region in the upper half of the larger ball that * *is outside the cone. Each segment of a ray from the south pole -en that lies in th* *at region is first projected to the annulus in the equatorial plane of the two bal* *ls; we then apply the previously defined map fn-1sto map the projected points to the equator of Dn. The lower half of the ball is mapped similarly. _____________________________________________________* *_____________________________________________________________________________* *_____________________________________________________________________________* *__________________________________________________ ______________________________________________________* *_______________________________________________________ ///___________________________________________________* *_____________________________________________________________________________* *_____________________________________________________________fflffl__________* *_____________________________________________________________________________* *_______________________________@ /___________________________________________________* *_________________________ffl__________________________________________________ ____________________________________________________* *__________//fflffl___________________________________________________________* *__ _____________________________________________________* *_____________________________________________________________________________* *_________________________________________________//fflffl____________________* *_____________________________________________________________________________* *_______________________________@ ______________________________________________________* *_________________________________________________________/ffl ______________________________________________________* *_____________________________________________________________________________* *__________//fflffl ______________________________________________________* *_____________________________________________________________________________* *_____________________________________________________________________________* *________________/fflsDn ______________________________________________________* *_____________________________________________________________________________* *_____________________________________________________________________________* *_____________________________________________________________________________* *_______________________________@ _____________________________________________________* *__________________________________________________________//fflffl _____________________________________________________* *_____________________________________________________________________________* *_________________________________________________/ffl ____________________________________________________* *__________//fflffl ___________________________________________________* *_________________________/ffl _________________________________________________* *_____________________________________________________________________________* *______________________________________________________________//fflfflffl -en Dn It is clear that fnsgives a homotopy from fn1=2to the identity and, given any d* *isk (Dn, q) in K =B, the map fnsinduces an h-equivalence from the f-disk (Dn, qOfn1* *=2) to the disk (Dn, q). 84 6. THE qf-MODEL STRUCTURE ON KB Finally we define the required cell structure on the domain of the desired m* *ap ~: (Sm , ~q) -! (Sm , q). For each n m, the boundary sphere (Sn, q O fn+11=2|* *Sn) is constructed from two copies of the f-disk (Dn, q O fn1=2) by gluing them along * *their boundary. The inclusions (Dn, q O fn1=2) -! (Sn, q O fn+11=2|Sn) of the two cel* *ls are given by projecting Dn to the upper hemisphere from the south pole -en+1 and, similarly, by projecting Dn to the lower hemisphere from the north pole en+1. T* *he map ~ = fm+11=2|Sm :(Sm , q O fm+11=2|Sm ) -! (Sm , q). is then the required f-cell sphere approximation. Proof of Lemma 6.3.2. Define s:Dn -! Dn for s 2 [1_2, 1] by 8 >>>s-1 . (x, t) if |(x, t)|, s <|(x, t)|-1 . (x, t) if |(x, t)| s, t 0 and,|x| s s(x, t) = > -1 n+1 n+1 >>:s . OE- (x, t) if |(x, t)| s, t 0 and |OE- (x,,t)| =* * s |OEn+1-(x, t)|-1 . OEn+1-(x,it)f |(x, t)| s, t 0 and |OEn+1-(x* *,,t)| s where OEn-is the projection as in the previous proof. Then s maps s . Dn home- omorphically to Dn, it is radially constant on the region in the upper half spa* *ce between the disks Dn and s . Dn with respect to projection from the origin, and it is radially constant on the region in the lower half space between the two d* *isks with respect to projection from the north pole. 6.4.The compatibility condition for the qf-model structure This section is devoted to the proof that K =B is a compactly generated model category. Since our generating sets IfBand JfBcertainly satisfy conditions (i) * *and (iii) of Theorem 5.5.1, it only remains to verify the compatibility condition (* *ii). That is, we must show that a map has the RLP with respect to IfBif and only if it is a q-equivalence and has the RLP with respect to JfB. Let p: E -! Y have the RLP with respect to IfB. Since all maps in JfBare relative IfB-cell complex* *es, p has the RLP with respect to JfB. To show that ssn(p) is injective, let ff: Sn -* *! E represent an element in ssn(E) such that p O ff: Sn -! Y is null-homotopic. Then there is a nullhomotopy fi :CSn -! Y that gives rise to a lifting problem Sn ___________ff__________//E i|| p|| fflffl| fflffl| Dn+1 _____//Dn+1 ~=CSn__fi_//Y where :Dn+1 -! Dn+1 is defined by ( 1 ,_ (x) = 2x if |x| 21 |x|-1 . xif |x| _2. Then i is an f-disk and we are entitled to a lift, which can be viewed as a nul* *lho- motopy of ff after we identify Dn+1 with CSn. To show that ssn(p) is surjective, choose a representative ff: Sn -! Y of an element in ssn(Y ). The projection of Y induces a projection q :Sn -! B and by Lemma 6.3.1 there is an h-equivalence ~: (Sn, ~q) -! (Sn, q) such that (Sn, ~q)* * is 6.4. THE COMPATIBILITY CONDITION FOR THE qf-MODEL STRUCTURE 85 an IfB-complex with two cells in each dimension. We may therefore assume that the source of ff is an IfB-cell complex. Inductively, we can then solve the li* *fting problems for the diagrams Sk-1D_____________//E | DDD | | DDD |p |fflfflD!! fflffl| Sk __i___//Sk____//Y, ff|Sk where Sk-1 -! Sk is the inclusion of the equator and i :Sk - ! Sk are the inclusions of the upper and lower hemispheres. We obtain a lift Sn -! E. Conversely, assume that p: E -! Y is an acyclic qf-fibration. We must show that p has the RLP with respect to any cell i in IfB. We are therefore faced wi* *th a lifting problem Sn ___ff_//E i|| |p| |fflffl |fflffl Dn+1 __fi_//Y. Identifying Dn+1 with CSn we see that fi gives a nullhomotopy of p O ff. Since ssn(p) is injective there is a nullhomotopy fl :CSn -! E such that ff = fl O i,* * but it may not cover fi. Gluing fi and p O fl along p O ff gives ffi :Sn+1 -! Y suc* *h that ffi|Sn+1+= fi and ffi|Sn+1-= pOfl. Surjectivity of ssn+1(p) gives a map : Sn+1* * -! E and a homotopy h: Sn+1 ^ I+ -! Y from p O to ffi. We now construct a diagram (-)=Sn n+1 [fl Sn+1+_____//Sn+1+[ H____//Sn+1 x 0 [ S- x_1__//E w | jwww | | | |p www | | | | --ww fflffl| fflffl| fflffl| fflffl| Dn+2 ____//_Dn+2__,___//Dn+2____OE____//Sn+1 ^ I+___h____//_Y where the downward maps, except p, are inclusions. Part of the bottom row of the diagram is drawn schematically below. Let H be the region on Sn+1-be- tween the equator Sn and the circle through e1 and -en+2 with center on the line R . (e1 - en+2). Let , be a homeomorphism whose restriction to Sn+1+maps it homeomorphically to Sn+1+[ H. Define OE: Dn+2 -! Dn+2=Sn ~=Sn+1 ^ I+ as the composite of the quotient map that identifies the equator Sn of Dn+2 to a point and a homeomorphism that maps the upper hemisphere Sn+1+to Sn+1 x0, maps H to Sn+1-x 1, and is such that (h O OE O ,)|Sn+1-= fi. The map is defined as a* *bove. _____________________________________________________________________* *_____________________________________________________________________________* *_____________________________________________________________________________* *_____________________________________________________________________________* *____ _______________________________________________________________________* *_______________________________pO____________________________________________* *____________________________________________________________________pO pOf_______________________________________________________________________* *_______________l____________________________________________pO_______________* *_____________________________________________________________________________* *________________fi ________________________________________________________________________* *____________________________________,________________________________________* *__________________OE_________________________________________________________* *_______________________________________________ ________________________________________________________________________* *_____________________________________________________________________________* *___pOff-!____________________________________________________________________* *___________________________________________-!________________________________* *_______________________________@ ________________________________________________________________________* *_____________________________________________________________________________* *_____________________________________________________________________________* *_____________________________________________________________________________* *_______________________________@ _______________________________________________________________________* *___________________________________________________________pOflpOffH_________* *_____________________________________________________________________________* *______________________ ______________________________________________________________________* *_____________________________________________________________________________* *__________________________________________________________________________pOfl fi ____________________________________________________________________* *_____________________________________________________________________________* *___________________________________________________________________________fi* *_____________________________________________________________________________* *_____________________ Dn+2 Dn+2 Sn+1 ^ I+ Since the restriction Sn -! Sn+1-~=Dn+1 of j agrees with the f-cofibration i in our original lifting problem, we see that j is a JfB-cell. Since p is a qf-fib* *ration 86 6. THE qf-MODEL STRUCTURE ON KB we get a lift in the outer trapezoid. Denote its restriction to Sn+1-~= Dn+1 by k :Dn+1 -! E. Then k solves our original lifting problem. 6.5.The quasifibration and right properness properties We have now established the qf-model structures on both K =B and KB . We will derive the right properness of K =B, and therefore of KB , from the fact t* *hat every qf-fibration is a quasifibration. Proposition 6.5.1. If p: E -! Y is a qf-fibration in K =B, then p is a quasi- fibration. Therefore, for any choice of e 2 E, there results a long exact seque* *nce of homotopy groups . .-.! ssn+1(Y, y) -! ssn(Ey, e) -! ssn(E, e) -! ssn(Y, y) -! . .-.! ss0(Y, y), where y = p(e) and Ey = p-1(y). Proof. We must prove that p induces an isomorphism ssn(p): ssn(E, Ey, e) -! ssn(Y, y) for all n 1 and verify exactness at ss0(E, e). We begin with the latter. Let * *e02 E and suppose that p(e0) is in the component of y0. Let fl :I -! Y be a path in Y from p(e0) to y0 such that fl is constant at p(e0) for time t 1_2. Let q b* *e the projection of Y . Then (I, q O fl) is a relative f-disk, and we obtain a lift ~* *fl:I -! E such that fl = p O ~fl. But then e0 is in the same component as the endpoint of* * ~fl, which lies in Ey. Now assume that n 1. Recall that an element of ssn(X, A, *) can be repre- sented by a map of triples (Dn, Sn-1, Sn-1+) -! (X, A, *). We begin by showing surjectivity. Let ff: (Dn, Sn-1) -! (Y, y) represent an element of ssn(Y, y). W* *e can view Dn as a disk over B, and Lemma 6.3.2 gives an approximation :Dn -! Dn by a relative f-disk. Then we can solve the lifting problem Sn-1+__ce_//E==- | -- |p | - f~f | |fflffl- fflffl| Dn __ffO_//Y, where the top map is the constant map at e 2 E. A lift is a map of triples ~ff:(Dn, Sn-1, Sn-1+) -! (E, Ey, e) such that p*([~ff]) = [ff]. For injectivity, let ff: (Dn, Sn-1, Sn-1+) -! (E, Ey, e) represent an elemen* *t of ssn(E, Ey, e) such that p*([ff]) = 0. Then there is a homotopy h: Dn x I -! Y r* *el Sn-1 such that h|Dn x 0 = p O ff and h maps the rest of the boundary of Dn x I * *to y. Let A = Dn x {0, 1} [ Sn-1+x I @(Dn x I) and define fi :A -! E by setting fi(x) = ff(x) if x 2 Dnx0 and fi(x) = e otherwise. We then have a homeomorphism of pairs OE: (Dn x I, A) -! (Dn+1, Sn+) and an approximation :Dn+1 -! Dn+1 by an f-disk by Lemma 6.3.2. We can now solve the lifting problem -1 Sn+fiO(OE|A)//_E== z | zz | | z ~ff | fflffl|z fflffl| Dn+1 hOOE-1O//_Y, 6.5. THE QUASIFIBRATION AND RIGHT PROPERNESS PROPERTIES 87 and this shows that [ff] = 0 in ssn(E, Ey, e). Corollary 6.5.2. The qf-model structure on K =B is right proper. Proof. Since qf-fibrations are preserved under pullbacks, this is a five le* *mma comparison of long exact sequences as in Proposition 6.5.1. CHAPTER 7 Equivariant qf -type model structures Introduction We return to the equivariant context in this chapter, letting G be a Lie gro* *up throughout. Actually, our definitions of the q and qf-model structures work for arbitrary topological groups G, but we must restrict to Lie groups to obtain st* *ruc- tures that are G-topological and behave well with respect to change of groups a* *nd smash products. A discussion of details special to the non-compact Lie case is given in x7.1, but after that the generalization from compact to non-compact Lie groups requires no extra work. However, we alert the reader that passage to sta* *ble equivariant homotopy theory raises new problems in the case of non-compact Lie groups that will not be dealt with in this book; see x11.6. The equivariant q-model structure on GKB is just the evident over and under * *q- model structure. However, the equivariant generalization of the qf-model struct* *ure is subtle. In fact, the subtlety is already relevant nonequivariantly when we s* *tudy base change along the projection of a bundle. The problem is that there are so * *few generating qf-cofibrations that many functors that take generating q-cofibratio* *ns to q-cofibrations do not take generating qf-cofibrations to qf-cofibrations. We* * show how to get around this in x7.2. For each such functor that we encounter, we find an enlargement of the obvious sets of (acyclic) generating qf-cofibrations on t* *he target of the functor so that it is still a model category, but now the functor* * does send generating (acyclic) qf-cofibrations to (acyclic) gf-cofibrations. The point is that there are many different useful choices of Quillen equiva- lent qf-type model structures, and they can be used in tandem. For all of our choices, the weak equivalences are the G -equivalences and all cofibrations are* * both q-cofibrations and f-cofibrations. Given a finite number of adjoint pairs with * *com- posable left adjoints such that each is a Quillen adjunction with its own choice of qf-type model structure, we can successively expand generating sets in target categories of the left adjoints to arrange that the composite be one of Quillen* * left adjoints with respect to well chosen qf-type model structures. In x7.2, we describe the qf(C )-model structure associated to a "generating * *set" C of G -complexes. Each such model structure is G-topological. In x7.3, we show that external smash products are Quillen adjunctions when C is a "closed" gener- ating set, as can always be arranged, and we show that all base change adjuncti* *ons (f!, f*) are Quillen adjunctions. We show further that there are generating set* *s for which (f*, f*) is a Quillen adjunction when f is a bundle with cellular fibers.* * In x7.4, we show similarly that various change of group functors are given by Quil* *len adjunctions when the generating sets are well chosen. In x7.5, we show that HoG* *KB has the properties required for application of the Brown representability theor* *em. 88 7.1. FAMILIES AND NON-COMPACT LIE GROUPS 89 Those adjunctions between our basic functors that are not given by Quillen adjo* *int pairs in any choice of qf-model structure are studied in Chapter 9. 7.1. Families and non-compact Lie groups Two sources of problems in the equivariant homotopy theory of general topolo* *gy groups G are that we only know that orbit types G=K are H-CW complexes for H G when G is a Lie group and K is a compact subgroup and we only know that a product of orbits G=H x G=K is a G-CW complex when G is a Lie group and K (or H) is a compact subgroup. This motivates us to restrict to Lie groups, for which these conclusions are ensured by Theorem 3.3.2 and Lemma 3.3.3. The compactness requirements force us to restrict orbit types when we prove properties of our model structures, and the family G of all compact subgroups of our Lie group G plays an important role. We recall the relevant definitions, wh* *ich apply to any topological group G and are familiar and important in a variety of contexts. They provide a context that allows us to work with non-compact Lie groups with no more technical work than is required for compact Lie groups. A family F in G is a set of subgroups that is closed under passage to subgro* *ups and conjugates. An F -space is a G-space all of whose isotropy groups are in F . An F -equivalence is a G-map f such that fH is a weak equivalence for all H 2 * *F . If X is an F -space, then the only non-empty fixed point sets XH are those for groups H 2 F . In particular, an F -equivalence between F -spaces is the same as a q-equivalence. For based G-spaces, the definition of an F -space must be alte* *red to require that all isotropy groups except that of the G-fixed base point must * *be in F . The notion of an F -equivalence remains unchanged. A map in GK =B or GKB is an F -equivalence if its map of total G-spaces is an F -equivalence. If B is an F -space, then so is any G-space X over B and any fiber Xb. The only orbits that can then appear in our parametrized theory a* *re of the form G=H with H 2 F and the only non-empty fixed point sets XH are those for groups H 2 F . In particular, H must be subconjugate to some Gb. An F -equivalence of G-spaces over an F -space B is the same as a q-equivalence. It is well-known that equivariant q-type model structures generalize natural* *ly to families. One takes the weak equivalences to be the F -equivalences, and one restricts the orbits G=H that appear as factors in the generating (acyclic) cof* *ibra- tions to be those such that H 2 F . The resulting cell complexes are called F -* *cell complexes. Restricting tensors from G-spaces to F -spaces, we obtain a restrict* *ion of the notion of a G-topological model category to an F -topological model cate* *gory that applies here; see Remark 10.3.5. Proper G-spaces are particularly well-behaved G -spaces, where G is the fami* *ly of compact subgroups of our Lie group G, and G -cell complexes are proper G- spaces. Restricting base G-spaces to be proper, or more generally to be G -spac* *es, has the effect of restricting all relevant orbit types G=H to ones where H is c* *ompact. However, this is too restrictive for some purposes. For example, we are interes* *ted in developing nonparametrized equivariant homotopy theory for non-compact Lie groups G. Here B = * is a G-space which, in the unbased sense, is not a G -spac* *e. We therefore do not make the blanket assumption that B is a G -space. We give the q-model structure in complete generality, in Theorem 7.2.3, but after * *that we restrict to G -model structures throughout. That is, our weak equivalences w* *ill be the G -equivalences. This ensures that, after cofibrant approximation, our t* *otal 90 7. EQUIVARIANT qf-TYPE MODEL STRUCTURES G-spaces are G -spaces. This convention enables us to arrange that all of our m* *odel categories are G-topological. Everything in this chapter applies more generally* * to the study of parametrized F -homotopy theory for any family F ; see Remark 7.2.* *14. The reader may prefer to think in terms of either the case when B = * or the case when B is proper. Indeed, in order to resolve the problems intrinsic to t* *he parametrized context that are described in the Prologue, which we do in Chapter 9, it seems essential that we restrict to proper actions on base spaces. The re* *ason is that Stasheff's Theorem 3.4.2 relating the equivariant homotopy types of fib* *ers and total spaces plays a fundamental role in the solution. Alternatively, the r* *eader may prefer to focus just on compact Lie groups, reading q-equivalence instead of G -equivalence and G-space instead of G -space. 7.2.The equivariant q and qf-model structures Recall from Definition 4.5.9 that the sets I and J of generating cofibrations and generating acyclic cofibrations of G-spaces are defined as the sets of all * *maps of the form G=H x i, where i is in the corresponding set I or J of maps of spac* *es. Definition 7.2.1. Starting from the sets I and J of maps of G-spaces, define sets IB and JB of maps of ex-G-spaces over B in exactly the same way that their nonequivariant counterparts were defined in terms of the sets I and J of maps of spaces in Definition 6.1.1. Note that if B is a G -space, then only orbits G=H * *with H compact appear in the sets IB and JB . Taking Y = B in the usual composite adjunction (7.2.2) GK (G=H x T, Y ) ~=HK (T, Y ) ~=K (T, Y H) for non-equivariant spaces T and G-spaces Y , we can translate back and forth between equivariant homotopy groups and cells for G-spaces over B on the one ha* *nd and nonequivariant homotopy groups and cells for spaces over BH on the other. Maps in each of the equivariant sets specified in Definition 7.2.1 correspond by adjunction to maps in the nonequivariant set with the same name. Systematically using this translation, it is easy to use Theorem 4.5.5 to generalize the q-mod* *el structures on K =B and KB to corresponding model structures on GK =B and GKB . We obtain the following theorem. Theorem 7.2.3 (q-model structure). The categories GK =B and GKB are compactly generated proper G -topological model categories whose q-equivalences* *, q- fibrations, and q-cofibrations are the maps whose underlying maps of total G-sp* *aces are q-equivalences, q-fibrations, and q-cofibrations. The sets IB and JB are t* *he generating q-cofibrations and generating acyclic q-cofibrations, and all q-cofi* *brations are ~h-cofibrations. If B is a G -space, then the model structure is G-topologi* *cal. To show that the q-model structures are G -topological, and G-topological if* * B is a G -space, we must inspect the maps i j in GK =B, where i is a generating q-cofibration in GK =B and j is a generating cofibration in GK . They have the form i j :G=H x G=K x @(Dm x Dn) -! G=H x G=K x Dm x Dn given by the product of G=H xG=K with the inclusion of the boundary of Dm xDn. By Lemma 3.3.3, G=H x G=K is a proper G-space if H or K is compact. Since we 7.2. THE EQUIVARIANT q AND qf-MODEL STRUCTURES 91 are assuming that G is a Lie group, we can then triangulate G=H x G=K as a G - CW complex and use the triangulation to write i j as a relative IB -cell comple* *x. The case when either i or j is acyclic works in the same way. As explained in Warning 6.1.7, there is no problem with projection maps in this external contex* *t. Moreover, if i is an f-cofibration, then so is i j, as we see from the fiberwis* *e NDR characterization. One might be tempted to generalize the qf-model structure to the equivariant context in exactly the same way as we just did for the q-model structure. This certainly works to give a model structure. However, there is no reason to think that it is either G or G -topological. The problem is that we need i j above to* * be a qf-cofibration when i is a generating qf-cofibration, and triangulations into f* *-cells are hard to come by. Therefore the G-CW structure on G=H x G=K will rarely produce a relative IfB-cell complex. This means that we must be careful when selecting the generating (acyclic) qf-cofibrations if we want the resulting mod* *el structure to be G-topological. We will build the solution into our definition * *of qf-type model structures, but we need a few preliminaries. We shall make repeated use of the adjunction (7.2.4) GK (C x T, Y ) ~=K (T, MapG (C, Y )) for non-equivariant spaces T and G-spaces C and Y . This is a generalization of (7.2.2). Taking Y = B, we note in particular that it gives a correspondence bet* *ween maps f :T -! T 0over Map G(C, B) and G-maps idx f :C x T -! C x T 0over B. Lemma 7.2.5. If C is a G -cell complex, then the functor MapG (C, -): GK - ! K preserves all q-equivalences. Proof. The functor Map (C, -) is a Quillen right adjoint since the q-model structure on GK is G -topological. The G-fixed point functor is also a Quillen right adjoint, for example by Proposition 7.4.3 below. The composite Map G(C, -) therefore preserves q-equivalences between q-fibrant G-spaces. However, every G- space is q-fibrant. Observe that Lemma 3.3.3 gives that the collection of G -cell complexes is c* *losed under products with arbitrary orbits G=H of G. Definition 7.2.6. Let OG denote the set of all orbits G=H of G. Any set C of G -cell complexes in GK that contains all orbits G=K with K 2 G and is closed under products with arbitrary orbits in OG is called a generating set. It is a * *closed generating set if it is closed under finite products. The closure of a generati* *ng set C is the generating set consisting of the finite products of the G -cell comple* *xes in C . We define sets of generating qf(C )-cofibrations and acyclic qf(C )-cofibra* *tions in GK =B associated to any generating set C as follows. (i)Let IfB(C ) consist of the maps (idx i)(d): C x Sn-1 -! C x Dn such that C 2 C , d: C x Dn -! B is a G-map, i is the boundary inclusion, and the corresponding map i over Map G(C, B) is a generating qf-cofibration in K =Map G(C, B); that is, i must be an f-cofibration. (ii)Similarly let JfB(C ) consist of the maps (idx i)(d): C x Sn+-! C x Dn+1 92 7. EQUIVARIANT qf-TYPE MODEL STRUCTURES such that C 2 C , d: C x Dn+1 -! B is a G-map, i is the inclusion, and the corresponding map i over Map G(C, B) is a generating acyclic qf-cofibration in K =Map G(C, B). Adjoining disjoint sections, we obtain the corresponding sets IfB(C ) and JfB(C* * ) in GKB . Fix a generating set C . We define a qf-type model structure based on C , called the qf(C )-model structure. Its weak equivalences are the G -equivalenc* *es, which are the same as the q-equivalences when B is a G -space. We define the qf(C )-fibrations. Definition 7.2.7. A map f in GK =B is a qf(C )-fibration if Map G(C, f) is a qf-fibration in K =Map G(C, B) for all C 2 C . A map in GKB is a qf(C )- fibration if the underlying map in GK =B is one. In either category, a map f is* * a G -quasifibration if fH is a quasifibration for H 2 G . Theorem 7.2.8 (qf-model structure). For any generating set C , the categories GK =B and GKB are well-grounded (hence G-topological) model categories. The weak equivalences and fibrations are the G -equivalences and the qf(C )-fibrati* *ons. The sets IfB(C ) and JfB(C ) are the generating qf(C )-cofibrations and the gen* *erating acyclic qf(C )-cofibrations. All qf(C )-cofibrations are both q-cofibrations a* *nd f~- cofibrations, and all qf(C )-fibrations are G -quasifibrations. Proof. Recall from Proposition 5.4.9 that the q-equivalences in GK =B and GKB are well-grounded with respect to the ground structure given in Defini- tion 5.3.6 and Proposition 5.3.7. It follows that the G -equivalences are also * *well- grounded. It suffices to verify conditions (i)-(iv) of Theorem 5.5.1. The acycl* *icity condition (i) is obvious. Consider the compatibility condition (ii). By the adjunction (7.2.4), a map f has the RLP with respect to IfB(C ) if and only if Map G(C, f) has the RLP with respect to IfMapG(C,B)for all C 2 C . By the compatibility condition for the nonequivariant qf-model structure, that holds if and only if Map G(C, f) is* * a q-equivalence and has the LLP with respect to JfMapG(C,B)for all C 2 C . By Lemma 7.2.5, Map G(C, f) is a q-equivalence if f is one. Conversely, if Map G(C* *, f) is a q-equivalence for all C 2 C , then the case C = G=K shows that fK is a q- equivalence for every compact K and thus f is a G -equivalence. By the adjuncti* *on again, we see that f has the RLP with respect to IfB(C ) if and only if f is a G -equivalence which has the RLP with respect to JfB(C ). The fiberwise NDR characterization of f~-cofibrations given in Lemma 5.2.4 shows that IfB(C ) and JfB(C ) consist of ~f-cofibrations, as stipulated in (ii* *i). More precisely, if (u, h), u: Dn -! I and h: Dn x I -! Dn, represents (Dn, Sn-1) as a fiberwise NDR-pair over MapG (C, B), then the map v = uOss :CxDn -! Dn -! I and the homotopy given by the maps idxhtover B corresponding to the htrepresent (C x Dn, C x Sn-1) as a fiberwise NDR pair over B. Since Map G(G=K, f) ~= fK is a nonequivariant qf-fibration for any qf(C )- fibration f, every qf(C )-fibration is a G -quasifibration by Proposition 6.5.1* *. That the model structure is right proper follows as in Corollary 6.5.2. Finally, we must verify the -product condition (iv). The relevant maps i j, i: C x Sm-1 - ! C x Dm and j :G=H x Sn-1 -! G=H x Dn, 7.2. THE EQUIVARIANT q AND qf-MODEL STRUCTURES 93 are of the form C x G=H x k :C x G=H x @(Dm x Dn) -! C x G=H x Dm x Dn, where k is the boundary inclusion. Now C x G=H 2 C by the closure property of the generating set, so we don't need to triangulate. The projection of the t* *ar- get factors through the projection of the target C x Dm of i. To see that the corresponding map k over Map G(C x G=H, B) is an ~f-cofibration, let (u, h) rep* *re- sent (Dm , Sm-1 ) as a fiberwise NDR-pair over Map G(C, B) and let (v, j) repre* *sent (Dn, Sn-1) as an NDR-pair; we can think of the latter as a fiberwise NDR-pair o* *ver * = Map G(G=H, *). Then the usual product pair representation (for example, [71, p. 43]) exhibits k as a fiberwise NDR over Map G(C, B) x MapG (G=H, *) and thus, by the factorization of the projection of i j, also over Map G(C x G=H, B x *). Theorem 7.2.9. If C C 0is an inclusion of generating sets, then the identi* *ty functor is a left Quillen equivalence from GK =B with the qf(C )-model structure to GK =B with the qf(C 0)-model structure. The identity functor is also a left Quillen equivalence from GK =B with the qf(C )-model structure to GK =B with the q-model structure. Both statements also hold for the identity functor on GK* *B . Proof. The first statement is obvious. For the second, if idCxi is a genera* *ting qf(C )-cofibration, then C is a G -cell complex and we can use the triangulatio* *n to write idCx i as a relative IB -cell complex. Theorem 7.2.10. For any C , the identity functor is a left Quillen adjoint from GK =B with the qf(C )-model structure to GK =B with the f-model structure. Similarly, the identity functor is a left Quillen adjoint from GKB with the qf(* *C )- model structure to GKB with the fp-model structure. The last result, which implements the philosophy of x4.1, is false for the q* *-model structures. Remark 7.2.11. The smallest generating set C is the set of all (non-empty) finite products of orbits G=H of G such that at least one of the factors has H compact. Clearly it is a closed generating set. Henceforward, by the qf-model structure, we mean the qf(C )-model structure associated to this choice of C . * *In the nonequivariant case, this is the qf-model structure of the previous chapter. Remark 7.2.12. In the nonparametrized setting, the G -model structure associ- ated to the q-model structure and the qf(C )-model structures on GK = GK =* co- incide, and similarly for GK*. This holds since the f-cofibrations and h-cofibr* *ations over a point coincide and since the C 2 C for any choice of C are G -cell compl* *exes. Of course, the qf(C )-model structures have more generating (acyclic) cofibrati* *ons. Remark 7.2.13. It might be useful to combine the various qf(C )-model struc- tures by taking the union of the qf(C )-cofibrations over some suitable collect* *ion of generating sets C and so obtain a "closure" of the qf-model structure whose cofibrations are as close as possible to being the intersection of the q-cofibr* *ations with the ~f-cofibrations. We do not know whether or not that can be done. Remark 7.2.14. As noted in the introduction, we can generalize the q and qf(C )-model structures to the context of families F . We generalize the q-model structure to the F -model structure by taking the F -equivalences and F -fibrat* *ions and by restricting the sets IB and JB to be constructed from orbits G=H with 94 7. EQUIVARIANT qf-TYPE MODEL STRUCTURES H 2 F . The resulting model structure will then be (F \ G )-topological and F - topological if the base space B is a G -space. To generalize the qf(C ) model structure, we take the weak equivalences to be the F \ G -equivalences and we require the generating set C to consist of F \ G* * - cell complexes, to contain the orbits G=K for K 2 F \ G , and to be closed under products with orbits G=K where K 2 F . With that modification, everything else above goes through unchanged. 7.3. External smash product and base change adjunctions The following results relate the q and qf(C )-model structures to smash prod- ucts and base change functors and show that various of our adjunctions are given by Quillen adjoint pairs and therefore induce adjunctions on passage to homotopy categories. For uniformity, we must understand the q-model structure to mean the associated G -model structure, although many of the results do apply to the full q-model structure. Those results that refer to q-equivalences by name work equa* *lly well for G -equivalences. Most of the results in this section and the next appl* *y both to the G -model structure and to the qf(C )-model structure for any generating * *set C . We agree to omit the q or qf(C ) from the notations in those cases. In other cases, we will have to restrict to well chosen generating sets C . With these conventions, our first result is clear from the fact that our mod* *el structures are G-topological. Proposition 7.3.1. For a based G-CW complex K, the functor (-) ^B K preserves cofibrations and acyclic cofibrations, hence the functor FB (K, -) pr* *eserves fibrations and acyclic fibrations. Thus ((-) ^B K, FB (K, -)) is a Quillen adjo* *int pair of endofunctors of GKB . For the rest of our results, recall from Lemma 5.4.6 that a left adjoint that takes generating acyclic cofibrations to acyclic cofibrations preserves acyclic* * cofi- brations. The following two results apply to the qf(C )-model structure for any closed generating set C . Proposition 7.3.2. If i: X -! Y and j :W -! Z are cofibrations over base G-spaces A and B, then i j :(Y Z W ) [XZW (X Z Z) -! Y Z Z is a cofibration over A x B which is acyclic if either i or j is acyclic. Proof. It suffices to inspect i j for generating (acyclic) cofibrations as * *was done for the case A = * in the proof of Theorem 7.2.8. For generating cofibrati* *ons, the argument there generalizes without change to this setting. The assumption t* *hat C is closed avoids the need for triangulations here. For the acyclicity, it su* *ffices to work in the q-model structure, for which the conclusion is both more general and easier to prove. There it is easily checked using triangulations of product* *s of G -cell complexes that if i is a generating cofibration and j is a generating a* *cyclic cofibration, then i j is an acyclic cofibration. Of course, by Warning 6.1.7, the analogue for internal smash products fails. Taking W = *B and changing notations, we obtain the following special case. Corollary 7.3.3. Let Y be a cofibrant ex-space over B. Then the functor (-) Z Y from ex-spaces over A to ex-spaces over A x B preserves cofibrations and 7.3. EXTERNAL SMASH PRODUCT AND BASE CHANGE ADJUNCTIONS 95 acyclic cofibrations, hence the functor F~(Y, -) from ex-spaces over A x B to e* *x- spaces over A preserves fibrations and acyclic fibrations. Thus ((-) Z Y, ~F(Y,* * -)) is a Quillen adjoint pair of functors between GKA and GKAxB . The next two results apply to the qf(C )-model structures for any C , provid* *ed that we use the same generating set C for both GKA and GKB . Proposition 7.3.4. Let f :A -! B be a G-map. Then the functor f!preserves cofibrations and acyclic cofibrations, hence (f!, f*) is a Quillen adjoint pair* *. The functor f! also preserves q-equivalences between well-sectioned ex-spaces. If f* * is a q-fibration, then the functor f* preserves all q-equivalences. Proof. If (D, p) is a space over A, then f!((D, p) q A) = (D, f O p) q B. Therefore f!takes generating (acyclic) q-cofibrations over A to such maps over * *B. If (u, h) represents (Dn, Sn-1) as a fiberwise NDR-pair over Map G(C, A), then, after composing the projection maps with Map G(C, A) -! Map G(C, B), it also represents (Dn, Sn-1) as a fiberwise NDR-pair over Map G(C, B). It follows that f! also preserves the generating (acyclic) qf-cofibrations. Recall that the we* *ll- sectioned ex-spaces are those that are ~f-cofibrant and that f-cofibrations are* * h- cofibrations. Since f!X is defined by a pushout in GK , the gluing lemma in GK implies that f!preserves q-equivalences between well-sectioned ex-spaces. If f is a q-fibration and k :Y -! Z is a q-equivalence of ex-spaces over B, consider the diagram f*k mmf*Z6_____________//Z677oo mmmmm jj okoooo'' f*Y m______jj____//_Yoo ''' 66 j 33 '' 666 jj 333 '' 66 jj 33 '' aejjae6 s'sss3 A _______f______//B. The relation (A xB Z) xZ Y ~= A xB Y shows that the top square is a pullback, and the pullback f*Z -! Z of f is a q-fibration. Since the q-model structure on the category of G-spaces is right proper, it follows that f*k is a q-equivalenc* *e. Proposition 7.3.5. If f :A -! B is a q-equivalence, then (f!, f*) is a Quill* *en equivalence. Proof. The conclusion holds if and only if the induced adjunction on homo- topy categories is an adjoint equivalence [44, 1.3.3], so it suffices to verify* * the usual defining condition for a Quillen adjunction in either model structure. The cond* *ition for the other model structure follows formally. We choose the q-model structure. Let X be a q-cofibrant ex-space over A and Y be a q-fibrant ex-space over B, so that A -! X is a q-cofibration and Y -! B is a q-fibration of G-spaces. Since t* *he model structure on the category of G-spaces is left and right proper, inspectio* *n of the defining diagrams in Definition 2.1.1 shows that the canonical maps X -! f!X and f*Y - ! Y of total spaces are q-equivalences. For an ex-map k :f!X -! Y 96 7. EQUIVARIANT qf-TYPE MODEL STRUCTURES with adjoint "k:X -! f*Y , the commutative diagram X ______//f!X "k|| |k| fflffl| fflffl| f*Y ______//Y of total spaces then implies that k is a q-equivalence if and only if "kis a q- equivalence. In view of Counterexample 0.0.1, we can at best expect only a partial and restricted analogue of Proposition 7.3.4 for (f*, f*). We first give a result f* *or the q-model structure and then show how to obtain the analogue for the qf(C )-model structures using well chosen generating sets C . Proposition 7.3.6. Let f :A -! B be a G-bundle such that B is a G -space and each fiber Ab is a Gb-cell complex. Then (f*, f*) is a Quillen adjoint pair* * with respect to the q-model structures. Moreover, if the total space of an ex-G-spac* *e Y over B is a G -cell complex, then so is the total space of f*Y . Proof. Since f is a q-fibration, f* preserves q-equivalences. It therefore * *suf- fices to show that f* takes generating cofibrations in IB to relative IA -cell * *com- plexes. Observe first that if OE: G=H -! B is a G-map with OE(eH) = b, then H Gb and the pullback G-bundle OE*f :f*(G=H, OE) -! G=H of f along OE is G-homeomorphic to G xH Ab -! G=H. We can triangulate orbits in a Gb-cell decomposition of Ab as H-CW complexes, by Theorem 3.3.2, and so give Ab the structure of an H-cell complex. Then G xH Ab has an induced structure of a G -c* *ell complex and thus so does f*(G=H, OE). For a space d: E -! B over B with associated ex-space E qB over B, we have f*(E q B) = f*E q A. Let E = G=H x Dn and let i: G=H -! G=H x Dn be the inclusion i(gH) = (gH, 0). The composite d O i is a map OE as above. Since * *the identity map on G=H x Dn is homotopic to the composite i O ss :G=H x Dn -! G=H x Dn, where ss is the projection, the pullback G-bundle d*f :f*(E, d) -! E is equivalent to the pullback bundle (OE O ss)*f :f*(E, OE O ss) -! E. But the * *latter is the product of OE*f :f*(G=H, OE) -! G=H and the identity map of Dn as we see from the following composite of pullbacks f*(G=H x Dn, OE O ss)___//f*(G=H, OE)__//f*(G=H x Dn, d)____//A (OEOss)*f|| OE*f|| d*f|| |f| fflffl| fflffl| fflffl| fflffl| G=H x Dn _____ss____//_G=H____i_____//_G=H x Dn__d___//_B. The G -cell structure on f*(G=H, OE) gives a canonical decomposition of the inc* *lusion f*(G=H, OE) x Sn-1 -! f*(G=H, OE) x Dn as a relative G -cell complex. The last statement follows by applying this analysis inductively to the cells of Y . The previous result fails for the qf-model structure. In fact, it already f* *ails nonequivariantly for the unique map f :A -! *, where A is a CW-complex. The proof breaks down when we try to use a cell decomposition of A (the fiber over * **) to decompose cells AxSn-1 -! AxDn over A as relative IfA-cell complexes. Similarly, the equivariant proof above breaks down when we try to use the G-cell structure 7.4. CHANGE OF GROUP ADJUNCTIONS 97 of f*(G=H, OE) to obtain a relative IfA-cell complex. Note, however, that there is no problem when the fibers are homogeneous spaces G=H; the nonequivariant analogue is just the trivial case when f is a homeomorphism, but principal bund* *les and projections G=H x B -! B give interesting equivariant examples. For the general equivariant case, we choose a closed generating set C (f) that depends on the G-bundle f and a given closed generating set C . Using the qf(C )-model structures on GKA and GKB , we then recover the Quillen adjunction. Construction 7.3.7. Let f :A -! B be a G-bundle such that B is a G -space and each fiber Ab is a Gb-cell complex and let C be a closed generating set. We construct the set C (f) inductively. Let C (f)0 = C and suppose that we have constructed a set C (f)n of G -cell complexes in GK that is closed under both f* *inite products and products with arbitrary orbits G=H of G. Let An = {f*(C, OE) | C 2 C (f)n and OE 2 GK (C,}B). Then let C (f)n+1 consist of all finite products of spaces in C (f)n [ An. Note* * that C (f)n+1 contains C (f)n and that the f*(C, OE) are GS-cell complexes by the la* *st statement of Proposition 7.3.6. Finally, let C (f) = C (f)n. Clearly C (f) C is a closed generating set that contains f*(C, OE) for all C 2 C (f) and all G-* *maps OE: C -! B. Proposition 7.3.8. Let f :A -! B be a G-bundle such that B is a G -space and all fibers Ab are Gb-cell complexes. Then (f*, f*) is a Quillen adjoint pai* *r with respect to the qf(C (f))-model structures on GKA and GKB . Proof. Reexamining the proof of Proposition 7.3.6, but starting with a map d: E = C x Dn -! B where C 2 C (f), we see that f*E ~=f*(C, OE) x Dn where OE = d O i. Since f*(C, OE) is a G -cell complex in C (f), it remains on* *ly to show that f*(C, OE) x Sn-1 -! f*(C, OE) x Dn is an f-cofibration. Let (u, h) represent (Dn, Sn-1) as a fiberwise NDR-pair over Map G(C, B). Applying f* to the corresponding maps ht:C x Dn - ! C x Dn over B, we obtain maps f*ht:f*E -! f*E over A. Under the displayed isomorphism, these maps give a homotopy f*h: Dn x I -! Dn that, together with u, represents (Dn, Sn-1) as a fiberwise NDR-pair over Map G(f*(C, OE), A). 7.4. Change of group adjunctions We consider change of groups in the q and the qf-model structures, starting with the former. The context of the following results is given in xx2.3 and 2.4. Proposition 7.4.1. Let ` :G -! G0 be a homomorphism of Lie groups. The restriction of action functor `*: G0KB -! GK`*B preserves q-equivalences and q-fibrations. If B is a G 0-space, then it also pr* *eserves q-cofibrations. Proof. Since (`*A)H = A`(H) for any subgroup H of G and a map f : X -! Y of G-spaces is a q-equivalence or q-fibration if and only if each fH is a q-equivalence or q-fibration, it is clear that `* preserves q-equivalences an* *d q- fibrations. To study q-cofibrations, recall that ` factors as the composite of* * a 98 7. EQUIVARIANT qf-TYPE MODEL STRUCTURES quotient homomorphism, an isomorphism, and an inclusion. If ` is an inclusion and H0 is a compact subgroup of G0, then we can triangulate G0=H0 as a G-CW complex by Theorem 3.3.2. If ` is a quotient homomorphism with kernel N and H0 is a subgroup of G0, then H0= H=N for a subgroup H of G and `*(G0=H0) = G=H so that no triangulations are required. Thus in both of these cases, `* takes g* *ener- ating q-cofibrations to q-cofibrations. Since `* is also a left adjoint in both* * cases, it preserves q-cofibrations in general. Remark 7.4.2. We did not require `*B to be a G -space in Proposition 7.4.1. However, if the kernel of ` is compact and B is a G 0-space, then `*B is a G -s* *pace. Indeed, ` is then a proper map and Gb = `-1(G0b) is compact since G0bis compact. The restriction to compact kernels is the price we must pay in order to stay in the context of compact isotropy groups. We might instead consider G0-spaces B such that the isotropy groups of both B as a G0-space and `*B as a G-space are compact, but the assumption on `*B would be unnatural. Note however that one of the main reasons for restricting to compact isotropy groups is to obtain G-CW structures. If X is a G0-CW complex where G0= G=N is a quotient group of G, then `*X is a G-CW complex with the same cells since the relevant orbits G0=H0 can be identified with G=H, where H0= H=N. For the qf-model structures, and to study adjunctions, it is convenient to c* *on- sider quotient homomorphisms and inclusions separately. For the former, we con- sider the adjunctions of Proposition 2.4.1. Proposition 7.4.3. Let ffl: G -! J be a quotient homomorphism of G by a normal subgroup N. For a G-space B, consider the functors (-)=N :GKB -! JKB=N and (-)N :GKB -! JKBN . Let j :BN - ! B be the inclusion and p: B -! B=N be the quotient map. Then ((-)=N, p*ffl*) and (j!ffl*, (-)N ) are Quillen adjoint pairs with respect to t* *he q-model structures on both GKB and JKB=N . Let CG and CJ be generating sets of G-cell complexes and J-cell complexes. Consider GKB with the qf(CG )-model structure and JKB=N and JKBN with the qf(CJ)-model structure. Then (i)((-)=N, p*ffl*) is a Quillen adjunction if C=N 2 CJ for C 2 CG . (ii)(j!ffl*, (-)N ) is a Quillen adjunction if "*C 2 CG for C 2 CJ. Proof. Since (j!, j*) and (p!, p*) are Quillen adjoint pairs in both the q * *and the qf contexts, it suffices to consider the case when N acts trivially on B, so that j and p are identity maps. Then "* is right adjoint to (-)=N and left adjo* *int to (-)N . The properties of "* in the previous result give the conclusion for * *the q-model structures. The functors "* and (-)N preserve q-equivalences. Since Map G(C, "*f0) ~=Map G(C=N, f0) and Map J(C0, fN ) ~=Map G(ffl*C0, f) for a J-map f0 and a G-map f, the conditions on generating sets in (i) and (ii) ensure that "* and (-)N preserve the relevant qf-fibrations. Remark 7.4.4. In (i), we can take CJ to consist of all finite products of qu* *o- tients C=N with C 2 CG and orbits J=H to arrange that CJ be closed and contain these C=N. In (ii), we can take CG to consist of all products of pullbacks "*C * *for C 2 CJ with finite products of orbits G=H. This set will be closed if CJ is clo* *sed since "* preserves products. 7.4. CHANGE OF GROUP ADJUNCTIONS 99 Using Proposition 7.4.3 in conjunction with the additional change of group relations of Propositions 2.4.2 and 2.4.3, we obtain the following compendium of equivalences in homotopy categories. Proposition 7.4.5. Let A and B be G-spaces. Let j :BN -! B be the in- clusion and p: B -! B=N be the quotient map, and let f :A -! B be a G-map. Then, for ex-G-spaces X over A and Y over B, (p!Y )=N ' Y=N, (f!X)=N ' (f=N)!(X=N), (j*Y )N ' Y N, (f*Y )N ' (fN )*(Y N), (p*Y )N ' Y=N, (f!X)N ' (fN )!(XN ), where, for the last equivalence on the left, B must be an N-free G-space. Proof. The equivalences displayed in the first line come from isomorphisms between Quillen left adjoints and are therefore clear. Similarly the equivalenc* *es in the second line come from isomorphisms between Quillen right adjoints. The first equivalence in the third line (in which we have changed notations from Proposi- tion 2.4.3) comes from an isomorphism between a Quillen right adjoint on the le* *ft hand side, by Proposition 7.3.6, and a Quillen left adjoint on the right hand s* *ide and therefore also descends directly to an equivalence on homotopy categories. For * *the last equivalence, note that (-)N preserves all q-equivalences and also preserve* *s well- grounded ex-spaces and that (fN )!preserves q-equivalences between well-grounded ex-spaces. Letting Q and R denote cofibrant and fibrant replacement functors, as usual, it follows that the maps (R(f!X))N - (f!X)N ~=(fN )!(XN ) - (fN )!(Q(XN )) are q-equivalences on ex-spaces X that are qf-fibrant and qf-cofibrant. As note* *d in the proof of Proposition 2.4.2, the point set level isomorphism (f!X)N ~=(fN )!* *(XN ) is only valid for an ex-space X whose section is a closed inclusion. However, i* *f X is qf-cofibrant, then it is compactly generated and this holds by Lemma 1.1.3(i* *). Thus the equivalence holds in general in the homotopy category. The context for the next result is given in Definition 2.3.7 and Proposition* * 2.3.9. Proposition 7.4.6. Let ': H -! G be the inclusion of a subgroup and let A be an H-space. The adjoint equivalence ('!, *'*) relating HKA and GK'!Ais a Quillen equivalence in the q-model structures and also in the qf(CH ) and qf(CG* * )- model structures for any generating sets CH and CG of H-cell complexes and G-ce* *ll complexes such that '!C = GxH C 2 CG for C 2 CH . If A is proper and completely regular, then the functor '!is also a Quillen right adjoint with respect to the* * q and qf-model structures. Proof. Recall that :A -! '*'!A = G xH A is the natural inclusion of H- spaces and that ( !, *) is a Quillen adjunction in both the q and qf contexts.* * The functor '* preserves q-equivalences and q-fibrations. It takes qf(CG )-fibratio* *ns to qf(CH )-fibrations when '!C 2 CG for C 2 CH since Map H (C, '*f) ~=Map G('!C, f). To show that ('!, *'*) is a Quillen equivalence, we may as well check the defining condition in the q-model structure. Let X be a q-cofibrant ex-H-space * *over A and Y be a q-fibrant ex-G-space over '!A. Consider a G-map f :'!X -! Y . We 100 7. EQUIVARIANT qf-TYPE MODEL STRUCTURES must show that f is a q-equivalence if and only if its adjoint H-map "f:X -! ** *'*Y is a q-equivalence. Since '! preserves acyclic q-cofibrations, we can extend f* * to f0:'!RX -! Y , where RX is a q-fibrant approximation. Since f0is a q-equivalence if and only if f is one, and similarly for their adjoints, we may assume withou* *t loss of generality that X is q-fibrant. Recall from Proposition 2.3.9 that '! and * **'* are inverse equivalences of categories and observe that *'*Y can be viewed as * *the restriction, Y |A , of Y along the inclusion of H-spaces :A -! G xH A. From t* *hat point of view, "f:X -! *'*Y is just the map X -! Y |A of ex-H-spaces over A obtained by restriction of '*f along . Now f is a q-equivalence if and only if f restricts to a q-equivalence f[g,a* *]on each fiber, meaning that this restriction is a weak equivalence after passage to fixed points under all subgroups of the isotropy group of [g, a]. For a 2 A, t* *he isotropy subgroup Ha H of a coincides with the isotropy subgroup G[e,a] G of [e, a] 2 G xH A. For g 2 G, the isotropy subgroup of [g, a] is gHag-1. Since the action by g 2 G induces a homeomorphism between the fibers over [e, a] and over [g, a], we see that f is a q-equivalence if and only if each of the restriction* *s f[e,a]is a q-equivalence. But that holds if and only if "fis a q-equivalence. For the last statement, recall the description of '!in Remark 2.4.4 as the c* *om- posite (p*ss*"*(-))H , where ": GxH -! H and ss :GxA -! A are the projections and p: G x A -! G xH A is the quotient map. Since G x A is completely regu- lar, p is a bundle with fiber G=Ha over [g, a], and Ha is compact since A is pr* *oper. Therefore, by Propositions 7.3.6 and 7.3.8, p* is a Quillen right adjoint with * *respect to the q and qf-model structures. In view of Proposition 7.4.3, this displays '* *!as a composite of Quillen right adjoints. Remark 7.4.7. We can take CG to consist of all finite products of the '!C wi* *th C 2 CH and orbits G=K to arrange that CG be closed and contain these '!C. We shall prove that ('!, *'*) descends to a closed symmetric monoidal equiv- alence of homotopy categories in Proposition 9.4.8 below. The first statement of Proposition 7.4.6 implies that the description of '* in terms of base change th* *at is given in Proposition 2.3.10 descends to homotopy categories. Corollary 7.4.8. The functor '*: HoGKB -! HoHK'*B is the composite * Ho GKB __~__//HoGK'!'*B' HoHK'*B 7.5. Fiber adjunctions and Brown representability For a point b in B, we combine the special case "b:G=Gb -! B of Proposi- tion 7.3.4 with the special case ': Gb -! G and A = *, hence :* -! G=Gb, of Proposition 7.4.6 to obtain the following result concerning passage to fibers. * *Re- call from Example 2.3.12 that the fiber functor (-)b:GKB -! GbK* is given by *'*"b*= b*'*. By conjugation, its left adjoint (-)b therefore agrees with "b!'* *!. Proposition 7.5.1. For b 2 B, the pair of functors ((-)b, (-)b) relating GbK* and GKB is a Quillen adjoint pair. We use this result to verify the formal hypotheses of Brown's representabili* *ty theorem [14] for the category HoGKB . Of course, this verification is independe* *nt of the choice of model structure. The category GKB has coproducts and homotopy pushouts, hence homotopy colimits of directed sequences. The usual constructions 7.5. FIBER ADJUNCTIONS AND BROWN REPRESENTABILITY 101 of homotopy pushouts as double mapping cylinders and of directed homotopy col- imits as telescopes makes clear that if the total spaces of their inputs are co* *mpactly generated, as they are after q-cofibrant approximation, then so are the total s* *paces of their outputs. We need a few preliminaries. Definition 7.5.2. For n 0, b 2 B, and H Gb, let Sn,bHbe the ex-G-space ((Gb=H xSn)+ )b over B. Explicitly, by Construction 1.4.6, Sn,bH= (G=H xSn)qB, with the obvious section and with the projection that maps G=H x Sn to the point b and maps B by the identity map. Equivalently, taking d to be the constant map at b, Sn,bHis the quotient ex-G-space associated to the generating cofibration * *i(d), i: G=H x Sn-1 -! G=H x Dn. Therefore, Sn,bHis cofibrant in both the q and the qf-model structures. Let DB be the "detecting set" of all such ex-G-spaces Sn,b* *H. Let [X, Y ]G,B denote the set of maps X -! Y in HoGKB . Lemma 7.5.3. Each X in DB is compact, in the sense that colim[X, Yn]G,B ~=[X, hocolimYn]G,B for any sequence of maps Yn -! Yn+1 in GKB . Proof. If X = Sn,bH, then [X, Y ]G,B ~=[Gb x Sn)+ , Yb]Gb. In GbK*, every object is fibrant and the target is the set of homotopy classes of Gb-maps (Gb x Sn)+ -! Yb, which is the set of unbased nonequivariant homotopy classes of maps Sn -! Yb. Using cofibrant replacement, we can arrange that the (Yn)b have total spaces in U . Then the conclusion follows from Lemma 1.1.5. The following result says that the set DB detects q-equivalences. Proposition 7.5.4. A map , :Y -! Z in GKB is a q-equivalence if and only if the induced map ,*: [X, Y ]G,B -! [X, Z]G,B is a bijection for all X 2 DB . Proof. We may assume that Y and Z are fibrant. By the evident long exact sequences of homotopy groups and the five lemma, , is a q-equivalence if and on* *ly if each Yb -! Zb is a q-equivalence. This is detected by the based Gb-spaces (Gb=H x Sn)+ and the conclusion follows by adjunction. Theorem 7.5.5 (Brown). A contravariant set-valued functor on the category HoGKB is representable if and only if it satisfies the wedge and Mayer-Vietoris axioms. CHAPTER 8 Ex-fibrations and ex-quasifibrations To complete the foundations of parametrized homotopy theory, we are faced with two problems that were discussed in the Prologue. In our preferred qf-model structure, the base change adjunction (f!, f*) is a Quillen pair for any map f * *and is a Quillen equivalence if f is an equivalence. As shown by Counterexample 0.0.1,* * this implies that the base change adjunction (f*, f*) cannot be a Quillen adjoint pa* *ir. Some such defect must hold for any model structure. Therefore, we cannot turn to model theory to construct the functor f* on the level of homotopy categories. T* *he same counterexample illustrates that passage to derived functors is not functor* *ial in general, so that a relation between composites of functors that holds on the point-set level need not imply a corresponding relation on homotopy categories. In any attempt to solve those two problems, one runs into a third one that concerns a basic foundational problem in ex-space theory. Model theoretical con- siderations lead to the use of Serre fibrations as projections, or to the even * *weaker class of qf-fibrations. However, only Hurewicz fibrations are considered in mos* *t of the literature. There is good reason for that. Fiberwise smash products, suspen- sions, cofibers, function spaces, and other fundamental constructions in ex-spa* *ce theory do not preserve Serre fibrations. The solutions to all three problems are obtained by the use of ex-fibrations. Recall that these are the well-sectioned h-fibrant ex-spaces. We study their pr* *op- erties in x8.1. They seem to give the definitively right kind of "fibrant ex-sp* *ace" from the point of view of classical homotopy theory, and they behave much better under the cited constructions than do Serre fibrations, as we show in x8.2. Many variants of this notion appear in the literature. Precisely this variant, with* * this name, appears in Monica Clapp's paper [18], and we are indepted to her work for an understanding of the centrality of the notion. Perversely, as we noted * *in Remark 5.2.7, it is unclear how it fits into the model categorical framework. We construct an elementary ex-fibrant approximation functor in x8.3. It plays a key role in bridging the gap between the model theoretic and classical worlds. In a different context, the classification of sectioned fibrations, the first a* *uthor introduced this construction in [64, x5]. We record some its properties in x8.4. We define quasifibrant ex-spaces and ex-quasifibrations and show that they inherit some of the good properties of ex-fibrations in x8.5. They will play a * *key role in the stable theory. Everything in this chapter works just as well equivariantly as nonequivarian* *tly for any topological group G of equivariance. 8.1. Ex-fibrations Under various names, the following notions were in common use in the 1970's. We shall see shortly that these definitions agree with those given in Definitio* *n 5.2.5. 102 8.1. EX-FIBRATIONS 103 Definition 8.1.1. Let (X, p, s) be an ex-space over B. (i)(X, p, s) is well-sectioned if s is a closed inclusion and there is a retra* *ction ae: X x I -! X [B (B x I) = Ms over B. (ii)(X, p, s) is well-fibered if there is a coretraction, or path-lifting func* *tion, ': Np = X xB BI -! XI under BI, where BI maps to Np via ff -! (sff(0), ff). (iii)(X, p, s) is an ex-fibration if it is both well-sectioned and well-fibered. The requirement in (i) that the retraction ae be a map over B ensures that it restricts on fibers to a retraction that exhibits the nondegeneracy of the base* *point s(b) in Xb for each b 2 B. In view of Theorem 5.2.8(i), we have the following characterization of well-sectioned ex-spaces, in agreement with Definition 5.2.* *5. Lemma 8.1.2. An ex-space X is well-sectioned if and only if X is ~f-cofibran* *t. We use the term "well-sectioned" since it goes well with "well-based". The category of well-sectioned ex-spaces is the appropriate parametrized generaliza* *tion of the category of well-based spaces, and restricting to well-sectioned ex-spac* *es is analogous to restricting to well-based spaces. Note that the section of X provides a canonical way of lifting a path in B t* *hat starts at b to a path in X that starts at s(b). The requirement in Definition 8* *.1.1(ii) that the path-lifting function ' be a map under BI says that '(sff(0), ff)(t) =* * s(ff(t)) for all ff 2 BI and t 2 I. That is, ' is required to restrict to the canonical lifts provided by the section, so that paths in X that start in s(B) remain in s(B). In contrast with Lemma 8.1.2, the well-fibered condition does not by itse* *lf fit naturally into the model theoretic context of Chapter 5. However, we have t* *he following characterization of ex-fibrations, which again is in agreement with t* *he original definition we gave in Definition 5.2.5. Lemma 8.1.3. If X is well-fibered, then X is h-fibrant. If X is well-section* *ed, then X is an ex-fibration if and only if X is h-fibrant. Proof. The first statement is clear since the coretraction ' is a path-lift* *ing function. This gives the forward implication of the second statement, and the converse is a special case of the following result of Eggar [38, 3.2]. Lemma 8.1.4. Let i: X -! Y be an ~f-cofibration of ex-spaces over B, where Y is h-fibrant. Then any map ': X xB BI -! Y Isuch that the composite X xB BI __'_//_Y_I__//Y xB BI is the inclusion can be extended to a coretraction Y xB BI -! Y I. Proof. The inclusion X xB BI -! Y xB BI is an ~h-cofibration by Proposi- tion 4.4.5. Therefore there is a lift in the diagram f (Y xB BI) x {0} [ (X xB BI) x I____//Y55kk | k k k | | k k | fflffl|kk fflffl| (Y xB BI) x I_______g_____//_B, 104 8. EX-FIBRATIONS AND EX-QUASIFIBRATIONS where f(y, !, 0) = y, f(x, !, t) = '(x, !)(t), and g(y, !, t) = !(t). The adjo* *int Y xB BI -! Y Iof is the required extension to a coretraction. Remark 8.1.5. We comment on the terminology. (1) We are following [29, 51] and others in saying that an ~f-cofibrant ex-s* *pace is well-sectioned; the term "fiberwise well-pointed" is also used. For a based * *space, the terms "nondegenerately based" and "well-based" or "well-pointed" are used interchangeably to mean that the inclusion of the basepoint is an h-cofibration* *. In contrast, for an ex-space, the term "fiberwise nondegenerately pointed" is used* * in [29, 51] to indicate a somewhat weaker condition than well-sectioned. (2) The term "well-fibered" is new but goes naturally with well-sectioned. T* *he concept itself is old. We believe that it is due to Eggar [38, 3.3], who calls* * a coretraction under BI a special lifting function. (3) Becker and Gottlieb [2] may have been the first to use the term "ex- fibration", but for a slightly different notion with sensible CW restrictions. * * As noted in the introduction, precisely our notion is used by Clapp [18]. Earlier, in [64, x5] and [65], the first author called ex-fibrations "T -fibrations", an* *d he studied their classification and their fiberwise localizations and completions.* * The equivariant generalization appears in Waner [97]. A more recent treatment of the classification of ex-fibrations has been given by Booth [9]. 8.2. Preservation properties of ex-fibrations We have a series of results that show that ex-fibrations behave well with re* *spect to standard constructions. In some of them, one must use the equivariant versio* *n of Lemma 5.2.4 to verify that the given construction preserves well-sectioned obje* *cts. In all of them, if we only assume that the input ex-spaces are well-sectioned, * *then we can conclude that the output ex-spaces are well-sectioned. It is the fact th* *at the given constructions preserve well-fibered objects that is crucial. Few if a* *ny of these results hold with Serre rather than Hurewicz fibrations as projections. Proposition 8.2.1. Ex-fibrations satisfy the following properties. (i)A wedge over B of ex-fibrations is an ex-fibration. (ii)If X, Y and Z are ex-fibrations and i is an f~-cofibration in the followi* *ng pushout diagram of ex-spaces over B, then Y [X Z is an ex-fibration. X ___i___//_Y | | | | fflffl| fflffl| Z ____//_Y [X Z (iii)The colimit of a sequence of ~f-cofibrations Xi- ! Xi+1 between ex-fibrati* *ons is an ex-fibration. If the input ex-spaces are only assumed to be well-sectioned, then the output e* *x- spaces are well-sectioned. Proof. The last statement is clear. Using it, we see that the colimits in (* *i), (ii), and (iii) are well-sectioned, hence it suffices to prove that they are h-* *fibrant. This is done by constructing path lifting functions for the colimits from path * *lifting functions for their inputs. In (i), we start with path lifting functions under* * BI for the wedge summands and see that they glue together to define a path lifting 8.2. PRESERVATION PROPERTIES OF EX-FIBRATIONS 105 function under BI for the wedge. Part (ii) is due to Clapp [18, 1.3], and we om* *it full details. She starts with a path lifting function for X and uses Lemma 8.1.* *4 to extend it to a path lifting function for Y . She also starts with a path liftin* *g function for Z. She then uses a representation (h, u) of (Y, X) as a fiberwise NDR pair * *to build a path lifting function for the pushout from the given path lifting funct* *ion for Z and a suitably deformed version of the path lifting function for Y . In (* *iii), Lemma 8.1.4 shows that we can extend a path lifting function for Xi to a path lifting function for Xi+1. Inductively, this allows the construction of compat* *ible path lifting functions for the Xi that glue together to give a path lifting fun* *ction for their colimit. Although of little use to us, since the f-homotopy category is not the right* * one for our purposes, many of our adjunctions give Quillen adjoint pairs with respe* *ct to the f-model structure. For example, the following result, which should be compa* *red with Proposition 7.3.4, implies that (f!, f*) is a Quillen adjoint pair in the * *f-model structures and that it is a Quillen equivalence if f is an h-equivalence. Proposition 8.2.2. Let f :A -! B be a map, let X be a well-sectioned ex- space over A, and let Y be a well-sectioned ex-space over B. Then f!X and f*Y a* *re well-sectioned. If Y is an ex-fibration, then so is f*Y , and the functor f* pr* *eserves f-equivalences. If f is an h-equivalence, then (f!, f*) induces an equivalence* * of f-homotopy categories. Proof. It is easy to check that representations of (X, A) and (Y, B) as fib* *er- wise NDR-pairs induce representations of (f!X, B) and (f*Y, A) as fiberwise NDR- pairs. As a pullback, the functor f* preserves both f-fibrant and h-fibrant ex- spaces, and f* preserve f-equivalences since it preserves f-homotopies. For the last statement, if f is a homotopy equivalence with homotopy inverse g, then st* *an- dard arguments with the CHP imply that f* induces an equivalence of f-homotopy categories with inverse g*; see, for example, [64, 2.5]. It follows that g* is * *equivalent to f!and that (f!, f*) is a Quillen equivalence. The following result appears in [38] and [64, 3.6]. It also leads to a Quil* *len adjoint pair with respect to the f-model structure; compare Corollary 7.3.3. Proposition 8.2.3. Let X and Y be well-sectioned ex-spaces over A and B. Then X Z Y is a well-sectioned ex-space over A x B. If X and Y are ex-fibration* *s, then X Z Y is an ex-fibration. Proof. Representations of (X, A) and (Y, B) as fiberwise NDR-pairs deter- mine a representation of (X Z Y, A x B) as a fiberwise NDR-pair, by standard formulas [71, p. 43]. Similarly, path lifting functions for X and Y can be used* * to write down a path lifting function for X Z Y . Corollary 8.2.4. If X and Y are ex-fibrations over B, then so is X ^B Y . Corollary 8.2.5. If X is an ex-fibration over B and K is a well-based space, then X ^B K is an ex-fibration over B. Proposition 8.2.6. Let X and Y be well-sectioned and let f :X -! Y be an ex-map that is an h-equivalence. Then f ^B id:X ^B Z -! Y ^B Z is an h-equivalence for any ex-fibration Z. In particular, f ^B id:X ^B K -! Y ^B K is an h-equivalence for any well-based space K. 106 8. EX-FIBRATIONS AND EX-QUASIFIBRATIONS Proof. As observed by Clapp [18, 2.7], this follows from the gluing lemma by comparing the defining pushouts. As in ordinary topology, function objects work less well, but we do have the following analogue of Corollary 8.2.5. Proposition 8.2.7. If X is an ex-fibration over B and K is a compact well- based space, then FB (K, X) is an ex-fibration over B. Proof. Let (h, u) represent (X, B) as a fiberwise NDR-pair. Then (j, v) rep- resents (FB (K, X), B) as a fiberwise NDR-pair, where v(f) = supk2Ku(f(k)) and jt(f)(k) = ht(f(k)) for f 2 FB (K, X). Note for this that FB (K, B) = B and that, by Proposi- tion 1.3.20, FB (K, X) is h-fibrant. 8.3. The ex-fibrant approximation functor We describe an elementary ex-fibrant replacement functor P . It is just the composite of a whiskering functor W with a version of the mapping path fibration functor N. The functor P replaces ex-spaces by naturally h-equivalent ex-fibrat* *ions. From the point of view of model theory, P can be thought of as a kind of q-fibr* *ant replacement functor that gives Hurewicz fibrations rather than just Serre fibra* *tions as projections. The nonequivariant version of P appears in [64, 5.3, 5.6], and the equivariant version appears in [97, x3]. With motivation from the theory of transports in fibrations, those sources work with Moore paths of varying length. Surprisingly, that choice turns out to be essential for the construction to wor* *k. We therefore begin by recalling that the space of Moore paths in B is given * *by B = {(~, l) 2 B[0,1]x [0, 1) | ~(r) = ~(l) for r} l with the subspace topology. We write ~ for (~, l) and l~ for l, which is the le* *ngth of ~. Let e: B -! B be the endpoint projection e(~) = ~(l~). The composite of Moore paths ~ and ~ such that ~(l~) = ~(0) is defined by l~~ = l~ + l~ and ( (~~)(r) = ~(r) if r l~, ~(r - l~) if r l~. Embed B and BI in B as the paths of length 0 and 1. For a Moore path ~ in B and real numbers u and v such that 0 u v, let ~|vudenote the Moore path r 7! ~(u + r) of length v - u. Definition 8.3.1. Consider an ex-space X = (X, p, s) over B. (i)Define the whiskering functor W by letting W X = (X [B (B x I), q, t), where the pushout is defined with respect to i0: B -! B x I. The projection q is given by the projection p of X and the projection B x I -! B, and the section t is given by t(b) = (b, 1). (ii)Define the Moore mapping path fibration functor L by letting LX = (X xB B, q, t), 8.3. THE EX-FIBRANT APPROXIMATION FUNCTOR 107 where the pullback is defined with respect to the map B -! B given by evaluation at 0. The projection q is given by q(x, ~) = e(~) and the sectio* *n t is given by t(b) = (s(b), b), where b is viewed as a path of length 0. Thus W X is obtained by growing a whisker on each point in the section of X, and the endpoints of the whiskers are used to give W X a section. Similarly, LX* * is obtained by attaching to x 2 X all Moore paths in B starting at p(x). The end- points of the paths give the projection. In the language of x4.3, W X is the st* *andard mapping cylinder construction of the section of X, thought of as a map in GK =B. The section t of W X is just the f-cofibration in the standard factorization ae* * O t of s through its mapping cylinder. In particular, W X is well-sectioned. Simila* *rly, LX is a modification of the mapping path fibration Np in GK . The projection p of X factors through the projection q of LX, which is an h-fibration; a path li* *fting function , :LX xB BI -! (LX)I is given by ,((x, ~), fl)(t) = (x, fl|t0~). Thus * *LX is h-fibrant, but it need not be well-fibered. We can display all of this conveniently in the following diagram. The third square on the top is a pushout and the second square on the bottom is a pullbac* *k. That defines the maps OE and ss, and the maps ae and ' are induced by the unive* *rsal properties from the identity map of X. B FFF FFFFF i1|| FFFFFFF fflffl|FFF B _______B_______B__i0//_B x Ipr_//_B | | | | | | | s| | | fflffl|'fflffl|sfflffl|sfflffl|OEfflffl|ae X ` ` `//LX____//_X____//W X` ` `//X | | | | | | | p| | | fflffl| fflffl|pfflffl|0fflffl| fflffl| B CCCC_// B____//_B_______B________B CCCCCCe| CCCCCCfflffl|| B Thus ae projects whiskers on fibers to the original basepoints and ' is the inc* *lusion x 7! (x, p(x)), where p(x) is the path of length zero. Note that OE is not a m* *ap under B and ss is not a map over B. They give an inverse f-equivalence to ae and an inverse h-equivalence to '. Proposition 8.3.2. The map ae: W X - ! X is a natural f-equivalence of ex-spaces and W X is well-sectioned. The map ': X -! LX is a natural h- equivalence of ex-spaces and LX is h-fibrant. Therefore W takes f-equivalences to fp-equivalences and L takes h-equivalences to f-equivalences. The last statement follows from Proposition 5.2.2. We think of ae and ' as g* *iving a well-sectioned approximation and an h-fibrant approximation in the category of ex-spaces. We will combine them to obtain the promised ex-fibrant approximation, but we first insert a technical lemma. Lemma 8.3.3. If X is an ex-space with a closed section, then W LX is an ex-fibration. If X is well-fibered, then W X is an ex-fibration. 108 8. EX-FIBRATIONS AND EX-QUASIFIBRATIONS Proof. A path lifting function , :NW LX = W LX xB BI -! (W LX)I for W LX is obtained by letting 8 ><(x, fl|t0~) 2 LX if z = (x, ~) 2,LX ,(z, fl)(t) = >(fl(t), u - t) 2 B xiIf z = (b, u) and,t u :(s(fl(u)), fl|t u) 2 LXif z = (b, u) and t u. It is easy to verify that, as a map of sets, , gives a well-defined section of * *the canon- ical retraction ss :(W LX)I -! W LX xB BI. Continuity is a bit more delicate, but if the section of X is closed, then one verifies that = {(z, fl) | z is the equivalence class of (s(b), b)}~ (b, 0) is a closed subset of WLX and hence N is a closed subset of NW LX. To see the implication, note that (-) x BI preserves closed inclusions and Z xB BI Z x BI is a closed inclusion because B is in U (see Remark 1.1.4). Continuity follows * *since we are then piecing together continuous functions on closed subsets. If X is well-fibered and , :X xB BI -! XI is a path-lifting function under BI, we can define a path lifting function ~,:W X xB BI -! (W X)I for W X by ( ~,(x, fl) = ,(x, fl)if x 2 X, (fl, u) if x = (b,.u) To check that ~,is continuous, we use the fact that the functor N(-) = BI xB (-) commutes with pushouts to write NW X as a pushout. We then see that ~,is the map obtained by passage to pushouts from a pair of continuous maps. Recall that the sections of ex-spaces in GUB are closed, by Lemma 1.1.3. Sin* *ce we shall only need to apply the constructions of this section to ex-spaces in G* *UB , the closed section hypothesis need not concern us. Definition 8.3.4. Define the ex-fibrant approximation functor P by the nat- ural zig-zag of h-equivalences OE = (ae, W ') displayed in the diagram ae W' X oo___W X _____//W LX = P X. By Proposition 8.3.2, P takes h-equivalences between arbitrary ex-spaces to fp- equivalences. If X has a closed section, then P X is an ex-fibration. If X is* * an ex-fibration, then it has a closed section, and the above display is a natural * *zig-zag of fp-equivalences between ex-fibrations. 8.4. Preservation properties of ex-fibrant approximation One advantage of ex-fibrant approximation over q or qf-fibrant approximation is that there are explicit commutation natural transformations relating it to m* *any constructions of interest. The following result is an elementary illustrative e* *xample. Lemma 8.4.1. Let D be a small category, X :D -! GKB be a functor, and ! :colimW Xd -! W colimXd and :colimLXd -! LcolimXd be the evident natural maps. Then ! is a map over colimXd and is a map under colimXd, so that the following diagrams commute. All maps in these diagrams are 8.4. PRESERVATION PROPERTIES OF EX-FIBRANT APPROXIMATION 109 h-equivalences. colimW XdN________!_________//NWpcolimXd NNNN pppppp colimaeN''NNNNwwppaepp colimXd colimXdNN colim'ppppppp NNN'NNN xxpppp NNN&& colimLXd _________________//_LcolimXd Let ~ = W O ! :colimP Xd -! P colimXd. Then the following diagram of h- equivalences commutes. colimae colimW' colimXd oo___colimW Xd _____//colimP Xd || | | || !| ~| || fflffl| fflffl| colimXd ooae_W colimXd _W'__//P colimXd The analogous statements for limits also hold. Proof. This is clear from the construction of limits and colimits in Propos* *i- tion 1.2.9. The relevant h-equivalences of total spaces are natural and piece t* *ogether to pass to limits and colimits. Warning 8.4.2. We would like an analogue of the previous result for tensors. In particular, we would like a natural map (LX) ^ K -! L(X ^ K) under X ^ K for ex-spaces X over B and based spaces K. Inspection of definitions makes clear that there is no such map. The obvious map that one might write down, as in the erroneous [64, 5.6], is not well-defined. In Part III, this complicates the ext* *ension of P to a functor on spectra over B. Lemma 8.4.3. Let f :A -! B be a map. (i)Let X be an ex-space over A. Then there are natural maps ! :f!W X -! W f!X and :f!LX -! Lf!X of ex-spaces over B such that ! is a map over f!X and is a map under f!X. Let ~ = W O ! :f!P X -! P f!X. Then the following diagram commutes. f!X of!aef!WoX__f!W'//_f!P X || | | || !| ~| || fflffl| fflffl| f!X ooae_W f!X _W'__//P f!X (ii)Let Y be an ex-space over B. Then there are natural maps ! :W f*Y -! f*W Y and :Lf*Y -! f*LY of ex-spaces over A, the first an isomorphism, such that ! is a map over f*Y and is a map under f*Y . Let ~ = ! O W :P f*Y - ! f*P Y . Then the 110 8. EX-FIBRATIONS AND EX-QUASIFIBRATIONS following diagram commutes. f*Y ooae_W f*Y _W'__//P f*Y || | | || !| |~ || fflffl| fflffl| f*Y of*aef*WoY__f*W'//_f*P Y If Y is an ex-fibration, then ~ is an fp-equivalence. (iii)Let X be an ex-space over A. Then there are natural maps ! :W f*X -! f*W X and :Lf*X -! f*LX of ex-spaces over B such that ! is a map over f*X and is a map under f*X. Let ~ = ! O W :P f*X -! f*P X. Then the following diagram commutes. f*X ooae_W f*X _W'__//P f*X || | | || !| |~ || fflffl| fflffl| f*X of*aef*WoX__f*W'//_f*P X Proof. Again, the proof is by inspection of definitions. Since f! does not preserve ex-fibrations, we do not have an analogue for f! of the last statement about f* in (ii). Warning 8.4.4. We offer another example of the technical dangers lurking in this subject. The maps ~ in the previous proposition are not h-equivalences in general, the problem in (ii), say, being that f* does not preserve h-equivalenc* *es in general. If ~: P f*Y - ! f*P Y were always an h-equivalence, then one could prove by the methods in x9.3 below that the relations (2.2.10) descend to homot* *opy categories for all pullbacks of the form displayed in Proposition 2.2.9. In vie* *w of Counterexample 0.0.1, that conclusion is false. This is another pitfall we fell* * into, and it invalidated much work in an earlier draft. 8.5. Quasifibrant ex-spaces and ex-quasifibrations By analogy with the fact that an ex-fibration is a well-sectioned h-fibrant * *ex- space, we adopt the following terminology. Definition 8.5.1. An ex-space X is quasifibrant if its projection p is a qua* *si- fibration. An ex-quasifibration is a well-sectioned quasifibrant ex-space. If X is quasifibrant, there is a long exact sequence of homotopy groups . .-.! ssHq+1(B, b) -! ssHq(Xb, x) -! ssHq(X, x) -! ssHq(B, b) -! . .-.! ssH0(B* *, b) for any b 2 B, x 2 Xb and H Gb. Using this and the long exact sequences of the pairs (X, Xb), five lemma comparisons give the following observations. Lemma 8.5.2. Let f :X -! Y be a q-equivalence of ex-spaces over B. Then each map of fibers f :Xb -! Yb is a q-equivalence if and only if each map of pairs f :(X, Xb) -! (Y, Yb) is a q-equivalence. If X and Y are quasifibrant, th* *en these maps of pairs are q-equivalences. Conversely, if these maps of pairs are* * q- equivalences and either X or Y is quasifibrant, then so is the other. 8.5. QUASIFIBRANT EX-SPACES AND EX-QUASIFIBRATIONS 111 Working in GUB , we obtain the following result. The same pattern of proof gives many other results of the same nature that we leave to the reader. Proposition 8.5.3. The following statements hold. (i)A wedge over B of ex-quasifibrations is an ex-quasifibration. (ii)If f : X -! Y is a map such that X is an ex-quasifibration and Y is quasif* *i- brant, then the cofiber CB f is quasifibrant. (iii)If X is an ex-quasifibration and K is a well-based space, then X ^B K is an ex-quasifibration. Proof. This follows from Lemma 8.5.2, the natural zig-zag X oo___W X _____//P X of h-equivalences, the corresponding preservation properties for ex-fibrations,* * and the properties of q-equivalences given by the statement that they are well-grou* *nded; see Definition 5.4.1 and Proposition 5.4.9. It is also relevant that in each c* *ase passage to fibers gives the nonparametrized analogue of the construction under consideration. Since this result plays a vital role in our work, we give more c* *omplete details of (ii) and (iii); (i) works the same way. The cofiber CB f is the pushout of the diagram f CB X oo___X _____//Y. If X is well-sectioned, then the left arrow is an h-cofibration and W X and P X* * are well-sectioned. Replacing f by W f and P f we obtain three such cofiber diagram* *s. Together with our original zig-zag this gives a 3 x 3-diagram. Applying the glu* *ing lemma, Definition 5.4.1(iii), we obtain a zig-zag of q-equivalences CB foo___ CB W f_____//CB P f. Similarly, on fibers we obtain zig-zags of q-equivalences Cfb oo___C(W f)b ____//_CW (Lf)b. There results a zig-zag of q-equivalences of pairs (CB f, Cfb)oo__(CB W f, CW fb)____//(CB P f, CW (Lf)b). Since CB P f is ex-fibrant and in particular quasifibrant, CB f is quasifibrant. Similarly, by Definition 5.4.1(v), we have natural zig-zags of q-equivalences X ^B K oo___W X ^B K _____//P X ^B K and Xb^ K oo___ W Xb^ K ____//_W (LX)b^ K. We therefore have a zig-zag of q-equivalences of pairs (X ^B K, Xb^ K) oo__(W X ^B K, W Xb^ K) ____//(P X ^B K, W (LX)b^ K). Since P X ^B K is ex-fibrant and in particular quasifibrant, X ^B K is quasifib* *rant. CHAPTER 9 The equivalence between Ho GKB and hGWB Introduction We developed the point-set level properties of the category GKB of ex-G-spac* *es over B in Chapters 1 and 2, and we developed those homotopical properties that are accessible to model theoretic techniques in Chapter 4 - 7. In this chapter,* * we use ex-fibrations to prove that certain structure on the point-set level that s* *eems inaccessible from the point of view of model category theory nevertheless desce* *nds to homotopy categories. In particular, we prove that Ho GKB is closed symmetric monoidal and that the right derived functor f* of the Quillen adjunction (f!, f* **) in the qf-model structure is closed symmetric monoidal and has a right adjoint f*. In x9.1 we use the ex-fibrant approximation functor to prove that our model theoretic homotopy category of ex-G-spaces over B is equivalent to the classical homotopy category of ex-G-fibrations over B. In x9.2, we discuss how to pass to derived functors on either side of that equivalence in certain general cases. R* *eplac- ing the model-theoretic method of constructing derived functors by a more class* *ical method given in terms of ex-fibrant approximation, we construct the functors f* and FB on homotopy categories in x9.3. By a combination of methods, we prove that Ho GKB is a symmetric monoidal category and that the base change func- tor f* descends to a closed symmetric monoidal functor on homotopy categories in x9.4. We also obtain such descent to homotopy categories results for change of group adjunctions and for passage to fibers in that section. These results * *are central to the theory, and there seem to be no shortcuts to their proofs. Everything is understood to be equivariant in this chapter, and we abbreviate ex-G-fibration and ex-G-space to ex-fibration and ex-space throughout. We shall retreat just a bit from all-embracing generality. We assume that G is a Lie gro* *up and that all given base G-spaces B are proper and are of the homotopy types of G-CW complexes. The reader may prefer to assume that G is compact, but there is no gain in simplicity. In view of the properties of the base change adjunct* *ion (f!, f*) given in Proposition 7.3.4, there would be no real loss of generality * *if we restricted further to base spaces that are actual G-CW complexes, but that would be inconveniently restrictive. 9.1.The equivalence of Ho GKB and hGWB Recall that X ^B I+ is a cylinder object in the sense of the qf-model struct* *ure. When we restrict to qf-fibrant and qf-cofibrant objects, homotopies in the qf- model sense are the same as fp-homotopies, by Lemma 5.6.1. The morphism set [X, Y ]G,B in Ho GKB is naturally isomorphic to [RQX, RQY ]G,B, and this is the set of fp-homotopy classes of maps RQX -! RQY . Here R and Q denote the functorial qf-fibrant and qf-cofibrant approximation functors obtained from the 112 9.1. THE EQUIVALENCE OF HoGKB AND hGWB 113 small object argument. The total space of RQX has the homotopy type of a G- CW complex since B does. This leads us to introduce the following categories. Definition 9.1.1. Define GVB to be the full subcategory of GKB whose ob- jects are well-grounded and qf-fibrant with total spaces of the homotopy types * *of G-CW complexes. Define GWB to be the full subcategory of GVB whose objects are the ex-fibrations over B. Let hGWB denote the category obtained from GWB by passage to fp-homotopy classes of maps. Note that the definition of GWB makes no reference to model category theory. Recall that well-grounded means well-sectioned and compactly generated. When B = *, GW* is just the category of well-based compactly generated G-spaces of t* *he homotopy types of G-CW complexes, and it is standard that its classical homotopy category is equivalent to the homotopy category of based G-spaces with respect * *to the q-model structure. We shall prove a parametrized generalization. We think of GVB as a convenient half way house between GKB and GWB . It turns out to be close enough to the category of qf-cofibrant and qf-fibrant objects in GKB to serve as such for our purposes, while already having some of * *the properties of GWB . The following crucial theorem fails for the q-model structu* *re. It is essential for this result that we allow the objects of VB to be well-sect* *ioned rather than requiring them to be qf-cofibrant. This will force an assymmetry wh* *en we deal with left and right derived functors in Proposition 9.2.2 below. Theorem 9.1.2. The qf-cofibrant and qf-fibrant approximation functor RQ and the ex-fibrant approximation functor P , together with the forgetful functo* *rs I and J, induce the following equivalences of homotopy categories. _RQ_//_ __P__// Ho GKB oo___Ho GVB oo___ hGWB I J Proof. For X in GKB , we have a natural zig-zag of q-equivalences in GKB X oo___QX _____//RQX. Therefore X and IRQX are naturally q-equivalent in GKB . If X is in GVB , then it is qf-fibrant and therefore so is QX. Then the above zig-zag is in GVB and t* *hus X and RQIX are naturally q-equivalent in GVB . Since q-equivalences in GVB are h-equivalences, and P takes h-equivalences to fp-equivalences, it is clear that P induces a functor on homotopy categories. C* *on- versely, since fp-equivalences are in particular q-equivalences, the forgetful * *functor J induces a functor in the other direction. For X in GVB we have the natural zig-zag of h-equivalences ae X oo___W X _W'__//P X of Definition 8.3.4. However W X may not be in GVB since it may not be qf-fibra* *nt. Applying qf-fibrant approximation, we get a natural zig-zag of q-equivalences in GVB connecting X and P X. It follows that X and JP X are naturally q-equivalent in GVB . Starting with X in GWB , the above display is a zig-zag of fp-equivale* *nces in GWB , by Proposition 8.3.2. It follows that X and JP X are naturally fp- equivalent in GWB . 114 9. THE EQUIVALENCE BETWEEN HoGKB AND hGWB 9.2. Derived functors on homotopy categories Model category theory tells us how Quillen functors V :GKA -! GKB induce derived functors on the homotopy categories on the left hand side of the equiva* *lence displayed in Theorem 9.1.2. We now seek an equivalent way of passing to derived functors on the right hand side. We begin with an informal discussion. We focus on functors of one variable, but functors of several variables work the same wa* *y. Following the custom in algebraic topology, we have been abusing notation by using the same notation for point-set level functors and for derived homotopy category level functors. We will continue to do so. However, the more accurate * *no- tations of algebraic geometry, LV and RV for left and right derived functors, m* *ight clarify the discussion. As we have already seen in Counterexample 0.0.1, passag* *e to derived functors is not functorial in general, so that a relation between compo* *sites of functors that holds on the point-set level need not imply a corresponding re* *lation on passage to homotopy categories. Recall that, model theoretically, if V is a Quillen right adjoint, then the * *right derived functor of V is obtained by first applying fibrant approximation R and then applying V on homotopy categories, which makes sense since V preserves weak equivalences between fibrant objects. The left derived functor of a Quill* *en left adjoint V is defined dually, via V Q. Problems arise when one tries to com* *pose left and right derived functors, which is what we must do to prove some of our compatibility relations. The equivalence of categories proven in Theorem 9.1.2 gives us a way of putt* *ing the relevant left and right adjoints on the same footing, giving a "straight" p* *assage to derived functors that is neither "left" nor "right". We need mild good behav* *ior for this to work. Definition 9.2.1. A functor V :GKA -! GKB is good if it is continu- ous, takes well-grounded ex-spaces to well-grounded ex-spaces, and takes ex-spa* *ces whose total spaces are of the homotopy types of G-CW complexes to ex-spaces with that property. Since V is continuous, it preserves fp-homotopies. Proposition 9.2.2. Let V :GKA -! GKB be a good functor that is a left or right Quillen adjoint. If V is a Quillen left adjoint, assume further that it p* *reserves q-equivalences between well-grounded ex-spaces. Then, under the equivalence of categories in Theorem 9.1.2, the derived functor Ho GKA - ! Ho GKB induced by V Q or V R is equivalent to the functor P V J :hGWA - ! hGWB obtained by passage to homotopy classes of maps. Proof. If V is a Quillen right adjoint, then it preserves q-equivalences be* *tween qf-fibrant objects. If V is a Quillen left adjoint, then we are assuming that* * it preserves q-equivalences between well-grounded objects. Since GVA consists of well-sectioned qf-fibrant objects, it follows in both cases that V :GVA -! GVB passes straight to homotopy categories to give V :Ho GVA -! HoGVB . Since V preserves G-CW homotopy types on total spaces, V takes q-equivalences to h- equivalences. Therefore P V takes q-equivalences to fp-equivalences and induc* *es a functor Ho GVA - ! hGWB . To show that P V J and either V Q or V R agree under the equivalence of categories, it suffices to verify that the following d* *iagram 9.3. THE FUNCTORS f* AND FB ON HOMOTOPY CATEGORIES 115 commutes. V Q orV R Ho GKA __________//_HoGKB RQ || PRQ|| fflffl| fflffl| Ho GVA ____PV_____//_hGWB We have a natural acyclic qf-fibration QX -! X and a natural acyclic qf- cofibration X - ! RX. If V is a Quillen left adjoint, then we have a zig-zag of natural q-equivalences RQV Q _____//RV Qoo___V Q ____//_V RQ because V preserves acyclic qf-cofibrations. If V is a Quillen right adjoint, t* *hen we have a zig-zag of natural q-equivalences RQV R oo___ RQV RQ ____//_RV RQoo___V RQ because V preserves q-equivalences between qf-fibrant objects. In both cases, a* *ll objects have total spaces of the homotopy types of G-CW complexes, so in fact we have zig-zags of h-equivalences. Therefore, applying P gives us zig-zags of * *fp- equivalences in GWB , by Proposition 8.3.2. Remark 9.2.3. When V preserves ex-fibrations, P V is naturally fp-equivalent to V on ex-fibrations, by Proposition 8.3.2. The derived functor of V can then * *be obtained directly by applying V and passing to equivalence classes of maps under fp-homotopy. 9.3. The functors f* and FB on homotopy categories We use the equivalence between Ho GKB and hGWB to prove that, for any map f :A -! B between spaces of the homotopy types of G-CW complexes, the (f*, f*) adjunction descends to homotopy categories. We begin by verifying that f* satisfies the hypotheses of 9.2.2. Proposition 9.3.1. Let f :A -! B be a map of base spaces. Then the base change functor f* restricts to a functor f* :GWB -! GWA . Proof. Consider Y in GWB . Since the total space of Y is of the homotopy ty* *pe of a G-CW complex, the fibers Yb are of the homotopy types of Gb-CW complexes by Theorem 3.4.2. The fiber (f*Y )a is a copy of Yf(a), and Ga acts through the evident inclusion Ga Gf(a). Therefore (f*Y )a is of the homotopy type of a Ga-CW complex. The total space of f*Y is therefore of the homotopy type of a G-CW complex, again by Theorem 3.4.2. Moreover, f*Y is an ex-fibration by Proposition 8.2.2. Thus f* restricts to a functor f* :GWB -! GWA . Theorem 9.3.2. For any map f :A -! B of base spaces, the right derived functor f* :HoGKB -! Ho GKA has a right adjoint f*, so that [f*Y, X]G,A ~=[Y, f*X]G,B for X in GKA and Y in GKB . 116 9. THE EQUIVALENCE BETWEEN HoGKB AND hGWB Proof. In view of the equivalence of categories in Theorem 9.1.2 and the fact that f* descends directly to a functor f* :hGWB - ! hGWB on homotopy categories, by Propositions 9.2.2 and 9.3.1, it suffices to construct a right a* *djoint f*: hGWA -! hGWB . We do that using the Brown representability theorem. By Theorem 7.5.5, Ho GKB satisfies the formal hypotheses for Brown representabilit* *y, and therefore so does hGWB . In fact GWB has all of the relevant wedges and hom* *o- topy colimits since these constructions preserve ex-fibrations by Proposition 8* *.2.1 and Corollary 8.2.5 and since they clearly preserve G-CW homotopy types on the total space level and stay within GUB . The objects in the detecting set DB of Definition 7.5.2 are not in GWB , but we can apply the ex-fibrant approximation functor P to them to obtain a detecting set of objects in hGWB . Therefore a co* *n- travariant set-valued functor on hGWB is representable if and only if it satisf* *ies the wedge and Mayer-Vietoris axoms. For a fixed ex-fibrant space X over A, consider the functor ss(f*Y, X)G,A on Y in GWB , where ss denotes fp-homotopy classes of maps. Since the functor ss(W, X)G,A on W in GWA is represented and is computed using homotopy classes of maps, it clearly satisfies the wedge and Mayer-Vietoris axioms. It therefore* * suf- fices to show that the functor f* preserves wedges and homotopy pushouts, since that will imply that the functor ss(f*Y, X)G,A of Y also satisfies the wedge a* *nd Mayer-Vietoris axioms. We can then conclude that there is an object f*X 2 GWB that represents this functor. It follows formally that f* is a functor of X and* * that the required adjunction holds. Because f* :GKB -! GKA is a left adjoint, it preserves colimits, and this im- plies that f* :GWB -! GWA preserves the relevant homotopy colimits. Moreover, f* preserves fp-homotopies and so induces a functor on homotopy categories that still preserves these homotopy colimits. We agree to write ' for natural equivalences on homotopy categories. Remark 9.3.3. For composable maps f and g, g* O f* ' (g O f)* on homotopy categories since f* O g* ' (g O f)* on homotopy categories. The latter equivale* *nce is clear since f* and g* are derived from Quillen right adjoints. More sophisti* *cated commutation laws are proven in the next section. Applying Theorem 9.3.2 to diagonal maps and composing with the homotopy category level adjunction between the external smash product and function ex-sp* *ace functors, we obtain the following basic result; compare Lemma 2.5.4. Theorem 9.3.4. Define ^B and FB on Ho GKB as the composite (derived) functors X ^B Y = *(X Z Y ) and FB (X, Y ) = ~F(X, *Y ). Then [X ^B Y, Z]G,B ~=[X, FB (Y, Z)]G,B for X, Y , and Z in Ho GKB . Proof. The displayed adjunction is the composite of adjunctions for the (de- rived) external smash and function ex-space functors and for the (derived) adjo* *int pair ( *, *). Remark 9.3.5. The referee points out that the ex-space analogue of [11, 7.2] shows that we can work directly with the point-set topology to show that the 9.4. COMPATIBILITY RELATIONS FOR SMASH PRODUCTS AND BASE CHANGE 117 (^B , FB ) adjunction on the original category GKB is continuous and so descends to (classical) fp-homotopy categories to give the adjunction hGKB (X ^B Y, Z) ~=hGKB (X, FB (Y, Z)). Presumably similar point-set topological arguments work to show that, for a map f :A -! B, we have an adjunction hGKA (f*X, Y ) ~=hGKB (X, f*Y ). These adjunctions do not imply our Theorems 9.3.2 and 9.3.4. By definition, our category hGWB is a full subcategory of hGKB , but it is not an equivalent full subcategory. The objects of GWB are very restricted, and general function ex- spaces FB (Y, Z) are not fp-homotopy equivalent to such objects. The force of o* *ur theorems is that, after restricting to our subcategories hGWB , we still have r* *ight adjoints in these categories. It is this fact that we need to obtain right adjo* *ints in our preferred homotopy categories Ho GKB . 9.4. Compatibility relations for smash products and base change We first prove that HoGKB satisfies the associativity, commutativity and uni* *ty conditions required of a symmetric monoidal category. We then show that all of the isomorphisms of functors in Proposition 2.2.1 and some of the isomorphisms of functors in Proposition 2.2.9 still hold after passage to homotopy categorie* *s. Finally, we relate change of groups and passage to fibers to the symmetric mono* *idal structure on homotopy categories. In some of our arguments, it is natural to wo* *rk in Ho GKB . In others, it is natural to work in the equivalent category hGWB . Proposition 9.4.1. For maps f :A -! B and g :A0 -! B0 of base spaces and for ex-spaces X over B and Y over B0, (9.4.2) (f*Y Z g*Z) ' (f x g)*(Y Z Z) in Ho GKA . For ex-spaces W over A and X over A0, (9.4.3) (f!W Z g!X) ' (f x g)!(W Z X) in Ho GKB . Proof. For (9.4.2), we work with ex-fibrations, starting in hGWBxB0 . By Propositions 8.2.2 and 8.2.3, the functors we are dealing with preserve ex-fibr* *ations and therefore descend straight to homotopy categories. The conclusion is thus i* *m- mediate from its point-set level analogue. For (9.4.3), we work with model theo* *retic homotopy categories, starting in Ho GKAxA0. Since (f x g)!' (f x id)!O (idx g)!, we can proceed in two steps and so assume that g = id. By Corollary 7.3.3 and Proposition 7.3.4, we are then composing Quillen left adjoints. Starting with q* *f- cofibrant objects, we do not need to apply qf-cofibrant approximation, and the conclusion follows directly from its point-set level analogue. We use this to complete the proof that Ho GKB is symmetric monoidal. Theorem 9.4.4. The category Ho GKB is closed symmetric monoidal under the functors ^B and FB . 118 9. THE EQUIVALENCE BETWEEN HoGKB AND hGWB Proof. In view of Theorem 9.3.4, we need only prove the associativity, com- mutativity, and unity of ^B up to coherent natural isomorphism. The external smash product has evident associativity, commutativity, and unity isomorphisms, and these descend directly to homotopy categories since the external smash prod* *uct of qf-cofibrant ex-spaces over A and B is qf-cofibrant over AxB. To see that th* *ese isomorphisms are inherited after internalization along *, we use (9.4.2). For * *the associativity of ^B , we have *( *(X Z Y ) Z Z) ' *( x id)*((X Z Y ) Z Z) ' (( x id) )*((X Z Y ) Z Z) ' ((idx ) )*(XZ(Y ZZ)) ' *(idx )*(XZ(Y ZZ)) ' *(XZ *(Y ZZ)). The commutativity of ^B is similar but simpler. For the unit, we observe that S0B' r*S0, r :B -! *. Therefore, since (idx r) = id, X ^B S0B' *(X Z r*S0) ' *(idx r)*(X Z S0) ' ((idx r) )*(X) = X. We turn next to the derived versions of the base change compatibilities of Propositions 2.2.1 and 2.2.9. Observe that the functor f!is good since the sect* *ion of a well-sectioned ex-space is an h-cofibration and since G-CW homotopy types are preserved under pushouts, one leg of which is an h-cofibration. Moreover, f! preserves q-equivalences between well-sectioned ex-spaces by Proposition 7.3* *.4. Therefore Proposition 9.2.2 applies to f!. Theorem 9.4.5. For a G-map f :A -! B, f* :Ho GKB - ! Ho GKA is a closed symmetric monoidal functor. Proof. Since f*S0B ~=S0Ain GKA and S0B is qf-fibrant, f*S0B ' S0Ain Ho GKA . We must prove that the isomorphisms (2.2.2) through (2.2.6) descend to equivalences on homotopy categories. Categorical arguments in [40, xx2, 3] show that it suffices to show that the two isomorphisms (2.2.2) and (2.2.5) descend * *to equivalences on homotopy categories. These two isomorphisms do not involve the right adjoints f* or * and are therefore more tractable than the other three. * *First consider (2.2.2): f*(Y ^B Z) ~=f*Y ^A f*Z. If Y and Z are in GWB , then the two sides of this isomorphism are both in GWA , by Proposition 8.2.2 and Proposition 8.2.3. Therefore the point-set level isomo* *r- phism descends directly to the desired homotopy category level equivalence. Nex* *t, consider (2.2.5): f!(f*Y ^A X) ~=Y ^B f!X. Assume that X is in GWA and Y is in GWB . The functor f!does not preserve ex- fibrations so, to pass to derived categories, we must replace it by P f!on both* * sides. By Proposition 8.2.6, the functor Y ^B (-) preserves h-equivalences between wel* *l- sectioned ex-spaces. Since P sends h-equivalences to fp-equivalences, we theref* *ore have fp-equivalences, natural up to fp-homotopy, P(id^BOE) OE P f!(f*Y ^A X) ~=P (Y ^B f!X)____//P (Y ^B P f!X)oo__Y ^B P f!X, where OE = (ae, W ') is the zigzag of h-equivalences of Definition 8.3.4. This * *implies the desired equivalence in the homotopy category. 9.4. COMPATIBILITY RELATIONS FOR SMASH PRODUCTS AND BASE CHANGE 119 The reader is invited to try to prove directly that the projection formula h* *olds in the homotopy category. Even the simple case of f :* -! B, the inclusion of a point, should demonstrate the usefulness of Proposition 9.2.2. Theorem 9.4.6. Suppose given a pullback diagram of G-spaces g C _____//D i|| |j| fflffl|fflffl| A __f__//B in which f (or j) is a q-fibration. Then there are natural equivalences of func* *tors on homotopy categories (9.4.7) j*f!' g!i*, f*j* ' i*g*, f*j!' i!g*, j*f* ' g*i*. Proof. As in Proposition 2.2.9 the second and fourth equivalences are conju- gate to the first and third. However, since the situation is no longer symmetri* *c, we must prove both the first and third equivalences, assuming f is a q-fibration. First consider the desired equivalence f*j!' i!g*. We work with ex-fibration* *s, starting with X 2 hGWD . We must replace j!and i!by P j!and P i!before passing to homotopy categories. By Proposition 7.3.4, f* preserves q-equivalences since* * f is a q-fibration. Moreover, our q-equivalences are h-equivalences since we are dea* *ling with total spaces of the homotopy types of G-CW complexes. By the diagram in Lemma 8.4.3(ii), we see that ~: P f* -! f*P is a natural h-equivalence here. Th* *is would be false for arbitrary maps f, as observed in Warning 8.4.4. Since ~ is an h-equivalence between ex-fibrations, it is an fp-equivalence. Therefore f*P j!X ' P f*j!X ~=P i!g*X. Now consider the desired equivalence j*f!X ' g!i*X in Ho GKD . Our assump- tion that f is a q-fibration gives us no direct help with this. However, we may factor j as the composite of a homotopy equivalence and an h-fibration. Expand- ing our pullback diagram as a composite of pullbacks, we see that it suffices to prove our commutation relation when j is an h-fibration and when j is a homotopy equivalence. The first case is immediate by symmetry from the first part. Thus assume that j is a homotopy equivalence. Then i is also a homotopy equivalence. By Proposition 7.3.4, (i!, i*) and (j!, j*) are adjoint equivalences of homotop* *y cat- egories. Therefore j*f!' j*f!i!i* ' j*j!g!i* ' g!i*. Finally, we turn to a promised compatibility relationship between products a* *nd change of groups. We observed in Proposition 7.4.6 that the point-set level clo* *sed symmetric monoidal equivalence of Proposition 2.3.9 is given by a Quillen equiv* *a- lence. The following addendem shows that the resulting equivalence on homotopy categories is again closed symmetric monoidal. Proposition 9.4.8. Let ': H - ! G be the inclusion of a subgroup and A be an H-space. The Quillen equivalence ('!, *'*) descends to a closed symmetric monoidal equivalence between HoHKA and HoGK'!A. Proof. Let : A -! A x A be the diagonal map. The isomorphisms '* *(X Z Y ) ~= *'*(X Z Y ) ~= *('*X Z '*Y ) 120 9. THE EQUIVALENCE BETWEEN HoGKB AND hGWB descend to equivalences on homotopy categories, the first since it is between Q* *uillen right adjoints, the second since '* preserves all q-equivalences. It follows t* *hat *'* is a symmetric monoidal functor on homotopy categories. Since it is also an equivalence, it follows formally that it is closed symmetric monoidal. Combined with Theorem 9.4.5 applied to the inclusion "b:G=Gb -! B, this last observation gives us the following conclusion. Theorem 9.4.9. The derived fiber functor (-)b: HoGKB -! Ho GbKb is closed symmetric monoidal, and it has a left adjoint (-)b and a right adjoint b* *(-). We emphasize that this innocent looking result packages highly non-trivial a* *nd important information. It gives in particular that, for ex-G-spaces X and Y , t* *he (derived) fiber FB (X, Y )b of the (derived) function space FB (X, Y ) is equiv* *alent in Ho GbKb to the (derived) function space F (Xb, Yb) of the (derived) fibers Xb a* *nd Yb. On the point set level, that is what motivated the definition of the inter* *nal function ex-space. That it still holds on the level of homotopy categories is a reassuring consistency result. Part III Parametrized equivariant stable homotopy theory Introduction We develop rigorous foundations for parametrized equivariant stable homotopy theory. The idea is to start with a fixed base G-space B and to build a good category, here denoted GSB , of G-spectra over B. We assume once and for all that our base spaces B must be compactly generated and must have the homotopy types of G-CW complexes. By "good" we mean that GSB is a closed symmet- ric monoidal topological model category whose associated homotopy category has properties analogous to those of the ordinary equivariant stable homotopy categ* *ory. Informally, the homotopy theory of GSB is specified by the homotopy theory seen on the fibers of G-spectra over B. One compelling reason for taking the parametrized stable homotopy category seriously, even nonequivariantly, is to b* *uild a natural home in which one can do stable homotopy theory while still keeping track of fundamental groups and groupoids. Stable homotopy theory has tended to ignore such intrinsically unstable data. This has the effect of losing contact * *with more geometric branches of mathematics in which the fundamental group cannot be ignored. For example, one basic motivation for the equivariant theory is that it gives a context in which to better understand equivariant orientations, Thom isomor- phisms, and Poincar'e duality. There is no problem for G-simply connected mani- folds M [59, IIIx6], but restriction to such M is clearly inadequate for applic* *ations to transformation group theory. Despite a great deal of work on the subject by Costenoble and Waner, and some by May, [24, 25, 26, 27, 69], this circle of ide* *as is not yet fully understood. Costenoble and Waner [28] use our work to study th* *is problem for ordinary equivariant theories, and for general theories this is wor* *k in progress by the second author. There are many problems that make the development far less than an obvi- ous generalization of the nonparametrized theory. Problems on the space level were dealt with in Parts I and II, and we deal with the analogous spectrum level problems here. We give some categorical preliminaries on enriched equivariant c* *at- egories in Chapter 10. We define and develop the basic properties of our prefer* *red category of parametrized G-spectra in Chapter 11, study its model structures in Chapter 12, and study adjunctions and compatibility relations in Chapter 13. All of the problems that we faced on the space level are still there, but their sol* *utions are considerably more difficult. In Chapter 14, we go on to study further such compatibilities that more fundamentally involve equivariance. The theory of highly structured spectra is highly cumulative. We build on the theory of equivariant orthogonal spectra of Mandell and May [61]. In turn, that theory builds on the theory of nonequivariant orthogonal spectra. A self-contai* *ned treatment of nonequivariant diagram spectra, including orthogonal spectra, is g* *iven by Mandell, May, Schwede, Shipley in [62]. The treatments of [61] and [62], like 122 INTRODUCTION 123 this one, are topological as opposed to simplicial. That seems to be essential * *when dealing with infinite groups of equivariance. It also allows use of orthogonal * *spectra rather than symmetric spectra. These are much more natural equivariantly and, even nonequivariantly, they have the major convenience that their weak equiva- lences are exactly the maps that induce isomorphisms of homotopy groups. The theory of equivariant parametrized spectra can be thought of as the push* *out over the theory of spectra of the theories of equivariant spectra and of nonequ* *iv- ariant parametrized spectra. However, there is no nonequivariant precursor of t* *he present treatment of parametrized spectra in the literature. There are prelimin* *ary forms of such a theory [2, 3, 18, 19, 29], but these either do not go beyond su* *s- pension spectra or are based on obsolescent technology. None of them go nearly far enough into the theory for the purposes we have in mind, although the early first approximation of Monica Clapp [18], written up in more detail with Dieter Puppe [19], deserves considerable credit. Clapp gave the strongest previous ver* *sion of our fiberwise duality theorem, and her emphasis on ex-fibrations, together w* *ith some key technical results about them, have been very helpful. The reader prima* *r- ily interested in classical homotopy theory should ignore all details of equiva* *riance in reading Chapters 11-13. In fact, given [61], the equivariance adds few serio* *us difficulties to the passage from spectra to parametrized spectra, although it d* *oes add many interesting new features. There are at least two possible alternative cumulative approaches. Rather th* *an building on the theory of orthogonal G-spectra of [61, 62], one can build on the theory of G-spectra of [59], the theory of S-modules of [39], and the pushout of these, the theory of SG -modules of [61]. Po Hu [47] began work on the first st* *age of a treatment along these lines, using parametrized G-spectra, but she did not address the foundational issues concerning smash products, function spectra, ba* *se change functors, and compatibility relations considered here. Moreover, followi* *ng the first author's misleading unpublished notes [72], she took the q-model stru* *cture on ex-G-spaces as her starting point, and the stable model structure cannot be made rigorous from there. It appears to us that resolving all of these issues i* *n that framework is likely to be more difficult than in the framework that we have ado* *pted. In particular, homotopical control of the parametrized spectrification functor * *and of cofiber sequences seems problematic. Alternatively, for finite groups G, one can build on the theory of symmetric spectra of Hovey, Smith, and Shipley [46] and its equivariant generalization du* *e to Mandell [60]. Such an approach would avoid the point-set topological technicali* *ties of the present approach and would presumably lead to rather different looking problems with fibrations and cofibrations. The problems with the stable homotopy category level adjunctions that involve base change functors, smash products, a* *nd function spectra are intrinsic and would remain. Our solutions to these problems do not seem to carry over to the simplicial context in an obvious way, and an alternative simplicial treatment could prove to be quite illuminating. In view of the understanding of unstable equivariant homotopy theory for proper actions of non-compact Lie groups that was obtained in Part II, it might seem that there should be no real difficulty in obtaining a good stable theory * *along the same lines as the theory for compact Lie groups. However, in contrast with * *the rest of this book, equivariant stable homotopy theory for non-compact Lie groups is in preliminary and incomplete form, with still unresolved technical problems* *. We 124 INTRODUCTION leave its study to future work, explaining in x11.6 where some of the problems * *lie. Except in that section, G is asssumed to be a compact Lie group from Chapter 11 onwards. A few other notes on terminology may be helpful. We shall not use the term "ex-spectrum over B" since, stably, there is no meaningful unsectioned theory. Instead, we shall use the term "spectrum over B". This is especially convenient when considering base change. We write out "orthogonal G-spectrum over B" until x11.4. However, since we consider no other kinds of G-spectra and work equivariantly throughout, we later abbreviate this to "spectrum over B" when there is no danger of confusion. That is, we work equivariantly throughout, but* * we only draw attention to this fact when it plays a significant mathematical role. CHAPTER 10 Enriched categories and G-categories Introduction To give context for the structure enjoyed by the categories of parametrized orthogonal G-spectra that we shall define, we first describe the kind of equiva* *riant parametrized enrichments that we shall encounter. In fact, our categories have * *sev- eral layers of enrichment, and it is helpful to have a consistent language, som* *ewhat non-standard from a categorical point of view, to keep track of them. In xx10.1* * and 10.2, we give some preliminaries on enriched categories, working nonequivariant* *ly in x10.1 and adding considerations of equivariance in x10.2. We discuss the rol* *e of the several enrichments in sight in our G-topological model G-categories in x10* *.3. In this chapter, G can be any topological group. 10.1.Parametrized enriched categories As discussed in x1.2, all of our categories C are topological, meaning that they are enriched over the category K* of based spaces (= k-spaces). In contrast with general enriched category theory and our further enrichments, the topologi* *cal enrichment is given just by a topology on the underlying set of morphisms, and * *we denote the space of morphisms X -! Y by C (X, Y ). We say that a topological category C is topologically bicomplete if it is bicomplete and bitensored over * *K*. In fact, we shall have enrichments and bitensorings over the category KB of ex- spaces over B that imply the topological enrichment and bitensoring by restrict* *ion to ex-spaces B x T for T 2 K*. Recall from x1.3 that KB is topologically bicomplete, with tensors and coten- sors denoted by K ^B T and FB (T, K) for T 2 K* and K 2 KB . (Since we shall use letters like X, Y , and Z for spectra, we have changed the letters that we * *use generically for spaces and ex-spaces from those that we used earlier). It is a* *lso closed symmetric monoidal under its fiberwise smash product and function space functors, which are also denoted by ^B and FB ; its unit object is S0B= B x S0. It is therefore enriched and bitensored over itself. The two enrichments are re* *lated by natural based homeomorphisms (10.1.1) KB (K, L) ~=KB (S0B, FB (K, L)). This is the case T = S0 of the more general based homeomorphism (10.1.2) K*(T, KB (K, L)) ~=KB (S0B^B T, FB (K, L)) for T 2 K* and K, L 2 KB . The Yoneda lemma, (10.1.1), and the bitensoring adjunctions imply that the two bitensorings are related by the equivalent natur* *al isomorphisms of ex-spaces (10.1.3) K ^B T ~=K ^B (S0B^B T ) and FB (T, K) ~=FB (S0B^B T, K). 125 126 10. ENRICHED CATEGORIES AND G-CATEGORIES These in turn imply the equivalent generalizations (10.1.4) K ^B (L ^B T ) ~=(K ^B L) ^B T and FB (T, FB (K, L)) ~=FB (K ^B T, L). Formally, rather than defining the enrichments and bitensorings over K* indepen- dently, we can take (10.1.2) and (10.1.3) as definitions of these structures in* * terms of the enrichment and bitensoring over KB . Then (10.1.4) and the bitensoring adjunction homeomorphisms (10.1.5) KB (K ^B T, L) ~=K*(T, KB (K, L)) ~=KB (K, FB (T, L)) follow directly. Remark 10.1.6. We shall be making much use of the functor S0B^B (-), and we henceforward abbreviate notation by setting TB = B x T = S0B^B T for a based space T , and similarly for maps. Observe that K ^B T and K ^B TB are two names for the same ex-space over B. When working on a formal conceptual level, it is often best to think in terms of tensors over K* and use the first * *name. However, on a pragmatic level, to avoid confusion, it is best to view based spa* *ces as embedded in ex-spaces via S0B^B (-) and to use the second notation, working only with tensors over KB . We generalize and formalize several aspects of the discussion above. Definition 10.1.7. A topological category C is topological over B if it is e* *n- riched and bitensored over KB . It is topologically bicomplete over B if it is* * also bicomplete. We write PB (X, Y ) for the hom ex-space over B, and we write X ^B K and FB (K, X) for the tensor and cotensor in C , where X, Y 2 C and K 2 KB . Explicitly, we require bitensoring adjunction homeomorphisms of based spaces (10.1.8) C (X ^B K, Y ) ~=KB (K, PB (X, Y )) ~=C (X, FB (K, Y )). By Yoneda lemma arguments, these imply unit and transitivity isomorphisms in C (10.1.9) X ~=X ^B S0B and X ^B (K ^B L) ~=(X ^B K) ^B L. and also bitensoring adjunction isomorphisms of ex-spaces (10.1.10) PB (X ^B K, Y ) ~=FB (K, PB (X, Y )) ~=PB (X, FB (K, Y )). Conversely, there is a natural homeomorphism (10.1.11) C (X, Y ) ~=KB (S0B, PB (X, Y )), and the isomorphisms (10.1.8) follow from (10.1.10) by applying KB (S0B, -). If we do not require C to be topological to begin with, we can take (10.1.11) as the definition of the space C (X, Y ) and so recover the topological enrichm* *ent. With the notation of Remark 10.1.6, we obtain tensors and cotensors with based spaces T by setting (10.1.12) X ^B T = X ^B TB and FB (T, X) = FB (TB , X). The adjunction homeomorphisms (10.1.13) C (X ^B T, Y ) ~=K*(T, C (X, Y )) ~=C (X, FB (T, Y )) are obtained by replacing K by TB in (10.1.8) and using (10.1.2) and (10.1.11). 10.2. EQUIVARIANT PARAMETRIZED ENRICHED CATEGORIES 127 In the cases of interest, C is closed symmetric monoidal, and then the hom ex-spaces PB (X, Y ) can be understood in terms of the internal hom in C by the following definition and result. Definition 10.1.14. Let C be a topological category over B with a closed symmetric monoidal structure given by a product ^B and function object functor FB , with unit object SB . We say that C is a topological closed symmetric mono* *idal category over B if the tensors and products are related by a natural isomorphism X ^B K ~=X ^B (SB ^B K) in C for K 2 KB and X 2 C . Proposition 10.1.15. Let C be a topological closed symmetric monoidal cate- gory over B. Then, for K 2 KB and X, Y , Z 2 C , there are natural isomorphisms FB (K, Y ) ~=FB (SB ^B K, Y ), PB (X, Y ) ~=PB (SB , FB (X, Y )), PB (X ^B Y, Z) ~=PB (X, FB (Y, Z)) in C and a natural homeomorphism of based spaces KB (K, PB (X, Y )) ~=C (SB ^B K, FB (X, Y )). 10.2.Equivariant parametrized enriched categories Turning to the equivariant generalization, we give details of the context of topological G-categories, continuous G-functors, and natural G-maps that we fir* *st alluded to in x1.4. The discussion elaborates on that given in [61, IIx1]. Gene* *rically, we use notations of the form CG and GC to denote a category CG enriched over the category GK* of based G-spaces and its associated "G-fixed category" GC with the same objects and the G-maps between them; GC is enriched over K*. We shall write (CG , GC ) for such a pair, and we shall refer to the pair as a "G-catego* *ry". In the terminology of enriched category theory, GC is the underlying topolog* *ical category of CG . The hom objects of CG are G-spaces CG (X, Y ); G-functors and natural G-maps just mean functors and natural transformations enriched over GK*. Consistently with enriched category theory, the space GC (X, Y ) = CG (X, Y )G * *can be identified with the space of G-maps S0 -! CG (X, Y ). We call the points of CG (X, Y ) "arrows" to distinguish them from the points of GC (X, Y ), which we call "G-maps", or often just "maps", with the equivariance understood. We cannot expect CG to have limits and colimits, but GC is usually bicomplet* *e. In many of our examples, both CG and GC are closed symmetric monoidal under functors ^B and FB . For example, we have the closed symmetric monoidal G- category (KG,B, GKB ) of ex-G-spaces over a G-space B described in x1.4. Definition 10.2.1. A G-category (CG , GC ) is G-topological over B if CG is enriched over GKB and bitensored over KG,B. It follows that GC is enriched over KB and bitensored over GKB . We say that (CG , GC ) is G-topologically bicomple* *te over B if, in addition, GC is bicomplete. We write PB (X, Y ) for the hom ex-G- space over B, and we write X ^B K and FB (K, X) for the tensor and cotensor in CG , where X, Y 2 CG and K 2 KG,B. Explicitly, we require bitensoring adjunction homeomorphisms of based G-spaces (10.2.2) CG (X ^B K, Y ) ~=KG,B(K, PB (X, Y )) ~=CG (X, FB (K, Y )). 128 10. ENRICHED CATEGORIES AND G-CATEGORIES There result coherent unit and transitivity isomorphisms in GC (10.2.3) X ~=X ^B S0B and X ^B (K ^B L) ~=(X ^B K) ^B L and also bitensoring adjunction isomorphisms of ex-G-spaces (10.2.4) PB (X ^B K, Y ) ~=FB (K, PB (X, Y )) ~=PB (X, FB (K, Y )). Conversely, there is a natural homeomorphism of based G-spaces (10.2.5) CG (X, Y ) ~=KG,B(S0B, PB (X, Y )), and the isomorphisms (10.2.2) follow from (10.2.4) by applying KG,B(S0B, -). Pa* *s- sage to G-fixed points from (10.2.2) gives the bitensoring adjunction homeomor- phisms of based spaces (10.2.6) GC (X ^B K, Y ) ~=GKB (K, PB (X, Y )) ~=GC (X, FB (K, Y )). We warn the reader that we shall not always adhere strictly to the notational pattern of Definition 10.2.1 for our several layers of enrichment. In particula* *r, in the domain categories for our equivariant diagram spaces and diagram spectra, o* *nly CG is of interest, not GC , and our notations will reflect that. On the other h* *and, when studying model categories, it is always the bicomplete category GC that is of fundamental interest. If (CG , GC ) is G-topological over B, then it is automatically G-topological (over *). This enrichment is recovered by taking (10.1.11), read equivariantly,* * as the definition of the based G-space CG (X, Y ). Just as in the nonequivariant c* *ase, for based G-spaces T and objects X of CG , the tensors and cotensors in CG and GC are given on objects by (10.2.7) X ^B T = X ^B TB and FB (T, X) = FB (TB , X), using the notation of Remark 10.1.6 equivariantly. The required G-homeomorphisms (10.2.8) CG (X ^B T, Y ) ~=KG,*(T, CG (X, Y )) ~=CG (X, FB (T, Y )) follow directly. We have equivariant analogues of Definition 10.1.14 and Proposition 10.1.15. Definition 10.2.9. Let (CG , GC ) be a G-topological G-category over B with a closed symmetric monoidal structure given by a product G-functor ^B and a function object G-functor FB , with unit object SB . We say that (CG , GC ) is a G-topological closed symmetric monoidal G-category over B if the tensors and products are related by a natural isomorphism X ^B K ~=X ^B (SB ^B K) in GC for K 2 GKB and X 2 GC . Proposition 10.2.10. Let (CG , GC ) be a G-topological closed symmetric mon- oidal G-category over B. Then, for K 2 KB and X, Y , Z 2 C , there are natural isomorphisms FB (K, Y ) ~=FB (SB ^B K, Y ), PB (X, Y ) ~=PB (SB , FB (X, Y )), PB (X ^B Y, Z) ~=PB (X, FB (Y, Z)) in GC and there is a natural homeomorphism of based G-spaces KG,B(K, PB (X, Y )) ~=CG (SB ^B K, FB (X, Y )). 10.3. G-TOPOLOGICAL MODEL G-CATEGORIES 129 10.3.G-topological model G-categories We explain what it means for a G-topological G-category (CG , GC ) over B to have a G-topological model structure. This structure implies in particular that* * the homotopy category HoGC is bitensored over the homotopy category HoGK . We need some notation. Throughout this section, we consider maps i: W -! X, j :V -! Z, and p: E -! Y in GC and a map k :K -! L in either GKB or GK*; in the latter case we apply the functor (-)B = B x (-) to k and so regard it as a map in GKB , as suggested in Remark 10.1.6. We shall define the notion of a G-topological model category * *in terms of the induced map (10.3.1) CG (i, p): CG (X, E) -! CG (W, E) xCG(W,Y )CG (X, Y ) of based G-spaces. Passing to G-fixed points, this gives rise to a map (10.3.2) GC (i, p): GC (X, E) -! GC (W, E) xGC(W,Y )GC (X, Y ) of based spaces, and we have the following motivating observation. Lemma 10.3.3. The pair (i, p) has the lifting property if and only if the fu* *nction GC (i, p) is surjective. Definition 10.3.4. Let (CG , GC ) be a G-topological G-category over B such that GC is a model category. We say that the model structure is G-topological if CG (i, p) is a fibration in GK* when i is a cofibration and p is a fibration an* *d is acyclic when, further, either i or p is acyclic. Remark 10.3.5. The definition must refer consistently to either h-type or q- type model structures. The resulting notions are quite different. We usually ha* *ve in mind a q-type model structure. In that case, the weak equivalences and fibratio* *ns are often characterized by conditions on the H-fixed point maps fH of a map f. If F is a family of subgroups of G, such as the family G of compact subgroups, we can restrict attention to those H 2 F . The resulting F -equivalences and F - fibrations usually specify another model structure on GC . In particular, we ha* *ve the F -model structure on GK*. For the qf-type model structures of x7.2, we must start with a generating set C that contains the orbits G=H with H 2 F \ G and consists of F \ G -cell complexes. We say that an F -model structure on GC is F -topological if the condition of the previous definition holds when we use th* *e F - notions of fibration, cofibration and weak equivalence throughout. The observat* *ions of this section generalize to F -topological model categories for any family F . In addition to the map of G-spaces displayed in (10.3.1), we have a map (10.3.6) PB (i, p): PB (X, E) -! PB (W, E) xPB(W,Y )PB (X, Y ) of ex-G-spaces over B. Warning 10.3.7. We can define what it means for (CG , GC ) to be G-topologic* *al over B, using the map PB (i, p) of ex-spaces rather than the map CG (i, p) of s* *paces. However, we know of no examples where this condition is satisfied. For example, (KG,B, GKB ) is G-topological, by Theorems 7.2.3 and 7.2.8, but, as Warning 6.1* *.7 makes clear by adjunction, we cannot expect it to be G-topological over B. 130 10. ENRICHED CATEGORIES AND G-CATEGORIES Just as in the classical theory of simplicial or topological model categorie* *s, there are various equivalent reformulations of what it means for GC to be G-topologic* *al. To explain them, observe that the tensors and cotensors with ex-G-spaces over B give rise to induced maps (10.3.8) i B k :(X ^B K) [W^BK (W ^B L) -! X ^B L and (10.3.9) FB (k, p): FB (L, E) -! FB (K, E) xFB(K,Y )FB (L, Y ) of ex-G-spaces over B. If (CG , GC ) is closed symmetric monoidal, then we also have the induced maps (10.3.10) i B j :(X ^B V ) [W^BV (W ^B Z) -! X ^B Z and (10.3.11) FB (j, p): FB (Z, E) -! FB (V, E) xFB(V,Y )FB (Z, Y ) in GC . We have various adjunction isomorphisms relating these various -product maps and -function object maps. Proposition 10.3.12. If k is a map of ex-G-spaces over B, then there are adjunction isomorphisms (10.3.13) PB (i B k, p) ~=FB (k, PB (i, p)) ~=PB (i, FB (k, p)) of maps of ex-G-spaces over B and (10.3.14) CG (i B k, p) ~=KG,B(k, PB (i, p)) ~=CG (i, FB (k, p)) of maps of based G-spaces. If k is a map of based G-spaces, then the last pair * *of isomorphisms can be rewritten as (10.3.15) CG (i B k, p) ~=KG,*(k, CG (i, p)) ~=CG (i, FB (k, p)). When (CG , GC ) is closed symmetric monoidal there are adjunction isomorphisms (10.3.16) PB (i B k, p) ~=PB (i, FB (k, p)) of maps of ex-G-spaces over B and (10.3.17) CG (i B k, p) ~=CG (i, FB (k, p)) of maps of based G-spaces. Together with Lemma 10.3.3, this implies the promised alternative equivalent conditions that describe when a model category is G-topological. Proposition 10.3.18. Let (CG , GC ) be a G-topological G-category over B such that GC has a model structure. Then the following conditions are equivalent. (i)The map i B k of (10.3.8) is a cofibration in GC if i is a cofibration in GC and k is a cofibration in GK*. It is acyclic if either i or k is acyclic. (ii)The map FB (k, p) of (10.3.9) is a fibration in GC if p is a fibration in * *GC and k is a cofibration in GK*. It is acyclic if either p or k is acyclic. (iii)The map CG (i, p) of (10.3.1) is a fibration in GK* if i is a cofibration * *in GC and p is a fibration in GC . It is acyclic if either i or p is acyclic. 10.3. G-TOPOLOGICAL MODEL G-CATEGORIES 131 Proof. The third condition is our definition of the model structure being G-topological. We prove that the first condition is equivalent to the third. A similar argument shows that the second condition is also equivalent to the thir* *d. The map CG (i, p) is a fibration if and only if (k, CG (i, p)) has the lifting * *property with respect to all acyclic cofibrations k in GK*. By Lemma 10.3.3 and the first adjunction isomorphism in (10.3.15), that holds if and only if (i B k, p) has t* *he lifting property, that is, if and only if i B k is an acyclic cofibration. If e* *ither i or p is acyclic, then we take k to be a cofibration in GK* and argue similarly. CHAPTER 11 The category of orthogonal G-spectra over B Introduction Intuitively, an orthogonal spectrum X over B consists of ex-spaces X(V ) ove* *r B and ex-maps oe :X(V )^B SW -! X(V W ) for suitable inner product spaces V and W . The orthogonal group O(V ) must act on X(V ), and oe must be (O(V )xO(W ))- equivariant. The orthogonal group actions enable the definition of a good exter* *nal smash product. Moreover, they will later allow us to define stable weak equival* *ences in terms of homotopy groups, as would not be possible if we only had actions by symmetric groups. Similarly, use of general inner product spaces allows us to build in actions by a compact Lie group G without difficulty. For non-compact Lie groups, we should ignore inner products and use linear isomorphisms, replacing the compact orthogonal group O(V ) by the general linear group GL(V ). However, as we expla* *in in x11.6, there are more serious problems in generalizing to non-compact Lie gr* *oups; except in that section, we require G to be a compact Lie group. Working equivariantly, we first describe X as a suitable diagram of ex-G-spa* *ces in x11.1. The domain category for our diagrams is denoted IG and is independent of B. We then build in the structure maps oe in x11.2, where we define the cate* *gory of orthogonal G-spectra over B. In x11.3, we show that it too can be described * *as a category of diagrams of ex-G-spaces. The domain category here is denoted JG,B. It does depend on B, as indicated by the notation. The formal properties of the category of ex-G-spaces over B carry over to the category of orthogonal G-spect* *ra over B, but there are some new twists. For example, our category of G-spectra over B is enriched not just over based G-spaces, but more generally over ex-G- spaces over B. We discussed the relevant formalities in the previous chapter. T* *his enhanced enrichment is essential to the definition of function G-spectra over B. We show in x11.4 that the base change functors and their properties also car* *ry over to these categories of parametrized G-spectra, and we discuss change of gr* *oup functors and restriction to fibers in x11.5. 11.1. The category of IG -spaces over B We recall the G-category (IG , GI ) from [61, II.2.1]. The objects and arrows of IG are finite dimensional G-inner product spaces and linear isometric isomor- phisms. The maps of GI are G-linear isometries. More precisely, as dictated by the general theory of [61, 62], we take IG (V, W ) as based with basepoint disj* *oint from the space of linear isometric isomorphisms V - ! W . As in [61, II.1.1], t* *he objects V run over the collection V of all representations that embed up to is* *o- morphism in a given "G-universe" U, where a G-universe is a sum of countably many copies of representations in a set of representations that includes the tr* *ivial 132 11.1. THE CATEGORY OF IG-SPACES OVER B 133 representation. We think in terms of a "complete G-universe", one that contains all representations of G, but the choice is irrelevant until otherwise stated. * *As in [61, II.2.2], we can restrict from V to any cofinal subcollection W that is c* *losed under direct sums. Based G-spaces are ex-G-spaces over *, and IG -spaces are defined in [61, II.2.3] as G-functors IG -! TG , where TG is the G-category of compactly gener- ated based G-spaces. One can just as well drop the weak Hausdorff condition, which plays no necessary mathematical role in [61, 62 ], and allow general k- spaces. With the notations of Part II, we can thus change the target G-category to KG,*. Then we generalize the definition to the parametrized context simply by changing the target G-category to the category KG,B of ex-G-spaces over a G- space B. Thus we define an IG -space X over (and under) B to be a G-functor X :IG -! KG,B. Using nonequivariant arrows and equivariant maps, we obtain the G-category (IG KB , GI KB ) of IG -spaces. To unravel definitions, for each representation V 2 V we are given an ex-G- space X(V ) over B, for each arrow (linear isometric isomorphism) f :V -! W we are given an arrow (non-equivariant map) X(f): X(V ) -! X(W ) of ex-G-spaces over B, and the continuous function X :IG (V, W ) -! KG,B(X(V ), X(W )) is a based G-map. An arrow ff: X -! Y is just a natural transformation, and a G-map is a G-natural transformation, for which each ffV :X(V ) -! Y (V ) is a G-map. For both arrows and G-maps, the naturality diagrams X(V )__ffV//_Y (V ) X(f)|| |Y|(f) fflffl| fflffl| X(W ) _ffW_//Y (W ) must commute for all arrows f :V -! W . The group G acts on the space IG KB (X, Y ) of arrows by levelwise conjugation. The G-fixed category is denot* *ed by GI KB . It has objects the IG -spaces X and maps the G-maps. To study the parametrized enrichment of the G-category of orthogonal G- spectra over B, it is convenient to extend the domain category IG , which is en* *riched over KG,*, to a new domain category IG,B that is enriched over KG,B. Departing from the notational pattern of Definition 10.2.1 and using Remark 10.1.6, we de* *fine the hom ex-G-spaces over B of IG,B by (11.1.1) IG,B(V, W ) = IG (V, W )B B x IG (V, W ). If X :IG -! KG,B is an IG -space, then the given based G-maps X :IG (V, W ) -! KG,B(X(V ), X(W )) correspond by adjunction (see (10.2.7) and (10.2.8)) to ex-G-maps X(V ) ^B IG,B(V, W ) -! X(W ). In turn, these correspond by the internal hom adjunction to ex-G-maps X :IG,B(V, W ) -! FB (X(V ), X(W )). 134 11. THE CATEGORY OF ORTHOGONAL G-SPECTRA OVER B These give an equivalent version of the original G-functor X, but now in terms * *of categories enriched over the category GKB . Lemma 11.1.2. The G-category (IG KB , GI KB ) of IG -spaces is equivalent to the G-category of IG,B-spaces, where an IG,B-space is a G-functor X :IG,B -! KG,B enriched over GKB . Proposition 11.1.3. The G-category (IG KB , GI KB ) is G-topological over B and thus also G-topological. Therefore the category GI KB is topologically bi- complete over B. Proof. We define tensor and cotensor IG -spaces over B X ^B K and FB (K, X) levelwise, where K is an ex-G-space and X is an IG -space. For IG -spaces X and Y , we must define a parametrized morphism ex-G-space PB (X, Y ) over B. Parallelling a standard formal description of the G-space IG KB (X, Y ), we def* *ine PB (X, Y ) to be the end Z (11.1.4) PB (X, Y ) = FB (X(V ), Y (V )). IG,B Explicitly, it is the equalizer displayed in the following diagram of ex-G-spac* *es. PB (X, Y ) | | Q fflffl| V FB (X(V ), Y (V )) "~||"|| Q ffflffl|flffl| V,W FB (IG,B(V, W ), FB (X(V ), Y (W ))). The products run over the objects and pairs of objects of a skeleton skIG of IG . The (V, W )th coordinate of "~is given by the composite of the projection * *to FB (X(W ), Y (W )) and the G-map FB (X(W ), Y (W )) -! FB (IG,B(V, W ), FB (X(V ), Y (W ))) adjoint to the composite ex-G-map FB (X(W ), Y (W )) ^B IG,B(V, W ) id^BX|| fflffl| FB (X(W ), Y (W )) ^B FB (X(V ), X(W )) O|| fflffl| FB (X(V ), Y (W )). The (V, W )th coordinate of "is the composite of the projection to FB (X(V ), Y* * (V )) and the G-map " V,W:FB (X(V ), Y (V )) -! FB (IG,B(V, W ), FB (X(V ), Y (W )) 11.1. THE CATEGORY OF IG-SPACES OVER B 135 adjoint to the composite ex-G-map IG,B(V, W ) ^B FB (X(V ), Y (V )) Y|^Bid| fflffl| FB (Y (V ), Y (W )) ^B FB (X(V ), Y (V )) O|| fflffl| FB (X(V ), Y (W )). Passage to ends from the isomorphisms of ex-G-spaces FB (X(V ) ^B K, Y (V )) ~=FB (K, FB (X(V ), Y (V ))) ~=FB (X(V ), FB (K, Y (V * *))) gives natural isomorphisms of ex-G-spaces (11.1.5) PB (X ^B K, Y ) ~=FB (K, PB (X, Y )) ~=PB (X, FB (K, Y )). With these constructions, we see that (IG KB , GI KB ) is G-topological over B; compare Definition 10.2.1 and the discussion following it. The last statement f* *ollows since GI KB is complete and cocomplete, with limits and colimits constructed levelwise from the limits and colimits in GKB . We have several kinds of smash products and function objects in this context. For IG -spaces X and Y over B, define the "external" smash product X ZB Y by X ZB Y = ^B O (X x Y ): IG x IG -! KG,B. Thus (X ZB Y )(V, W ) = X(V ) ^B Y (W ). Here we have used the word "external" to refer to the use of pairs of representations, as is usual in the theory of d* *iagram spectra. It is standard category theory [30, 62] to use left Kan extension to i* *nter- nalize this external smash product over B. This gives the internal smash product X ^B Y of IG -spaces over B, which is again an IG -space over B. For an IG -spa* *ce Y over B and an (IG xIG )-space Z over B, define the external function IG -space over B, denoted ~FB(Y, Z), by F~B(Y, Z)(V ) = PB (Y, Z), where Z(W ) = Z(V, W ). It is mainly to allow this definition that we need * *the morphism ex-G-spaces PB (-, -). It is also formal to obtain an internal function IG -space functor FB on IG -spaces over B by use of right Kan extension [30, 62* *]. Using these internal smash product and function IG -space functors, we obtain t* *he following result. Recall Definition 10.2.9 and Proposition 10.2.10. Theorem 11.1.6. (IG KB , GI KB ) is a G-topological closed symmetric mon- oidal G-category over B. Remark 11.1.7. In the theory of ex-spaces, we also have the "external smash product" of ex-spaces over different base spaces defined in x2.5. Using the two different notions of "external" together, we obtain the definition of the "exte* *rnal external smash product" of an IG -space X over A and an IG -space Y over B; it is an (IG x IG )-space over A x B. We write X Z Y for the left Kan extension internalization of this smash product. Thus X Z Y is an IG -space over A x B. Similarly, using the external function ex-space construction ~Fof x2.5, for an * *IG - space Y over B and an IG -space Z over AxB, we obtain the "internalized external 136 11. THE CATEGORY OF ORTHOGONAL G-SPECTRA OVER B function IG -space" F~(Y, Z) over A. Notationally, use of Z and F~ without an ensuing subscript always denotes these internalized external operations with re* *spect to varying base spaces. We shall return to these functors in Proposition 11.4.1* *0. Similarly, but more simply, we have the "external tensor" K Z Y of an ex-G- space K over A and an IG -space Y over B, which again is an IG -space over AxB. When A = *, this is just the tensor of based G-spaces with IG -spaces over B. T* *he case B = * shows how to construct an IG -space over A from an ex-G-space over A and an IG -space. Since these external tensors can be view as special cases of external smash products, via variants of Definition 10.2.9 and (11.2.6) below, * *we shall not treat them formally and shall not repeat the definitions on the G-spe* *ctrum level. However, we shall find several uses for them. 11.2. The category of orthogonal G-spectra over B For a representation V of G and an IG -space X, we define (11.2.1) VBX = X ^B SVB and VBX = FB (SVB, X), where SV is the one-point compactification of V . Definition 11.2.2. Define the G-sphere SB , written SG,B when necessary for clarity, to be the IG -space over B that sends V to SVB. Clearly SVB^B SWB ~=SVB W , and the functor SB is strong symmetric monoidal, where the monoidal structure on IG is given by direct sums. It follows that SB * *is a commutative monoid in the symmetric monoidal category GI KB , and we can define SB -modules X in terms of (right) actions X^B SB -! X. These SB -modules are our orthogonal G-spectra over B, but it is more convenient to give the defi* *nition using the equivalent reformulation in terms of the external smash product. Definition 11.2.3. An IG -spectrum, or orthogonal G-spectrum, over B is an IG -space X over B together with a structure G-map oe :X ZB SB -! X O such that the evident unit and associativity diagrams commute. Thus we have compatible equivariant structure maps oe : WBX(V ) = X(V ) ^B SWB -! X(V W ). Let SG,B denote the topological G-category of IG -spectra over B and arrows f :X -! Y that commute with the structure maps, with G acting by conjugation on arrows. Let GSB denote the topological category of IG -spectra over B and G-maps (equivariant arrows) between them. Definition 11.2.4. Define the suspension orthogonal G-spectrum functor and the 0th ex-G-space functor 1B:KG,B -! SG,B and 1B:SG,B -! KG,B by ( 1BK)(V ) = VBK, with the evident isomorphisms as structure maps, and 1BX = X(0). Then 1B and 1B give left and right adjoints between KG,B and SG,B and, on passage to G-fixed points, between GKB and GSB . The category GSB is our candidate for a good category of parametrized G- spectra over B. It inherits all of the properties of the category GI KB of IG -* *spaces that were discussed in the previous section and, in the case B = *, it is exact* *ly 11.2. THE CATEGORY OF ORTHOGONAL G-SPECTRA OVER B 137 the category GS of orthogonal G-spectra that is studied in [61]. We summarize its formal properties in the following omnibus theorem. In the language of x10.* *2, much of it can be summarized by the assertion that the G-category (SG,B, GSB ) is a G-topological closed symmetric monoidal G-category over B, but we prefer to be more explicit than that. Theorem 11.2.5. The G-category SG,B is enriched over GKB and is ten- sored and cotensored over KG,B. The category GSB is enriched over KB and is tensored and cotensored over GKB . The G-category SG,B and the category GSB admit smash product and function spectrum functors ^B and FB under which they are closed symmetric monoidal with unit object SB . Let X and Y be orthogonal G-spectra over B and K be an ex-G-space over B. The morphism ex-G-spaces PB (X, Y ) can be specified by PB (X, Y ) = 1BFB (X, Y ), and there are natural isomorphisms 1BK ~=SB ^B K and 1BX ~=PB (SB , X). The tensors and cotensors are related to smash products and function G-spectra * *by natural isomorphisms (11.2.6) X ^B K ~=X ^B 1BK and FB (K, X) ~=FB ( 1BK, X) of orthogonal G-spectra. There are natural isomorphisms (11.2.7) PB ( 1BK, X) ~=FB (K, 1BX) and (11.2.8) PB (X ^B K, Y ) ~=FB (K, PB (X, Y )) ~=PB (X, FB (K, Y )) of ex-G-spaces, (11.2.9) SG,B(X ^B K, Y ) ~=KG,B(K, PB (X, Y )) ~=SG,B(X, FB (K, Y )) of based G-spaces, and (11.2.10) GSB (X ^B K, Y ) ~=GKB (K, PB (X, Y )) ~=GSB (X, FB (K, Y )) of based spaces. Moreover, GSB is G-topologically bicomplete over B. Proof. For the enrichment, the G-space SG,B(X, Y ) is the evident sub G- space of IG KB (X, Y ), and the space GSB (X, Y ) is the evident sub space of GI KB (X, Y ). The tensors and cotensors in SG,B are constructed in IG KB and given induced structure maps. The limits and colimits in GSB are constructed in the same way. As in [61, IIx3], we think of orthogonal G-spectra over B as SB -modules, and we construct the smash product and function spectra functors by passage to coequalizers and equalizers from the smash product and function IG -space functors, exactly as in the definition of tensor products and hom fun* *ctors in algebra. We have defined PB (X, Y ) in the statement, but we shall give a mo* *re intrinsic alternative description later. The first isomorphism of (11.2.6) is g* *iven by unit and associativity relations X ^B K ~=(X ^B SB ) ^B K ~=X ^B 1BK. 138 11. THE CATEGORY OF ORTHOGONAL G-SPECTRA OVER B The second follows from the Yoneda lemma since GSB (X, FB (K, Y ))~=GSB (X ^B K, Y ) ~=GSB (X ^B 1BK, Y ) ~=GSB (X, FB ( 1BK, Y )). Now (11.2.7) and (11.2.8) follow from already established adjunctions. For part* * of the latter, we apply 1B to the composite isomorphism FB (X ^B K, Y )~=FB (X ^B 1BK, Y ) ~=FB (X, FB ( 1BK, Y )) ~=FB (X, FB (K, Y )). Comparisons of definitions, seen more easily from (11.3.2) below, give (11.2.11) SG,B(X, Y ) = KG,B(S0B, PB (X, Y )) and (11.2.12) GSB (X, Y ) ~=GKB (S0B, PB (X, Y )). Therefore the isomorphisms (11.2.9) and (11.2.10) follow from (11.2.8). As noted in x10.1, we obtain the following corollary by replacing K with TB for a based G-space T in the tensors and cotensors of the theorem. Of course, t* *hese tensors and cotensors with G-spaces could just as well be defined directly. It * *will be important in our discussion of model category structures to keep separately * *in mind the tensors and cotensors over ex-G-spaces over B and over based G-spaces. Corollary 11.2.13. The G-category SG,B is enriched over GK* and is ten- sored and cotensored over KG,*. The category GSB is enriched over KG,* and is tensored and cotensored over GK*. Thus, for orthogonal G-spectra X and Y and based G-spaces T , (11.2.14) SG,B(X ^B T, Y ) ~=KG,*(T, SG,B(X, Y )) ~=SG,B(X, FB (T, Y )) and (11.2.15) GSB (X ^B T, Y ) ~=GK*(T, SG,B(X, Y )) ~=GSB (X, FB (T, Y )). We have the parallel definition of G-prespectra over B. Definition 11.2.16. A G-prespectrum X over B consists of ex-G-spaces X(V ) over B for V 2 V together with structure G-maps oe : WBX(V ) -! X(V W ) such that oe is the identity if W = 0 and the following diagrams commute. ~= W Z ZB WBX(V )_______// B X(V ) ZBoe|| |oe| fflffl| fflffl| ZBX(V W ) __oe//_X(V W Z) Let PG,B denote the G-category of G-prespectra and nonequivariant arrows, and let GPB denote its G-fixed category of G-prespectra and G-maps. There result forgetful functors U: SG,B -! PG and U: GSB -! GPB . 11.3. ORTHOGONAL G-SPECTRA AS DIAGRAM EX-G-SPACES 139 The categories PG,B and GPB enjoy the same properties that were speci- fied for SG,B and GSB in Theorem 11.2.5 and Corollary 11.2.13, except for the statements about smash product and function spectra. Here, since we do not have the internal hom functor FB , we must give an alternative direct description of PB (X, Y ), as in (11.3.2) below. 11.3.Orthogonal G-spectra as diagram ex-G-spaces Arguing as in [62, x2] and [61, IIx4], we construct a new domain category JG* *,B which has the same object set V as IG and, like IG,B, is enriched over GKB . It builds in spheres in such a way that the category of IG -spectra over B is equi* *valent to the category of JG,B-spaces over B. Here, just as for IG,B in Lemma 11.1.2, we understand a JG,B-space to be an enriched G-functor X :JG,B -! KG,B. Thus it is specified by ex-G-spaces X(V ) and ex-G-maps X :JG,B(V, W ) -! FB (X(V ), X(W )). To construct JG,B, recall from [61, IIx4] that we have a topological G-category JG with object set V such that the category of IG -spectra is equivalent to the category of JG -spaces. We define (11.3.1) JG,B(V, W ) = JG (V, W )B , just as we defined IG,B in (11.1.1), and the desired equivalence of categories * *follows. Rather than repeat either of the different constructions of JG given in [62] and [61], we shall shortly give a direct description of JG,B. The intuition is that* * an extension of an IG,B-space to a JG,B-space builds in an action by SB . The alternative description of GSB as the category of enriched G-functors JG,B -! KG,B and enriched G-natural transformations leads to a more concep- tual proof of Theorem 11.2.5: it is a specialization of general results about d* *iagram categories of enriched functors. In analogy with (11.1.4) we could have defined PB (X, Y ) to be the end Z (11.3.2) PB (X, Y ) = FB (X(V ), Y (V )) JG,B and derived the isomorphism (11.2.8) just as we derived (11.1.5) in the previous section. By the Yoneda lemma, the two definitions of PB (X, Y ) agree. With this description of PB , some of the adjunctions in Theorem 11.2.5 become more transparent. This leads to an alternative description of JG,B in terms of IG,B, following* * [62, 2.1]. We have the represented functors V *:IG -! KG,B specified by V *(W ) = IG,B(V, W ). If X is an IG -space, such as V *, then the smash product X ^B SB in the category of IG -spaces is a "free" orthogonal G-spectrum over B. Let (11.3.3) JG,B(V, W ) = PB (W *^B SB , V *^B SB ), with the evident composition. Then we can mimic the arguments of [62, xx2, 23] * *to check that the category of JG,B-spaces is equivalent to the category of IG -spe* *ctra over B. An enriched Yoneda lemma argument [53, 2.4] shows that this description of JG,B coincides up to isomorphism with our original one. Although we will not have occasion to quote it formally, we record the follo* *wing consequence of the identification of IG -spectra over B with JG,B-spaces. 140 11. THE CATEGORY OF ORTHOGONAL G-SPECTRA OVER B Lemma 11.3.4. For any enriched G-functor T :KG,B -! KG,B and orthogonal G-spectrum X over B, the composite functor T OX is an orthogonal G-spectrum over B. Similarly, an enriched natural transformation , :T -! T 0induces a natural G- map , :T O X -! T 0O X. Proof. The enriched functor T is given by maps T :FB (K, L) -! FB (T (K), T (L)). Composing levelwise with X gives maps JG,B(V, W ) -! FB (T (X(V )), T (X(W ))) that specify T O X. It is a direct categorical implication of the fact that T i* *s an enriched functor that there are natural maps of ex-G-spaces T (K) ^B L -! T (K ^B L) and T FB (K, L) -! FB (K, T (L)) for ex-G-spaces K and L. This explains more concretely why the structure maps of X induce structure maps for T O X. Similarly, since , is enriched, it is giv* *en by maps from the unit ex-G-space S0Bto FB (T (K), T 0(K)) such that the appropriate diagrams commute. We specialize to K = X(V ) to obtain , :T O X -! T 0O X. The following functors relating ex-G-spaces to orthogonal G-spectra over B play a central role in our theory. In particular, they give "negative dimension* *al" spheres 1VS0B= S-VB. Definition 11.3.5. Let V *= VB*denote the represented JG,B-space specified by V *(W ) = JG,B(V, W ). Define the shift desuspension functor FV :KG,B -! SG,B by letting FV K = V *^B K for an ex-G-space K. Let EvV :SG,B -! KG,B be the functor given by evaluation at V . The alternative notations 1VK = FV K and 1VK = EvV are often used. In particular, F0 = 10= 1B and Ev0 = 10= 1B. Lemma 11.3.6. The functors FV and EvV are left and right adjoint, and there is a natural isomorphism FV K ^B FW L ~=FV W(K ^B L). Proof. The first statement is clear, and the verification of the second sta* *te- ment is formal, as in [62, x1]. 11.4.The base change functors f*, f!, and f* From now on, we drop the adjective "orthogonal" (or prefix IG ), and we gen- erally take the equivariance for granted, referring to orthogonal G-spectra ove* *r B just as spectra over B. We return G to the notations when considering change of groups, or for emphasis, but otherwise G-actions are tacitly assumed throughout. We first show that the results on base change functors proven for ex-spaces in x2.2 extend to parametrized spectra. We then show that the results in x2.5 relating external and internal smash product and function ex-spaces also extend* * to parametrized spectra. Let A and B be base G-spaces. 11.4. THE BASE CHANGE FUNCTORS f*, f!, AND f* 141 Theorem 11.4.1. Let f :A -! B be a G-map. Let X be in SG,A and let Y and Z be in SG,B. There are G-functors f!:SG,A -! SG,B, f* :SG,B -! SG,A, f*: SG,A -! SG,B and G-adjunctions SG,B(f!X, Y ) ~=SG,A(X, f*Y ) and SG,A(f*Y, X) ~=SG,B(Y, f*X). On passage to G-fixed points levelwise, there result functors f!:GSA -! GSB , f* :GSB -! GSA , f*: GSA -! GSB and adjunctions GSB (f!X, Y ) ~=GSA (X, f*Y ) and GSA (f*Y, X) ~=GSB (Y, f*X). The functor f* is closed symmetric monoidal. Therefore, by definition and impli- cation, f*SB ~=SA and there are natural isomorphisms (11.4.2) f*(Y ^B Z) ~=f*Y ^A f*Z, (11.4.3) FB (Y, f*X) ~=f*FA (f*Y, X), (11.4.4) f*FB (Y, Z) ~=FA (f*Y, f*Z), (11.4.5) f!(f*Y ^A X) ~=Y ^B f!X, (11.4.6) FB (f!X, Y ) ~=f*FA (X, f*Y ). Proof. We define the functors f*, f!, and f* levelwise. This certainly gives well-defined functors on IG -spaces that satisfy the appropriate adjunctions th* *ere. We shall show shortly that these functors preserve IG -spectra. For a based G- space T , f*(TB ) ~=TA , and this implies f*SB ~= SA . If we replace IG -spect* *ra by IG -spaces and replace the internal smash product and function object functo* *rs (^ and F ) by their external precursors (Z and ~F), then everything is immediate by levelwise application of the corresponding results for ex-spaces. Still wor* *king with IG -spaces, we first show how to internalize the isomorphisms (11.4.2) and (11.4.5) by use of the universal property of left Kan extension. Indeed, noting* * that (f*X) O ~=f*(X O ), and similarly for f* and f!, we have IG KA (f*(Y ^B Z), X)~=IG KB (Y ^B Z, f*X) ~=(IG x IG )KB (Y ZB Z, f*X O ) ~=(IG x IG )KA (f*(Y ZB Z), X O ) ~=(IG x IG )KA (f*Y ZA f*Z, X O ) ~=IG KA (f*Y ^A f*Z, X) and IG KB (f!X ^B Y, Z)~=(IG x IG )KB (f!X ZB Y, Z O ) ~=(IG x IG )KB (f!(X ZA f*Y ), Z O ) ~=(IG x IG )KA (X ZA f*Y, f*Z O ) ~=IG KA (X ^A f*Y, f*Z) ~=IG KA (f!(X ^A f*Y ), Z). 142 11. THE CATEGORY OF ORTHOGONAL G-SPECTRA OVER B As explained in [40, xx2-3], the remaining isomorphisms on the IG -space level follow formally. We must show that our functors on IG -spaces preserve IG -spectra. The given structure map oe :Y ZB SB -! Y O gives rise via the external version of (11.4* *.2) to the required structure map f*Y ZA SA ~=f*(Y ZB SB ) -! f*Y O . Similarly, the given structure map oe :X Z SA -! X O gives rise to the requir* *ed structure map f!X ZB SB ~=f!(X ZA SA ) -! f!X O . As in [40, (3.6)], there is a canonical natural map, not usually an isomorphism, ss :f*X ZB Y -! f*(X ZA f*Y ). Taking Y = SB , we see that oe also induces the required structure map f*X ZB SB -! f*(X ZA SA ) -! f*X O . Now the spectrum level adjunctions follow directly from their IG -space analogu* *es. The spectrum level isomorphisms (11.4.2) and (11.4.5) follow from their IG -spa* *ce analogues by comparisons of coequalizer diagrams, and the remaining isomorphisms again follow formally. Remark 11.4.7. Since the base change functors are defined levelwise, they commute with the evaluation functors EvV . These commutation relations for the right adjoints f* and f* imply conjugate commutation isomorphisms f*FV ~=FV f* and f!FV ~=FV f! of left adjoints. In particular, f* 1B~= 1Af* and f! 1A~= 1Bf!. Via (11.2.6), these isomorphisms and the isomorphisms of the theorem imply iso- morphisms relating base change functors to tensors and cotensors. For example (11.4.5) implies isomorphisms f!(f*Y ^A K) ~=Y ^B f!K and f!(f*L ^A X) ~=L ^B f!X. Here K and L are ex-spaces over A and B and X and Y are spectra over A and B. The following result is immediate from its precursor Proposition 2.2.9 for e* *x- spaces. Proposition 11.4.8. Suppose given a pullback diagram of G-spaces g C _____//D i|| |j| fflffl|fflffl| A __f__//B. Then there are natural isomorphisms of functors (11.4.9) j*f!~=g!i*, f*j* ~=i*g*, f*j!~=i!g*, j*f* ~=g*i*. Returning to Remark 11.1.7, we have the following important results on exter- nal smash product and function spectra and their internalization by means of ba* *se change along diagonal maps. 11.5. CHANGE OF GROUPS AND RESTRICTION TO FIBERS 143 Proposition 11.4.10. Let X be a spectrum over A, Y be a spectrum over B, and Z be a spectrum over A x B. There is an external smash product functor that assigns a spectrum X ZY over AxB to X and Y and an external function spectrum functor that assigns a spectrum ~F(Y, Z) over A to Y and Z, and there is a natu* *ral isomorphism GSAxB (X Z Y, Z) ~=GSA (X, ~F(Y, Z)). The internal smash products are determined from the external ones via X ^B Y ~= *(X Z Y ) and FB (X, Y ) ~=~F(X, *Y ), where X and Y are spectra over B and : B -! B x B is the diagonal map. Proof. It is not hard to start from Remark 11.1.7 and construct these func- tors directly. We instead follow Lemma 2.5.3 and observe that the spectrum level external functors can and, up to isomorphism, must be defined in terms of the internal functors as X Z Y ~=ss*AX ^AxB ss*BY and F~(Y, Z) ~=ssA *FAxB (ss*BY, Z), where ssA :A x B -! A and ssB :A x B -! B are the projections. The dis- played adjunction is immediate from the adjunctions (ss*A, ssA *), (ss*B, ssB ** *), and (^AxB , FAxB ). The second statement follows formally, as in Lemma 2.5.4. Proposition 11.4.11. For ex-spaces K over A and L over B, there is a natural isomorphism 1AxB(K Z L) ~= 1AK Z 1BL. Proof. This is most easily seen using adjunction and the Yoneda lemma. Us- ing external function objects, we see that ~F( 1BL, Z) ~=~F(L, Z) for Z 2 GSAxB* * . This has zeroth ex-space ~F(L, Z(0)) over A. 11.5. Change of groups and restriction to fibers We give the analogues for parametrized spectra of the results concerning cha* *nge of groups and restriction to fibers that were given for parametrized ex-spaces * *in x2.3. We shall say more about change of groups in Chapter 14. Fix an inclusion ': H -! G of a (closed) subgroup H of G and let A be an H-space and B be a G-space. We index H-spectra over A on the collection '*V of H-representations '*V with V 2 V . As we discuss in xx14.2 and 14.3, when V is the collection of * *all representations of G, we can change indexing to the collection of all represent* *ations of H since our assumption that G is compact ensures that every representation of H is a direct summand of a representation '*V . We have an evident forgetful functor (11.5.1) '*: GSB -! HS'*B. On the space level, we write '!ambiguously for both the based and unbased induc- tion functors G+ ^H (-) and G xH (-), and similarly for coinduction '*. Context should make clear which is intended. Applying the unbased versions to retracts, we defined induction and coinduction functors '! and '* on ex-spaces in Defini- tion 2.3.7. These functors extend to the spectrum level. Recall that SG,B denot* *es the G-sphere spectrum over B. Proposition 11.5.2. Levelwise application of '!and '* gives functors '!:HSA -! GS'!A and '*: HSA -! GS'*A. 144 11. THE CATEGORY OF ORTHOGONAL G-SPECTRA OVER B Proof. We must show that the structure H-maps oe :X ZSH,A -! X O of an H-spectrum X over A induce structure G-maps for the IG -spaces '!X and '*X. It is clear that '!(X O ) ~='!X O and '*(X O ) ~='*X O . Using (2.3.4), we see th* *at SG,'!A~='!SH,A. Since the functor '!on the ex-space level is symmetric monoidal by Proposition 2.3.9, its levelwise IG -space analogue commutes up to isomorphi* *sm with the external smash product Z. Thus oe induces a structure G-map '!X Z'!ASG,'!A~='!(X ZA SH,A) -! '!(X O ) ~='!X O . For '*, let ~ : '*'* -! Id be the counit of the space level adjunction ('*, '*) (see (2.3.2)). For an H-space A, ~ is the H-map Map H(G, A) -! A given by evaluation at the identity element of G. Applied to an ex-space K over A, thoug* *ht of as a retract, ~ gives a map '*'*K -! K of total spaces over and under the map ~ : '*'*A -! A of base spaces in the category of retracts of x2.5. We can apply this to X levelwise. We also have the projection pr : ~*SH,A -! SH,A over ~. Together, these maps give '*('*X Z'*ASG,'*A) ~='*'*X Z'*'*A~*SH,A~Zpr//_X ZA SH,A. For the isomorphism, we have used the facts that '* is strong monoidal and that '*SG,'*A~= SH,'*'*A~=~*SH,A. The adjoint of the composite of this map with the structure map oe : X ZA SH,A -! X O gives the required structure map '*X Z'*ASG,'*A-! '*X O . As on the ex-space level, the categories HSA and GSGxHA can be used inter- changeably. The following result is immediate from Proposition 2.3.9. Proposition 11.5.3. Let :A -! '*'!A be the natural inclusion of H-spaces. Then '!:HSA -! GS'!Ais a closed symmetric monoidal equivalence of categories with inverse the composite * O '*: GS'!A-! HS'*'!A-! HSA . In particular, if A = * then maps * to the identity coset eH 2 G=H and we * *see that HS and GSG=H can be used interchangeably. Arguing as in Proposition 2.3.1, we could more easily prove this directly. Corollary 11.5.4. The category HS is equivalent as a closed symmetric monoidal category to GSG=H . Under this equivalence, '* ~=r*, '!~=r!, and '* ~=r*, where r :G=H -! *. Looking at the fiber Xb(V ) = X(V )b over b of a G-spectrum X over B, we see a Gb-spectrum Xb of the sort that has been studied in [61], where Gb is the isotropy group of b. Our homotopical analysis of parametrized G-spectra will be based on the idea of applying the results of [61] fiberwise. By the previous re* *sult, we can think of this fiber as a G-spectrum over G=Gb. The following spectrum le* *vel analogues of Example 2.3.12 and Example 2.3.13 analyze the relationships among passage to fibers, base change, and change of groups. Example 11.5.5. For b 2 B, we write b: * -! B for the Gb-map that sends * to b and "b:G=Gb -! B for the induced inclusion of orbits. Under the equivalence GSG=Gb ~=GbS , "b*may be interpreted as the fiber functor GSB -! GbS that sends Y to Yb. Its left and right adjoints "b!and "b*may be interpreted as the * *functors 11.6. SOME PROBLEMS CONCERNING NON-COMPACT LIE GROUPS 145 that send a Gb-spectrum X to the G-spectra Xb and bX over B obtained by level- wise application of the corresponding ex-space level adjoints of Construction 1* *.4.6 and Example 2.3.12. With these notations, the isomorphisms of Theorem 11.4.1 specialize to the following natural isomorphisms, where Y and Z are in GSB and X is in GbS . (Y ^B Z)b ~=Yb^ Zb, FB (Y, bX) ~=bF (Yb, X), FB (Y, Z)b ~=F (Yb, Zb), (Yb^ X)b ~=Y ^B Xb, FB (Xb, Y ) ~=bF (X, Yb). Example 11.5.6. Let f :A -! B be a G-map and let ib:Ab -! B be the inclusion of the fiber over b, which is a Gb-map. As in Example 2.3.13, we have* * the compatible pullback squares fb GxGbfb Ab ____//_{b} G xGb Ab _____//G=Gb ib|| |b| "-b|| "b|| fflffl| fflffl| fflffl| fflffl| A ___f__//B A ____f____//_B. Applying Proposition 11.4.8 to the right-hand square and interpreting the concl* *u- sion in terms of fibers, we obtain canonical isomorphisms of Gb-spectra (f!X)b ~=fb!i*bX and (f*X)b ~=fb*i*bX, where X is a G-spectrum over A, regarded on the right-hand sides as a Gb-spectr* *um over A by pullback along ': Gb -! G. 11.6.Some problems concerning non-compact Lie groups In equivariant stable homotopy theory, the key idea is that the one-point co* *m- pactification of a representation V of dimension n is a G-sphere and that smash* *ing with that sphere should be a self-equivalence of the equivariant stable homotopy category. That is, the idea is to invert G-spheres in just the way that we in- vert spheres when constructing the nonequivariant stable homotopy category. For compact Lie groups of equivariance, the philosophy and its implementation and applications are well understood. When we invert representation spheres, we inv* *ert other homotopy spheres as well, and the relevant Picard group is analyzed in [4* *1]. For non-compact Lie groups, the present work seems to be the first attempt to consider foundations for equivariant stable homotopy theory. The philosophy is * *less clear, and its technical implementation is problematic. The need for such a the* *ory is evident, however. The focus on finite dimensional representations is intrinsic * *to the philosophy but fails to come to grips with basic features of the representation* * theory of non-compact Lie groups. A theory based on finite dimensional representations should still have its uses, but there are real difficulties to obtaining even t* *hat much. In particular, a focus on spheres associated to linear representations, rather * *than on less highly structured homotopy spheres, may be misplaced. 146 11. THE CATEGORY OF ORTHOGONAL G-SPECTRA OVER B A non-compact semi-simple Lie group will generally have no non-trivial finite dimensional unitary or orthogonal representations, hence our theory of "orthogo- nal" G-spectra is clearly too restrictive. This is easily remedied. The use of * *linear isometries in the definition of orthogonal spectra is a choice dictated more by* * the history than by the mathematics. In the alternative approach to equivariant sta* *ble homotopy theory based on Lewis-May spectra and EKMM [39, 59, 61], use of orthogonal complements is certainly convenient and perhaps essential. However, the diagram orthogonal spectra of [61, 62] could just as well have been develop* *ed in terms of diagram "general linear spectra". In the few places where complemen* *ts are used, they can by avoided. For consistency with the previous literature, we* * have chosen to give our exposition in the compact case using the word "orthogonal" a* *nd the language from the cited references, but for general Lie groups of equivaria* *nce, we should eliminate all considerations of isometries. More precisely, for the complete case, we redefine I by taking V to be the collection of all finite dimensional representations V of G. More generally, we* * can index on any subcollection that contains the trivial representation and is clos* *ed under finite direct sums. Since we are only interested in a skeleton of I , we * *may as well restrict to orthogonal representations in V when G is compact. We repla* *ce linear isometries by linear isomorphims when defining the G-spaces I (V, W ). T* *hus we replace orthogonal groups by general linear groups. Otherwise, the formal de* *fi- nitional framework developed in this chapter (or, in the nonparametrized case, * *[61, II]) goes through verbatim for general topological groups G. However, we emphasize the formality. When considering change of groups, for example, the significance changes drastically. As noted at the start of the pre* *vious section, for an inclusion ': H - ! G of a (closed) subgroup H of G, we index H-spectra on the collection '*V of H-representations '*V with V 2 V . We also pointed out the relevance of the compact case of the following result. Proposition 11.6.1. If G is either a compact Lie group or a matrix group and W is a representation of a subgroup H, then there is a representation V of G and an embedding of W as a subrepresentation of '*V . This is clear in the compact case and is given by [81, 3.1] for matrix group* *s. However, the following striking counterexample, which we learned from Victor Ginzburg, shows just how badly this basic result fails in general. Counterexample 11.6.2 (Ginzburg). Let H be the Heisenberg group of 3 x 3 matrices 0 1 1 a c @ 0 1 b A 0 0 1 where a, b, and c are real numbers. Embed R in H as the subgroup of matrices with a = b = 0. Embed Z in R as usual. Then R is a central subgroup of H. Define G = H=Z. Then T = R=Z is a circle subgroup of G. Moreover, T is the center of G and coincides with the commutator subgroup [G, G]. Let V be any finite dimensional (complex linear) representation of G. Since T is compact, t* *he action of T on V is semisimple, and since T is central, any weight space of T i* *s a G-submodule. Therefore V is a direct sum of G-submodules Visuch that T acts on each Viby scalar matrices. Since T = [G, G], this scalar action of T on Viis tr* *ivial: the determinant of g is 1 for any g 2 [G, G]. Therefore no nontrivial 1-dimensi* *onal 11.6. SOME PROBLEMS CONCERNING NON-COMPACT LIE GROUPS 147 character of T can embed in V . Reinterpreting in terms of real representations* *, as we may, we conclude that, for ': T -! G, '*V is the trivial T -universe. For a compact Lie group G and inclusion ': H G, '*X is a dualizable H- spectrum if X is a dualizable G-spectrum, and an H-spectrum indexed on the triv* *ial H-universe is dualizable if and only if it is a retract of a finite H-CW spectr* *um built up from trivial orbits. We conclude that duality theory (in the nonparametrized context) cannot work as one would wish in the context of the previous example. Looking ahead, much of the theory of the following three chapters also works formally in the context of non-compact Lie groups. However, there is at least o* *ne serious technical difficulty. Our theory is based on the use of one-point compa* *ctifi- cations SV . If V is a linear representation of a non-compact Lie group G, ther* *e is no reason to think that G acts smoothly and properly on SV , even if the isotro* *py groups of V are compact. In fact, if Illman's Theorem 3.3.2 were to apply, then SV would be a G-cell complex, hence it would be built up from non-compact orbits G=H given by compact subgroups H. However, as closed subsets of SV , the closed cells would have to be compact. That is, the putative G-CW structure would con- tradict the compactness of SV . Said another way, we see no reason to believe t* *hat the SV are q-cofibrant G-spaces. Therefore, the functors (-) ^ SV need not be Quillen left adjoints and the functors V and VBneed not preserve fibrant obje* *cts in the relevant model structures. Compare, for example, Proposition 12.2.2 and * *the derivation of the long exact sequences (12.3.2) and (12.3.3) below. What seems * *to be needed, for a start, is something like a model structure on G-spaces such th* *at X is cofibrant if '*X is H-cofibrant for all inclusions ': H ! G of compact subgr* *oups. CHAPTER 12 Model structures for parametrized G-spectra Introduction We define and study two model structures on the category GSB of (orthogonal) G-spectra over B. We emphasize that, except for the theory of smash products, everything in this chapter applies equally well to the category GPB of G-prespe* *ctra over B. That fact will become important in the next chapter. We start in x12.1 by defining a "level model structure" on GSB , based on the qf-model structure on GKB . In x12.2, we record analogues for this model struct* *ure of the results on external smash product and base change functors that were giv* *en for GKB in x7.2. The level model structure serves as a stepping stone to the st* *able model structure, which we define in x12.3. It has the same cofibrations as the * *level model structure, and we therefore call these "s-cofibrations". An essential po* *int in our approach is a fiberwise definition of the homotopy groups of a parametri* *zed G-spectrum that throws much of our work onto the theory of nonparametrized orthogonal G-spectra developed by Mandell and the first author in [61]. We defi* *ne homotopy groups using the level qf-fibrant replacement functor provided by the level model structure, and we define stable equivalences to be the ss*-isomorph* *isms. It is essential to think in terms of fibers and not total spaces since the tota* *l spaces of a parametrized spectrum do not assemble into a spectrum. We show in x12.4 that the ss*-isomorphisms give a well-grounded subcategory of weak equivalences, and we complete the proofs of the model axioms in x12.5. We return to the conte* *xt of x12.2 in x12.6, where we prove that various Quillen adjoint pairs in the lev* *el model structures are also Quillen adjoint pairs in the stable model structures. The basic conclusion is that GSB is a well-grounded model category under the stable structure. Although not very noticeable on the surface, essential use is* * made of the qf-model structure on GKB throughout this chapter. It is possible to obt* *ain a level model structure on GSB from the q-model structure on GKB , as we explain in Remark 12.1.8. However this model structure is not well-grounded and therefo* *re does not provide the necessary tools to work out the technical details of x12.4* *. The results there are crucial to prove that the relative cell complexes over B defi* *ned in terms of the appropriate generating acyclic s-cofibrations are acyclic.1 It was* * our fruitless attempt to obtain a stable model structure starting from the level q-* *model structure that led us to the construction of the qf-model structure on GKB and * *to the notion of a well-grounded model category. When there are no issues of equivariance, we generally abbreviate G-spectrum over B, ex-G-space, and G-space to spectrum over B, ex-space, and space; G is a compact Lie group throughout. ____________ 1In [47, 3.4], such acyclicity of relative cell complexes is assumed withou* *t proof. 148 12.1. THE LEVEL MODEL STRUCTURE ON GSB 149 12.1. The level model structure on GSB After changing the base space from * to B, the level model structure works in much the same way as in the nonparametrized case of [61]. Definition 12.1.1. Let f :X -! Y be a map of spectra over B. With one exception, for any type of ex-space and any type of map of ex-spaces, we say th* *at X or f is a level type of spectrum over B or a level type of map of spectra ove* *r B if each X(V ) or f(V ): X(V ) -! Y (V ) is that type of ex-space or that type of m* *ap. Thus, for example, we have level h, level f and level fp-fibrations, cofibratio* *ns and equivalences from x5.1 together with the corresponding fibrant and cofibrant objects. We have level q-equivalences and level q and qf-fibrations from x7.1 a* *nd we have level ex-fibrations and level ex-quasifibrations from x8.1 and x8.5. T* *he exceptions concern cofibrations and cofibrant objects. We shall never be intere* *sted in "level q-cofibrations" or "level qf-cofibrations", nor in "level q-cofibrant* *" or "level qf-cofibrant" objects, since these do not correspond to cofibrations and cofibr* *ant objects in the model structures that we consider. Instead we have the following definitions. (i)f is an s-cofibration if it satisfies the LLP with respect to the level acy* *clic qf-fibrations. (ii)f is a level acyclic s-cofibration if it is both a level q-equivalence and* * an s-cofibration. To reiterate, in the phrase "level acyclic qf-fibration", the adjective "level"* * applies to "acyclic qf-fibration", but in the phrase "level acyclic s-cofibration" it a* *pplies only to "acyclic"; the cofibrations are not defined levelwise. Definition 12.1.2. A spectrum X over B is well-sectioned if it is level well- sectioned, so that each ex-space X(V ) is ~f-cofibrant. It is well-grounded if* * it is level well-grounded, so that each X(V ) is well-sectioned and compactly generat* *ed. The discussion of x4.3 applies to the category GSB of G-spectra over B with homotopies defined in terms of the cylinders X ^B I+ . In particular, we have t* *he notion of a Hurewicz cofibration in GSB , abbreviated cyl-cofibration, defined * *in terms of these_cylinders, and we also have the notion of strong Hurewicz cofibr* *ation, abbreviated cyl-cofibration. Lemma 12.1.3. A cyl-cofibration of spectra over B is a level fp-cofibration * *and a cyl-fibration of spectra over B is a level fp-fibration. A cyl-cofibration be* *tween well-sectioned spectra over B is a level f-cofibration and therefore both a lev* *el h- cofibration and a level fp-cofibration. Proof. By the mapping cylinder retraction criterion of Hurewicz cofibration* *s, a cyl-cofibration of spectra over B is a level fp-cofibration. The statement ab* *out fi- brations follows similarly from the path lifting function characterization of H* *urewicz fibrations. An fp-cofibration between well-sectioned ex-spaces is an f-cofibrat* *ion by Proposition 5.2.3, and all f-cofibrations are h-cofibrations. Recall the notions of a ground structure and of a well-grounded subcategory * *of weak equivalences from Definitions 5.3.2, 5.3.3, and 5.4.1. Proposition 12.1.4. The well-grounded spectra over B give GSB a ground structure whose ground cofibrations, or g-cofibrations, are the level h-cofibra* *tions. 150 12. MODEL STRUCTURES FOR PARAMETRIZED G-SPECTRA The level q-equivalences specify a well-grounded subcategory of weak equivalenc* *es with respect to this ground structure. In the gluing and colimit lemmas, one ne* *ed only assume that the relevant maps are level h-cofibrations, not necessarily al* *so cyl-cofibrations. Proof. That we have a ground structure follows levelwise from the ground structure on ex-spaces in Proposition 5.3.7. That the level q-equivalences are * *well- grounded follows levelwise from Proposition 5.4.9. We construct the level model structure on GSB from the qf-model structure on GKB specified in Remark 7.2.11, but all results apply verbatim starting from the qf(C )-model structure for any closed generating set C (as defined in Defin* *i- tion 7.2.6). We shall need the extra generality for the reasons discussed in Ch* *apter 7. Recall that IfBand JfBdenote the sets of generating qf-cofibrations and gene* *r- ating acyclic qf-cofibrations in GKB . We use the shift desuspension functors FV of Definition 11.3.5 to obtain corresponding sets on the spectrum level. We need the following observations. Lemma 12.1.5. The functor FV enjoys the following properties. (i)If K is a well-grounded ex-space over B, then FV K is well-grounded. If K is an ex-fibration, then FV K is a level ex-fibration. (ii)If i: K -! L is an h-equivalence between well-grounded ex-spaces over B, then FV i is a level h-equivalence. (iii)If i: K -! L is an fp-cofibration, then FV i is a cyl-cofibration and ther* *efore a level fp-cofibration. If, further, K and L are well-sectioned, then FV i * *is a level f-cofibration_and_therefore a level h-cofibration._ (iv)If i: K -! L is an fp-cofibration,_then FV i is a cyl-cofibration. (v)If i: K -! L is_an_f-cofibration between well-grounded_ex-spaces over B, then FV i_is_a cyl-cofibration which_is a level f-cofibration and therefore* * both a level fp-cofibration and a level h-cofibration. Proof. By Definition 11.3.5, (FV K)(W ) = JG (V, W )B ^B K, and the G- space JG (V, W ) is well-based. Now (i) holds by Corollary 8.2.5 and (ii) holds by Proposition 8.2.6. Since FV is left adjoint to the evaluation functor Ev V and since cyl-fibrations are level fp-fibrations, (iv) and the first statement * *of (iii) follow from the definitions by adjunction. The second statement of (iii)_follo* *ws from_Proposition_5.2.3. The first half of (v) follows from (iv) since f-cofibra* *tions are fp-cofibrations, and the second half follows from (iii) since_FV i is a lev* *el f- cofibration between well-grounded spectra and therefore a level f-cofibration by Theorem 5.2.8(ii). Definition 12.1.6. Define F IfBto be the set of maps FV i with V in a skelet* *on skIG of IG and i in IfB. Define F JfBto be the set of maps FV j with V in skIG and j in JfB. Recall the notion of a well-grounded model structure from Definition 5.5.2. Among other properties, such model structures are compactly generated, proper, and G-topological. Theorem 12.1.7. The category GSB is a well-grounded model category with respect to the level q-equivalences, the level qf-fibrations and the s-cofibrat* *ions. The 12.1. THE LEVEL MODEL STRUCTURE ON GSB 151 sets F IfBand F JfBgive the generating s-cofibrations_and the generating level_* *acyclic s-cofibrations._ All s-cofibrations are level f-cofibrations, hence level fp an* *d level h-cofibrations, and all s-cofibrant spectra over B are well-grounded. ___ Proof. By Lemma 12.1.5, the maps_in F IfB and F JfB are cyl-cofibrations between well-grounded objects and f-cofibrations. Moreover, the maps in F JfB are level acyclic. Therefore, to prove the model axioms, we need only verify t* *he compatibility condition (ii) in Theorem 5.5.1. Adjunction arguments show that a map is a level qf-fibration if and only if it has the RLP with respect to F JfB* *and that it is a level acyclic q-fibration if and only if it has the RLP with respe* *ct to F IfB. This implies that the classes of s-cofibrations and of F IfB-cofibratio* *ns (in the sense of Definition 4.5.1(iii)) coincide. Therefore, if a map has the RLP w* *ith respect to F IfB, then it is a level acyclic qf-fibration. The required compati* *bility condition now follows from its analogue for GKB . Condition (iv) in Theorem 5.5* *.1 holds by its ex-space level analogue and the fact that (FV K) ^B T ~=FV (K ^B T* * ) for an ex-space K over B and a based space T . Right properness follows directly from the space level analogue. Remark 12.1.8. Just as in Definition 12.1.6, we can also define sets F IB and F JB based on the generating sets IB and JB for the q-model structure on GKB . * *We can then use Theorem 4.5.5 to prove the analogue of Theorem 12.1.7 stating that GSB is a cofibrantly generated model category under the level q-model structure. Since the compatibility condition holds by the same proof as for the level qf-m* *odel structure, we need only verify the acyclicity condition to show this. For a generating acyclic q-cofibration j 2 JB , we have FV j = V *^B j, where V *(W ) = JG,B(V, W ). This map is a level h-equivalence by Lemma 12.1.5(ii). Although j is an h-cofibration, it is not immediate that FV j is a level h-cofi* *bration. (This holds for j 2 JfBby Lemma 12.1.5(iii), since j is then an fp-cofibration). Indeed, for general ex-spaces K and h-cofibrations f, K ^B f need not be an h- cofibration. However, since JG,B(V, W ) = JG (V, W )B , we see directly that FV* * j is indeed a level h-cofibration. By inspection of the definition of wedges over* * B in terms of pushouts, the gluing lemma in K then applies to show that wedges over B of maps in F JB are level acyclic h-cofibrations. Since pushouts and colimits i* *n SB are constructed levelwise on total spaces, it follows that relative F JB comple* *xes are acyclic h-cofibrations since the q-model structure on K is well-grounded. Remark 12.1.9. As in the nonparametrized case [61], "positive" model struc- tures would be needed to obtain a comparison with the as yet undeveloped alter- native approach to parametrized stable homotopy theory based on [39, 59]. Such model structures can be defined as in [61, p. 44], starting from the subsets (F* * IfB)+ and (F JfB)+ that are obtained by restricting to those V such that V G 6= 0. One then defines the positive level versions of all of the types of maps specified * *in Def- inition 12.1.1 by restricting to those levels V such that V G 6= 0. The posit* *ive level analogue of Theorem 12.1.7 holds, where the positive s-cofibrations are t* *he s-cofibrations that are isomorphisms at all levels V such that V G = 0; compare [61, III.2.10]. However, we shall make no use of the positive model structure i* *n this paper, and we will make little further reference to it. The same proof as in [61, I.2.10, II.4.10, III.2.12] gives the following res* *ult. 152 12. MODEL STRUCTURES FOR PARAMETRIZED G-SPECTRA Theorem 12.1.10. The forgetful functor U from spectra over B to prespectra over B has a left adjoint P such that (P, U) is a Quillen equivalence. 12.2.Some Quillen adjoint pairs relating level model structures This section gives the analogues for the level model structure of some of the ex-space level results in xx7.2-7.4. These results are also analogues of resul* *ts in [61, III.x2], which in turn have non-equivariant precursors in [62, x6]. They a* *dmit essentially the same proofs as in Chapter 7 or in the cited references. The lev* *el qf- model structure is understood throughout. More precisely, where a qf(C )-model structure was used in Chapter 7, we must use the corresponding level qf(C )-mod* *el structure here. Since we want our model structures to be G-topological, we only use generating sets C that are closed under finite products. Our first observation is immediate from the fact that equivalences and fibra- tions are defined levelwise, the next follows directly from its ex-space analog* *ue Proposition 7.3.1, and the third and fourth are proven in the same way as their ex-space analogues 7.3.2 and Corollary 7.3.3. All apply to the level qf(C )-mod* *el structures for any choice of C . Proposition 12.2.1. The pair of adjoint functors (FV , EvV) between GKB and GSB is a Quillen adjoint pair. Proposition 12.2.2. For a based G-CW complex T , ((-) ^B T, FB (T, -)) is a Quillen adjoint pair of endofunctors of GSB . Proposition 12.2.3. If i: X - ! Y and j :W - ! Z are s-cofibrations of spectra over base spaces A and B, then i j :(Y Z W ) [XZW (X Z Z) -! Y Z Z is an s-cofibration over A x B which is level acyclic if either i or j is acycl* *ic. As in x7.2, we cannot expect this result to hold for internal smash products* * over B. The case A = *, which relates spectra to spectra over B, is particularly imp* *or- tant. As we explain in x14.1, it leads to a fully satisfactory theory of parame* *trized module spectra over nonparametrized ring spectra. Corollary 12.2.4. If Y is s-cofibrant over B, then the functor (-) Z Y from GSA to GSAxB is a Quillen left adjoint with Quillen right adjoint ~F(Y, -). Again the next result is a direct consequence of its ex-space analogue Propo* *si- tion 7.3.4 and applies with any choice of C . Proposition 12.2.5. Let f :A -! B be a G-map. Then (f!, f*) is a Quillen adjoint pair. The functor f! preserves level q-equivalences between well-secti* *oned G-spectra over B. If f is a qf-fibration, then f* preserves all level q-equival* *ences. Proposition 12.2.6. If f :A -! B is a q-equivalence, then (f!, f*) is a Quil* *len equivalence. Proof. We mimic the proof of Proposition 7.3.5, but with X and Y taken to be an s-cofibrant G-spectrum over A and a level qf-fibrant G-spectrum over B. It is clear that f*Y - ! Y is a level q-equivalence since A -! B is a q-equivalenc* *e. Since X is s-cofibrant, *A -! X is a level h-cofibration. Note that it is essen* *tial for this statement that we start from the qf and not the q-model structure on ex-sp* *aces. 12.3. THE STABLE MODEL STRUCTURE ON GSB 153 Since pushouts along level h-cofibrations preserve level q-equivalences, X -! f* *!X is a level q-equivalence. The conclusion follows as in Proposition 7.3.5. Proposition 12.2.7. Let f :A -! B be a G-bundle whose fibers Ab are Gb- CW complexes. Then f* preserves level q-equivalences and s-cofibrations. There- fore (f*, f*) is a Quillen adjoint pair. Proof. Here we must use a generating set C (f) as specified in Proposi- tion 7.3.8. The proof that f* preserves s-cofibrations reduces to showing that the maps f*FV i ~=FV f*i are s-cofibrations for generating s-cofibrations i. Si* *nce FV is a Quillen left adjoint it takes qf-cofibrations to s-cofibrations, so we * *are reduced to the ex-space level, where f*i is shown to be a qf-cofibration in Pro* *po- sition 7.3.8. Now consider the change of groups functors of x11.5. The following result sh* *ows that the equivalence of Proposition 11.5.3 descends to homotopy categories. It * *is proven by levelwise application of its ex-space analogue Proposition 7.4.6, tog* *ether with change of universe considerations that are deferred until x14.2 and x14.3. Proposition 12.2.8. Let ': H -! G be the inclusion of a subgroup. The pair of functors ('!, *'*) relating HSA and GS'!Agive a Quillen equivalence. If A is completely regular, then '!is also a Quillen right adjoint. For a point b in B, we combine the special case "b:G=Gb -! B of Propo- sition 12.2.5 with Proposition 12.2.8, where ': Gb -! G and :* -! G=Gb, to obtain the following analogue of Proposition 7.5.1. Recall from Example 11.5.5 * *that the fiber functor (-)b:GSB -! GbS is given by *'*"b*= b*'*. Its left adjoint (-)b therefore agrees with "b!'!. Proposition 12.2.9. For b 2 B, the pair of functors ((-)b, (-)b) relating GbS* and GSB is a Quillen adjoint pair. 12.3. The stable model structure on GSB The essential point in the construction of the stable model structure is to * *define the appropriate (stable) homotopy groups. The weak equivalences will then be the maps of parametrized spectra that induce isomorphisms on all homotopy groups. We refer to them as the ss*-isomorphisms or s-equivalences, using these terms i* *n- terchangeably. There are several motivating observations for our definitions. We return the group G to the notations for the moment. First, a G-spectrum X over B is level qf-fibrant if and only if each projec- tion X(V ) -! *B (V ) = B is a qf-fibration of ex-G-spaces. It is equivalent th* *at each fixed point map X(V )H - ! BH be a non-equivariant qf-fibration, and, by Proposition 6.5.1, we have resulting long exact sequences of homotopy groups (12.3.1) . .-.! ssHq+1(B) -! ssHq(Xb(V )) -! ssHq(X(V )) -! ssHq(B) -! . . . for each b 2 BH . Here, for a G-space T , ssHq(T ) denotes ssq(T H). Second, as we have already discussed in x11.4, the fibers Xb of a G-spectrum* * X are Gb-spectra, and our guiding principle is to use these nonparametrized spect* *ra to encode the homotopical information about our parametrized spectra. Proposi- tion 12.2.9 allows us to encode levelwise information in the level homotopy gro* *ups of fibers, and it is plausible that we can similarly encode the full structure * *of our 154 12. MODEL STRUCTURES FOR PARAMETRIZED G-SPECTRA parametrized G-spectrum X in the spectrum level homotopy groups of the fiber Gb-spectra Xb. However, we can only expect to do so when X is level qf-fibrant and we have the long exact sequences (12.3.1). Recall that the homotopy groups ssHq(Y ) of a nonparametrized G-spectrum Y are defined in [61, III.3.2] as the colimits of the groups ssHq( V Y (V )), * *where the maps of the colimit system are induced in the evident way by the adjoint structure maps "oe:Y (V ) -! W-V Y (W ) of Y . The functor V on based G- spaces preserves q-fibrations and the functor VB= FB (SV , -) on G-spectra ove* *r B preserves level qf-fibrations. Formally, these hold since SV is a q-cofibrant G* *-space and the relevant model structures are G-topological. This leads to two families* * of long exact sequences relating the homotopy groups ssHq( V Xb(W ) of fibers to t* *he homotopy groups of the base space B and of the total spaces X(W ). First, if X * *is a level q-fibrant G-spectrum over B, then, using basepoints determined by a poi* *nt b 2 BH for any H Gb, the q-fibrations V X(W ) -! V B of based G-spaces with fibers V Xb(W ) induce long exact sequences (12.3.2) . .-.! ssHq+1( V B) -! ssHq( V Xb(W )) -! ssHq( V X(W )) -! ssHq( V B) -! . ... Second, if X is level qf-fibrant, then the qf-fibrations ( VBX)(W ) -! *B of ex* *-G- spectra over B with fibers V Xb(W ) induce long exact sequences (12.3.3) . .-.! ssHq+1(B) -! ssHq( V Xb(W )) -! ssHq(( VBX)(W )) -! ssHq(B) -! . ... The first allows us to relate the homotopy groups of the Xb to the homotopy gro* *ups of the ordinary loops V X(W ) on total spaces. The second allows us to relate * *the homotopy groups of the Xb to the homotopy groups of the parametrized loop ex- spaces ( VBX)(W ). It is the second that is most relevant to our work. Definition 12.3.4. The homotopy groups of a level qf-fibrant G-spectrum over B, or of a level qf-fibrant G-prespectrum X, are all of the homotopy groups ssHq(Xb) of all of the fibers Xb, where H Gb. The homotopy groups of a general G-spectrum, or G-prespectrum, X over B are the homotopy groups ssHq((RX)b) of a level qf-fibrant approximation RX to X. We still denote these homotopy groups by ssHq(Xb). In either category, a map f :X -! Y is said to be a ss*-isomorphism o* *r, synonymously, an s-equivalence, if, after level qf-fibrant approximation, it in* *duces an isomorphism on all homotopy groups. There are also homotopy groups specified in terms of maps out of sphere spec* *tra over B, but we choose to ignore them in setting up our model theoretic foundati* *ons. Our choice captures the intuitive idea that spectra over B should be parametriz* *ed spectra: the fiber spectra should carry all of the homotopy theoretical informa* *tion. With this choice, a good deal of the work needed to set up the stable model str* *ucture reduces to work that has already been done in [61]. The following observation i* *s a starting point that illustrates the pattern of proof. Now that we have seen how* * the equivariance appears in the definition of homotopy groups, we revert to our cus* *tom of generally deleting G from the notations. Lemma 12.3.5. A level q-equivalence of spectra over B is a ss*-isomorphism. Proof. A level qf-fibrant approximation to the given level q-equivalence is* * a level acyclic qf-fibration, and it induces a level q-equivalence on fibers over* * points 12.3. THE STABLE MODEL STRUCTURE ON GSB 155 of B by Proposition 12.2.9. This allows us to apply [61, III.3.3], which gives * *the same conclusion for nonparametrized spectra, one fiber at a time. To exploit our definition of homotopy groups, we need the following accompa- nying definition and proposition. Definition 12.3.6. An -prespectrum over B is a level qf-fibrant prespectrum X over B such that each of its adjoint structure maps "oe:X(V ) -! W-VBX(W ) is a q-equivalence of ex-spaces over B, that is, a q-equivalence of total space* *s. An (orthogonal) -spectrum over B is a level qf-fibrant spectrum over B such that each of its adjoint structure maps is a q-equivalence; equivalently, its underl* *ying prespectrum must be an -prespectrum over B. Since we are omitting the adjective "orthogonal" from "orthogonal spectrum over B", we must use the term " -prespectrum over B" on the prespectrum level to avoid confusion; the more standard term " -spectrum" was used in [61]. Proposition 12.3.7. A level fibrant G-spectrum X over B is an -G-spectrum over B if and only if each fiber Xb is an -Gb-spectrum. The G-prespectrum ana- logue also holds. Proof. By the five lemma, this is immediate from a comparison of the long exact sequences in (12.3.1) and (12.3.3). This result leads to the following partial converse to Lemma 12.3.5. Theorem 12.3.8. A ss*-isomorphism between -spectra over B is a level q- equivalence. Proof. The analogue for nonparametrized -spectra is [61, III.3.4]. In view of Proposition 12.3.7, we can apply that result on fibers and then use that -s* *pectra over B are required to be level qf-fibrant to deduce the claimed level q-equiva* *lence on total spaces from (12.3.1). Technically, the real force of our definition of homotopy groups is that this result describing the ss*-isomorphisms between -spectra over B is an immediate consequence of the work in [61]. Given this relationship between -spectra and homotopy groups, many of the arguments of [61] apply fiberwise to allow the de- velopment of the stable model structure. However, as discussed in the next sect* *ion, careful use of level fibrant approximation is required. We shall use the terms * *"stable model structure" and "s-model structure" interchangeably. The s-cofibrations are the same as those of the level qf-model structure and the s-fibrant spectra ove* *r B turn out to be the -spectra over B. Definition 12.3.9. A map of spectra or prespectra over B is (i)an acyclic s-cofibration if it is a ss*-isomorphism and an s-cofibration, (ii)an s-fibration if it satisfies the RLP with respect to the acyclic s-cofib* *rations, (iii)an acyclic s-fibration if it is a ss*-isomorphism and an s-fibration. We shall prove the following basic theorem in the next two sections. Theorem 12.3.10. The categories GSB and GPB are well-grounded model categories with respect to the ss*-isomorphisms (= s-equivalences), s-fibration* *s and s-cofibrations. The s-fibrant objects are the -spectra over B. 156 12. MODEL STRUCTURES FOR PARAMETRIZED G-SPECTRA Remark 12.3.11. Recall Remark 12.1.9. We can define positive -prespectra and positive analogues of our s-classes of maps, starting with the positive lev* *el qf- model structure. As in [61, IIIx5], the positive analogue of the previous theor* *em also holds, with the same proof. The identity functor is the left adjoint of a * *Quillen equivalence from GSB or GPB with its positive stable model structure to GSB or GPB with its stable model structure. The proof of the following result is virtually the same as the proof of its * *non- parametrized precursor [61, III.4.16 and III.5.7] and will not be repeated. Theorem 12.3.12. The adjoint pair (P, U) relating the categories GPB and GSB of prespectra and spectra over B is a Quillen equivalence with respect to e* *ither the stable model structures or the positive stable model structures. As in [61, III.x6], Theorem 12.3.10 leads to the following definition and th* *eorem, whose proof is the same as the proof of [61, III.6.1]. Definition 12.3.13. Let [X, Y ]` denote the morphism sets in the homotopy category associated to the level qf-model structure on GPB or GSB . A map f :X -! Y is a stable equivalence if f* :[Y, E]` -! [X, E]` is an isomorphism f* *or all -spectra E over B. Define the positive analogues similarly. Let [X, Y ] de* *note the morphism sets in the stable homotopy category Ho GSB of spectra over B. Theorem 12.3.14. The following are equivalent for a map f :X - ! Y of spectra or prespectra over B. (i)f is a stable equivalence. (ii)f is a positive stable equivalence. (iii)f is a ss*-isomorphism. Moreover [X, E] = [X, E]` if E is an -spectrum. Lemma 12.6.1 below should make it clear why the last statement is true. 12.4.The ss*-isomorphisms In the main, the proof of Theorem 12.3.10 is obtained by applying the result* *s in [61] fiberwise. Since total spaces are no longer assumed to be weak Hausdorff, * *we have to be a little careful: we are quoting results proven for T and using them* * for K*. However, we can just as well interpret [61] in terms of K*. The total spaces X(V ) of an s-cofibrant spectrum over B are weak Hausdorff, hence s-cofibrant approximation places us in a situation where total spaces are in U and therefore fibers are in T . There is a more substantial technical problem to overcome in adapting the proofs of [61, 62] to the present setting. In the situations encountered in th* *ose references, all objects were level q-fibrant, and that simplified matters consi* *derably. Here, level qf-fibrant approximation entered into our definition of homotopy gr* *oups, and for that reason the results of this section are considerably more subtle th* *an their counterparts in the cited sources. We begin by noting that any level ex-quasifibrant approximation, not neces- sarily a qf-fibrant approximation, can be used to calculate the homotopy groups* * of parametrized spectra. Lemma 12.4.1. A zig-zag of level q-equivalences connecting a spectrum X over B to a level ex-quasifibrant spectrum Y over B induces an isomorphism between t* *he 12.4. THE ss*-ISOMORPHISMS 157 homotopy groups of X and of Y , and the latter can be computed directly in terms of the fibers of Y . Proof. This follows from Lemma 12.3.5 by applying a level qf-fibrant approx- imation functor to the zig-zag. Theorem 12.4.2. Let f :X -! Y be a map between G-spectra over B. For any H G and b 2 BH , there is a natural long exact sequence . .-.! ssHq+1(Yb) -! ssHq((FB f)b) -! ssHq(Xb) -! ssHq(Yb) -! . . . and, if X is well-sectioned, there is also a natural long exact sequence . .-.! ssHq(Xb) -! ssHq(Yb) -! ssHq((CB f)b) -! ssHq-1(Xb) -! . ... Proof. For the first long exact sequence, let R be a level qf-fibrant appro* *xi- mation functor and consider Rf. We claim that the induced map FB f -! FB Rf is a level q-equivalence and that FB Rf is level qf-fibrant. This means that FB* * Rf is a level qf-fibrant approximation to FB f, so that the homotopy groups of the fi* *bers (FB Rf)b ~=F ((Rf)b) are the homotopy groups of FB f. When restricted to fibers over b, the parametrized fiber sequence RX -! RY -! FB Rf of spectra over B gives the nonparametrized fiber sequence (RX)b -! (RY )b -! F ((Rf)b), and the long exact sequence follows from [61, III.3.5]. To prove the claim, observe* * that since FB (I, Y ) -! Y is a Hurewicz fibration, it has a path-lifting function w* *hich levelwise shows that FB (I, Y ) -! Y is a level fp-fibration and therefore a le* *vel qf-fibration (since all qf-cofibrations are fp-cofibrations in GKB ). The dual * *gluing lemma (see Definition 5.4.1(iii)) then gives that the induced map FB f -! FB Rf is a level q-equivalence. Since FB (I, -) preserves level qf-fibrant objects an* *d since pullbacks of level qf-fibrant objects along a level qf-fibration are level qf-f* *ibrant, FB Rf is level qf-fibrant. Since the maps X -! CB X and RX -! CB RX are cyl-cofibrations between well-sectioned spectra and therefore level h-cofibrations by Lemma 12.1.3, the * *glu- ing lemma gives that CB f -! CB Rf is a level q-equivalence. Since RX and RY are level well-sectioned and level qf-fibrant, they are level ex-quasifibration* *s. It follows from Proposition 8.5.3 that CB Rf is a level ex-quasifibration. We cann* *ot conclude that CB Rf is level qf-fibrant, but by Lemma 12.4.1 we can nevertheless use CB Rf to calculate the homotopy groups of CB f. On fibers over b, the cofib* *er sequence of Rf is just the cofiber sequence of (Rf)b, and the long exact sequen* *ce follows from [61, III.3.5]. Recall Proposition 12.1.4, which specifies the ground structure in GSB and shows that the level q-equivalences give a well-grounded subcategory of weak eq* *uiv- alences; the g-cofibrations are just the level h-cofibrations. The following r* *esult shows that the same is true for the ss*-isomorphisms. However, in contrast to Proposition 12.1.4, it is crucial to assume that the relevant maps in the gluing and colimit lemmas are both cyl-cofibrations and g-cofibrations, as prescribed * *in Definition 5.4.1. Theorem 12.4.3. The ss*-isomorphisms in GSB give a well-grounded subcat- egory of weak equivalences. In detail, the following statements hold. (i)A homotopy equivalence is a ss*-isomorphism. (ii)The homotopy groups of a wedge of well-grounded spectra over B are the dir* *ect sums of the homotopy groups of the wedge summands. 158 12. MODEL STRUCTURES FOR PARAMETRIZED G-SPECTRA (iii)The ss*-isomorphisms are preserved under pushouts along maps that are both cyl and g-cofibrations. (iv)Let X be the colimit of a sequence in :Xn -! Xn+1 of maps that are both cyl and g-cofibrations and assume that X=BX0 is well-grounded. Then the homotopy groups of X are the colimits of the homotopy groups of the Xn. (v)For a map i: X -! Y of well-grounded spectra over B and a map j :K -! L of well-based spaces, i j is a ss*-isomorphism if either i is a ss*-isomorp* *hism or j is a q-equivalence. Proof. The conclusion that the ss*-isomorphisms give a well-grounded subcat- egory of weak equivalences, as prescribed in Definition 5.4.1, follows directly* * from the listed properties, using Lemma 5.4.2 to derive the gluing lemma. Since leve* *l q- equivalences are ss*-isomorphisms, s-cofibrant approximation in the level qf-mo* *del structure gives the factorization hypothesis Lemma 5.4.2(ii). A homotopy equivalence of spectra is a level fp-equivalence and hence a level q-equivalence, so (i) follows from Lemma 12.3.5. For finite wedges, (ii) is im* *me- diate from the evident split cofiber sequences and Theorem 12.4.2. For arbitrary wedges of well-grounded spectra over B, _B Xi- ! _B RXi is a level q-equivalence since the level q-equivalences are well-grounded and _B RXiis level quasifibran* *t by Proposition 8.5.3. By Lemma 12.4.1 we can use _B RXito calculate the homotopy groups of _B Xi. Over a point b in B, _B RXiis just _(RXi)b and the result foll* *ows from the nonparametrized analogue [61, III.3.5]. Now consider (iii). Let i: X -! Y be both a cyl-cofibration and a g-cofibrat* *ion and let f :X -! Z be a ss*-isomorphism. Since i and its s-cofibrant approxima- tion Qi are both cyl and g-cofibrations and since the level q-equivalences give* * a well-grounded subcategory of weak equivalences, the gluing lemma shows that we may approximate our given pushout diagram by one in which all objects are well- sectioned. Let j :Z -! Y [X Z be the pushout of i along f. Since i and j are cyl-cofibrations and j is the pushout of i, their cofibers are homotopy equival* *ent. Comparing the long exact sequences of homotopy groups associated to the cofiber sequences of i and j gives that the pushout Y - ! Y [X Z of f along i is a ss*- isomorphism. For (iv), we may use s-cofibrant approximation in the level model structure to replace our given tower by one in which all objects are well-sectioned. We note as in the proof of Lemma 5.6.5 that the natural map TelXn -! colimXn is a level q-equivalence and therefore a ss*-isomorphism. Relating the telescope * *to a classical homotopy coequalizer as in the cited proof, we reduce the calculati* *on of the homotopy groups of the telescope to an algebraic inspection based on (ii* *). Alternatively, one can commute double colimits to reduce the verification to its space level analogue. For (v), it suffices to show that the tensor X ^B T preserves ss*-isomorphis* *ms in either variable, by Lemma 5.4.4. That follows from Proposition 12.4.4 below. Proposition 12.4.4. Let f :X -! Y be a map between well-grounded spectra over B. (i)If f is a level q-equivalence and g :T -! T 0is a q-equivalence of well-bas* *ed spaces, then id^B g :X ^B T -! X ^B T 0 is a level q-equivalence and therefore a ss*-isomorphism. 12.5. PROOFS OF THE MODEL AXIOMS 159 (ii)If f is a ss*-isomorphism, then f ^B id:X ^B T -! Y ^B T is a ss*-isomorphism for any well-based space T and FB (id, f): FB (T, X) -! FB (T, Y ) is a ss*-isomorphism for any finite based CW complex T . (iii)For a representation V in V , the map f is a ss*-isomorphism if and only if VBf is a ss*-isomorphism. Proof. Part (i) holds since the level q-equivalences are well-grounded. The* *re- fore, for the first part of (ii), we may assume by q-cofibrant approximation in* * the space variable that T is a based CW complex. Using Proposition 8.5.3, it also implies that - ^B T preserves approximations of well-grounded spectra over B by level ex-quasifibrations. Now the first part of (ii) follows fiberwise from* * its nonparametrized analogue [61, III.3.11] and (iii) follows fiberwise from its no* *n- parametrized analogue [61, III.3.6]. Since FB (-, X) takes cofiber sequences of based spaces to fiber sequences of spectra over B, the second part of (iii) fol* *lows from the first exact sequence in Theorem 12.4.2, as in the proof of [61, III.3.* *9]. This leads to the following result, which shows that we are in a stable situ* *ation. Proposition 12.4.5. For all well-grounded spectra X over B and all repre- sentations V in IG , the unit j :X -! VB VBX and counit ": VB VBX -! X of the ( VB, VB) adjunction are ss*-isomorphisms. Therefore, if f :X -! Y is a map between well-grounded spectra over B, then the natural maps j :FB f -! B CB f and ffl: B FB f -! CB f are ss*-isomorphism. Proof. For j, after approximation of X by an ex-quasifibration, the conclus* *ion follows fiberwise from its nonparametrized analogue [61, III.3.6]. Using the tw* *o out of three property and the triangle equality for the adjunction, it follows that* * VB" is a ss*-isomorphism, hence so is ". For the last statement, the maps j and " a* *re the parametrized analogues of the maps defined for ordinary loops and suspensio* *ns in [71, p. 61], and they fit into diagrams relating fiber and cofiber sequences* * like those displayed there. Now the last statement follows from the five lemma and t* *he exact sequences in Theorem 12.4.2. 12.5. Proofs of the model axioms We need some G-spectrum level recollections from [61] and their analogues for G-spectra over B to describe the generating acyclic s-cofibrations. Let (SG , G* *S ) denote the G-category of G-spectra. To keep track of enrichments, we return G to the notations for the moment. We have a shift desuspension functor FV from based G-spaces to G-spectra given by FV T = V *^ T , where V *(W ) = JG (V, W ) [61, III.4.6]. It is left a* *djoint to evaluation at V . For G-spectra X, the adjoint structure G-map "oe:X(V ) -! W X(V W ) may be viewed by adjunction as a G-map "oe:SG (FV S0, X) -! SG (FV WSW , X). 160 12. MODEL STRUCTURES FOR PARAMETRIZED G-SPECTRA Passing to G-fixed points and taking X = FV S0, the image of the identity map gives a map of G-spectra ~V,W :FV WSW -! FV S0. (The notation ~V,W was used in [61], but we need room for a subscript). A Yoneda lemma argument then shows that the map of G-spaces SG (~V,W, id): SG (FV S0, X) -! SG (FV WSW , X) can be identified with "oe:X(V ) -! W X(V W ). We need the analogue for G-spectra over B. Recall from Definition 11.3.5 tha* *t, for an ex-G-space K over B, (FV K)(W ) = V *(W ) ^B K, where V *(W ) = JG,B(V, W ) = JG (V, W )B = (FV S0)(W ) ^B S0B. It follows that we can identify FV K with the evident external tensor FV S0^B K* * of the G-spectrum FV S0 and the ex-G-space K over B; compare Remark 11.1.7. We have used the notation ^B for this generalized tensor, but viewing it as a spec* *ial case of the external smash product of spectra over * and over B would suggest t* *he alternative notation Z. Definition 12.5.1. For ex-G-spaces K over B, we define a natural map ~V,WB:FV W WBK -! FV K. Namely, identifying the source and target with external tensor products, define ~V,WB= ~V,W ^B id:(FV WSW ) ^B K -! (FV S0) ^B K. We can describe the adjoint structure maps of G-spectra over B in terms of these maps ~V,WB. Lemma 12.5.2. Under the adjunctions PB (FV S0B, X) ~=FB (S0B, X(V )) ~=X(V ) and PB (FV WSWB, X) ~=FB (S0B, WBX(V W )) ~= WBX(V W ), the map PB (~V,WB, id): PB (FV S0B, X) -! PB (FV WSWB, X) corresponds to "oe:X(V ) -! WBX(V W ). Proof. When X = FV S0B, the conclusion holds by comparison with the case of G-spectra. The general case follows from the Yoneda lemma of enriched catego* *ry theory. See, for example, [12, 6.3.5]. We could have started off by defining ~V,WBin a conceptual manner analogous to our definition of ~V,W, but we want the explicit description of ~V,WBin term* *s of ~V,W in order to deduce homotopical properties in the parametrized context from homotopical properties in the nonparametrized context. For that and other pur- poses, we need the following observation. We return to our convention of deleti* *ng G from the notations, on the understanding that everything is equivariant. Lemma 12.5.3. If OE: X -! Y is an s-equivalence of level well-based nonpara- metrized spectra and K is a well-grounded ex-space with total space of the homo* *topy type of a G-CW complex, then OE ^B id:X ^B K -! Y ^B K is an s-equivalence. 12.5. PROOFS OF THE MODEL AXIOMS 161 Proof. We use the ex-fibrant approximation functor P of Definition 8.3.4. We have a natural zig-zag of h-equivalences between K and P K. By Proposition 8.2.* *6, it induces a zig-zag of level h-equivalences between X ^B K and X ^B P K and, by Corollary 8.2.5, X ^B P K is a level ex-fibration. Therefore, by Lemma 12.4.1, * *it suffices to consider the case when K is an ex-fibration. Since (X ^B K)b = X ^ * *Kb and Kb is of the homotopy type of a Gb-CW complex, by Theorem 3.4.2, each (OE ^B id)b is an s-equivalence by [61, III.3.11]. The following result is crucial. Proposition 12.5.4. Let K be a well-grounded ex-space with total space of the homotopy type of a CW complex. Then ~V,WB:FV W WBK -! FV K and ~V,W Z id:FV WSW Z FZK -! FV S0 Z FZK are ss*-isomorphisms of spectra over B. Proof. Since ~V,WB= ~V,W ^B id, Lemma 12.5.3 and the corresponding non- parametrized statement [61, III.4.5] imply the first statement. For the second statement, observe that for spectra X we have the associativity relation X Z FZK ~=X Z (FZS0 ^B K) ~=(X ^ FZS0) ^B K. Taking X = FV T for a based space T and using Lemma 11.3.6, we see that FV T Z FZK ~=FV Z(T ^B K). Using equivalences of this form and checking definitions, we conclude that the * *map ~V,W Z idof the statement can be identified with the map ~V Z,W^B id:(FV Z W SW ) ^B K -! (FV ZS0) ^B K. Thus the second ss*-isomorphism is a special case of the first. From here, the proof of Theorem 12.3.10 closely parallels arguments in [61, III.x4], but simplified a little by Theorem 5.5.1. The generating set of s-cofi* *brations is again F IfB. The generating set F KfBof acyclic s-cofibrations is given by a* * variant of the definition in the nonparametrized case [61, III.4.6]. Definition 12.5.5. Recall the factorization of ~V,W through the mapping cylinder (in the category of spectra) as V,W rV,W ~V,W :FV WSW k___//_M~V,W____//_FV S0. Here kV,W is an s-cofibration and rV,W is a deformation retraction. For i: C -!* * D in IfB, the map i kV,W :C ^B M~V,W [C^BFV WSW D ^B FV WSW -! D ^B M~V,W is an s-cofibration in GSB by Proposition 12.2.3, and it is therefore also a cy* *l- cofibration by Theorem 12.1.7. It is a ss*-isomorphism by Proposition 12.5.4 and inspection of definitions. The s-cofibrations in F JfBare level acyclic and are* * there- fore also ss*-isomorphisms. Restricting to V and W in skIG , define the generat* *ing set F KfBof acyclic s-cofibrations to be the union of F JfBand the set of all m* *aps of the form i kV,W with i 2 IfB. 162 12. MODEL STRUCTURES FOR PARAMETRIZED G-SPECTRA A fortiori, the following result identifies the s-fibrations, but it must be* * proven a priori as a first step towards the verification of the model axioms. Proposition 12.5.6. A map f :X -! Y satisfies the RLP with respect to F KfBif and only if f is a level qf-fibration and the diagrams (12.5.7) X(V ) _o"e//_ WBX(V W ) f(V )|| ||WBf(V W) fflffl| fflffl| Y (V )_o"e//_ WBY (V W ) are homotopy pullbacks for all V and W . Proof. As in [61, III.4.7], the homotopy pullback property must be inter- preted as requiring a q-equivalence from X(V ) into the pullback in the display* *ed diagram. Recall that F JfBis contained in F KfBand that a map has the RLP with respect to F JfBif and only if it is a level qf-fibration. This gives part of * *both implications. It remains to show that a level qf-fibration f has the RLP with r* *e- spect to i kV,W for all i 2 IfBif and only if the displayed diagram is a homoto* *py pullback. This is a formal but not altogether trivial exericise from the fact * *that the level qf-model structure is G-topological in the sense characterized in Pro* *po- sition 10.3.18. Notice that the map i kV,W is isomorphic to the map i kV,WB, where kV,WB= kV,W ^B S0B. With notation as in (10.3.6), f has the RLP with respect to i kV,WBfor all i 2 IfBif and only if the pair (i, PB (kV,WB, f)) has* * the lifting property for all i 2 IfB, which holds if and only if the map PB (kV,WB,* * f) of ex-spaces over B is an acyclic qf-fibration. This map is a qf-fibration since,* * for j 2 JfB, the map j kV,W ~=j kV,WBis a level acyclic s-cofibration of spectra ov* *er B by Proposition 12.2.3. Since f is a level qf-fibration, (j kV,WB, f) has the * *lift- ing property, hence, by adjunction, so does (j, PB (kV,WB, f)). Finally, PB (kV* *,WB, f) is homotopy equivalent to PB (~V,WB, f) so one is a q-equivalence if and only i* *f the other is. Under the isomorphisms in Lemma 12.5.2, the map PB (~V,WB, f) coincid* *es with the map from X(V ) into the pullback in the displayed diagram and is thus a q-equivalence if and only if that diagram is a homotopy pullback. Let *B be the terminal spectrum over B, so that each *B (V ) is the terminal ex-space *B . Observe that *B is an -spectrum with trivial homotopy groups. Corollary 12.5.8. The terminal map F -! *B satisfies the RLP with respect to F KB if and only if F is an -spectrum over B. Corollary 12.5.9. If f :X -! Y is a ss*-isomorphism that satisfies the RLP with respect to F KB , then f is a level acyclic qf-fibration. Proof. Since f is a level qf-fibration by Proposition 12.5.6, the dual of t* *he gluing lemma applied to the diagram *B _______//_Yofo_X | || || | || || fflffl| || || FB (I, Y_)___//Yoof__X 12.6. SOME QUILLEN ADJOINT PAIRS RELATING STABLE MODEL STRUCTURES 163 gives that the induced map F -! FB f of pullbacks is a level q-equivalence. Sin* *ce f has the RLP with respect to F KB , so does its pullback F - ! *B . By the previous corollary, F is thus an -spectrum over B. In particular, it is level * *qf- fibrant. We conclude that F is a level qf-fibrant approximation for FB f. Since* * f is a ss*-isomorphism, Theorem 12.4.2 gives that F is acyclic. By Theorem 12.3.8, this implies that F - ! *B is a level q-equivalence. Thus the fibers F (V )b a* *ll have trivial homotopy groups. We conclude (with a bit of extra argument as in [62, 9.8] to handle ss0) that each map of fibers f(V )b induces an isomorphism * *on homotopy groups. Therefore, since each f(V ) is a qf-fibration, each f(V ) indu* *ces an isomorphism on homotopy groups. The proof of the model axioms for the stable model structure is now immediat* *e. Proof of Theorem 12.3.10. The ss*-isomorphisms give a well-grounded sub- category of weak equivalences, by Theorem 12.4.3. Conditions (i), (iii), and (* *iv) in Theorem 5.5.1 are clear from our specification of the generating acyclic s- cofibrations and the result for the level qf-model structure. For condition (i* *i), a ss*-isomorphism that satisfies the RLP with respect to F KB has the RLP with respect to F IB by Corollary 12.5.9. Conversely, a map that has the RLP with re- spect to F IB is a level acyclic qf-fibration and therefore has the RLP with re* *spect to F KB by Proposition 12.5.6. It is a ss*-isomorphism since it is level acyclic. * *Since all s-fibrations are level qf-fibrations, right properness follows from the sli* *ghtly stronger observation in the following result. Proposition 12.5.10. The ss*-isomorphisms in GSB are preserved under pull- backs along level qf-fibrations. Proof. Let g be the pullback of a level qf-fibration f along a ss*-isomorph* *ism. Then g is a level qf-fibration and the fibers of g(V ) are isomorphic to the fi* *bers of f(V ). Therefore the homotopy fibers FB g are level q-equivalent to the homotopy fibers FB f. The result follows by comparison of the first long exact sequence* * in Theorem 12.4.2 for f and g. 12.6. Some Quillen adjoint pairs relating stable model structures We prove here that all of the adjoint pairs that were shown to be Quillen adjoints with respect to the level model structure in x12.2 are still Quillen a* *djoints with respect to the stable model structure. In view of the role played by level qf-fibrant approximation in our definition of homotopy groups, it is helpful to first understand the relationship between s-fibrant approximation and level qf- fibrant approximation. Now that the model structures have been established, we henceforward use the term s-equivalence rather than the synonymous term ss*- isomorphism. Lemma 12.6.1. Let :X - ! RX and `:X - ! R`X be an s-fibrant ap- proximation of X and a level qf-fibrant approximation of X. Then there is an s-equivalence , :R`X -! RX under X. Proof. Since ` is a level acyclic s-cofibration, it is an acyclic s-cofibr* *ation by Lemma 12.3.5. Since RX is s-fibrant, the RLP gives a map , under X, and it is an s-equivalence since and ` are s-equivalences. 164 12. MODEL STRUCTURES FOR PARAMETRIZED G-SPECTRA We have the following relationship between the homotopy categories of ex- spaces over B and of spectra over B. Proposition 12.6.2. The pair ( 1B, 1B) is a Quillen adjunction relating GSB and GKB . More generally, ( 1V, 1V) = (FV , EvV ) is a Quillen adjunction for * *any representation V 2 V . Proof. The maps 1Vi, where i 2 IfBis a generating cofibration for the qf- model structure on GKB , are among the generating cofibrations of the s-model structure on GSB , and it follows that 1V preserves cofibrations. Since 1V ta* *kes acyclic qf-cofibrations to level acyclic qf-cofibrations, and these are acyclic* * by Lemma 12.3.5, 1V also preserves acyclic cofibrations. Now consider an adjoint pair (F, V ) between categories of parametrized spec* *tra that is a Quillen adjunction with respect to the level model structures. Since * *the cofibrations are the same in the level model structure and in the stable model structure, the left adjoint F certainly preserves cofibrations. Thus, to show t* *hat (F, V ) is also a Quillen adjunction with respect to the stable model structure* *s, we need only show that F carries acyclic s-cofibrations to s-equivalences. When F preserves all s-equivalences, this is obvious; otherwise, by Lemma 5.4.6, it su* *ffices to verify this for the generating acyclic s-cofibrations. The cited result appl* *ies in general to subcategories of well-grounded weak equivalences, and in our context it applies to both the level q-equivalences and the s-equivalences. Recall tha* *t a Quillen left adjoint in any model structure preserves weak equivalences between cofibrant objects, by Ken Brown's lemma [44, 1.1.12]. The following parenthetic* *al observation applies to give a stronger conclusion for the Quillen left adjoints* * that we shall encounter. It will play a crucial role in exploiting the equivalence * *of homotopy categories that we will establish in the next chapter. Note that the s- cofibrant spectra are the cofibrant objects in both the level and the stable mo* *del structures, and they are well-grounded. Proposition 12.6.3. Let F be a Quillen left adjoint between categories of parametrized spectra with their stable model structures and suppose that F pres* *erves level q-equivalences between well-grounded spectra. Then F preserves s-equivale* *nces between well-grounded spectra. Proof. If g :X -! Y is an s-equivalence, where X and Y are well-grounded, factor g in the level model structure as g0 g00 X _____//W_____//Y, where g0 is an s-cofibration and g00is a level acyclic qf-fibration. Then W is * *well- grounded and F g00is a level q-equivalence by assumption. Since F is a Quillen * *left adjoint in the s-model structures, F g0is an s-equivalence. Since level q-equiv* *alences are s-equivalences it follows that F g = F g00O F g0 is an s-equivalence. The following sequence of results consists of analogues for the stable model structures of results proven for the level model structures in x12.2. Recall th* *at we actually have well-grounded stable model structures s(C ) for any closed genera* *ting set C . As in x12.2, wherever a qf(C )-model structure was used in Chapter 7 for some particularly well chosen C , we must use the corresponding s(C )-model structure here. 12.6. SOME QUILLEN ADJOINT PAIRS RELATING STABLE MODEL STRUCTURES 165 Proposition 12.6.4. Let T be a based G-CW complex. Then (-^B T, FB (T, -)) is a Quillen adjunction on GSB . When T = SV , it is a Quillen equivalence. Proof. This is immediate from the fact that the stable model structure is G-topological, together with Propositions 12.4.4 and 12.4.5. Proposition 12.6.5. If i: X - ! Y and j :W - ! Z are s-cofibrations of spectra over base spaces A and B, then i j :(Y Z W ) [XZW (X Z Z) -! Y Z Z is an s-cofibration over A x B which is s-acyclic if either i or j is s-acyclic. Proof. The statement about s-cofibrations is part of the analogue, Proposi- tion 12.2.3, for the level model structure. As usual, it suffices to show that* * i j is an s-equivalence if i 2 F IfBand j 2 F KfB, where F KfBis the set of generat* *ing acyclic s-cofibrations specified in Definition 12.5.5. Arguing as in Lemma 5.4.* *4 and using properness, this will hold if smashing the source and the target of i wit* *h j give s-equivalences. The reduction so far would work just as well for internal * *smash products. The required last step reduces via inspection of Definition 12.5.5 to* * an application of Proposition 12.5.4, with base space taken to be A x B. The reason that this last step works for external smash products but fails for internal sm* *ash products is made clear in Warning 6.1.7. Corollary 12.6.6. If Y is an s-cofibrant spectrum over B, then the functor (-) Z Y from GSA to GSAxB is a Quillen left adjoint with Quillen right adjoint ~F(Y, -). Proposition 12.6.7. Let f :A -! B be a G-map. Then (f!, f*) is a Quillen adjoint pair. If f is a q-equivalence, then (f!, f*) is a Quillen equivalence. Proof. We must show that f! takes acyclic s-cofibrations to s-equivalences. Since f! preserves well-grounded objects and level q-equivalences between well- grounded objects by Proposition 12.2.5, it suffices by Lemma 5.4.6 to prove tha* *t f!k is an s-equivalence for each map k in F KfA. This follows from the corresponding Quillen adjunction with respect to the level model structure if k 2 F JfA, so a* *ssume that k is of the form i kV,W ~=i kV,WA. We claim that f!k is a map in F KfBand * *is therefore an s-equivalence. Observe that kV,WA~=f*kV,WB. Using (11.4.5) and the fact that f! preserves pushouts, we see from the definition of the -product th* *at f!(i f*kV,WB) ~=(f!i) kV,WB. Since i is obtained from a map over A by adjoining* * a disjoint section, f!i is obtained from a map over B by adjoining a disjoint sec* *tion and is thus in IfB. Now assume that f is a q-equivalence. By [44, 1.3.16], (f!, f*) is a Quillen equivalence if and only if f* reflects s-equivalences between s-fibrant objects* * and the composite X - ! f*f!X - ! f*Rf!X given by the unit of the adjunction and s-fibrant approximation is an s-equivalence for all s-cofibrant X. Since th* *e s- fibrant objects are the -spectra over B and the s-equivalences between -spect* *ra over B are the level q-equivalences, the reflection property follows directly f* *rom the corresponding Quillen equivalence with respect to the level model structure. That result also gives that the composite X -! f*f!X -! f*R`f!X is a level q- equivalence and hence an s-equivalence. Applying Lemma 12.6.1 with X replaced by f!X and observing that f* preserves s-equivalences between level qf-fibrant 166 12. MODEL STRUCTURES FOR PARAMETRIZED G-SPECTRA G-spectra over B since (f*Y )a ~= Yf(a), a little diagram chase shows that the composite X -! f*f!X -! f*Rf!X is an s-equivalence. Observe that Proposition 12.6.3 applies to f!. Proposition 12.6.8. Let f :A -! B be a G-bundle whose fibers Ab are Gb- CW complexes. Then (f*, f*) is a Quillen adjoint pair. Proof. We must show that f* preserves acyclic s-cofibrations. Again it suf- fices by Lemma 5.4.6 to prove that f*k is an s-equivalence between well-grounded spectra for each map k 2 F KfB. That f*k is a map between well-grounded spectra follows from the fact that if K q B is a space over B with a disjoint section, * *then f*FV (KqB) = FV f*KqA is well-grounded. To see that f*k is an s-equivalence, it is enough, as in the proof of Proposition 12.6.7, to consider k = i kV,WBwith i* * 2 IfB. We have that f*kV,WB= kV,WAand, since f* preserves pushouts, smash products, and factorizations through mapping cylinders, we see as in the cited proof that f*k ~=f*i kV,WA, which is an acyclic s-cofibration. Proposition 12.6.9. Let ': H -! G be the inclusion of a subgroup. The pair of functors ('!, *'*) relating HSA and GS'!Agives a Quillen equivalence. If A * *is completely regular, then '!is also a Quillen right adjoint. Proof. By Proposition 14.3.1 below, ('!, *'*) is a Quillen adjoint pair. T* *he proof that it is a Quillen equivalence is the same as the proof of the ex-space level analogue in Proposition 7.4.6. The last statement is less obvious. As in the proof of the corresponding statement in Proposition 7.4.6, it follows from * *the spectrum level analogue of Remark 2.4.4, which in turn requires the spectrum le* *vel analogue of Proposition 2.4.3, and the analogue in the stable model structure of Proposition 7.4.3. The required analogues are proven in x14.4 below. We shall see that ('!, *'*) descends to a closed symmetric monoidal equival* *ence of homotopy categories in Proposition 13.7.9 below. Corollary 12.6.10. The functor '*: HoGSB -! HoHS'*B is the composite * Ho GSB __~__//HoGK'!'*B' HoHK'*B Using Example 11.5.5 as in Proposition 12.2.9, the following result is now a special case of Propositions 12.6.9 and 12.6.7. Proposition 12.6.11. For b 2 B, the pair of functors ((-)b, (-)b) relating GbS and GSB is a Quillen adjoint pair. CHAPTER 13 Adjunctions and compatibility relations Introduction The utility of the stable homotopy category Ho GSB depends on the fact that the usual functors and adjunctions descend to it and still satisfy appropriate * *com- mutation relations. We consider such matters in this chapter. Many of our basic adjunctions are Quillen adjunctions in the stable model structure. We recorded those in x12.6. The crucial adjunction missing from x12.6 is (f*, f*) for a gen* *eral map f of base spaces. This cannot be a Quillen adjoint pair by the argument in Counterexample 0.0.1. We used Brown representability to construct the right ad- joint f* between homotopy categories of ex-spaces in Theorem 9.3.2. Analogously, in x13.1 we use Brown representability to construct f* between homotopy categor* *ies of parametrized spectra, and we use base change along diagonal maps to internal* *ize smash products and function spectra. There is an interesting twist here. It is * *not easy to verify the Mayer-Vietoris axiom directly. Rather, we use the triangulat* *ed category variant of the Brown representability theorem, whose hypotheses turn o* *ut to be easier to check. In x13.7, we complete the proof that our stable homotopy categories are sym- metric monoidal and prove some basic compatibility relations among smash prod- ucts and base change functors. These results involve commutation of Quillen left and right adjoints, and we would not know how to prove them using only model theoretic fibrant and cofibrant replacement functors. Rather, their proo* *fs depend on an equivalence between our model theoretic stable homotopy category of parametrized G-prespectra and a classical homotopy category of what we call "excellent" parametrized G-prespectra. We used an analogous, but more elemen- tary, equivalence of categories in Chapter 9. It is essential to use parametri* *zed G-prespectra rather than parametrized G-spectra to make the comparison since the relevant constructions do not all preserve functoriality on linear isometri* *es; that is, they do not preserve IG -spaces. Results proven using the comparison a* *re then translated to parametrized G-spectra along the Quillen equivalence between parametrized G-prespectra and parametrized G-spectra. These equivalences of categories allow us to use a prespectrum level analogue T of the ex-fibrant approximation functor P to study derived functors. We define excellent parametrized G-prespectra in x13.2. We lift the ex-fibrant approximat* *ion functor P from ex-G-spaces to parametrized G-spectra in x13.3. There are several further twists here. First, the functor P on ex-G-spaces does not behave well w* *ith respect to tensors, so extending it to a functor on parametrized G-prespectra is subtle. Second, with the extension, the zig-zag of h-equivalences connecting P * *to the identity functor is no longer given by honest maps of parametrized G-prespe* *ctra, only weak maps. Third, the functor P does not take parametrized G-prespectra to 167 168 13. ADJUNCTIONS AND COMPATIBILITY RELATIONS excellent ones. To remedy this, we introduce two auxiliary functors K and E in x13.4. The composite T = KEP does land in excellent parametrized G-prespectra, and K converts weak maps to honest maps. In xx13.5 and 13.6 we use T to prove the promised equivalence of homotopy categories and show how to study derived funtors in this context. There are few issues of equivariance in this chapter, and we generally conti* *nue to omit the (compact Lie) group G from the notations. We adopt the convention of calling isomorphisms in homotopy categories equivalences and we denote them by ' rather than ~=. 13.1. Brown representability and the functors f* and FB We need some preliminaries about the two versions of Brown representability that are applicable in stable situations. Recall Example 11.5.5. Definition 13.1.1. For n 2 Z and H G, we have an s-cofibrant sphere G-spectrum SnHsuch that ssHn(X) = [SnH, X]G for all G-spectra X. Explicitly, ( 1 n SnH= (G=H+ ^ S ) if n 0, F-n (G=H+ ^ S0) if n <,0 as in [61, II.4.7], where F-n is the shift desuspension by Rn. We may allow the ambient group to vary. Replacing G by Gb for b 2 B and letting H Gb, define Sn,bHto be the G-spectrum (SnH)b over B. Note that Sn,bHis s-cofibrant, by Prop* *o- sition 12.2.9. By adjunction, for G-spectra X over B, ssHn(Xb) is isomorphic to [Sn,bH, X]G,B. Let DB be the set of all such G-spectra Sn,bHover B. From here, the following three results work in exactly the same way as their ex-space analogues in x7.4. Observe that the category Ho GKB has coproducts and homotopy pushouts, hence homotopy colimits of directed sequences. Lemma 13.1.2. Each X 2 DB is compact, in the sense that colim[X, Yn]G,B ~=[X, hocolimYn]G,B for any sequence of maps Yn -! Yn+1 in GSB . Proposition 13.1.3. A map , :Y - ! Z in GSB is an s-equivalence if and only if the induced map ,*: [X, Y ]G,B -! [X, Z]G,B is a bijection for all X 2 * *DB . Proof. This is a tautology since as X ranges through the Sn,bH, [X, Y ]G,B ranges through the homotopy groups ssHn(Yb) that define the s-equivalences. Theorem 13.1.4 (Brown). A contravariant set-valued functor on the category Ho GSB is representable if and only if it satisfies the wedge and Mayer-Vietoris axioms. Since we have the Quillen adjoint pair (f!, f*), we have the right derived f* *unc- tor f* :HoGSB -! Ho GSA . As in the proof of the analogous result on the level of ex-spaces, Theorem 9.3.2, we can obtain the desired right adjoint f* to f* b* *y use of Brown's theorem provided that we can show that f* preserves the relevant ho- motopy colimits. However, since f* :GSB -! GSA does not preserve s-cofibrant objects, this is not obvious. We will later give results that would allow us to* * carry 13.1. BROWN REPRESENTABILITY AND THE FUNCTORS f* AND FB 169 out the proof in a manner analogous to the proof of Theorem 9.3.2, but it is in- structive to switch gears and give a more direct proof. It is based on the use* * of triangulated categories and would not have applied on the ex-space level. Lemma 13.1.5. The category Ho GSB is triangulated. Proof. The treatment of triangulated categories in [74] gives a general pat* *tern of proof for showing that homotopy categories associated to appropriate model categories are triangulated. It applies here. The distinguished triangles are t* *hose equivalent in HoGSB to cofiber sequences that start with a well-grounded spectr* *um or, equivalently by Proposition 12.4.5, those equivalent to the negatives of fi* *ber sequences. Note that, by the proof of Theorem 12.4.2, every cofiber sequence is equivalent in Ho GSB to a cofiber sequence of level ex-quasifibrations. In triangulated categories, there is an alternative version of Brown's repre- sentability theorem due to Neeman [80]. It requires a "detecting set of compact objects". In triangulatedLcategoriesLwith coproducts (or sums), an object X is * *said to be compact if [X, Yi] ~=[X, Yi] for any set of objects Yi. In our topolo* *gi- cal situations, this reduces to the compactness of spheres, exactly as the proo* *f of Lemma 13.1.2. A detecting set of objects is one that detects equivalences, in t* *he sense suggested by Proposition 13.1.3. We have the following result. Lemma 13.1.6. DB is a detecting set of compact objects in Ho GSB . Recall that an additive functor between triangulated categories is said to be exact if it commutes with up to a natural equivalence and preserves distingui* *shed triangles. The following theorems are proven in [80, 3.1, 4.1]; they are discus* *sed with an eye to applications such as ours in [40, x8]. Theorem 13.1.7. Let A be a compactly detected triangulated category. A functor H :A op-! A b that takes distinguished triangles to long exact sequences and converts coproducts to products is representable. Theorem 13.1.8. Let A be a compactly detected triangulated category and B be any triangulated category. An exact functor F :A -! B that preserves coproducts has a right adjoint G. Theorem 13.1.9. For any G-map f :A -! B, there is a right adjoint f* to the functor f* :HoGSB -! Ho GSA , so that [f*Y, X]G,A ~=[Y, f*X]G,B for X in GSA and Y in GSB . Proof. The left adjoint f!commutes with and preserves cofiber sequences, and this remains true after passage to derived homotopy categories. Therefore t* *he derived functor f! is exact. Since f* is Quillen right adjoint to f!, the deri* *ved functor f* is right adjoint to f!and is therefore also exact; see, for example,* * [79, 3.9]. If X is in DA , then f!X is compact in Ho GSB , as we see from commutation relations between relevant Quillen left adjoints given in Remark 11.4.7. It fol* *lows formally that f* preserves coproducts, by [80, 5.1] or [40, 7.4]. Remark 13.1.10. For composable maps f and g, there is a natural equivalence g* O f* ' (g O f)* on homotopy categories since f* O g* ' (g O f)*. 170 13. ADJUNCTIONS AND COMPATIBILITY RELATIONS Exactly as for ex-spaces in Theorem 9.3.4, we apply change of base along the diagonal map : B -! B x B to obtain internal smash product and function spectra functors in Ho GSB . Theorem 13.1.11. Define ^B and FB on HoGSB to be the composite (derived) functors X ^B Y = *(X Z Y ) and FB (X, Y ) = ~F(X, *Y ). Then [X ^B Y, Z]G,B ~=[X, FB (Y, Z)]G,B for X, Y and Z in Ho GSB . Proof. The displayed adjunction is the composite of the adjunction for the external smash product and function spectra functors given by Corollary 12.6.6 * *and the adjunction ( *, *). 13.2. The category GEB of excellent prespectra over B We must still prove that Ho GSB is a closed symmetric monoidal category un- der the derived internal smash product, that the derived functor f* is closed s* *ym- metric monoidal, and that various compatibility relations that hold on the poin* *t-set level descend to homotopy categories. In particular, since our right adjoints f* **, *, and therefore FB come from Brown's representability theorem, it is not at all o* *bvi- ous how to prove that they are well-behaved homotopically. In Chapter 9, we sol* *ved the corresponding ex-space level problems by proving that Ho GKB is equivalent to the more classical and elementary homotopy category hGWB . Here GWB is the category of ex-fibrations over B whose total spaces are compactly generated and* * of the homotopy types of G-CW complexes, and hGWB is obtained from GWB simply by passage to homotopy classes of maps. This equivalence allowed us to exploit * *the ex-fibrant approximation functor P of x8.3 to resolve the cited problems. We shall resolve our spectrum level problems similarly, and the following de* *fi- nitions give the appropriate analogues of GWB and hGWB . However, to keep closer to the ex-space level, it is essential to work with parametrized prespectra rat* *her than parametrized spectra. It is safe to do so in view of the Quillen equivalen* *ce (P, U) of Theorem 12.3.12 relating GPB and GSB . Definition 13.2.1. Let X be a G-prespectrum over B. (i)X is well-structured if each level X(V ) is in GWB . (ii)X is -cofibrant if it is well-grounded and each structure map oe : WBX(V ) -! X(V W ) is an fp-cofibration. We can now give the definition of excellent G-prespectra over B and of the associated classical homotopy category. Working with classical nonequivariant a* *nd nonparametrized coordinatized prespectra {En}, it has been known since the 1960* *'s that the following definition gives the simplest quick and dirty rigorous const* *ruction of the stable homotopy category. Definition 13.2.2. The category GEB of excellent G-prespectra over B is the full subcategory of GPB whose objects are the well-structured -cofibrant -G- prespectra over B. Let hGEB denote the classical homotopy category obtained from GEB by passage to homotopy classes of maps. 13.2. THE CATEGORY GEB OF EXCELLENT PRESPECTRA OVER B 171 We comment on the conditions we require of excellent prespectra over B. We require that they be well-structured so that we can exploit levelwise our equiv* *alence of homotopy categories on the ex-space level. We require that they be -cofibra* *nt since that provides "homotopical glue" that is necessary for the transition from the known equivalence on the ex-space level to the desired equivalence on the p* *re- spectrum level. We shall make this idea precise shortly, in Proposition 13.2.5.* * We require that they be -prespectra over B since it is clearly sensible to restri* *ct at- tention to s-fibrant objects in GSB if we hope to compare homotopy categories. Recall that X is an -prespectrum if it is a level qf-fibrant prespectrum over B whose adjoint structure maps "oe:X(V ) -! W-VBX(W ) are q-equivalences. Since excellent prespectra over B are required to be level * *ex- fibrations, they are automatically level qf-fibrant. The condition on the adjo* *int structure maps is stronger than it appears on the surface. Lemma 13.2.3. For excellent G-prespectra X over B, the adjoint structure maps "oe:X(V ) -! WBX(V W ) are fp-equivalences. Proof. The "oeare q-equivalences between G-CW homotopy types and are therefore h-equivalences. Since they are maps between ex-fibrations, they are f* *p- equivalences by Proposition 5.2.2. This implies, for example, that homotopy-preserving functors GEB -! GPB that may not preserve level q-equivalences nevertheless do preserve the equival* *ence property required of the adjoint structure maps. Remark 13.2.4. Our definition of excellent parametrized prespectra is close * *to that used by Clapp and Puppe [18, 19], who in turn were influenced by definitio* *ns in [66]. Curiously, while Clapp [18] focuses on ex-fibrations, Clapp and Puppe [19] never mention fibration conditions. These papers are nonequivariant, but t* *he second is written in terms of what the authors call "coordinate-free spectra" o* *ver B. These are the same as our nonequivariant prespectra over B, except that their adjoint structure maps "oeare required to be closed inclusions, which holds aut* *omat- ically for -cofibrant prespectra. Clapp and Puppe [19] use the term "cofibrant" for our notion of -cofibrant. A crucial result of Clapp and Puppe makes the idea of homotopical glue preci* *se. It is stated nonequivariantly in [19, 6.1], but it works just as well equivaria* *ntly. Translated to our language, it reads as follows. Proposition 13.2.5 (Clapp-Puppe). If f :X -! Y is a level fp-equivalence between -cofibrant prespectra over B, then f is a homotopy equivalence of pre- spectra over B. Therefore, if f :X -! Y is a level h-equivalence between well- structured -cofibrant prespectra over B, then f is a homotopy equivalence of p* *re- spectra over B. Sketch proof. The proof is analogous to the proof that a ladder of homotopy equivalences connecting sequences of cofibrations induces a homotopy equivalence on passage to colimits. The point is that, for -cofibrant parametrized prespec* *tra 172 13. ADJUNCTIONS AND COMPATIBILITY RELATIONS Y , we can carry out inductive arguments just as if Y were just such a colimit.* * Us- ing standard cofibration arguments, carried over to the parametrized case, we c* *an extend an fp-homotopy inverse of WiBX(Vi) -! WiBY (Vi) to an fp-homotopy inverse of X(Vi+1) -! Y (Vi+1) and proceed inductively. The last statement fol- lows by Corollary 5.2.6(i), which shows that a level h-equivalence between well- structured prespectra over B is a level fp-equivalence. 13.3.The level ex-fibrant approximation functor P on prespectra We seek an approximation functor to play the role on the parametrized pre- spectrum level that the functor P played on the ex-space level functor. We shall introduce three approximation functors, P , E and K, that successively build in* * the properties of being well-structured, being an -prespectrum, and being -cofibr* *ant, each preserving the properties already obtained. We define P in this section and E and K in the next. Lifting the ex-space level functor P of x8.3 to the prespectrum level requir* *es care. Recall that P is the composite of the whiskering functor W and the Moore mapping path space functor L, together with the natural zig-zag of h-equivalenc* *es (13.3.1) K ooae_W K __W'_//W LK = P K of Definition 8.3.4 for ex-spaces K over B. The functors W and L do not com- mute with tensors with based spaces, hence cannot be enriched over GKB , by Lemma 11.3.4. There is therefore no canonical way of inducing structure maps after applying P levelwise to a prespectrum, as one might at first hope. We sha* *ll resolve this by constructing by hand certain non-canonical but natural maps (13.3.2) ffV :W K ^B SV -! W (K ^B SV ) and (13.3.3) fiV :LK ^B SV -! L(K ^B SV ) such that ff0 = id, fi0 = id and the following associativity diagram commutes, where (F, fV ) stands for either (W, ffV ) or (L, fiV ). (13.3.4) fV^id fV00 0 F K ^B SV ^B SV 0____//_F (K ^B SV ) ^B SV___//F (K ^B SV ^B SV ) ~=|| |~=| fflffl|0 fV V 0 fflffl| 0 F K ^B SV V ________________________________//F (K ^B SV V ) The definitions of these maps and the proofs that these diagrams commute depend on chosen decompositions of V and V 0as direct sums of indecomposable representations, and we cannot choose compatible decompositions for all represe* *n- tations V and V 0at once. For this reason, and for other reasons that will beco* *me apparent later, we must switch gears and work with sequentially indexed prespec* *tra. Thus, to be precise about the constructions in this section and the next, we restrict our original collection V of indexing representations to a countable c* *ofinal sequence W of expanding representations in our given universe U. More precisely, W consists of representations Vi for i 0 such that V0 = 0 and Vi Vi+1. We set Wi= Vi+1-Vi. Such a sequence can be chosen in any universe. We could just as we* *ll start with representations Wiand define Viinductively by Vi+1= Vi Wi. There is 13.3. THE LEVEL EX-FIBRANT APPROXIMATION FUNCTOR P ON PRESPECTRA 173 no need to use orthogonal complements. We shall write in terms of complements, but on the understanding that that is just a notational convenience. Remark 13.3.5. There is a small quibble here since we originally defined our categories of parametrized prespectra only on collections of representations th* *at are closed under finite direct sums, which W clearly is not. However, if we l* *et W 0consist of all finite sums of the Wi, then0we recover such a collection. As* * in x11.3 (or [62, x2]), we can interpret0GPWB as a diagram category indexed on a certain small category, say DWG , with object set W 0, and we can0interpret GPWB as a diagram category indexed on the full subcategory0DWG of DWG whose object set is W . This gives a restriction functor U: GPWB -! GPWB that is right adjoi* *nt to a prolongation functor P [62, x3], and (P, U) induces an adjoint equivalence of homotopy categories. We shall study such "change of universe" adjunctions in x14.2. They allow us to lift all results we prove about the categories of param* *etrized prespectra indexed on cofinal sequences to our usual ones indexed on collection* *s of representations closed under direct sums. Definition 13.3.6. Let X be a prespectrum over B indexed on the countable cofinal sequence W = {Vi}, where V0 = 0 and Vi+1= Vi Wi. Let X have structure maps oei: WiBX(Vi) -! X(Vi+1). Then the maps W oeiO ff: W X(Vi) ^B SWi -! W X(Vi+1) and LoeiO fi :LX(Vi) ^B SWi -! LX(Vi+1) specify structure maps for prespectra W X and LX over B. Therefore P X = W LX is a prespectrum over B. Unfortunately, as will be clear from the following construction, the maps in* * the zig-zag (13.3.1) do not lift to the prespectrum level. They only induce weak ma* *ps of prespectra, that is, levelwise maps that only commute with the structure maps up to (canonical) fp-homotopy. Fortunately, the last approximation functor K, which arranges -cofibrancy and will be discussed in the next section, turns we* *ak maps into honest ones. Construction 13.3.7. We define ffV and fiV . Fix a decomposition of V into irreducible representations and let PV be the set of the projections from V to * *the irreducible subrepresentations in this fixed decomposition. Define three equiva* *riant maps from V to the real numbers by setting Y Y kvkV = maxss2P|ssv|, ~V (v) = (1 - |ssv|), V (v) = max(1, |ssv|). V ss2PV ss2PV Applying the same definitions to another representation V 0and to V V 0with its induced decomposition as a sum of irreducible representations, we see that * *the following equations hold. kv v0kV V 0= max{kvkV , kv0kV 0}, ~V V 0(v v0) = ~V (v)~V 0(v0), V V 0(v v0) = V (v) V 0(v0). Define a natural map hV :W K ^B SV ^B [1, 1)+ -! W (K ^B SV ), 174 13. ADJUNCTIONS AND COMPATIBILITY RELATIONS by setting ( -1 -1 hV (x ^ v ^ t) = x ^ ~(t v) . v if kvk t, (p(x), 1 - (t-1v)-1)if kvk t, ( hV ((b, s) ^ v ^ t) = (b, s) if kvk t, (b, 1 - (1 - s) (t-1v)-1)if kvk t. At time t = 1 this specifies ffV and it is easy to verify that the associativi* *ty diagram (13.3.4) commutes. Further, the map ae O hV extends to t = 1 to give an fp-homotopy from aeOffV to ae^B id. It follows that ae induces levelwise a weak* * map of prespectra W X -! X. Similarly define kV :LK ^B SV ^B [1, 1)+ -! L(K ^B SV ), by setting ( kV ((x, ~) ^ v ^ t) = (x ^ v, ~) if kvk t, (x ^ v, (t-1v)~if kvk t. Here, if 1 a < 1, and ~ 2 B, then a~ denotes the Moore path of length l~=a given by a~(t) = ~(at). At time t = 1 this specifies fiV , and it is again easy* * to check the required associativity. The map kV O (' ^ id) extends to an fp-homotopy from fiV O (' ^B id) to ', hence ' induces levelwise a weak map of prespectra X -! L* *X, to which we can apply W to obtain a weak map W X -! W LX = P X. In view of Definition 8.3.4, naturality arguments from Definition 13.3.6 and Construction 13.3.7 prove the following theorem. Theorem 13.3.8. There are functors L, W , and P = W L on GPB that are given levelwise by the functors L, W , and P on GKB . There are natural weak ma* *ps ae: W X -! X and ': X -! LX that are given levelwise by the ex-space maps ae and '. Therefore, there is a natural zig-zag of weak maps OE = (ae, W ') as dis* *played in the diagram ae W' X oo___W X _____//W LX = P X. These maps are level h-equivalences, and P converts level h-equivalences to lev* *el fp-equivalences. If each X(V ) is compactly generated and of the homotopy type of a G-CW complex, then P X is well-structured. If X is well-structured, then t* *he weak maps in the above display are level fp-equivalences between well-structure* *d G- prespectra over B. If, further, the adjoint structure maps of X are h-equivalen* *ces or q-equivalences, then so are the adjoint structure maps of LX, W X, and P X. Proof. The only point that may need elaboration is the last clause. For a weak map f :X -! Y , we have a homotopy commutative diagram X(V )__"oe//_ WBX(V W ) f|| |WBf| fflffl| fflffl| Y (V )_"oe//_ WBY (V W ). The functor WB preserves fp-equivalences. Therefore, if f is an fp-equivalenc* *e, then the "oefor X are h-equivalences or q-equivalences if and only if the "oefo* *r Y are so. We apply this to f = ae and f = W '. 13.4. THE AUXILIARY APPROXIMATION FUNCTORS K AND E 175 13.4. The auxiliary approximation functors K and E We begin with the parametrized -prespectrum approximation functor E. This is a folklore construction when B is a point. In the parametrized context, the * *proof of the following result makes essential use of Stasheff's theorem, Theorem 3.4.* *2, and therefore depends on our standing assumption that G acts properly on B. Proposition 13.4.1. There is a functor E :GPB - ! GPB and a natural map ff: X -! EX with the following properties. (i)The functor E preserves level fp-equivalences and well-grounded prespectra. (ii)If X is well-structured, then EX is a well-structured -prespectrum and the map ff: X -! EX is an s-equivalence. Proof. Define EX by letting EX(Vi) be the telescope over j i of the ex- spaces Vj-ViBX(Vj) with respect to the adjoint structure maps Vj-ViB"oe: Vj-ViBX(Vj) -! Vj-ViB WjBX(Vj+1) ~= Vj+1-ViBX(Vj+1). Since the functor WiBcommutes with telescopes, WiBEX(Vi+1) is isomorphic to t* *he telescope over j i+1 of the ex-spaces Vj-Vi+1BX(Vj). The adjoint structure m* *ap EX(Vi) -! WiBEX(Vi+1) is induced by the maps Vj-ViB"oejfor j i. The map ff: X -! EX is given by the inclusion of the bases of the telescopes. If f :X -* *! Y is a level fp-equivalence, then Ef :EX -! EY is a level fp-equivalence since a standard inductive argument (applicable in any topologically bicomplete categor* *y) shows that the telescope of a ladder of fp-equivalences is an fp-equivalence. If X is well-grounded or level ex-fibrant, then so is EX since the construct* *ion clearly stays in the category of compactly generated spaces and since it preser* *ves the conditions of being well-sectioned or level ex-fibrant by results in x8.2. * * To show that E preserves well-structured prespectra, it remains to show that if X * *has total spaces of the homotopy types of G-CW complexes, then so does EX. By Stasheff's theorem (Theorem 3.4.2), the fibers X(V )b = Xb(V ) have the homotopy types of Gb-CW complexes. We have the analogous construction E in the category of Gb-prespectra and, by Milnor's theorem (Theorem 3.3.5) and standard facts about telescopes, the (E(Xb))(V ) have the homotopy types of Gb-CW complexes. It is clear from the definition of E that (E(Xb))(V ) = ((EX)(V ))b. That is, t* *he Gb-prespectrum E(Xb) is the fiber (EX)b of the G-prespectrum EX over B. By Stasheff's theorem again, it follows that the (EX)(V ) have the homotopy types * *of G-CW complexes. To check that the adjoint structure maps are q-equivalences when X is well- structured, it suffices to check that they induce q-equivalences on the fibers * *over b for all b 2 B. That holds by inspection of the homotopy groups of the colimits that define (EX)b ~=E(Xb). Similarly, we see that ff is a ss*-equivalence when X is well-structured by fiberwise comparison of the colimits of homotopy groups of fibers that define the homotopy groups of X and EX. To approximate parametrized prespectra by level fp-equivalent -cofibrant pr* *e- spectra, we use the elementary cylinder construction K that was first defined in [63] and has been used in various papers since. We recall the construction and * *its main properties from [59, 6.8], which carries over verbatim to the parametrized context. A more sophisticated but less convenient treatment is given in [39]. 176 13. ADJUNCTIONS AND COMPATIBILITY RELATIONS Proposition 13.4.2. There is a functor K :GPB - ! GPB and a natural level fp-equivalence ss :KX -! X. Therefore K preserves level fp-equivalences. If X is well-grounded, then KX is -cofibrant. If X is well-structured, then KX* * is well-structured. If X is a well-structured -prespectrum, then so is KX and thus KX is excellent. There is a natural weak map ': X -! KX that is a right inverse of ss, and K takes weak maps f to honest maps Kf such that ' O f = Kf O '. Proof. Define KX, a level inclusion ': X -! KX, and a level fp-deformation retraction ss :KX -! X right inverse to ' as follows. Let KX(0) = X(0) and '(0) = ss(0) = id. Inductively, suppose given KX(Vi), an inclusion '(Vi): X(Vi)* * -! KX(Vi) and an inverse fp-deformation retraction ss(Vi): KX(Vi) -! X(Vi). Let KX(Vi+1) be the double mapping cylinder in GKB of the pair of maps WiB'(Vi) oe WiBKX(Vi) oo______ WiBX(Vi)________//X(Vi+1) in GKB . Let oe : WiBKX(Vi) -! KX(Vi+1) be the inclusion of the left base of the double mapping cylinder, which is an fp-cofibration and let '(Vi+1): X(Vi+1) -! KX(Vi+1) be the inclusion of the right base. Let ss(Vi+1): KX(Vi+1) -! X(Vi+1) be the map obtained by first using the fp-equivalence WiBss(Vi) on the left ba* *se to map to the mapping cylinder of oe and then using the evident deformation retrac* *tion to the right base. There is an equivalent description as a finite telescope. Ce* *rtainly ss is a map of prespectra over B and a level fp-deformation retraction with lev* *el inverse the weak map '. The functoriality of the construction is clear. If X is well-grounded, then KX is clearly also well-grounded and thus KX is -cofibrant. If X is well-structured, then so is KX by Propositions 8.2.1 and 8* *.2.3. If, further, the adjoint structure maps of X are q-equivalences, then they are * *fp- equivalences since X is well-structured. Since K preserves fp-homotopies, it fo* *llows that KX is also an -prespectrum. Alternatively, since VBis a Quillen right ad* *joint in the qf-model structure, it preserves q-equivalences between qf-fibrant ex-sp* *aces. In particular, the maps WBss(Vi) are q-equivalences. If f :X -! Y is a weak map with fp-homotopies hi: WiBX(Vi) ^B I+ -! Y (Vi+1) from oeY O Wif(Vi) to f(Vi+1) O oeX , define Kf inductively by setting Kf(0) = f(0) and letting Kf(Vi+1) be WjBKf(Vi) on the left end of the mapping cylinder, f(Vi+1) on the right end and as follows on the cylinder itself: ( Wi 1 , _ Kf(Vi+1)[x, t] = [ B f(Vi)(x), 2t]if10 t 2 hi(x, 2t - 1) if _2 t 1. Then Kf is a map of prespectra over B and ' O f = Kf O '. The composite approximation functor T = KEP has various good preservation properties. The ex-space level properties of P recorded in x8.4 are inherited o* *n the prespectrum level, and we have the following sample result for E and K. Lemma 13.4.3. For a G-map f :A -! B, a prespectrum Y over B and a prespectrum X over A, there are natural isomorphisms f*EY ~=Ef*Y, f*KY ~=Kf*Y and Kf!X ~=f!KX. 13.5. THE EQUIVALENCE BETWEEN HoGPB AND hGEB 177 Proof. The relevant telescopes commute with f* since it is a symmetric monoidal left adjoint and with f!since it is a left adjoint and the projection * *formula (2.2.5) holds. 13.5. The equivalence between Ho GPB and hGEB We can now extend the results of x9.1 to parametrized prespectra. As in the previous section, our parametrized prespectra are indexed on a countable cofinal sequence of expanding representations in our given universe. We begin by collat* *ing the results of the previous two sections. Theorem 13.5.1. Let X be a well-grounded G-prespectrum over B whose total spaces are of the homotopy types of G-CW complexes and define T X = KEP X. (i)T X is an excellent G-prespectrum. (ii)T takes level q-equivalences between G-prespectra over B that satisfy the * *hy- potheses on X to homotopy equivalences of G-prespectra. (iii)There is a zig-zag of s-equivalences between X and T X. (iv)If X is an excellent G-prespectrum over B, then the zig-zag consists of lev* *el fp-equivalences, and it gives rise to a zig-zag of homotopy equivalences of G-prespectra over B connecting X and T X. Proof. We have that P X is well-structured by Theorem 13.3.8, EP X is a well-structured -prespectrum by Proposition 13.4.1, and T X is excellent by Pr* *opo- sition 13.4.2. In (ii), a level q-equivalence is a level h-equivalence. By the * *results just quoted, P takes level h-equivalences to level fp-equivalences, which are p* *re- served by E, and K takes level fp-equivalences to homotopy equivalences. Since K converts weak maps to genuine maps, we have the following diagram of maps of G-presepectra over B. (13.5.2) KX ooKaeKW_X W KX _WK'_//W KLX KEP X ss|| |ss| Wss || Wss|| |ss| fflffl| fflffl| fflffl| fflffl| fflffl| X W X ________W X W LX __ff__//_EP X The vertical maps ss, hence also the vertical maps W ss, are level fp-equivalen* *ces. The map ae is a level f-equivalence. The map ' is a level h-equivalence, hence * *so is W K'. The map ff is an s-equivalence because P X is well-structured. Since leve* *l q- equivalences are also s-equivalences, the diagram displays a zig-zag of s-equiv* *alences between X and T X. For the last statement, observe that all prespectra in the diagram are well- structured -prespectra over B. Moreover, ff is a level q-equivalence by Theo- rem 12.3.8. It is therefore a level h-equivalence since our total spaces have * *the homotopy types of G-CW complexes. Since all prespectra in our diagram are well- structured, our level h-equivalences are level fp-equivalences, by Proposition * *8.3.2. Applying K where needed, we can expand the diagram to a zig-zag of level fp- equivalences between -cofibrant prespectra. By Proposition 13.2.5, this gives* * a zig-zag of homotopy equivalences connecting X and T X. We introduce a category that is intermediate between GPB and GEB . Definition 13.5.3. Define GQB to be the full subcategory of GPB consisting of the well-grounded -prespectra over B whose total spaces are of the homotopy 178 13. ADJUNCTIONS AND COMPATIBILITY RELATIONS types of G-CW complexes. Define HoGQB to be the homotopy category obtained by inverting the s-equivalences in GQB ; by the proof of the next theorem, ther* *e are no set-theoretic problems in defining HoGQB . Define T = KEP :GQB -! GEB . Since the -prespectra over B are the s-fibrant prespectra over B and since s-cofibrant spectra are well-grounded and have total spaces of the homotopy typ* *es of G-CW complexes, all G-prespectra over B that are s-cofibrant and s-fibrant a* *re in GQB . We prove that Ho GPB is equivalent to hGEB by proving that these categories are both equivalent to HoGQB . Theorem 13.5.4. The canonical s-cofibrant and s-fibrant approximation func- tor RQ and the composite approximation functor T = KEP , together with the forgetful functors, induce the following equivalences of homotopy categories. _RQ__// __T__// Ho GPB oo___Ho GQB oo___hGEB I J Proof. For X in GPB , we have a natural zig-zag of s-equivalences in GPB X oo___QX _____//RQX. Therefore X and IRQX are naturally s-equivalent in GPB . If X is in GQB , then it is s-fibrant and therefore so is QX. Then the above zig-zag is in GQB so X a* *nd RQIX are naturally s-equivalent in GQB . By Theorem 12.3.8, s-equivalences in GQB are level q-equivalences, and T tak* *es level q-equivalences to homotopy equivalences by Theorem 13.5.1. Conversely, si* *nce homotopy equivalences are s-equivalences, the forgetful functor J induces a fun* *ctor in the other direction. For X in GQB we have the natural zig-zag of s-equivalences displayed in (13.5.2). Applying s-fibrant approximation, we get a natural zig-zag of s-equiv* *alences in GQB so X and JT X are naturally s-equivalent in GQB . Starting with X in GEB , the last statement of Theorem 13.5.1 shows that X and T JX are naturally homotopy equivalent in GEB . 13.6.Derived functors on homotopy categories With P replaced by T , the discussion of derived functors in x9.2 carries ov* *er from the level of ex-spaces to the level of parametrized prespectra indexed on * *cofinal sequences. In x13.7 and x14.2 we will discuss how to pass from there to conclus* *ions on the level of parametrized spectra indexed on our usual collections of repres* *en- tations closed under direct sums. We must show that if V is a Quillen left or right adjoint, then its model theoretic left or right derived functor agrees un* *der our equivalences of categories with the functor obtained simply by passing to homot* *opy classes of maps from the composite T V . As on the ex-space level, we need some mild good behavior for this to work. Definition 13.6.1. A functor V :GPA - ! GPB is good if it is continu- ous, preserves well-grounded parametrized prespectra, and takes prespectra over A whose levelwise total spaces are of the homotopy types of G-CW complexes to prespectra over B with that property. Since V is continuous, it preserves homo- topies. There are evident variants for functors V with source or target GK*: V must be continuous, preserve well-grounded objects, and preserve G-CW homotopy type conditions on objects. 13.6. DERIVED FUNCTORS ON HOMOTOPY CATEGORIES 179 Note that a good functor V need not take -G-prespectra to -G-prespectra and recall that a Quillen right adjoint must preserve fibrant objects and thus,* * in our context, must preserve -G-prespectra. Proposition 13.6.2. Let V :GPA -! GPB be a good functor that is a part of a Quillen adjoint pair. If V is a Quillen left adjoint, assume further that * *it pre- serves level q-equivalences between well-grounded objects. Then the derived fu* *nc- tor Ho GPA - ! Ho GPB , induced by V Q or V R, is equivalent to the functor T V J :hGEA -! hGEB under the equivalence of categories in Theorem 13.5.4 Proof. If V is a Quillen right adjoint, then it preserves s-equivalences be* *tween s-fibrant objects. If V is a Quillen left adjoint, then it preserves s-equivale* *nces be- tween well-grounded objects by Proposition 12.6.3. Therefore, since GQA consists of well-sectioned s-fibrant objects, the functor V :GQA -! GPB passes straight to homotopy categories to give V :HoGQA -! HoGPB in both cases. If V is a Quillen right adjoint, then it takes an s-equivalence f in GQA to an s-equivalence since the objects of GQA are s-fibrant. Then V f is a level q- equivalence by Theorem 12.3.8 and, since V is good, it is a level h-equivalence* *. On the other hand, if V is a Quillen left adjoint, then Theorem 12.3.8 gives that * *f is a level q-equivalence and, by assumption, V f is then a level q-equivalence. Si* *nce V is good, V f is actually a level h-equivalence. In both cases it follows tha* *t V takes s-equivalences to level h-equivalences and therefore T V passes to a func* *tor Ho GQA -! hGEB . To show that T V J and either V Q or V R agree under the equivalence of cate- gories, it suffices to verify that the following diagram commutes. V Q orV R Ho GPA ___________//HoGPB RQ || TRQ|| fflffl| fflffl| Ho GQA ____TV______//hGEB We have functorial s-cofibrant and s-fibrant approximation functors Q and R, with natural acyclic s-fibrations QX -! X and acyclic s-cofibrations X -! RX. Clearly Q and R preserve s-equivalences. If V is a Quillen left adjoint, then * *we have a zig-zag of natural s-equivalences RQV Q _____//RV Qoo___V Q ____//_V RQ because V preserves acyclic s-cofibrations. If V is a Quillen right adjoint, th* *en we have a zig-zag of natural s-equivalences RQV R oo___ RQV RQ ____//_RV RQoo___V RQ because V preserves s-equivalences between s-fibrant objects. In both cases, * *all objects have total spaces of the homotopy types of G-CW complexes, hence we have zig-zags of level h-equivalences. Applying T , we obtain a zig-zag of homo* *topy equivalences in GEB by Theorem 13.5.1. Remark 13.6.3. If V preserves excellent parametrized prespectra, then T V is naturally homotopy equivalent to V on excellent parametrized prespectra. The derived functor of V can then be obtained directly by applying V and passing to homotopy classes of maps. 180 13. ADJUNCTIONS AND COMPATIBILITY RELATIONS 13.7.Compatibility relations for smash products and base change This section is parallel to x9.3. The main change is just that we must repla* *ce the functor P used there with the functor T = KEP that we have here. This gives us results for the categories GPWB of parametrized prespectra indexed on a collection W consisting of a cofinal sequence in some universe U. In order * *to obtain statements about GSBV, where V = V (U), we have two pairs of Quillen equivalences, both of which can be viewed as consisting of a prolongation funct* *or left adjoint to a forgetful functor that creates the weak equivalences; see [61* *, 1.2]. __j*_// __P__// GPWB oo___GPVB oo___GSBV j* U We postpone until x14.2 consideration of the pair (j*, j*) and the extension fr* *om GPWB to GPVBand deal with the extension from GPVBto GSBV in this section. One general remark is in order, though. The forgetful functors j* and U crea* *te weak equivalences and therefore pass directly to homotopy categories. If they c* *om- mute on the point set level with a functor V which is a part of a Quillen adjoi* *nt pair, then they will also commute with its derived functor on the level of homo* *topy categories. It follows formally that the derived prolongation functors P and j** * then also commute with the derived functor V and its adjoints. This holds in particu* *lar for the base change functor V = f*. Extending commutation results for such func- tors from GPWB to GSBV is therefore easy. However, some of the functors V that we need to consider only exist on some of the categories in the above display, * *and such functors require special care. These include the change of universe functo* *rs that we discuss in x14.2, which don't exist on the level of GPWB, and the smash product ^B , which we have only specified on the spectrum level and which we now discuss on the prespectrum level. Remark 13.7.1. Because the domain category for the diagram category of (equivariant and parametrized) prespectra is only monoidal, not symmetric mon- oidal, we cannot use left Kan extension to internalize "external" smash product* *s of prespectra; see [62, 4.1]. Here "external" is understood in the sense of indexi* *ng on pairs of representations. Therefore, on the equivariant parametrized prespectrum level, when we write X Z Y for prespectra X over A and Y over B, we should understand the external external smash product, in the sense of Remark 11.1.7. When passing from prespectrum level arguments to spectrum level conclusions usi* *ng (P, U), we are implicitly using composites of the general form PV U, and simila* *rly for functors of several variables involving smash products. We can carry out the several variable arguments externally on the prespectrum level, only internaliz* *ing with left Kan extension after passage to spectra, where we have good homotopical control by Corollary 12.6.6. Alternatively, we can make use of classical "handicrafted smash products" of prespectra, which are defined by use of arbitrary choices of sequences of repre* *senta- tions. As discussed on the nonequivariant nonparametrized level in [62, x11], h* *and- icrafted smash products of prespectra agree under the adjoint equivalence (P, U) with the internalized smash products. Provided that we use external parametrized handicrafted smash products over varying base spaces, only internalizing along diagonal maps at the end, the discussion there adapts readily to give the same * *con- clusion for homotopy categories of equivariant parametrized prespectra and spec* *tra. 13.7. COMPATIBILITY RELATIONS FOR SMASH PRODUCTS AND BASE CHANGE 181 The advantage of handicrafted smash products is that their definition involves * *only direct use of ex-space level constructions that enjoy good preservation propert* *ies with respect to ex-fibrations. This often allows direct transposition of ex-spa* *ce level arguments in hGWB to parametrized prespectrum level arguments in hGEB . We state the following results in terms of parametrized spectra, and we indi* *cate which parts of the proofs require the use of hGEB and which parts work directly* * in the stable homotopy category HoGSB . Proposition 13.7.2. Let f :A -! B and g :A0- ! B0 be G-maps. If W and X are spectra over A and A0, then f!W Z g!X ' (f x g)!(W Z X) in Ho GSBxB0 . If Y and Z are spectra over B and B0, then f*Y Z g*Z ' (f x g)*(Y Z Z) in Ho GSAxA0. Proof. Working directly in Ho GSBxB0 , the first equivalence reduces to its point-set level analogue by consideration of Quillen left adjoints, as in the c* *orre- sponding proof of Proposition 9.4.1. We work in hGEAxA0 to prove the second equivalence. Here f* and Z (understood in the external or handicrafted sense) are both good, and both preserve excellent prespectra. Indeed, they preserve we* *ll- structured prespectra by levelwise application of Propositions 8.2.2 and 8.2.3,* * they preserve -cofibrant prespectra since f* and Z on ex-spaces preserve fp-cofibra* *tions because they are left adjoints that commute with fp-homotopies, and they preser* *ve -prespectra by Lemma 13.2.3 since they preserve fp-homotopies. Therefore, using excellent prespectra, we can pass straight to homotopy categories, without use * *of T , as in the corresponding proof of Proposition 9.4.1. Theorem 13.7.3. The category Ho GSB is closed symmetric monoidal under the functors ^B and FB . Proof. Working in HoGSB , the associativity, commutativity, and unity of ^B follow by pullback along diagonal maps from their easily proven external analog* *ues and the second equivalence in the previous result, exactly as in Theorem 9.4.4. We have a commutation relation between change of base and suspension spec- trum functors that is analogous to the relation between change of base and smash products recorded in Proposition 13.7.2. Proposition 13.7.4. For a G-map f :A -! B, there are natural equivalences 1Bf!' f! 1A and 1Af* ' f* 1B of (derived) functors. The same conclusion holds more generally for the shift d* *esus- pension functors FV = 1V. Proof. Working in Ho GSB , the first equivalence is clear since it is a com- parison of Quillen left adjoints that commute on the point-set level. For the s* *econd equivalence, we start in hGWB and end in hGEA . For K 2 GWB , the point set lev* *el suspension prespectrum 1BK is both -cofibrant and well-structured, by Corol- lary 8.2.5, but of course it is not an -prespectrum over B. Since 1B is good * *and takes well-grounded q-equivalences to well-grounded level q-equivalences, T 1B* * is 182 13. ADJUNCTIONS AND COMPATIBILITY RELATIONS equivalent to the model theoretic left derived functor of the Quillen left adjo* *int 1B. Here we may omit P from the composite functor T and, since f* commutes with both K and E, the conclusion follows on passage to homotopy categories. Applying this to : B -! B x B and using Proposition 11.4.11, we obtain the following consequence. Proposition 13.7.5. For ex-spaces K and L over B, 1B(K ^B L) ' 1BK ^B 1BL in Ho GSB . For f :A -! B, evident properties of the functor f! on ex-spaces imply that the functor f!:GPA -! GPB is good, and f! satisfies the other hypotheses of Proposition 13.6.2 by Proposition 12.2.5. We use this to prove the following ba* *sic result. Theorem 13.7.6. For a G-map f :A -! B between base spaces, the derived functor f* :HoGSB -! Ho GSA is closed symmetric monoidal. Proof. Since SB is not s-fibrant, the isomorphism f*SB ~=SA in GSB does not immediately imply the required equivalence f*SB ' SA in Ho GSA , where f*SB means f*RSB . However, Proposition 13.7.4 specializes to give this equiva- lence. For the rest, we must show that the isomorphisms (11.4.2) through (11.4.* *6) descend to equivalences on homotopy categories. By category theory in [40], it suffices to consider (11.4.2) and (11.4.5), and the proofs are similar to those* * in Theorem 9.4.5. Since Z and * both preserve excellent prespectra, so do the int* *er- nalized smash products ^A and ^B . For excellent prespectra Y and Z over B, it follows that both sides of f*(Y ^B Z) ~=f*Y ^A f*Z are excellent prespectra over A, hence the point-set level isomorphism descends directly to the desired equivalence on the homotopy category level. Next consid* *er f!(f*Y ^A X) ~=Y ^B f!X, where X is an excellent prespectrum over A. Here we must replace f! by T f! on both sides. By Theorem 13.5.1 we have a natural zig-zag OE of level h-equivalen* *ces connecting T to the identity functor which, when applied to excellent parametri* *zed prespecra gives rise to a zig-zag _ of actual homotopy equivalences. We obtain * *the following zig-zag. T(id^BOE) _ T f!(f*Y ^A X) ~=T (Y ^B f!X)_________//_To(Yo^B_T f!X)_//_Yo^BoT_f!X. Using handicrafted products with their termwise construction in terms of smash products of ex-spaces, it follows from Proposition 8.2.6 that id^B - preserves * *level h-equivalences between well-sectioned spectra. Thus id^B OE is a zig-zag of le* *vel h-equivalences and T (id^B OE) is a zig-zag of actual homotopy equivalences. 13.7. COMPATIBILITY RELATIONS FOR SMASH PRODUCTS AND BASE CHANGE 183 Theorem 13.7.7. Suppose given a pullback diagram of G-spaces g C _____//D i|| |j| fflffl|fflffl| A __f__//B in which f (or j) is a q-fibration. Then there are natural equivalences of (der* *ived) functors on stable homotopy categories (13.7.8) j*f!' g!i*, f*j* ' i*g*, f*j!' i!g*, j*f* ' g*i*. Proof. Working in hGEB , the proof is similar to that of Theorem 9.4.6 but with P replaced by T . Again it suffices to consider the first equivalence, and* *, as explained there, since f is a q-fibration there is a level fp-equivalence ~: P * *f* -! f*P . Since f* commutes with both K and E, we obtain a level fp-equivalence ~: T f* -! f*T between -cofibrant prespectra over A so it is in fact a homotopy equivalence by Proposition 13.2.5. Then f*T j!X ' T f*j!X ~=T i!g*X. The following observation holds by the same proof as the analogous ex-space level result Proposition 9.4.8. Proposition 13.7.9. Let ': H -! G be the inclusion of a subgroup and A be an H-space. The closed symmetric monoidal Quillen equivalence ('!, *'*) descen* *ds to a closed symmetric monoidal equivalence between HoHSA and HoGS'!A. Combined with Theorem 13.7.6, applied to the inclusion "b:G=Gb -! B, and Proposition 13.7.4, this last observation gives us the following stable analogu* *e of Theorem 9.4.9. Theorem 13.7.10. The derived fiber functor (-)b:Ho GKB -! Ho GbK* is closed symmetric monoidal and it has both a left adjoint (-)b and a right adjoi* *nt b(-). Moreover, the derived fiber functor commutes with the derived suspension spectrum functor, ( 1BK)b ' 1 (Kb) as Gb-spectra. For emphasis, we repeat a remark that we made after the analogous ex-space level result. This innocent looking result packages highly non-trivial and impo* *rtant information. In particular, it gives that FB (X, Y )b ' F (Xb, Yb) in Ho GbS f* *or X, Y 2 Ho GSB , where the fiber and function object functors are understood in the derived sense. This reassuring consistency result is central to our applica* *tions in the last two chapters, where parametrized duality is studied fiberwise. CHAPTER 14 Module categories, change of universe, and change of groups Introduction We first give a discussion of module categories of parametrized spectra over nonparametrized ring spectra. Although we shall not go into these applications here, one basic motivation for our work is to set up the homotopical foundations for studying the generalized homology and cohomology theories of parametrized spectra that are represented by such nonparametrized ring spectra. The good behavior of the external smash product GS x GSB -! GSB makes it easy to do this. While the mathematics here is evident, it deserves emphasis since the ide* *as are likely to be central to future applications. In the rest of the chapter, we focus on problems that are special to the equ* *ivari- ant context. We give the parametrized generalization of some of the work in [61] concerning change of universe, change of groups, and fixed point and orbit spec* *tra. As usual, an essential point is to determine which of the standard adjunctions * *are given by Quillen adjoint pairs and to prove that other adjunctions and compatib* *il- ities that are evident on the point set level also descend to homotopy categori* *es. We discuss change of universe in x14.2. Here the use of prespectra indexed on cofinal sequences in the previous chapter introduces some minor difficulties th* *at were not studied in the nonparametrized theory of [61, Vx1] and are already rel* *e- vant nonequivariantly. We study subgroups and fixed point spectra in x14.3. We study quotient groups and orbit spectra in x14.4. Aside from some analogues for parametrized spectra of earlier results for parametrized spaces, these sections* * are precisely parallel to [61, Vxx2 and 3]. We have not written down the parametriz* *ed analogue of [61, Vx4], which gives the theory of geometric fixed point spectra,* * since it would be tedious to repeat the constructions given there. It will be apparen* *t to the interested reader that, mutatis mutandis, the definitions and results in [6* *1, Vx4] generalize to the parametrized context. 14.1. Parametrized module G-spectra We can define a parametrized (strict) ring G-spectrum R over B to be a monoid in the symmetric monoidal category GSB , and we can then define parametrized R-modules and R-algebras in the usual way, as has become standard in stable homotopy theory [39, 46, 61, 62]. However, even though the smash product ^B in GSB gives a point-set level symmetric monoidal structure, we cannot expect to obtain Quillen model structures on the categories of such R-modules or R-algebr* *as, as was done for orthogonal G-spectra in [61, IIIxx7,8]. To do that, we would ne* *ed better homotopical behavior than we can prove here. We have only set up adequate foundations for the classical style theory of up to homotopy parametrized module 184 14.1. PARAMETRIZED MODULE G-SPECTRA 185 spectra over up to homotopy parametrized ring spectra. From that point of view, our homotopical foundations are entirely satisfactory. The source of the proble* *m is Warning 6.1.7, which implies that X ^B (-) in GSB cannot be a Quillen functor. However, in applications, it is natural to start with a nonparametrized orth* *og- onal ring G-spectrum R. We are then interested in understanding the R-homology and R-cohomology theories of G-spectra over B and their relationships with the R-homology and R-cohomology of the fibers. For this study, just as in the non- parametrized work of [39, 46, 61, 62], one is interested in the theory of R-mod* *ules. The external smash product Z: GS x GSB - ! GSB has enough of the good properties of the nonparametrized smash product GS x GS -! GS to give us homotopical control over parametrized module spectra over nonparametrized ring spectra. We devote this section to developing the relevant theory, which is par* *allel to [61, IIIx7]. Let R be a ring spectrum in GS which is well-grounded when view* *ed as a spectrum, meaning that each R(V ) is well-based and compactly generated. Definition 14.1.1. A (left) R-module over B is a G-spectrum M over B to- gether with a left action R Z M -! M satisfying the usual associativity and unit conditions. The category GRMB of left R-modules over B consists of the G-spectra M over B and the maps of G-spectra over B that preserve the action by R. Since (R Z X)b = R ^ Xb, a parametrized R-module over B is precisely that: each Xb is an R-module Gb-spectrum. More formally, we have the G-category (RMG,B, GRMB ), as discussed in xx1.4 and 12.2, and the following result is cle* *ar. Proposition 14.1.2. The G-category (RMG,B, GRMB ) is G-topologically bi- complete in the sense of Definition 10.2.1. All of the required limits, colimi* *ts, tensors, and cotensors are constructed in the underlying G-category (SG,B, GSB ) and then given induced R-module structures in the evident way. A cyl-cofibration of R-modules is a cyl-cofibration of underlying G-spectra over B. The last statement holds by the retract of mapping cylinders characterizatio* *n of cyl-cofibrations. This immediately implies that GRMB inherits a ground structure from GSB , in the sense of Definition 5.3.2. Recall that the well-grounded G-sp* *ectra over B are those that are level well-grounded (well-sectioned and compactly gen* *er- ated) and that the g-cofibrations of G-spectra over B are the level h-cofibrati* *ons; see Definition 12.1.2 and Proposition 12.1.4. Definition 14.1.3. An R-module over B is well-grounded if its underlying G- spectrum over B is well-grounded. A map of R-modules over B is a g-cofibration, level q-equivalence, or s-equivalence if its underlying map of G-spectra over B* * is such a map. Also recall the notion of a subcategory of well-grounded weak equivalences from Definition 5.4.1. Since colimits and tensors for R-modules are defined in terms of the underlying G-spectra over B, the following theorem is immediate fr* *om its counterpart for G-spectra over B, which is given by Proposition 12.1.4 and Theorem 12.4.3. Theorem 14.1.4. Definition 14.1.3 specifies a ground structure on GRMB such that the level q-equivalences and the s-equivalences both give subcategori* *es of well-grounded weak equivalences. 186 14. MODULE CATEGORIES, CHANGE OF UNIVERSE, AND CHANGE OF GROUPS Finally, recall the definition of a well-grounded model structure from Defin* *i- tion 5.5.2. Such model structures are compactly generated, and we must define t* *he generators of GRMB . The free R-module functor FR = R Z -: GSB -! GRMB is left adjoint to the forgetful functor U: GRMB -! GSB . Adjunction_arguments from the definitions show that FR preserves cyl-cofibrations and cyl-cofibratio* *ns. Definition 14.1.5. Define FR F IfB, FR F JfBand FR F KfBby applying the free R-module functor to the maps in the sets specified in Definition 12.1.6 and Def* *ini- tion 12.5.5. A map of R-modules over B is (i)a level qf-fibration or an s-fibration if it is one in GSB , (ii)an s-cofibration if it satisfies the LLP with respect to the level acyclic* * qf- fibrations, Theorem 14.1.6. The category GRMB is a well-grounded model category with respect to the level q-equivalences, the level qf-fibrations, and the s-cofibra* *tions. The sets FR F IfBand FR F JfBgive the generating s-cofibrations and generating * *level acyclic s-cofibrations. All s-cofibrations of R-modules over B are s-cofibratio* *ns of G-spectra over B. We omit the proof since it is virtually the same as the proof of the followi* *ng theorem, which gives the starting point for serious work on the homology and cohomology theory of parametrized G-spectra. Theorem 14.1.7. The category GRMB is a well-grounded model category with respect to the s-equivalences, the s-fibrations, and the s-cofibrations; FR F K* *fBgives the generating acyclic s-cofibrations. Proof. The compatibility condition is automatic by adjunction from the para- metrized spectrum level,_and we have already observed that the free R-module functor FR preserves cyl-cofibrations. It also preserves the relevant -produc* *ts, and FR FV K = (R ^ FV S0) Z K is well-grounded if K is a well-grounded ex-space. Only the acyclicity condition remains. If R is s-cofibrant as a ring spectrum, * *then R is also s-cofibrant as a spectrum, by [61, III.7.6(iv) and (v)]. In that case* *, the functor R Z (-) = UFR is a Quillen left adjoint by Corollary 12.6.6 and therefo* *re preserves level acyclic s-cofibrations. It follows that the maps in FR KfBare * *s- equivalences. The case of a general well-grounded R reduces to the cofibrant ca* *se by use of the next result; compare Proposition 14.1.9 below. Proposition 14.1.8. The following statements hold. (i)For an s-cofibrant spectrum X over B, the functor - Z X :GS - ! GSB preserves s-equivalences between well-grounded spectra in GS . (ii)If Y is well-grounded in GS , j :A -! X is an acyclic s-cofibration in GSB* * , and A is well-grounded, then Y Z j :Y Z A -! Y Z X is an s-equivalence. Proof. Let OE: Y - ! Z be an s-equivalence between well-grounded spectra. By parts (ii)-(iv) of Definition 5.4.1, it suffices to show that OE Z FV K is a* *n s- equivalence if K is the source or target of a map in IfB. This map is isomorphi* *c to the map (OE ^ FV S0) ^B K, where FV S0 is the shift desuspension in GS , not GS* *B . Here OE ^ FV S0 is an s-equivalence by the nonparametrized analogue [61, III.7.* *3], and the conclusion follows from Lemma 12.5.3. (The comment on the notations Z and ^B above Definition 12.5.1 is relevant: the former is an external smash pro* *duct and the latter is a tensor). 14.1. PARAMETRIZED MODULE G-SPECTRA 187 For (ii), we apply an argument from [62, 12.5]. We let Z = X=BA, which is s-cofibrant, and we let QY - ! Y be an s-cofibrant approximation. Since j is an s-cofibration, it is a cyl-cofibration and Cj is homotopy equivalent to Z. Sin* *ce A is well-grounded, we can apply the long exact sequence of homotopy groups of Theorem 12.4.2 to conclude that Z is s-acyclic. The map Z -! *B is then an s-equivalence between s-cofibrant spectra over B. Since QY Z - is a Quillen left adjoint, by Proposition 12.2.3, QY ZZ -! QY Z*B ~=*B is an s-equivalence. Since QY Z Z -! Y Z Z is an s-equivalence by part (i), Y Z Z is s-acyclic. Since the functor Y Z - preserves cofiber sequences, another application of Theorem 12.4.2 shows that Y Z j is an s-equivalence. Proposition 14.1.9. If OE: Q -! R is an s-equivalence of well-grounded ring spectra, then the functors OE* = R ^Q (-): GQMB -! GRMB and OE*: GRMB -! GQMB given by extension of scalars and restriction of action define a Quillen equiva* *lence (OE*, OE*) between the categories of Q-modules and of R-modules over B. Proof. Since s-fibrations and s-equivalences are created in the underlying category of spectra over B, it is clear that they are preserved by OE*, so that* * we have a Quillen pair. If M is an s-cofibrant Q-module, then, by the previous res* *ult, the unit map OE ^ id:M ~= Q ^Q M -! OE*(R ^Q M) of the adjunction is an s- equivalence of spectra over B. Therefore, if N is an s-fibrant R-module, then a map M -! OE*N of Q-modules is an s-equivalence if and only if its adjoint map R ^Q M -! N of R-modules is an s-equivalence. Implicitly, we have been dealing all along with the case when R is the sphere spectrum S, and we can mimic all of the model theoretic work that we have done in that case. The results of x12.6 and x13.1 carry over directly. For f :A -! B, base change preserves R-modules, (f!, f*) gives a Quillen adjoint pair relating* * the categories of R-modules over A and over B, and we obtain a Quillen equivalence * *if f is a q-equivalence. If f is a bundle with CW fibres, we obtain a Quillen pair (* *f*, f*), and we can apply the triangulated category version of Brown representability to construct a right adjoint f* in general. However, we do not know how to general* *ize the rest of Chapter 13 to the module context since we have not worked out a the* *ory of excellent R-modules with an accompanying excellent R-module approximation functor. In view of the retreat to prespectra with their primitive handicrafted* * smash products in that theory, it seems unlikely to us that any such construction can* * be expected. We also have the notion of a right R-module over a nonparametrized ring spectrum R. If M and N are right and left R-modules over A and B and L is a left R-module over A x B, then we define spectra M ZR N over A x B and ~FR(N, L) over A by the usual coequalizer M Z R Z N ____//_//_M_Z_N_//M ZR N and equalizer ~FR(N, L)____//~F(N, L)___////_~F(R Z N, L). If R is commmutative, then M ^R N and FR (N, L) are naturally R-modules. We have good homotopical control over these external constructions, as in Propositions 12.2.3 and 12.6.5. For example, if we take A = *, then we have good 188 14. MODULE CATEGORIES, CHANGE OF UNIVERSE, AND CHANGE OF GROUPS homotopical control over the smash product spectrum M ^R N over B and the non-parametrized function spectrum FR (N, L), where M is a non-parametrized right R-module and N and L are left R-modules over B. However, if we take A = B and internalize M ZR N along the diagonal : B -! B x B by setting M ^R N = *M ZR N and FR (M, N) = ~FR(M, *N), we lose homotopical control. Similarly, when R is commutative, RMB has the structure of a closed sym- metric monoidal category, and that allows us to define (commutative) R-algebras over B to be (commutative) monoids in RMB . However, because of the lack of homotopical control, in the absence of the theory of Chapter 13, we cannot give* * the categories of R-algebras and of commutative R-algebras over B model structures. Remark 14.1.10. Although we have not pursued the idea, it seems highly likely that there are interesting examples of rings and modules that allow varying base spaces and are defined in terms of the external smash product. For example, one might consider G-spectra Rn over Bn with products Rm Z Rn -! Rm+n , or one might consider "globally defined" parametrized ring spectra R consisting of spe* *ctra RB over B for all B together with appropriate products RA Z RB -! RAxB . The RB would in particular be module spectra over the nonparametrized ring spectrum R*. As in the nonparametrized theory, one must use the positive stable model structures to study such ring objects model theoretically when R* is commutativ* *e. The essential point is that the external smash product is sufficiently well-beh* *aved homotopically that there is no obstacle to passage from point-set level constru* *ctions to homotopy category level conclusions. 14.2. Change of universe Recall that G-spectra over B are defined in terms of a chosen collection V of representations of G. As usual in equivariant stable homotopy theory, we must introduce functors that allow us to change the collection V . Such functors are usually referred to as "change of universe" functors, since V is often given as* * the collection V (U) of all representations that embed up to isomorphism in a given G-universe U. It is however often convenient to restrict V to be a cofinal subc* *ol- lection of V (U) that is closed under direct sums, and when we dealt with excel* *lent prespectra it became essential to restrict V further to a countable cofinal seq* *uence of expanding representations in U. In both cases it is usual to insist that the* * trivial representation R is included in V . In order to deal with the change functors * *in all of the above cases at once, we adopt a slightly different approach from the* * one that was used in [61, V.x1]. We then explain how it specializes to the more exp* *licit approach given there. Let GSBV denote the category of G-spectra over B indexed on V . If V is not closed under direct sums, then we are thinking0of GSBV as the restriction of the diagram category corresponding to GSBV, where V 0is the closure of V under sums, as discussed in Remark 13.3.5. Let i: V V 0be the inclusion of one collection of representations in anoth* *er. Thinking of parametrized spectra as diagram ex-spaces, we see that the evident forgetful functor 0 i*: GS V -! GS V has a left adjoint i* given by the prolongation, or expansion of universe, func* *tor 0 0 (i*X)(V 0) = JGV(-, V ) JGV X. 14.2. CHANGE OF UNIVERSE 189 Such prolongation functors are discussed in detail in [62, Ix3] and [61, Ix2]. * * By [61, I.2.4], the unit Id-! i*i* of the adjunction is a natural isomorphism. We have more concrete descriptions of the functor i* when V consists of a cofinal sequence of representations in some universe U. Recall that JGV(V, V ) * *is the orthogonal group O(V ) with a disjoint base point. Lemma 14.2.1. If V = {Vi} V 0is a countable expanding sequence in some G-universe U, then 0 0 (i*X)(V 0) ~=JGV(Vi, V ) ^O(Vi)X(Vi) where i is the largest natural number such that there is a linear isometry Vi- * *! V 0. Proof. The forgetful functor i* is restriction along a functor ': JGV- ! JG* *V 0 and (i*X)(V 0) is constructed as the coequalizer of the pair of parallel maps W V 0 0 V ____//_W V 0 0 j,kJG (Vj, V ) ^B JG (Vk, Vj) ^B X(Vk)__//_jJG (Vj, V ) ^B X(Vj) given by composition in JGV 0and by the evaluation maps associated to the diagr* *am X. A cofinality argument that0is easily made precise by use of the explicit des* *crip- tion of the category JGV given in [61, II.x4] shows that the above coequalizer agrees with the coequalizer of the subdiagram JGV(0Vi, V 0) ^B JGV(Vi, Vi) ^B X(Vi)//_//_JGV(0Vi, V 0) ^B X(Vi). This coequalizer is the space that we have denoted by JGV(0Vi, V 0)^O(Vi)X(Vi). Remark 14.2.2. The argument above works in the same way for prespectra. It gives the conclusion that, for parametrized prespectra X in GPVB, (i*X)(V ) ~= VB-ViX(Vi). Remark 14.2.3. Assume that V and V 0are closed under finite sums and con- tain the trivial representation. We can then define the change of universe func* *tors 0 V IVV 0= i*i0*:GSBV -! GSB where i: {Rn} V and i0:{Rn} V 0. Explicitly (IVVX0)(V ) ~=JGV(Rn, V ) ^O(n)X(Rn). This is the definition given in [61, V.1.2]. These change of universe functors * *IVV 0 are exceptionally well behaved on the point set level and agree with those we a* *re using when V V 0. They are symmetric monoidal equivalences of categories. For collections V , V 0and V 00, they satisfy 0 V V V 0 V V IVV 0O VB ~= B , IV 0O IV 00~=IV 00, IV ~=Id. Moreover, IVV 0is continuous and commutes with smash products with ex-spaces. In particular, it is homotopy preserving and therefore induces equivalences of * *the classical homotopy categories. Unfortunately, however, the functors IVV 0are as poorly behaved on the homotopy level as they are well behaved on the point set * *level. They do not preserve either level q-equivalences or s-equivalences in general a* *nd the point set level relations above do not descend to the model theoretic homot* *opy categories that we are interested in. Furthermore, these functors IVV 0do not e* *xist if V is a cofinal expanding sequence. We shall therefore not make much use of them. 190 14. MODULE CATEGORIES, CHANGE OF UNIVERSE, AND CHANGE OF GROUPS Returning to our full generality, let i: V V 0. The adjoint pair (i*, i*)* * has good homotopical properties. Theorem 14.2.4. Let i: V V 0. Then i* preserves level q-equivalences, level qf-fibrations, s-fibrations, and s-acyclic s-fibrations. Therefore (i*, i*) is * *a Quillen adjoint pair in the level qf-model structure and in the s-model structure. More* *over, i* on homotopy categories is symmetric monoidal. If V is cofinal in V 0, then * *i* creates the weak equivalences and (i*, i*) is a Quillen equivalence. Proof. It is clear from its levelwise definition that i* preserves level q-* *equi- valences and level qf-fibrations. It follows that its left adjoint i* preserve* *s s- cofibrations and level acyclic s-cofibrations. This in turn implies that i* pre* *serves s-acyclic s-fibrations, since those are the maps that satisfy the RLP with resp* *ect to the s-cofibrations. The levelwise description of s-fibrations in Proposition* * 12.5.6 implies that i* preserves s-fibrations. The last statement follows from the def* *ini- tion of homotopy groups and the fact that the unit id-! i*i* is an isomorphism. The functor i* commutes with Z on the point set level, by [61, I.2.14], and this commutation relation descends directly to homotopy categories. Applying Propo- sition 14.2.8 below to the diagonal map of B, it follows that the derived funct* *or i* is symmetric monoidal. We have constructed the change of universe functors on both the spectrum and prespectrum level and they are compatible with the restriction functors U. Howe* *ver, in order to make use of excellent parametrized prespectra, we must restrict to parametrized prespectra indexed on cofinal sequencess j :W V and j0:W 0 V 0 of indexing representations in the given universes U U0. But then there need * *not be an induced inclusion i: W W 0. We therefore also define change of universe functors for prespectra indexed on cofinal sequences. Definition 14.2.5. Let i: V V 0and choose cofinal sequences W = {Vi} and W 0= {Vi0} in V and V 0such that Vi+1 = Vi Wi and Vi0= Vi Zi, where Zi+1= Zi Wi0and thus Vi0+1= Vi0 Wi Wi0. Define a pair of adjoint functors __~-*//_ 0 GPWB oo___GPWB ~-* by setting (~-*X)(Vi0) = ZiBX(Vi) and (~-*Y )(Vi) = ZiBY (Vi0). The structure maps are induced from the given structure maps in the evident way. Proposition 14.2.6. The pair (~-*,~-*) is a Quillen adjoint pair with respect to both the level qf-model structure and the stable model structure. The follow* *ing diagram commutes when the vertical arrows point in the same direction. j* HoGPWBOOoo____Ho GPVBOO ~-*~-*|||| i*||i*|| |fflffl|0 |fflffl|0 Ho GPWB o(j0)*HoGPVBo_ 14.2. CHANGE OF UNIVERSE 191 Proof. This is clearly a Quillen adjunction in the level qf-model structure, and to show that it is a Quillen adjunction in the stable model structure it th* *erefore suffices to verify the condition of Proposition 12.5.6. The homotopy pullback 1* *2.5.7 associated to the pair (Vi, Wi) and an s-fibration f :X -! Y is still a homotopy pullback after we apply ZiBto it and displays the required diagram 12.5.7 for * *the map ~-*f. We have that 0-Vi (~-*j*X)(Vi0) = ZiBX(Vi) ~= ViB X(Vi) = ((j0)*i*X)(Vi0) and this point set level isomorphism descends to homotopy categories since the functors0j* and (j0)* preserve all s-equivalences. The adjoint structure maps * *of X 2 GPVB induce maps (j*i*X)(Vi) = X(Vi) -! ZiBX(Vi0) = (~-*(j0)*X)(Vi). When X is s-fibrant, its adjoint structure maps are level q-equivalences, and we thus obtain an equivalence j*i* ' ~-*(j0)* on homotopy categories. On the point-set level, we have the following commutation relations between change of universe functors and change of base functors. Lemma 14.2.7. Let i: V V 0and let f :A -! B be a G-map. Then i* commutes up to natural isomorphism with the change of base functors f!, f*, and f*, and i* commutes up to natural isomorphism with f!and f*. Proof. The first statement is clear from the levelwise constructions of the base change functors, and the second statement follows by conjugation since i*,* * f!, and f* are left adjoints of i*, f*, and f*. The missing relation, i*f* ~=f*i*, would hold with the alternative point-set level definitions of Remark 14.2.3, where i* and i* are inverse equivalences. H* *ow- ever, these are point-set level relationships that need not descend to model th* *eoretic homotopy categories. With our preferred definition of i* in terms of prolongati* *on, the following result shows that i*f* ' f*i* on homotopy categories even though we need not have an isomorphism on the point-set level. Proposition 14.2.8. Let i: V V 0and let f :A -! B be a G-map. Then there are natural equivalences of derived functors i*f* ' f*i*, i*f!' f!i*, i*f* ' f*i*, i*f* ' f*i*, i*f!' f!i* in the relevant homotopy categories. Proof. The first two equivalences are clear since we are commuting Quillen right adjoints and their corresponding Quillen left adjoints. The fourth will f* *ollow by adjunction from the third. If f is a homotopy equivalence, then f* ' (f!)-1 * *and in this case the third follows from the second and the fifth from the first. Fa* *ctoring f as the composite of an h-fibration and a homotopy equivalence, we see that the third will hold in general if it holds when f is an h-fibration. Similarly, fac* *toring f as the composite of an h-cofibration and a homotopy equivalence, we see that the fifth will hold in general if it holds when f is an h-cofibration. Further, for the third equivalence, it suffices to show that ~-*f* ' f*~-*si* *nce Proposition 14.2.6 then gives that i*f* ' i*j*j*f* ' (j0)*~-*f*j* ' (j0)*f*~-*j* ' f*(j0)*(j0)*i* ' f*i*. 192 14. MODULE CATEGORIES, CHANGE OF UNIVERSE, AND CHANGE OF GROUPS Similarly, for the fifth equivalence, it suffices to show that ~-*f!' f!~-*, fo* *r then i*f!' i*(j0)*(j0)*f!' j*~-*f!(j0)* ' j*f!~-*(j0)* ' f!(j0)*(j0)*i* ' f!i*. We have reduced the proof of the third equivalence to the situation when f i* *s an h-fibration and i* is replaced by ~-*. The functor f* preserves excellent presp* *ectra over B, but we must apply T to ~-*before passing to homotopy categories. As in * *the proof of Theorem 13.7.7, since f is assumed to be an h-fibration we have a natu* *ral homotopy equivalence ~: T f* -! f*T in our categories indexed on W or on W 0. Therefore T~-*f* ~=T f*~-*' f*T~-*. Similarly, we have reduced the proof of the fifth equivalence to the situati* *on when f is an h-cofibration and i* is replaced by ~-*. Then f! preserves level * *h- equivalences, and so does ~-*since it preserves level q-equivalences and preser* *ves objects whose total spaces are of the homotopy types of G-CW complexes. Since T takes zig-zags of level h-equivalences to homotopy equivalences, T f!T~-*oo'//_T f!~-*~=T~-*f!oo'//_T~-*T f! displays a zig-zag of homotopy equivalences showing that f!~-*' ~-*f!. 14.3.Restriction to subgroups Let ` :G0 -! G be a homomorphism and let `*V be the collection of G0- representations `*V for V 2 V , where V is our chosen collection of indexing G- representations. We have implicitly used the following result in our earlier re* *sults on change of groups. Proposition 14.3.1. The functor `*: GSB -! G0S``*V*Bpreserves level q- equivalences, level qf-fibrations, s-fibrations, and s-equivalences provided th* *at the model structures are defined with respect to generating sets CG and CG0 of G-ce* *ll complexes and G0-cell complexes such that `!C = G xG0C 2 CG for C 2 CG0. Proof. Since (`*A)H = A`*(H)for a G-space A and a subgroup H of G0, this is clear from the definitions of homotopy groups and from the characterizations* * of fibrations given in Definition 7.2.7 and Proposition 12.5.6. Note in particular* * that `* preserves the level qf-fibrant approximations that are used in the definitio* *n of the stable homotopy groups. For the remainder of this section fix a subgroup H of G and consider the inclusion ': H G. For an H-space A, we simplify notation by letting HSAV denote the category of H-spectra over A indexed on '*V . Clearly, we then have the restriction of action functor '*: GSBV- ! HS'V*B. For i: V V 0, we have '*i* = i*'* since with either composite we are just re- stricting from the representations in V 0to the representations in V and viewin* *g all G-spaces in sight as H-spaces. When V = V (U) for a G-universe U, there is a quibble here (as was discussed in [61, V.10]). We are using '*V as the corresponding indexing collection for * *H. However, if V is an irreducible representation of G, '*V is generally not an ir* *re- ducible representation of H and we should expand '*V to include all representat* *ions 14.3. RESTRICTION TO SUBGROUPS 193 that embed up to isomorphism in '*U to fit the definitions into our usual frame- work. However, there is a change of universe functor associated to the inclusi* *on i: '*V (U) V ('*U) that fixes this. The functor i* preserves all s-equivalenc* *es and descends to an equivalence on homotopy categories. We can and should use these rectifications when restricting to H-spectra over '*B for a fixed chosen H. Remark 14.3.2. Consider passage to fibers and recall Proposition 12.6.11. (i)Applied to inclusions of orbits, Proposition 14.2.8 implies that the functo* *rs i* for i: V V 0are compatible with passage to fibers, in the sense that (i*X)b ~=i*(Xb) forb 2 B, where i* on the right is the change of universe functor on Gb-spectra. (ii)When V = V (U), we can view the fiber functor (-)b:GSB -! GbS as landing in spectra indexed on V ('*U), ': Gb -! G by composing with i* for i: '*V (U) V ('*U). However, these change of universe functors must be used with caution since they are not compatible as b and therefore Gb vary. Recall from Propositions 12.6.9 and 13.7.9 that the equivalence of categories ('!, *'*) between HSA and GS'!Ainduces a closed symmetric monoidal equiva- lence of categories between HoHSA and HoGS'!A. By Corollary 12.6.10, we can interpret the restriction functor '*: HoGSB - ! Ho HS'*B as the composite of base change ~* along ~: '!'*B -! B and this equivalence applied to A = '*B. The following spectrum level analogue of Proposition 2.3.11 gives compatibility rel* *ations between change of base functors and these results on change of groups. Proposition 14.3.3. Let f :A -! '*B be a map of H-spaces and "f:'!A -! B be its adjoint map of G-spaces. Then the following diagrams commute up to natur* *al isomorphism, where ~: '!'*B -! B and :A -! '*'!A are the counit and unit of the adjunction ('!, '*). "f! "f* GS'!AO_____//_GSBOOO GSB ______//GS'!A '!|| |~!O'!| '*|| ||*O'* | | fflffl| fflffl| HSA __f!_//_HS'*B HS'*B __f*_//_HSA These diagrams descend to natural equivalences of composites of derived functors on homotopy categories. Proof. The point set level diagrams commute by Proposition 2.3.11, ap- plied levelwise. The left diagram is one of Quillen left adjoints and the right diagram is one of Quillen right adjoints, by Propositions 12.6.7 and 12.6.9 and Corollary 12.6.10. We now define a parametrized fixed point functor associated to the inclusion ': H -! G. Its target is a category of nonequivariant parametrized spectra. In * *the next section we will consider a fixed point functor that takes values in a cate* *gory of parametrized W H-spectra, where W H = NH=H is the Weyl group. Write GSBtrivfor G-spectra over B indexed on trivial representations. These are "naive" parametrized G-spectra. As usual, to define fixed point spectra, we 194 14. MODULE CATEGORIES, CHANGE OF UNIVERSE, AND CHANGE OF GROUPS must change to the trivial universe before taking fixed points levelwise. Thus * *let V G = {V G | V 2 V }. It is contained in V if V = V (U) for some universe U. Definition 14.3.4. The G-fixed point functor (-)G :GSB - ! SBG is the composite of i*, i: V G V , and levelwise passage to fixed points. For a subg* *roup H of G the H-fixed point functor (-)H :GSB - ! SBH is the composite of '*, ': H G, and (-)H . Since the homotopy groups of a level qf-fibrant G-spectrum X over B are the homotopy groups ssHq(Xb), we see from the nonparametrized analogue [61, V.3.2] that these are then the homotopy groups of XH . Recall in particular that the s-fibrant G-spectra over B are the -G-spectra over B, which are level qf-fibra* *nt. Therefore, for all subgroups H of G, the homotopy groups of a parametrized G- spectrum X are the nonequivariant homotopy groups of the nonequivariant spectra XH , provided that (-)H is understood to mean the derived fixed point functor. On the point-set level, the functor (-)G is a right adjoint. Thinking of the homomorphism ": G -! e to the trivial group, let "*: SA - ! GS"triv*Abe the functor that sends spectra over a space A to G-trivial G-spectra over A regarded as a G-trivial G-space. The following result is immediate by passage to fibers * *from its nonparametrized special case [61, V.3.4]. Let A`` denote the collection of * *all representations of G. Proposition 14.3.5. Let A be a space. Let Y be a naive G-spectrum over "*A and X be a spectrum over A. There is a natural isomorphism GS"triv*A("*X, Y ) ~=SA (X, Y G). For (genuine) G-spectra Y over "*A, there is a natural isomorphism GS"*A(i*"*X, Y ) ~=SA (X, (i*Y )G ), where i: triv A``. Both of these adjunctions are given by Quillen adjoint pairs relating the respective level and stable model structures. Returning to G-spaces B and comparing Definition 11.3.5 with the proof of [61, V.3.5-3.6], we obtain the following curious results. Proposition 14.3.6. For a representation V and an ex-G-space K, we have that (FV K)G = *BG unless G acts trivially on V , when (FV K)G ~=FV (KG ) as a nonequivariant spectrum over BG . The functor (-)G preserves s-cofibrations, but it does not preserve acyclic s-cofibrations. Corollary 14.3.7. For ex-G-spaces K, ( 1BK)G ~= 1B(KG ). This isomorphism of spectra over BG does not descend to the homotopy cate- gory Ho GSBG . The reader is warned to consult [61, Vx3] for the meaning of the* *se results. There is also an analogue of the comparison between G-fixed points and smash products in [61, V.3.8], but only when B = BG and only with good behavior with respect to cofibrant objects when external smash products are used. We sha* *ll not state the result formally. 14.4. NORMAL SUBGROUPS AND QUOTIENT GROUPS 195 14.4.Normal subgroups and quotient groups We now turn to quotient homomorphisms and associated orbit and fixed point functors. The material of this section generalizes a number of results from x2* *.4, x7.3, and x9.5 to the level of parametrized spectra. Just as we have been using ' generically for inclusions of subgroups, we sha* *ll use " generically for quotient homomorphisms. In particular, for an inclusion ': H * * G, we let W H = NH=H, where NH is the normalizer of H in G, and we have the quotient homomorphism ": NH -! W H. We can study this situation by first restricting from G to NH, thus changing the ambient group. Therefore, there is * *no loss of generality if we focus attention on a normal subgroup N of G with quoti* *ent group J = G=N, as we do throughout this section. Definition 14.4.1. Let GSBN-trivbe the category of G-spectra over B indexed on the N-trivial representations of G. Regard representations of J as N-trivial representations of G by pullback along ": G -! J. For a J-space A, define "*: JSA -! GS"N-triv*A levelwise by regarding ex-J-spaces over A as N-trivial G-spaces over "*A. For a G-space B, define (-)=N :GSBN-triv-! JSB=N and (-)N :GSBN-triv-! JSBN by levelwise passage to orbits over N and to N-fixed points. Lemma 14.4.2. The N-fixed point functor (-)N preserves level q-equivalences, level qf-fibrations, s-fibrations, and s-equivalences, provided that the model * *struc- tures are defined with respect to generating sets CG and CJ of G-cell complexes* * and J-cell complexes such that C=N 2 CJ for C 2 CG . Proof. This is a special case of Proposition 14.3.1; it also follows direct* *ly from the ex-space level analogue in Proposition 7.4.3, the characterization of s-fib* *rations in Proposition 12.5.6, and inspection of the definition of the s-equivalences. Proposition 14.4.3. Let j :BN - ! B be the inclusion and p: B -! B=N be the quotient map. Then the following factorization diagrams commute (-)=N (-)N GSBN-triv____//JSB=N99 and GSBN-triv____//_JSBN sss ss99s p!|| ssss j*|| ssss N fflffl|(-)=Nsss fflffl|(-)sss GSBN-triv=N GSBN-trivN and they descend to natural equivalences on homotopy categories (p!X)=N ' X=N and (j*X)N ' XN for X in Ho GSBN-triv. The following adjunction isomorphisms follow. (i)For Y 2 GSBN-trivand X 2 JSB=N , JSB=N (Y=N, X) ~=GSBN-triv(Y, p*"*X). (ii)For Y 2 GSBN-trivand X 2 JSBN , GSBN-triv(j!"*X, Y ) ~=JSBN (X, Y N). 196 14. MODULE CATEGORIES, CHANGE OF UNIVERSE, AND CHANGE OF GROUPS (iii)For (genuine) G-spectra Y 2 GSB and X 2 JSBN , GSB (i*j!"*X, Y ) ~=JSBN (X, (i*Y )N ), where i: triv A``. All of these adjunctions are Quillen adjoint pairs with respect to both the lev* *el and the stable model structures and so descend to homotopy categories. Proof. The factorizations follow from the ex-space level analogue Proposi- tion 2.4.1. The statement about Quillen adjunctions holds since (-)N , ffl* and* * i* preserve level q-equivalences, level fibrations, s-equivalences and level s-fib* *rations, by Lemma 14.4.2, Proposition 14.3.1 and Theorem 14.2.4. The behavior of the orbit and fixed point functors with respect to base chan* *ge is recorded in the following result. Proposition 14.4.4. Let f :A -! B be a map of G-spaces. Then the following diagrams commute up to natural isomorphism f! N-triv N-trivf* N-triv N-triv f! N-triv GSAN-triv___//GSB GSB _____//GSA GSA ____//_GSB (-)=N|| |(-)=N|(-)N|| (-)N||(-)N|| |(-)N| fflffl| fflffl| fflffl| fflffl| fflffl| |fflffl JSA=N (f=N)_//_JSB=N JSBN _______//JSAN JSAN ______//_JSBN ! (fN )* (fN )! and they descend to the following natural equivalences on homotopy categories (f!X)=N ' (f=N)!(X=N), (f*X)N ' (fN )*(Y N), (f!X)N ' (fN )!(X=N) for X 2 Ho GSAN-trivand Y 2 Ho GSBN-triv. Proof. The first statement follows levelwise from the ex-space level analog* *ue Proposition 2.4.2. The proof that it descends to equivalences on homotopy cate- gories is the same as for the ex-space level analogue Proposition 7.4.5. Specializing to N-free G-spaces, we obtain a factorization result that is an* *al- ogous to those in Proposition 14.4.3, but is less obvious. It is a precursor of* * the Adams isomorphism. Proposition 14.4.5. Let E be an N-free G-space, let B = E=N, and let p: E -! B be the quotient map. Then the diagram (-)=N GSEN-triv_____//JSB:: ttt p*|| tttt N |fflffl(-)ttt GSBN-triv commutes up to a natural isomorphism, and it descends to a natural equivalence X=N ' (p*X)N in GSEN-trivfor X 2 Ho JSB . Therefore the left adjoint (-)=N of the functor p** *"* is also its right adjoint. Proof. The point set level result follows levelwise from the ex-space level result Proposition 2.4.3. Since it is an isomorphism between a Quillen left adj* *oint on the left hand side and a composite of Quillen right adjoints on the right ha* *nd side, it descends directly to homotopy categories. Part IV Duality, transfer, and base change isomorphisms CHAPTER 15 Fiberwise duality and transfer maps Introduction We put the foundations of Part III to use in the two chapters of this last part. Unless otherwise stated, we work in the derived homotopy categories, and all functors should be understood in the derived sense. For example, we have the derived fiber functor (-)b:Ho GSB -! Ho GbS . Since passage to fibers is a Quillen right adjoint, this means that we replace * *G- spectra X over B by s-fibrant approximations before taking point-set level fibe* *rs. For emphasis, and to make the notation Xb clear and unambiguous, we may assume that X is s-fibrant, but there is no loss of generality. A map f in Ho GSB is an equivalence if and only if fb is an equivalence for all b 2 B, and that allows * *us to transer information back and forth between the parametrized and unparametrized homotopy categories with impunity. Here we use the word "equivalence" to mean an isomorphism in HoGSB , and we use the notation ' for this relation. We reser* *ve the symbol ~=to mean an isomorphism on the point set level. We have proven that the basic structure enjoyed by the category GSB of parametrized spectra descends coherently to the homotopy category Ho GSB . In particular, Ho GSB is closed symmetric monoidal, and the derived fiber functor * *is closed symmetric monoidal. In any symmetric monoidal category, we have standard categorical notions of dualizable and invertible objects. In x15.1, we prove t* *he fiberwise duality theorem, which says that a G-spectrum X over B is dualizable * *or invertible if and only if each fiber Xb is dualizable or invertible. This allow* *s us to recognize dualizable or invertible G-spectra over B when we see them. In x15.2, we explain how the fiberwise duality theorem leads to a simple and general conceptual definition of trace and transfer maps with good properties. * *To define the transfer, we regard a Hurewicz fibration p: E -! B with stably duali* *z- able fibers as a space over B. We adjoin a copy of B to obtain a section, and we suspend to obtain a G-spectrum over B. It is dualizable since its fibers are du* *aliz- able, hence it has a transfer map defined by categorical nonsense. Pushing down* * to G-spectra by base change along the map r :B -! *, we obtain the transfer map of G-spectra 1 B+ -! 1 E+ . This construction is a generalization of various earlier constructions of the transfer [2, 3, 15, 18, 97], most of which restric* *t to finite dimensional base spaces and are nonequivariant. An essential point is th* *at the homotopy category of G-spectra over B is closed symmetric monoidal with a "compatible triangulation", in the sense specified in [74]. We defer the proof* * of the required compatibility relations to x15.6. This point implies that our tra* *ces and transfers satisfy additivity relations as well as the more elementary stand* *ard properties. 198 15.1. THE FIBERWISE DUALITY THEOREM 199 Some of the classical constructions of the transfer work only for bundles, b* *ut have various properties that are inaccessible to the more general construction * *and are important in calculations. These transfers also admit a perhaps more satisf* *ying construction. Rather than relying on duality on the level of parametrized spect* *ra, they are obtained by inserting duality maps for fibers fiberwise into bundles. * *In the literature, the construction again usually requires finite dimensional base spa* *ces and is nonequivariant. We give a general conceptual version of this alternative construction in x15.5. As a first preliminary, in x15.3 we show how to insert parametrized spectra fiberwise into the standard construction of equivariant bundles associated to p* *rin- cipal bundles. The general construction is of considerable interest nonequivari* *antly. The construction on the ex-space level is easy enough, but even here many of the properties that we describe seem to be new. The construction is likely to have many further applications. The idea is to generalize the standard construction * *of the bundle of tangents along the fibers of a bundle by replacing the tangent bu* *ndle of the fiber by any spectrum over the fiber. In more detail, we consider G-bund* *les p: E -! B with fibers F . We allow the structure group and ambient group G to be related by an extension 1 -! -! -! G -! 1, and we take F to be a - space. The bundle p has an associated principal ( ; )-bundle ss :P -! B, where P is a -free -space and B = P= . We show how to construct a G-spectrum P x X over E from a -spectrum X over F . As a second preliminary, in x15.4 we develop the theory of -free parametriz* *ed -spectra. This is a direct generalization of the nonparametrized theory and is important in many contexts. In particular, it will play a role in our proof of * *the Adams isomorphism in x16.4. The application to transfer maps in x15.5 can be described as follows. When F is dualizable, we have a transfer map o :S -! 1 F+ of -spectra. We insert this into the functor P x (-) to obtain a map P x o :P x S -! P x 1 F+ of G-spectra over B. Again pushing down to a map of G-spectra along r :B -! *, we obtain the transfer G-map 1GB+ -! 1GE+ . This description hides a subtlety. The construction involves a change of universe functor, and the key point is th* *at this functor is a symmetric monoidal equivalence between categories of parametrized * * - free -spectra. This makes it transparent from the naturality of transfer maps * *with respect to symmetric monoidal functors that the fiberwise transfer map of a bun* *dle agrees with its transfer map as a Hurewicz fibration. We assume throughout that all given groups G are compact Lie groups and all given base G-spaces are of the homotopy types of G-CW complexes. 15.1. The fiberwise duality theorem We characterize the dualizable and invertible G-spectra over B. A recent ex- position of the general theory of duality in closed symmetric monoidal categori* *es appears in [73], to which we refer the reader for discussion of the relevant ca* *tegori- cal definitions and arguments. The following theorem is a substantial generaliz* *ation of various early results of the same nature about ex-fibrations. These are due,* * for example, to Becker and Gottlieb [2, x4], Clapp [18, 3.5], and Waner [97, 4.6]. 200 15. FIBERWISE DUALITY AND TRANSFER MAPS Theorem 15.1.1 (The fiberwise duality theorem). Let X be an (s-fibrant) G- spectrum over B. Then X is dualizable (respectively, invertible) if and only if* * Xb is dualizable (respectively, invertible) as a Gb-spectrum for each b 2 B. Proof. By definition, X is dualizable if and only if the natural map :DB X ^B X -! FB (X, X) in Ho GSB is an equivalence. Passing to (derived) fibers, this holds if and onl* *y if the resulting map DXb^ Xb ' (DB X ^B X)b __b_//_FB (X, X)b ' F (Xb, Xb) in HoGbS is an equivalence for all b 2 B. By the categorical coherence observat* *ion Remark 2.2.8, the latter map is the corresponding natural map in HoGbS . Again by definition, that map is an equivalence if and only if Xb is dualizable. Similarly, X is invertible if and only if the evaluation map ev:DB X ^B X -! SB in Ho GSB is an equivalence. Passing to (derived) fibers, this holds if and onl* *y if the resulting map DXb^ Xb ' (DB X ^B X)b _evb//_(SB )b ' S in Ho GbS is an equivalence for all b 2 B. Again by Remark 2.2.8, the latter map is the evaluation map for Xb in Ho GbS , and that map is an equivalence if and only if Xb is invertible. Therefore, to recognize parametrized dualizable and invertible G-spectra, it suffices to recognize nonparametrized dualizable and invertible G-spectra. As we now recall from [41], these are well-understood. Recall that a G-space X is dominated by a G-space Y if X is a retract up to homotopy of Y , so that the identity map of X is homotopic to a composite X -! Y -! X. If Y has the homotopy type of a G-CW complex, then so does X. We say that X is finitely dominated if it is dominated by a finite G-CW complex. This does not imply that X has the homotopy type of a finite G-CW complex, even when X and all of its fixed point spaces XH are simply connected and therefore, since they are finitely dominated, homotopy equivalent to finite CW complexes. For example, a G-space X is a G-ENR (Euclidean neighborhood retract) if it can be embedded as a retract of an open subset of some representation V . Such open subsets are triangulable as G-CW complexes, so X has the homotopy type of a G-CW complex. A compact G-ENR is a retract of a finite G-CW complex and is thus finitely dominated, but it need not have the homotopy type of a finite G-CW complex. Non-smooth topological G-manifolds give examples of such non-finite compact G-ENRs. The following result is [41, 2.1]. Theorem 15.1.2. Up to equivalence, the dualizable G-spectra are the G-spectra of the form -V 1 X where X is a finitely dominated based G-CW complex and V is a representation of G. Definition 15.1.3. A generalized homotopy representation X is a finitely dom- inated based G-CW complex such that, for each subgroup H of G, XH is equivalent to a sphere Sn(H). A stable homotopy representation is a G-spectrum of the form 15.2. DUALITY AND TRANSFER MAPS 201 -V 1 X, where X is a generalized homotopy representation and V is a represen- tation of G. The following result is [41, 0.5]. Theorem 15.1.4. Up to equivalence, the invertible G-spectra are the stable homotopy representations. Combining results, we obtain the following conclusion about ex-G-fibrations. Theorem 15.1.5. Let E be an ex-G-fibration over B. If each fiber Eb is a finitely dominated Gb-space, then 1BE is a dualizable G-spectrum over B. If ea* *ch Eb is a generalized homotopy representation of Gb, then 1BE is an invertible G- spectrum over B. Proof. Since the derived suspension spectrum functor commutes with passage to derived fibers, by Theorem 13.7.10, the derived fiber ( 1BE)b is equivalent * *to 1 Eb. The conclusion follows directly from Theorems 15.1.1, 15.1.2, and 15.1.4. In particular, sphere G-bundles and, more generally, spherical G-fibrations * *over B, have invertible suspension G-spectra over B. 15.2. Duality and transfer maps Since the stable homotopy category Ho GSB is closed symmetric monoidal, we have the following generalized trace maps at our disposal. We state the definit* *ion and recall its properties in full generality, and we then specialize to show ho* *w it gives a simple conceptual definition of the transfer maps associated to equivar* *iant Hurewicz fibrations. Definition 15.2.1. Let C be any closed symmetric monoidal category with unit object S. For a dualizable object X of C with a "coaction" map X :X -! X ^ CX for some object CX 2 C , define the trace o(f) of a self map f of X by t* *he diagram j fl S _____//_X ^ DX_______//_DX ^ X o(f)|| |Df^|X fflffl| fflffl| CX oo~=_S_^ CX ooffl^DX1^_X ^ CX . Remark 15.2.2. Such a categorical description of generalized trace maps was first given by Dold and Puppe [35], where they showed that it gives the right framework for trace maps in algebra, the transfer maps of Becker and Gottlieb [* *2, 3], and the fixed point theory of Dold [34]. These early constructions of transfer * *maps had finiteness conditions that were first eliminated by Clapp [18, 19]. Indeed, she gave an early construction of a parametrized stable homotopy category and proved a precursor of our fiberwise duality theorem. The equivariant analogue of the attractive space level treatment of Spanier-Whitehead duality given by Dold and Puppe was worked out in [59], and a recent categorical exposition of duality has been given in [73]. Two cases are of particular interest. The first is when CX = S and X is the unit isomorphism. Then o(f) is called the Lefschetz constant of f and is denoted 202 15. FIBERWISE DUALITY AND TRANSFER MAPS by O(f); in the special case when f = idit is called the Euler characteristic o* *f X and denoted by O(X). The second is when CX = X. We then think of X as a diagonal map, and oX = o(id) is called the transfer map of X. Remark 15.2.3. If CX comes with a "counit" map , :CX -! S such that the composite id^, X _____//X ^ CX____//X is the identity, then O(f) = , O o(f) by a little diagram chase. The reason for the terminology "coaction" and "counit" for the maps X and , is that in many situations CX is a comonoid and X is a coaction of CX on X. The following basic properties of the trace are proven in [59, IIIx7] and in* * [74], where more detailed statements are given. Define a map (f, ff): (X, X ) -! (Y, Y ) to be a pair of maps f :X - ! Y and ff: CX -! CY such that the following diagram commutes. X X _____//X ^ CX f|| |f^ff| |fflffl |fflffl Y __Y__//Y ^ CY Proposition 15.2.4. The trace satisfies the following properties, where X and Y are dualizable and X and Y are given. (i)(Naturality) If C and D are closed symmetric monoidal categories and F :C -! D is a lax symmetric monoidal functor such that F SC ~=SD , then o(F f) = F o(f), where CFX = F CX and FX = F X . (ii)(Unit property) If f is a self map of the unit object, then O(f) = f. (iii)(Fixed point property) If (f, ff) is a self map of (X, X ), then ff O o(f) = o(f). (iv)(Invariance under retracts) If X -i! Y -r! X is a retract, f is a self map * *of X, and (i, ff) is a map (X, X ) -! (Y, Y ), then ff O o(f) = o(ifr). (v)(Commutation with ^) If f and g are self maps of X and Y , then o(f ^ g) = o(f) ^ o(g), where X^Y = (id^ fl ^ id) O ( X ^ Y ) with fl the transposition. (vi)(Commutation with _) If C is additive and h: X _ Y -! X _ Y induces f :X -! X and g :Y -! Y by inclusion and retraction, then o(h) = o(f) + o(g), where CX = CY = CX_Y and X_Y = X _ Y . (vii)(Anticommutation with suspension) If C is triangulated, then o( f) = -o(f) for all self maps f, where X = X . 15.2. DUALITY AND TRANSFER MAPS 203 In the triangulated context, there is another and very much deeper property. Theorem 15.2.5 (Additivity). Let C be a closed symmetric monoidal category with a "compatible triangulation". Let X and Y be dualizable and let X and Y be given, where C = CX = CY . Let (f, id) be a map (X, X ) -! (Y, Y ) and extend f to a distinguished triangle f g h X _____//Y____//Z____//_ X. Assume given maps OE and _ that make the left square commute in the first of the following two diagrams. f g h X _____//Y____//Z____//_ X OE|| |_| |!| ||OE fflffl|fflffl|fflffl| fflffl| X __f__//Y_g__//Z_h__//_ X f g h X ________//_Y_______//_Z________//_ X X || |Y| |Z| || X fflffl| fflffl| fflffl| fflffl| X ^ C _f^id//_Y ^_Cg^id//_Z ^hC^id//_ (X ^ C) Then there are maps ! and Z such that the diagrams commute and o(_) = o(!) + o(OE). The most important case starts with only the distinguished triangle (f, g, h) and concludes with the fundamental additivity relation O(Y ) = O(X) + O(Z). The additivity of traces was studied in [59, IIIx7] in the equivariant stable h* *omotopy category, but the proof there is incorrect. A thorough investigation of precise* *ly what is needed to prove the additivity of traces is given in [74], where the axioms * *for a "compatible triangulation" are formulated. These axioms hold in all situations previously encountered in algebraic topology and algebraic geometry. However, the model theoretic method of proof described in [74] assumes the usual model theoretic compatibilities, such as the pushout-product axiom of [86], and these* * fail to hold in the present context. Since the proof of the following result only ma* *kes sense by close comparison with the proof in [74], we shall defer it to x15.6. Theorem 15.2.6. The category Ho GSB is a closed symmetric monoidal cat- egory with a compatible triangulation. With these foundations in place, we can now generalize the classical constru* *c- tion of transfer maps. The results above specialize to give more information ab* *out them than is to be found in the literature. If X is a dualizable G-spectrum over B with a diagonal map X :X - ! X ^B X, then we have the transfer map oX :SB -! X. We shall apply this to suspension G-spectra associated to G- fibrations p: E -! B, but we do not assume that p has a section. We need some notation. It has been the custom since the beginnings of algebraic topology to * *use the same letter E for a bundle and for its underlying total space. It seems to * *us that this standard abuse of notation seriously obscures the literature of parametriz* *ed ho- motopy theory, and for that reason we shall be very pedantic at this point. 204 15. FIBERWISE DUALITY AND TRANSFER MAPS Notation 15.2.7. For a G-space E over B, let (E, p)+ denote the ex-G-space E qB over B, with section at the disjoint copy of B. The usual notation is E+ ,* * but we shall reserve that notation for the union of the total G-space E with a disj* *oint basepoint. Observe that if p is a Hurewicz G-fibration, then (E, p)+ is an ex-* *G- fibration. Except where otherwise indicated, we agree to write r for the unique map B -! * for any based G-space B. Recall the desription of the base change functors associated to r from Exam- ple 2.1.7. The spectrum level versions of these functors are central to the ded* *uction of results in classical stable homotopy theory from results in parametrized sta* *ble homotopy theory. The following observation is particularly relevant. Lemma 15.2.8. For a G-map p: E -! B, thought of as a G-space over B, r! 1B(E, p)+ ' 1 E+ , where r :B -! *. In particular, r!SB ' 1 B+ . Proof. We have r! 1B ' 1 r!. This is a commutation relation between Quillen left adjoints, and the corresponding commutation relation for right adj* *oints holds since r* 1 X = B x X0 ~= 1Br*X for a G-spectrum X. It therefore suffices to show that r!(E, p)+ is equivalent* * to E+ , where r! denotes the functor on derived categories. By Proposition 7.3.4, r! preserves q-equivalences between well-sectioned ex-spaces and it follows that r!Q(E, p)+ ' r!(E, p)+ ~=E+ where the first equivalence is induced by qf-cofibr* *ant approximation of (E, p)+ . To be precise about diagonal maps on the parametrized level, we consider base change along : B -! B x B. We have the obvious commutative diagram E _____//E x E p || |pxp| fflffl| fflffl| B _____//B x B. We consider E as a space over B x B via this composite. The diagonal map of E then specifies a natural map !((E, p)+ ) = (E, O p)+ -! (E x E, p x p)+ ~=(E, p)+ Z (E, p)+ of ex-spaces over B x B. This is a comparison map between Quillen left adjoints and therefore descends to a natural map in HoGKBxB . Its adjoint is a natural m* *ap (E, p)+ -! (E, p)+ ^B (E, p)+ in HoGKB . Apply the (derived) suspension functor 1Bto this map and note that the target is equivalent to 1B(E, p)+ ^B 1B(E, p* *)+ , by Proposition 13.7.5. This gives the required natural diagonal map (E,p)+: 1B(E, p)+ -! 1B(E, p)+ ^B 1B(E, p)+ in Ho GSB . Definition 15.2.9 (The transfer map). Let p: E -! B be a Hurewicz G- fibration over B such that each fiber Eb is homotopy equivalent to a retract of 15.3. THE BUNDLE CONSTRUCTION ON PARAMETRIZED SPECTRA 205 a finite Gb-CW-complex. Then 1B(E, p)+ is a dualizable G-spectrum over B by Theorem 15.1.5 and we obtain the transfer map o(E,p)+:SB -! 1B(E, p)+ in Ho GSB . Define the transfer map of E to be the map oE = r!o(E,p)+: 1 B+ ~=r!SB -! r! 1B(E, p)+ ~= 1 E+ in Ho GS . With this definition, all of the standard properties of transfer maps are di- rect consequences of the general categorical results Proposition 15.2.4 and The* *o- rem 15.2.5 and the properties of r!. 15.3.The bundle construction on parametrized spectra The construction of the transfer in the previous section works "globally", s* *tart- ing on the parametrized spectrum level. We now give a fiberwise construction of "stable bundles" that leads to an alternative fiberwise perspective. However, i* *t is natural to work in greater generality than is needed for the construction of the transfer. The extra generality will be needed in the proof of the Wirthm"uller * *iso- morphism in x16.3 and will surely find other applications. The relevant bundle theoretic background was recalled in x3.2. Let be a normal subgroup of a compact Lie group such that = = G and let q : -! G be the quotient homomorphism. Let p: E -! B be a ( ; )-bundle with fiber a -space F and with associated principal ( ; )-bundle ss :P -! B. Then P is a -free -space, ss is the quotient map to the orbit G-space B = P= , and p is the associated G-bundle E ~=P x F -! B. To simplify the homotopical analysis, we assume for the rest of this section that F and P are -CW complexes such that P is -free. We let E = P x F and B = P x *. Note that B is a G-CW complex. We are thinking of the cases when F is a point or when F is a smooth -manifold. On the ex-space level, application of P x (-) to retracts g* *ives the functor P~F = P x (-): KF -! GKE . Thus, for an ex- -space K over F , the ex-G-space P x K over P x F has section and projection induced by the section and projection of K. Observe that if F is a smooth manifold and So is the sphere bundle obtained by fiberwise one-point compactification of the tangent bundle of F , then P x So is the G-bundle of spherical tangents along the fiber associated to p. We can extend the functor ~PF from ex-spaces to ex-spectra. Change of uni- verse must enter since -spectra are indexed on representations of and G-spec* *tra are indexed on representations of G. We view representations of G as -trivial representations of . This gives i: q*VG -! V . Implicitly applying the functor i* to -spectra, we agree to index both G-spectra and -spectra on VG for the r* *est of the section. We are interested in -spectra indexed on a complete universe, * *and we shall return to this point in the next section. Since acts trivially on o* *ur representations V , we have ~PFK ^E SV ~=P~F(K ^F SV ). Therefore, for a -spectrum X over F , the ex-G-spaces ~PFX(V )over E inherit structure maps from X, so that ~PFX is a G-spectrum over E. We have the same 206 15. FIBERWISE DUALITY AND TRANSFER MAPS definition on the prespectrum level. These functors P~F are exceptionally well- behaved, as the following results show. Proposition 15.3.1. The functor ~PF: SF -triv-! GSE is both a left and a right Quillen adjoint with respect to the level and stable model structures. Mo* *reover, the functor ~PF: PF-triv-! GPE takes excellent -prespectra over F to excellent G-prespectra over E = P x F . Proof. Let ss :P x F -! F be the projection. Clearly ~PFis the composite of ss*: SF -! SPxF and (-)= : SPxF -! GSE . By Propositions 12.2.5, 12.2.7, 12.6.7, and 12.6.8, ss* is both a left and a right Quillen adjoint, pro* *vided we use appropriate generating sets in our definitions of the model structures. By * *Propo- sition 14.4.3, the functor (-)= is a Quillen left adjoint. By Proposition 14.* *4.5, it coincides with the right adjoint (-) O p*, where p here is the quotient map P x F - ! P x F = E. Using Lemma 3.2.1, we see that p: E -! B is a G- bundle with CW fibers. Therefore p* is a Quillen right adjoint by Propositions 12.2.7 and 12.6.8, and (-) is a Quillen right adjoint by Proposition 14.4.3. T* *he last statement is easily checked from Definition 13.2.2 and Lemma 13.2.3. We need an observation about the behavior of ~PFon fibers. Lemma 15.3.2. Fix b 2 B. Let ': Gb -! G and aeb:Gb -! be the inclusion and the homomorphism of Lemma 3.2.1. Let b: {b} -! B and ib:Eb -! E denote the evident inclusions of Gb spaces. The following diagrams commute, and these commutation relations descend to homotopy categories. ae*b ae*b S* -triv____//GbSbOOand SF -triv____//GbSEbOO ~P*|| |b*| P~F|| i*b|| fflffl| | fflffl| | GSB ___'*_//_GbSB GSE __'*__//_GbSE Proof. On the level of ex-spaces, this is immediate by inspection. The di- agrams extend levelwise to parametrized spectra, and passage to homotopy cate- gories is clear from the previous result. Writing 1 for suspension spectra functors indexed on complete universes, we have that i* 1 ,F, where i: q*VG V , is the suspension -spectrum functor indexed on the -trivial -universe q*VG . Theorem 15.3.3. There is a natural isomorphism of functors ~PFi* 1 ,F~= 1G,E~PF: KF -! GSE , and this isomorphism descends to homotopy categories. The functor ~PF:Ho SF -triv-! Ho GSE is closed symmetric monoidal. Proof. Let K be an ex- -space over F . Since we are indexing on representa- tions V of G, we have isomorphisms (P~Fi* 1 ,FK)(V ) = P x (K ^F SVF)~=(P x K) ^E SV = ( 1G,E~PFK)(V ). This gives a natural isomorphism of G-spectra over E, and it descends to homoto* *py categories since it is a comparison of composites of Quillen left adjoints. Not* *e in 15.3. THE BUNDLE CONSTRUCTION ON PARAMETRIZED SPECTRA 207 particular that ~PFi*S ,Fis isomorphic to SG,E. We must show that the functor ~* *PF commutes up to coherent natural isomorphism with smash products and function objects. For ex- -spaces K and L over F , it is easy to check that there is a n* *atural isomorphism ~PF(K ^F L)- ! ~PFK ^E ~PFL of ex-G-spaces over E. This isomorphism extends levelwise to external smash pro* *d- ucts (external in the sense of pairs of representations). However, since exter* *nal pairings (in the sense of pairs of base spaces) do not naturally come into play* * here, to retain homotopical control it seems simplest to just extend levelwise to han* *di- crafted smash products of -prespectra; compare Remark 13.7.1. Using excellent prespectra to pass to homotopy categories of prespectra and then using the equi* *v- alence (P, U) to pass to homotopy categories of spectra, we obtain the required natural equivalence ~PF(X ^F Y )-! ~PFX ^E ~PFY in Ho GSE for -spectra X and Y over F . The adjoint of the composite P~FFF (X, Y^)E~PFX ' ~PF(FF (X, Y ) ^F X)_P~F(ev)_//_~PFY is a natural map P~FFF (X, Y )-! FE (P~FX , ~PFY) in Ho GSE , and we must show that it is an equivalence. This will hold if it ho* *lds when restricted to fibers over points of E. Since each point is in some Eb, it suffices to show that the restriction to each Eb is an equivalence. However, us* *ing Lemma 15.3.2, we see that the restriction to Eb is the adjoint to the Gb-map ev:FEb(ae*bX, ae*bY ) ^Eb ae*bX -! ae*bY , and is thus the identity map. We have the following relations between ~PFand base change functors. Proposition 15.3.4. Consider r :F -! * and p = P x r :E -! B. For Y 2 S* -trivand X 2 SF -triv, there are natural isomorphisms p!~PFX -! ~P*r!X, ~PFr*Y-! p*P~*Y, and ~P*r*X-! p*P~FX , and these isomorphisms induce natural equivalences on homotopy categories. Proof. We first work on the ex-space level. Let T be a based G-space and K be an ex-G-space over F . Applying the functor P x (-) to the maps of retracts that define r!K and r*T (see Definition 2.1.1), we immediately obtain the first* * two maps. The first is the natural isomorphism (P x K) [E B ~=P x (K=F ) in which the section F is collapsed to a point in K on both sides. The second is the evident natural isomorphism P x (F x T ) ~=(P x F ) xB (P x T ). For the third map, recall that r*K = Sec(F, K). The adjoint of (P x Sec(F, K)) xB E ~=P x (Sec(F, K) x F_)Px_ev__//_P x K gives a map P~*r*K = P x Sec(F, K)-! Map B(E, P x K). 208 15. FIBERWISE DUALITY AND TRANSFER MAPS Together with the projection of the source to B, it induces an isomorphism to p*P~FK , which is the pullback along B -! Map B(E, E) of the projection of the target induced by the projection P x K -! P x F = E. Applied levelwise, these point-set level isomorphisms carry over directly to parametrized prespect* *ra and spectra. We must show that they descend to equivalences in homotopy cate- gories. Since Proposition 12.6.8 applies to show that both p* and r* are Quillen right adjoints (and we have no need to use Brown representability here), the fi* *rst commutation relation is between composites of left Quillen adjoints, the second is between functors that are both left and right Quillen adjoints, and the thir* *d is between Quillen right adjoints, so descent to homotopy categories is immediate. 15.4. -free parametrized -spectra We retain the notations of the previous section in this section and the next. In the next section, we show that the bundle construction on parametrized spect* *ra leads to a fiberwise generalization of the restriction to bundles of the trace * *and transfer maps for fibrations that we described in x15.2. The definition depends* * on a result that is proven by use of the theory of -free -spectra that we presen* *t here. We first recall what it means to say that a -spectrum X (indexed on any universe) is -free. Let F ( ; ) be the family of subgroups of such that \ = e. A -CW complex T is -free if and only if the only orbit types = that appear in its construction have 2 F ( ; ). We then say that T is an F ( ; )-CW complex. We can make the same definitions for -CW spectra, and in general we say that a -spectrum is -free if it is isomorphic in Ho S * *to an F ( ; )-CW spectrum. There is a more conceptual homotopical reformulation that is the one relevant to the parametrized point of view and that does not de* *pend on the theory of -CW spectra. Let E( ; ) be the universal -free -space, so that E( ; ) is contractible if \ = e and is empty otherwise. We may take E( ; ) to be an F ( ; )-CW complex. Let B( ; ) = E( ; )= and observe that B is a G-CW complex and therefore also a -CW complex. We note parenthetically that the quotient map p : E( ; ) -! B( ; ) is the universal principal ( ; )-bundle. That is, pullb* *ack along p gives a bijection [X, B( ; )]G -! B( ; )(X), where B( ; )(X) denotes the set of equivalence classes of principal ( ; )-bun* *dles over the G-space X; see [55] or [68, VIIx2]. Definition 15.4.1. Let r :E( ; ) -! * be the projection and let oe be the counit of the (derived) adjunction (r!, r*). A -spectrum X is said to be -fre* *e if oe :r!r*X -! X is an equivalence. The definition should seem reasonable since r!r*T ~=E( ; )+ ^T for a -space T . It is equivalent to the original definition in terms of an equivalence in H* *o GS to an F ( ; )-CW spectrum; see [59, II.2.12] or [61, VIx4]. The definition genera* *lizes readily to the parametrized context. Definition 15.4.2. Let ss :E( ; ) x F -! F be the projection and let oe be the counit of the (derived) adjunction (ss!, ss*). An ex- -space or -spectrum* * X over a -space F is said to be -free if oe :ss!ss*X -! X is an equivalence. 15.5. THE FIBERWISE TRANSFER FOR ( ; )-BUNDLES 209 Since the fiber (ss!ss*X)f is E( ; )+ ^ Xf, the definition should seem reas* *on- able. Since equivalences are detected fiberwise, we have the following results. Lemma 15.4.3. A -spectrum X over F is -free if and only if each of its fib* *ers Xf is a ( \ f)-free f-spectrum. Proof. The fiber of E( ; )xF -! F over f 2 F is the -space E( ; ) with the action restricted along ': f -! . It is a model of the universal ( \ f)* *-free f-space E( \ f, f). Applying (-)f to the counit ss!ss*X -! X and using Theorem 13.7.7 we obtain the counit r!r*Xf -! Xf where r :'*E( ; ) -! *. Lemma 15.4.4. If P is a -free -space and X is any ex- -space or -spectrum over F , then P x X is a -free ex- -space or -spectrum over P x F . A useful slogan asserts that " -free -spectra live in the -trivial univers* *e". To explain it, consider the inclusion i: q*VG -! V of the complete G-universe * *VG as the universe of -trivial representations in the complete -universe V . Th* *en the slogan is given meaning by the following result. In the nonparametrized case F = *, it is proven in [59, IIx2] and is discussed further in [61, VIx4]. Since* * the parametrized case presents no complications and the proof is quite easy, we only give a sketch. Proposition 15.4.5. The change of universe adjunction (i*, i*) descends to a symmetric monoidal equivalence between the homotopy categories of -free - spectra over F indexed on -trivial representations of on the one hand and in* *dexed on all representations of on the other. For -free -spectra X over F indexed* * on V , there is a natural equivalence i*(E( ; )+ ^ i*X) ' X. Sketch Proof. If \ = e, then the quotient map q : -! G maps isomorphically onto a subgroup of G. Any representation V of is therefore of the form q*W for a representation W of q( ). It follows that the restrictions * *to of the universes V and q*VG have the same representations. This makes clear that, on -free -spectra over F , the unit and counit of the adjunction (i*, i* **) must be F ( ; )-equivalences, in the sense that they are -equivalences for any in F ( ; ). Smashing the unit and counit with E( ; )+ , which has trivial fix* *ed point sets for subgroups not in F ( ; ), we obtain natural equivalences, and it follows from Definition 15.4.1 that the unit and counit are themselves equivale* *nces when applied to -free -spectra. Alternatively, restricting to s-fibrant -spe* *ctra over F , the conclusion follows fiberwise from its nonparametrized precursor. S* *ince i* is symmetric monoidal, by Theorem 14.2.4, so is the equivalence. The last statement holds since i*(E( ; )+ ^ i*X) ' E( ; )+ ^ i*i*X ' X. 15.5. The fiberwise transfer for ( ; )-bundles We consider a fixed given principal ( ; )-bundle P , where is a normal su* *b- group of with quotient group G and quotient map q : -! G. We also consider a -space F and the associated ( ; )-bundle p: E = P x F -! P x * = B. We have the inclusion i: q*VG -! V of the complete G-universe VG as the univer* *se of -trivial representations in the complete -universe V . 210 15. FIBERWISE DUALITY AND TRANSFER MAPS The change of universe functor i*: SF -! SF -trivis not symmetric mon- oidal, and it does not preserve dualizable objects. For example, with F = * and = e, the orbit spectrum i* 1 = is not dualizable if is a non-trivial subg* *roup of . The bundle theoretic study of transfer maps is based on the following res* *ult, whose proof is based on the theory of -free -spectra given in the previous se* *ction. Theorem 15.5.1. The composite functor ~PFi*:Ho SF -! Ho GSE is sym- metric monoidal. Proof. Let ss :P x F -! F be the projection and note that ss*X = P x X. The functor ~PF is the composite of the symmetric monoidal Quillen left adjoint ss* and the Quillen left adjoint (-)= . By Theorem 15.3.3, the functor P~F on homotopy categories is also symmetric monoidal since the -space P is -free. We observe first that the composite ss*i* is symmetric monoidal. Indeed, for -spe* *ctra X and Y over F , we have ss*i*(X ^F Y')i*ss*(X ^F Y ) by Proposition 14.2.8 ' i*(ss*X ^PxF ss*Y )by Theorem 13.7.6 ' i*ss*X ^PxF i*ss*Y by Lemma 15.4.4 and Proposition 15.4.5 ' ss*i*X ^PxF ss*i*Y by Proposition 14.2.8. It follows directly that ~PFi*is symmetric monoidal: ~PFi*(X ^F Y )= (ss*i*(X ^F Y ))= by definition ' (ss*i*X ^PxF ss*i*Y )= by the previous display ' (ss*(i*X ^F i*Y ))= by Theorem 13.7.6 = ~PF(i*X ^F i*Y) by definition ' ~PFi*X^E ~PFi*Y by Theorem 15.3.3. Now Proposition 15.2.4(i) shows that ~PFi*commutes with trace maps. Theorem 15.5.2. Let X 2 Ho SF be dualizable. Then P~Fi*X 2 Ho GSE is dualizable. Suppose given a coaction map X : X ! X ^F CX and a self map OE: X -! X. Then o(P~Fi*OE) ' ~PFi*o(OE): SE -! ~PFi*CX, where ~PFi*X is given the coaction map ~PFi*( X ) : ~PFi*X -! ~PFi*(X ^F CX ) ' ~PFi*X ^E ~PFi*CX . These trace maps are maps of G-spectra over E, rather than over B. We can apply r!, r :E -! *, to obtain trace maps of nonparametrized spectra. This kind* * of trace map can be viewed as a fiberwise generalization of the kind of nonparamet* *rized trace map that is defined bundle theoretically in the literature. To connect up* * with the latter, we specialize and change our point of view so as to arrive at bundle theoretic trace maps over B. Specializing further to transfer maps, we obtain t* *he promised comparison with the transfer maps of Definition 15.2.9. With these goals in mind, we now focus on the case F = *, so that E above becomes B, with p the identity map, and our trace maps are parametrized over B. We study our original fixed given ( ; )-bundle p: E -! B in a different fashio* *n. We rename its fiber M to avoid confusion with respect to the role that space is 15.5. THE FIBERWISE TRANSFER FOR ( ; )-BUNDLES 211 playing. In the theory above, F was a base space for paramentrized spectra and there was no need for F to be dualizable. We now consider the case when M is stably dualizable, so that 1 M+ is dualizable, and we write oM for the transf* *er map S -! 1 M+ in S , as defined in and after Definition 15.2.1. We apply Theorem 15.5.2 with F = * and X = 1 M+ to obtain the following special case. Here we use the diagonal map induced by the diagonal map of M. Observe that, by Theorem 15.3.3, ~P*i* 1 M+' 1 ~P*M+ = 1B(E, p)+ . Theorem 15.5.3. Let M be a compact -ENR and let p: E -! B be a ( ; )- bundle with fiber M and associated principal ( ; )-bundle P . Let OE be a self* *-map of 1 M+ . Then o(P~*i*OE) ' ~P*i*(o(OE)) : SB ! 1B(E, p)+ . Therefore, taking OE = idand applying r!, r :B -! *, oE ' r!~P*i*oM : 1 B+ -! 1 E+ . This result gives a clear and precise comparison between the specialization to bundles of the globally defined transfer map for Hurewicz fibrations and the fiberwise transfer map for bundles. Effectively, we have inserted the transfer * *map for M+ fiberwise into P x (-) to obtain an alternative description of the tran* *sfer map for the dualizable G-spectrum 1 (E, p)+ over B. There is a useful reinterpretation of the description of transfer maps given by Theorem 15.5.3. Consider ss :P - ! *. Observe that, by Proposition 14.4.4, instead of applying r!, r :B -! *, to orbit spectra under the action of , we c* *ould first apply ss! and then pass to orbits. For a -spectrum X, we have a natural isomorphism ss!ss*i*X ~=P+ ^ i*X and a natural equivalence i*(P+ ^ i*X) ' P+ ^ X. Corollary 15.5.4. let M be a compact -ENR and let p: E -! B be a ( ; )- bundle with fiber M and associated principal ( ; )-bundle P . Then the transf* *er oE : 1 B+ -! 1 E+ is obtained by passage to orbits over from the map "o= id^ i*oM :P+ ^ i*S -! P+ ^ i* 1 M+ , and i*"ocan be identified with id^ oM :P+ ^ S -! P+ ^ 1 M+ . Remark 15.5.5. The corollary gives exactly the transfer map as defined by Lewis and May [59, IV.3.1]. Working in the nonparametrized context, they tried in vain to obtain a spectrum level transfer map for Hurewicz fibrations over ge* *neral base spaces. The comparison here also sheds light on the relationship between the two constructions of Becker and Gottlieb [2, 3], both of which require fini* *te dimensional base spaces. The first is bundle theoretic and is easily seen to be equivalent to the construction in this section by using Atiyah duality to inter* *pret oM for a -manifold M. Precisely, by [59, IV.2.3], if M is embedded in V with normal bundle and o is the tangent bundle of M, then the transfer map oM is homotopic to the map obtained by applying the functor -V 1 to the composite of the Pontryagin-Thom map SV -! T and the map T -! T ( o) ~=M+ ^ SV 212 15. FIBERWISE DUALITY AND TRANSFER MAPS induced by the inclusion -! o. The second, which is generalized to the equivariant setting by Waner [97], is fibration theoretic and is easily seen to* * be equivalent to the construction of x15.2. Another approach to the comparison is to show that suitable Hurewicz fibrations are equivalent to bundles, as is done* * by Casson and Gottlieb in [15]. Remark 15.5.6. Since our definition coincides with that of [59, IV.3.1], the properties of the transfer catalogued in [59, IVxx3-7] apply verbatim. Many of these properties generalize directly to the parametrized trace and transfer map* *s of Theorem 15.5.2. Actually, the definition of [59, IV.3.1] works more generally w* *ith P , or rather i* 1 P+ , replaced by a general -free -spectrum P indexed on VG* * . The constructions here admit similar generalizations. One way to achieve this w* *ith minimal work is to use the case P = E( ; ) of the construction already on hand. Thus, for a -free -spectrum P over F indexed on VG , we can define ~PFi*X = _______E(F;(P)^F i*X) and develop parametrized trace and transfer maps from there. We leave the furth* *er development of the theory to the interested reader. 15.6. Sketch proofs of the compatible triangulation axioms We must explain why Ho GSB is a closed symmetric monoidal category with a compatible triangulation, in the sense specified in [74]. We have the closed symmetric monoidal structure and the triangulation, the latter by Lemma 13.1.5. We must prove the compatibility axioms (TC1)-(TC5) of [74, x4]. The essential idea is to verify the axioms using external smash products and function objects and then pull back along diagonal maps to obtain the conclusions. The axiom (TC1) only involves suspension, in our case B , and is thus easily checked usi* *ng Proposition 12.6.4. For (TC2), we must show that the functors X^B (-), FB (X, -* *), and FB (-, Y ) preserve distinquished triangles, where X and Y are G-spectra ov* *er B. Either model theoretically or by standard topological arguments with cofiber sequences and fiber sequences, it is easy to see that these conclusions hold wi* *th ^B and FB replaced by the external functors Z and ~F. Since * and therefore its right adjoint * are exact, the conclusion internalizes directly. Similarly, th* *e braid axiom (TC3) and additivity axiom (TC4) hold for Z by the arguments explained in [74, x6], and they pull back along * to give these axioms internally in Ho GSB* * . The braid duality axiom (TC5) is more subtle because it involves simultaneous use of ^B and FB . Externally, we can work over B x B, using Z. Inspecting the argument in [74, x7], we see that the only internal homs used in the verificati* *on of the braid duality axiom are duals of the form F (-, T ) for a suitable approxim* *ation T of the unit object. In our context, it turns out that we need to use two anal* *ogues of this functor, one to mimic the proof of (TC5a) given in [74, pp 62-64] and another to mimic the proof of (TC5b) given in [74, pp 65-67]. For the first, l* *et T 2 GSBxB be a fibrant model of the derived *SB , so that ~F(X, T ) is a model for DX = FB (X, SB ) in Ho GSB . With this replacement for F (X, T ), the cited proof of (TC5a) goes through, first working externally and then internalizing a* *long *. The cited proof of (TC5b) relies on a natural point-set level map (15.6.1) F (X, T ) ^ F (Y, T ) -! F (X ^ Y, T ), 15.6. SKETCH PROOFS OF THE COMPATIBLE TRIANGULATION AXIOMS 213 and this makes no sense in our external context. Working internally, in Ho GSB , we have such a map (15.6.2) FB (X, SB ) ^B F (Y, SB ) -! FB (X ^B Y, SB ), but we need a point-set level external model for it to carry out the cited argu* *ment. Let U be a fibrant model for *SBxB in GSBxBxBxB . Replacing the functor D0(-) = F (X, T ) used in [74, pp. 66-67] with the functor D0(-) = ~F(-, U): GSBxB -! GSBxB , we find that the cited argument goes through verbatim on the external level, wo* *rk- ing in the category GSBxB , once we construct a natural map (15.6.3) ~F(X, T ) Z ~F(Y, T ) -! ~F(X Z Y, U) in GSBxB to substitute for the pairing (15.6.1). Starting from the (Z, ~F) adj* *unc- tion, we obtain an external pairing (15.6.4) ~F(X, T ) Z ~F(Y, T ) -! ~F(X Z Y, T Z T ). We also have the natural map *X Z *Y j|| fflffl| * *( *X Z *Y ) ~=|| fflffl| *( * *X Z * *Y ) |*("Z")| fflffl| *(X Z Y ). Applying this with X = Y = SB and using that SB Z SB is isomorphic to SBxB , we obtain a lift , in the diagram *SB Z *SB _____// *(SB Z SB ) ~= *SBxB______//U22eeeeeeee eeeee | ,eeeeeeee | | eeeeeeee | fflffl|eeeeeeeeee fflffl| T Z T ___________________________________//_*BxB Composing ~F(X Z Y, ,) with the pairing (15.6.4), we obtain the required pairing (15.6.3). Internalization along * is then a not altogether trivial exercise w* *hich shows that, on passage to homotopy categories, application of * to the pairing (15.6.3) gives a model for the pairing (15.6.2). The latter pairing can be view* *ed as a map *(F~(X, T ) Z ~F(Y, T )) -! ~F( *(X Z Y ), T ), and the essential point of the exercise is to verify that *F~(X Z Y, U) is equ* *ivalent to ~F( *(XZY ), T ). Using that *SBxB ~=SB and looking at represented functor* *s, we see that a Yoneda lemma argument reduces the verification to the proof of a derived analogue of (11.4.5) that is proven in the same way as Theorem 13.7.6. CHAPTER 16 The Wirthm"uller and Adams isomorphisms Introduction This chapter consists of variations on a theme. For a G-map f :A -! B, the base change functor f* from G-spectra over B to G-spectra over A has a left adj* *oint f!and a right adjoint f*. We study comparisons between f!and f*. As preamble, we show in x16.1 that there is always a natural map OE: f!- ! f* that relates t* *he two adjunctions. It is an equivalence when f is a homotopy equivalence, but not* * in general. This comparison is largely formal and applies to analogous sheaf theor* *etic contexts. In the rest of the chapter, we use our foundations together with formal ar- guments developed in [40] to obtain a simple proof of a general version of the Wirthm"uller isomorphism and to reprove the Adams isomorphism as a special case. This material constitutes a considerably simplified version of work of Po Hu on* * the same topic [47]. We consider G-bundles p: E -! B, as in x3.2 and x15.3. We assume that the fiber M is a smooth closed -manifold; manifolds with boundary work similarly. The generalized Wirthm"uller isomorphism computes the relatively mysterious right adjoint p* of the functor p* as a suitable shift of the relati* *vely familiar left adjoint p!. We explain the result in the special case when E -! B is M -! * in x16.2, but we defer the proof to x16.5. We also show how to relate the Wirthm"uller isomorphisms for M and N when N is smoothly embedded in M. When M = G=H, the result specializes under the equivalence between the category of G-spectra * *over G=H and the category of H-spectra to the Wirthm"uller isomorphism in the form proven by Lewis and May [59, IIx6]. As explained in [75], the categorical analy* *sis in [40] allows considerable simplification of that proof. Our proof for general* * M follows the same pattern, but it is quite different in detail since the special* * case M = G=H has certain simplifying features. For example, when G is finite, that case follows formally from Atiyah duality for G=H and the trivial observation t* *hat H=K+ is an H-retract of G=K+ for K H G. In x16.3, we show that the general case of G-bundles p: E - ! B reduces fiberwise to the special case M -! *. The proof is an immediate application of * *the construction P x (-) on parametrized -spectra that was studied in x15.3. This allows a simple fiberwise construction of the G-spectrum over E by which one mu* *st shift p!to obtain the desired isomorphism. With this construction, it is immedi* *ate that the map of G-spectra over B that we wish to prove to be an equivalence coincides on the fiber over b with a map that we know to be an equivalence by t* *he case M -! *. Since equivalences are detected fiberwise, that proves the result. In turn, we prove in x16.4 that the Adams isomorphism relating orbit spectra and fixed point spectra that was proven by Lewis and May in [59, IIx8] is a vir* *tually 214 16.1. A NATURAL COMPARISON MAP f!-! f* 215 immediate special case of our generalized Wirthm"uller isomorphism. These resul* *ts complete the program originated in [61] of reproving conceptually all of the ba* *sic foundational results that were first proven in a less satisfactory ad hoc way i* *n [59]. The pioneering work of Po Hu [47] paved the way but, in the absence of adequate foundations, the bundle construction of x15.3, and the simplifying framework of [40], her arguments were very long and difficult. Our work recovers variant ver* *sions of all of her results. The basic idea that parametrized G-spectra should clarif* *y and simplify the Wirthm"uller and Adams isomorphisms is due to Gaunce Lewis [58]. Again, we assume throughout that all given groups G are compact Lie groups and all given base G-spaces are of the homotopy types of G-CW complexes. 16.1.A natural comparison map f!-! f* The Wirthm"uller isomorphism that is the subject of the next few sections gi* *ves an equivalence between f* and a shift of f! for certain equivariant bundles f. * *In the course of our work on that, we came upon a curious natural comparison map f!-! f* for any map f whatever. We have no current applications for it, but sin* *ce the relationships among base change functors are so central to the theory and i* *ts applications, we shall describe that map in this digressive section. It works j* *ust as well on the level of ex-spaces and indeed quite generally in other contexts whe* *re one has analogous base change adjunctions. Theorem 16.1.1. Let f :A -! B be a G-map and let X be a G-spectrum over A. Let ": f*f* -! Idand oe :f!f* -! Iddenote the counits of the adjunc- tions (f*, f*) and (f!, f*) relating Ho GSA and Ho GSB . There is a natural map OE: f!X -! f*X in Ho GSB such that the following diagram commutes: (16.1.2) f!f*f*XJ f!"uuuu JJJoeJ uu JJJ zzuuu $$J f!X_________OE_______//f*X. Proof. Let K = (K, p, s) be an ex-space over A. Then f!K = K [A B and f*f!K = f!K xB A. Here points (f(a), a) in B xB A are identified with points (s(a), a) in K xB A, and we see that f*f!K can be identified with the pullback K xB A. The projection to K is then a map _ :f*f!K -! K of ex-spaces over A. When K = f*L for an ex-space L over B, _ = f*oe :f*f!f*L -! f*L since f*oe is also given by the projection f*L xB A -! f*L. Passing to spectra over A levelwise, we obtain a natural map _ :f*f!X -! X of spectra over A such that _ = f*oe when X = f*Y . To pass to homotopy categories, we take two steps. Factoring f as a composite of a homotopy equivalence and an h-fibration, we see that we may assume that f is either a homotopy equivalence or an h-fibration. In the former case, f* must be inverse to the equivalence f* and thus equivalent to f!. Here " and oe are equivalences and we may as well define OE by the commutativity of (16.1.2). In * *the latter case, we may work in GEA . Since f is an h-fibration, we have a natural homotopy equivalence ~: T f*Y -! f*T Y for Y 2 GEB . The derived functor f!is induced by T f!, and T X is naturally homotopy equivalent to X when X 2 GEA . The composite f*T f!X ' T f*f!X_T__//T X ' X 216 16. THE WIRTHM"ULLER AND ADAMS ISOMORPHISMS gives a natural map _ :f*f!X -! X in hGEA . When X = f*Y , we have the commutative naturality diagram *oe T f*f!f*Y__T_=Tf____//T f*Y ~|| ~|| fflffl| fflffl| f*T f!f*Y___f*Toe__//_f*T Y. The bottom arrow is the derived version of f*oe and the composite around the top is the derived version of _. Using the equivalences of categories of x13.5, we obtain a natural map _ :f*f!X -! X in Ho GSA . Let j :Id- ! f*f* be the unit of the (derived) adjunction (f*, f*) and define OE: f!X - ! f*X to be the adjoint of _ in Ho GSA , so that OE = f*_ O j. For Y 2 Ho GSB , we have f*oe = _ :f*f!f*Y -! f*Y . It follows formally that (16.1.2) commutes. Indeed, " O f*oe = " O _ = _ O f*f!". The adjoint of " O f*oe is oe since f*(" O f*oe) O j = f*" O f*f*oe O j = f*" O j O oe = oe, while the adjoint of _ O f*f!" is OE O f!" since f*(_ O f*f!") O j = f*_ O f*f*f!" O j = f*_ O j O f!" = OE O f!". 16.2. The Wirthm"uller isomorphism for manifolds The classical Wirthm"uller isomorphism in the equivariant stable homotopy ca* *t- egory relates induction and coinduction, the left and right adjoints of the res* *triction functor from G-spectra to H-spectra. More precisely, it says that for H-spectra* * X, there is a natural equivalence of G-spectra (16.2.1) FH (G+ , X) ' G+ ^H (X ^ S-L ), where L is the tangent representation at the identity coset in G=H and S-L is the inverse of the invertible H-spectrum 1 SL . Here again, "equivalence" means isomorphism in the relevant stable homotopy category and is denoted by '. One can also think of this in terms of base change functors. Recall from Cor* *ol- lary 11.5.4 that the category of H-spectra is equivalent to the category of G-s* *pectra over G=H. The equivalence is given in one direction by applying the functor GxH* * -, and in the other by taking the fiber over the identity coset. This equivalence * *pre- serves all structure in sight, including the symmetric monoidal and model struc- tures. The map r :G=H -! * induces a pullback functor r* from G-spectra to G-spectra over G=H, and it has left and right adjoints r! and r*. The functor r* corresponds under the equivalence to the restriction functor and therefore r! a* *nd r* correspond to the induction and coinduction functors. In this terminology, t* *he Wirthm"uller isomorphism (16.2.1) takes the form (16.2.2) r*X ' r!(X ^G=H CG=H ) for G-spectra X over G=H, where CG=H = '!S-L , ' : H G (see Proposi- tion 11.5.2). We think of G=H -! * as the simplest kind of a bundle with a compact man- ifold as fiber, and we generalize (16.2.2) to maps p: E -! B that are equivaria* *nt bundles with a smooth closed manifold M as a fiber. We discuss the case B = * 16.2. THE WIRTHM"ULLER ISOMORPHISM FOR MANIFOLDS 217 in this section and prove the general case in the next. However, it is convenie* *nt to begin by describing the form of the map that gives the equivalence in general. * *For that, we require a G-spectrum Cp over E together with an equivalence (16.2.3) ffp: p!Cp_'__//D(p!SE ) that identifies the dual of p!SE . We call Cp, together with ffp, a Wirthm"ulle* *r object. In [40], Fausk, Hu, and May give a categorical discussion of equivalences of Wirthm"uller type, including a simplifying formal analysis that describes the m* *in- imal amount of information that is needed to prove such a result. In particula* *r, given a Wirthm"uller object Cp, they define a canonical candidate !p: p*X -! p!(X ^E Cp) for an equivalence, namely the composite displayed in the commutative diagram id^BD(oe) (16.2.4) p*X ' p*X ^B D(SB ) __________//_p*X ^B D(p!SE ) | OO | ' |id^Bffp | || !p || p*X ^B p!Cp | OO | | | |' fflffl| | p!(X ^E Cp)oo___p_________ p!(p*p*X ^E Cp). !("^Eid) The maps oe :p!SE ' p!p*SB -! SB and ": p*p*X -! X are given by the counits of the adjunctions (p!, p*) and (p*, p*). The arrow labelled ' is an equivalen* *ce given by the derived version of the projection formula (11.4.5) that is proven * *in Theorem 13.7.6. When M is a smooth closed G-manifold and r is the map :M -! *, we write CM for a Wirthm"uller object Cr and we write !M for !r. It is easy to describe * *CM . Let o be the tangent G-bundle of M. Embed M in a G-representation V and let be the normal G-bundle of the embedding. By Atiyah duality, the union M+ of M and a disjoint basepoint is V -dual to the Thom G-space T . A detailed equivar* *iant proof is given in [59, IIIx5], but we require little beyond the mere statement. For a G-vector bundle , over a G-space B, let S, denote the fiberwise one-po* *int compactification of ,, with section given by the points at infinity. This ex-G-* *space over B must not be confused with the Thom complex T ,. The latter is obtained by identifying the section to a point and is precisely r!S,, r :B -! *. Definition 16.2.5. Define CM to be the G-spectrum -VM 1MS over M. Remark 16.2.6. By Theorem 15.1.5, the suspension G-spectrum 1MSo is in- vertible. Visibly, CM is its inverse. Lemma 16.2.7. There is an equivalence ffM :r!CM - ! D(r!SM ), r :M -! *. Proof. Since SM (V ) = MxSV , r!SM = 1 M+ . Since r!S = T and r!com- mutes with shift desuspension functors, r!CM is equivalent to -V 1 T . Ther* *e is a canonical evaluation map ev:T ^M+ -! SV of a duality [59, p.152]. Explicitl* *y, using the diagonal of M and the zero section of we obtain an embedding of M in x M with trivial normal bundle M x V , and evis composite of the Pontryagin- Thom map associated to this embedding and the projection M+ ^ SV -! SV . We 218 16. THE WIRTHM"ULLER AND ADAMS ISOMORPHISMS apply the functor -V 1 to obtain -V 1 T ^ 1 M+ ' -V 1 (T ^ M+ ) -! -V 1 SV ' S. Atiyah duality states that the adjoint of this map is an equivalence from -V * *1 T to D(M+ ). This is the required map ffM . We shall prove the following result in x16.5. Theorem 16.2.8 (The Wirthm"uller isomorphism for manifolds). For G-spectra X over M and r :M -! *, the map !M :r*X -! r!(X ^M CM ) is a natural equivalence in the homotopy category Ho GS of G-spectra. In an earlier draft of this paper, we thought we could reduce the general ca* *se of Theorem 16.2.8 to the special case M = G=H. However, instead of leading to a simplifiction, the argument we had in mind leads to an interesting relative ver* *sion of the Wirthm"uller isomorphism. Its starting point is the following observatio* *n. Lemma 16.2.9. Let i: N -! M be an embedding of smooth closed G-manifolds and let M,N be the normal bundle of i. Then CN is equivalent to S M,N ^N i*CM . Proof. An embedding of M in a representation V restricts along i to an embedding of N in V , and i* M M,N ~= N . Commutation relations in Proposi- tion 13.7.4 give that i* -VM 1MS M ' -VN 1Ni*S M . The conclusion follows after smashing with S M,N. Corollary 16.2.10 (The relative Wirthm"uller isomorphism). Let i: N -! M be a smooth embedding of closed G-manifolds. For G-spectra X over N, there is a natural equivalence !M,N :r*i*X -! r*i!(X ^N S M,N). Proof. Here r :M -! *. Write q = r O i: N -! *. Then q* ' r*i* and q!' r!i!. Define !M,N by commutativity of the diagram of equivalences q*X _________!N______//q!(X ^N CN ) ' || '|| fflffl| fflffl| r*i*X r!i!(X ^N S M,N ^N i*CM ) !M,N || '|| fflffl| fflffl| r*i!(X ^N S M,N)_!M__//r!(i!(X ^N S M,N) ^M CM ). Here the derived version of the projection formula (11.4.5) gives the lower rig* *ht equivalence. We explain the strategy that we have not implemented for deducing the Wirth- m"uller isomorphism for M from the Wirthm"uller isomorphism for orbits. 16.3. THE FIBERWISE WIRTHM"ULLER ISOMORPHISM 219 Remark 16.2.11. One can use relative Atiyah duality to define an intrinsic map ffM,N and use ffM,N to define a map !M,N directly. One can then obtain the displayed diagram by a chase. If one could prove directly that !M,N was an equivalence, then, using the invertibility of 1NS M,N, one could deduce that !M is an equivalence on all i!X if !N is an equivalence. By Proposition 13.1.3, * *!M is an isomorphism for all Y if it is an isomorphism for all Y in the detecting * *set DM of Definition 13.1.1. Those Y , namely the Sn,bHfor b 2 M and H Gb, are of the form "b!X, where "b:G=Gb -! M is the inclusion of the orbit of b and X is a G-spectrum over G=Gb. Thus the Wirthm"uller isomorphism for orbits would imply the Wirthm"uller isomorphism for M. 16.3. The fiberwise Wirthm"uller isomorphism As in x15.3, let G be a quotient = , where is a normal subgroup of a compact Lie group . Let M be a smooth closed -manifold and let p: E -! B be a ( ; )-bundle with fiber M. This means that p has an associated principal ( ;* * )- bundle ss :P -! B and p is the associated G-bundle E = P x M -! P= = B. We apply the functor ~PM to the Wirthm"uller object CM to obtain the Wirthm"ull* *er object Cp, and we apply ~PM to ffM to obtain the required equivalence ffp. This means that the Wirthm"uller object for p is obtained by inserting the Wirthm"ul* *ler object for M fiberwise into the functor P x (-). Definition 16.3.1. Define Cp to be the G-spectrum ~PMi*CM over E, where i* is the change of universe functor associated to the inclusion of the -trivi* *al -universe in the complete -universe. Remark 16.3.2. Recall Remark 16.2.6. By Theorem 15.1.5, the suspension G-spectrum 1E(P x So) is invertible. The Wirthm"uller object Cp is its invers* *e. Lemma 16.3.3. There is an equivalence ffp: p!Cp -! D(p!SE ). Proof. We define ffp to be the composite (16.3.4) ~P*i*ffM p!~PMi*CM_____//~P*i*r!CM_________//_~P*i*D(r!SM_)_//_D(p!SE ). The left arrow is given by the first equivalence of Proposition 15.3.4 and the * *last equivalence of Proposition 14.2.8. The middle arrow is an equivalence since ffM* * is one. The right arrow is the following composite equivalence, ~P*i*D(r!SM )' ~P*D(i*r!SM ) ' ~P*D(r!i*SM) by Propositions 14.2.8 and 15.4.5 ' D(P~*r!i*SM) by Theorem 15.3.3 ' D(p!~PMi*SM) by Proposition 15.3.4 ' D(p!SE ) by Theorem 15.3.3. For the first displayed equivalence, r!SM ' 1 M+ by Lemma 15.2.8, hence D(r!SM ) ' D( 1 M+ ) ' F (M+ , S). For based -spaces T , i*F (T, S) ~= F (T, i*S) by inspection. If T is a based* * - CW complex, this is an isomorphism of Quillen right adjoints and so descends to 220 16. THE WIRTHM"ULLER AND ADAMS ISOMORPHISMS homotopy categories. Again by inspection, i* 1 ~= 1 :GK* -! GS -triv. This isomorphism passes to homotopy categories since both sides take q-equiva- lences to level q-equivalences. Therefore i*S ' S and F (T, i*S) ' D(i* 1 T ). Theorem 16.3.5 (The fiberwise Wirthm"uller isomorphism). For G-spectra X over E, the map !p: p*X -! p!(X ^E Cp) is a natural equivalence of G-spectra over B. Proof. The action of Gb on the fiber Eb ~=ae*bM of b 2 B is smooth, hence the Wirthm"uller isomorphism for manifolds gives the result for r :Eb -! *. We claim that the restriction ffb of ffp to the fiber over b is an equivalence ffEb: r!CEb -! D(r!SEb) of Gb-spectra of the form used to prove Theorem 16.2.8 for r. Indeed, with pb =* * r, i: Eb E and ': Gb G, the derived version of Example 11.5.6 and Lemma 15.3.2 give that the source of ffb is (p!~PMCM )b ' r!i*'*P~MCM ' r!ae*bCM ~= r!CEb. For the last isomorphism we must view the representation V of that appears in the definition of CM as a representation of Gb by pullback along aeb. Similarly* *, using Theorem 13.7.10 and the derived version of Example 11.5.6, the target of ffb is D(p!SG,E)b ' D((p!SG,E)b) ' D(r!i*'*SG,E) ' D(r!SGb,Eb). In view of the role of ffM in the definition of ffp, diagram chases from the de* *finitions show that ffb agrees under these equivalences with the Gb-equivalence ffEb. Now, looking at the definition of !p (16.2.4), we see that, aside from the e* *quiv- alence ffp, its constituent maps are just counits of adjunctions and derived is* *omor- phisms coming from the closed symmetric monoidal structures. By Theorem 13.7.10 and the derived versions of commutation relations in Example 11.5.5, these maps restrict on fibers to maps of the same form. Therefore the restriction !b:(p*X)b -! (p!(X ^E Cp))b of !p to the fiber over b can be identified with the map of Gb-spectra !Eb: r*Xb -! r!(Xb^Eb CEb). This map is an equivalence of Gb-spectra by Theorem 16.2.8. Since equivalences * *of G-spectra over B are detected fiberwise, this implies that !p is an equivalence. Remark 16.3.6. When = G x and only acts on M, one can think of p: E -! B as a topological G-bundle with a reduction of its structural group to a suitably large compact subgroup of the group of diffeomorphisms of M. Our fiberwise Wirthm"uller isomorphism theorem is a variant of the main theorem, [4* *7, 4.8], of a paper of Po Hu. She worked with Diff(M) itself as an implicit struct* *ure group, without use of an auxiliary group and without an ambient group . That bundle theoretic framework leads to formidable complications, hence her argumen* *ts are very much more difficult than ours. Her result is both more and less general than the specialization of ours to the case = G x : it allows bundles that m* *ight not admit a single compact structure group , but it requires the base spaces to 16.4. THE ADAMS ISOMORPHISM 221 be G-CW complexes with countably many cells. It does not handle more general group extensions. 16.4.The Adams isomorphism Let N be a normal subgroup of G and let ffl: G -! J be the quotient by N. The conjugation action of G on N induces an action of G on the tangent space of N at the identity element, giving us the adjoint representation A = A(N; G). Let (i** *, i*) be the change of universe adjunction associated to the inclusion i: q*VJ -! VG of the complete J-universe VJ as the universe of N-trivial representations in t* *he complete G-universe VG . Recall the discussion of N-free G-spectra from x15.4, where and played the roles of N and G. Theorem 16.4.1 (Adams isomorphism). For N-free G-spectra X in GS N-triv, there is a natural equivalence X=N ' (i* -A i*X)N in Ho JS N-triv. We shall derive this by applying the fiberwise Wirthm"uller isomorphism to t* *he quotient G-map p: E(N; G) -! B(N; G), where E(N; G) is the universal N-free G-space and B(N; G) = E(N; G)=N. To place ourselves in the bundle theoretic context of the previous section, we give another description of p, following [6* *8, IIx7]. It is formal and would similarly identify p: E -! E=N for any N-free G-space E. Let = GnN be the semi-direct product of G and N, where G acts by conjugation on N. Write for the normal subgroup {e} n N of . We then have an extension 1 -! -! -`!G -! 1, where `(g, n) = gn. Give N the -action (g, n) . m = gnmg-1. Then N ~= =G as -spaces, where we view G as the subgroup G n {e} of . The composite E(N; G) ~=`*E(N; G) x ( =G) -! `*E(N; G) x * ~=B(N; G) induced by =G -! * is p. Since `*E(N; G) is a -free -space, we see that p is a bundle with fiber =G ~=N to which the fiberwise Wirthm"uller isomorphism applies. We must identify the relevant Wirthm"uller object. We write r for the * *map E(N; G) -! *. Proposition 16.4.2. The Wirthm"uller object Cp is r*S-A . Proof. The tangent bundle of =G ~= N is the trivial bundle N x A [59, p. 99]. Indeed, let act on A via the projection ffl: -! G, "(n, g) = g. We obtain a -trivialization of the tangent bundle of =G by sending (n, a) 2 N x A to deLn(a), where deLn is the differential at e of left translation by n. It fo* *llows that the tangent bundle along the fibers of p is also trivial: `*E(N; G) xN ( =G x A) ~=(`*E(N; G) xN =G)) x A ~=E(N; G) x A. Thus the spherical bundle of tangents along the fiber is E(N; G) x SA = r*SA , and the inverse of its suspension G-spectrum over E(N; G) is r*S-A . In view of Remark 16.3.2, this gives the conclusion. 222 16. THE WIRTHM"ULLER AND ADAMS ISOMORPHISMS Proof of the Adams isomorphism. Let X 2 GS N-trivbe N-free. Apply- ing the fiberwise Wirthm"uller isomorphism to the G-spectrum r*i*X over E(N; G) and using that Cp is r*S-A , we obtain a natural equivalence p*r*i*X ' p!(r*i*X ^E(N;G)r*S-A ) of G-spectra over B(N; G). Write ~rfor the map B(N; G) -! *, so that ~rO p = r. Applying the functor ~r!((i*(-))N ) to the displayed equivalence, we obtain a n* *atural equivalence ~r!((i*p*r*i*X)N ) ' ~r!((i*p!(r*i*X ^E(N;G)r*S-A ))N ) in Ho JS N-triv. We proceed to identify both sides. The source is ~r!((i*p*r*i*X)N')~r!((p*r*i*i*X)N ) by Proposition 14.2.8 ' ~r!((p*r*X)N ) by Proposition 15.4.5 ' ~r!((p!r*X)=N) by Proposition 14.4.5 ' (~r!p!r*X)=N by Proposition 14.4.4 ' (r!r*X)=N by functoriality ' X=N. by Definition 15.4.1. The target is ~r!((i*p!(r*i*X^E(N;G)r*S-A ))N ) ' ~r!((i*p!r* -A i*X)N )by Theorem 13.7.3 ' (~r!i*p!r* -A i*X)N by Proposition 14.4.4 ' (i*~r!p!r* -A i*X)N by Propositions 14.2.8 and 15.4.5 ' (i*r!r* -A i*X)N by functoriality ' (i* -A i*X)N by Definition 15.4.1. Remark 16.4.3. In outline, the proof just given is essentially that indicated by Po Hu [47, pp 81-99]. However, her argument, although more conceptual, is a good deal longer and more complicated than the original proof in [59, pp 96-102* *]. 16.5. Proof of the Wirthm"uller isomorphism for manifolds We prove Theorem 16.2.8 here. Thus consider r :M -! * for a smooth com- pact G-manifold M. With CM = -VM 1MS , the diagram (16.2.4) displays a canonical map ! = !M :r*(X) -! r!(X ^M CM ) of G-spectra, where X is a G-spectrum over M. We must show that ! is an equiva- lence. In outline, we follow the pattern of proof explained in [40] and illustr* *ated in the case M = G=H in [75], but the details are very different from those applica* *ble in that special case. We first describe a formal reduction implied by the results of [40]. Consider the set DM of detecting objects in Ho GSM that is specified in Definition 13.* *1.1. The objects in DM are compact, by Lemma 13.1.2, and dualizable. We have the analogous detecting set D* of compact objects in Ho GS . For Y in D*, r*Y is dualizable and it follows formally, by [45, 2.1.3(d)], that r*Y is compact (in * *the sense of Lemma 13.1.2). Therefore r*, as well as r!, preserves coproducts [40, * *7.4]. 16.5. PROOF OF THE WIRTHM"ULLER ISOMORPHISM FOR MANIFOLDS 223 This verifies the hypotheses of the formal Wirthm"uller isomorphism theorem, [4* *0, 8.1], and that result shows that ! will be an equivalence for all G-spectra X o* *ver M if it is an equivalence for those X in DM . Such X are of the form Sn,mH= em!'!SnH, where n 2 Z, m 2 M, H Gm , and ' is the inclusion of Gm in G. By commutation with suspension, we can assume that n 0. Then X is of the form 1MK for an ex-G-space K over M, and X can be any such G-spectrum over M in the rest of the proof. By [40, 6.3], it suffices * *to construct a map ,X: r*r!(X ^M CM ) -! X such that certain diagrams commute. To be precise, let oe and i be the counit and unit of the (r!, r*) adjunction, * *note that r*S ~=SM , and define maps o = oS and , = ,SM by commutativity of the diagrams (16.5.1) S ___o__//_r!CM and r*r!CM _________,_________//_SM ' || |ffM| r*ffM|| |'| fflffl| fflffl| fflffl| fflffl| DS _Doe//_Dr!r*S r*Dr!SM __'__//Dr*r!SM_Di_//_DSM Then define oY for a general G-spectrum Y to be the composite (16.5.2) oY :Y ' Y ^ S id^o//_Y ^ r!CM ' r!(r*Y ^M CM ) and define ,r*Y for the G-spectrum r*Y over M to be the composite (16.5.3) ,r*Y: r*r!(r*Y ^M CM ) ' r*Y ^M r*r!CMid^,//_r*Y ^M SM ' r*Y. Here the equivalences are given by the derived versions of (11.4.2) and the pro* *jection formula (11.4.5). With these notations, we shall prove the following result. Proposition 16.5.4. For X = 1MK, there is a map ,X :r*r!(X ^M CM ) -! X such that the composite (16.5.5) r!(X ^M CM ) or!(X^M|CM|) fflffl| r!(r*r!(X ^M CM ) ^M CM ) r!(,X^M|id)| fflffl| r!(X ^M CM ) is the identity map (in Ho GS ) and, for any map ` :r*Y -! X of G-spectra over M, the following diagram commutes in Ho GSM . ,r*Y (16.5.6) r*r!(r*Y ^M CM )____//r*Y r*r!(`^id)|| |`| fflffl| fflffl| r*r!(X ^M CM )__,X___//X This will complete the proof of the theorem by the cited reduction from [40]. 224 16. THE WIRTHM"ULLER AND ADAMS ISOMORPHISMS Corollary 16.5.7. For X in DM , !M :r*(X) -! r!(X ^M CM ) is an equiv- alence with inverse the adjoint of ,X . Proof. Taking Y to be r*X and ` to be the counit of the (r*, r*) adjunction in (16.5.6), the conclusion is a direct application of [40, 6.3]. Thus it suffices to prove Proposition 16.5.4. We shall construct the map ,X * *and prove that it satisfies the stated properties by reducing to space level consid* *erations. We begin with a space level description of the maps o and , displayed in (16.5.* *1), and we need some space level notations. Notations 16.5.8. Recall that denotes the normal bundle of M and that we have the duality map ev:T ^ M+ -! SV specified in the proof of Lemma 16.2.7. Also, recall that r!K = K=s(M), r*T = TM = M x T, and r*K = Sec(M, K) for any based G-space T and any ex-G-space (K, p, s) over M. In particular, r!S = T , r!S0M= M+ , and r*r*T ~=F (M+ , T ). Therefore the adjoint eev:T -! F (M+ , SV ) is a map r!S - ! r*SVM. Let t: SV -! r!S be the Pontryagin-Thom construction and k be the composite *"ev " k :r*r!S_r___//r*r*SVM___//_SVM, where ": r*r* -! idis the counit of the adjunction (r*, r*); note that, in gene* *ral, " is just the evaluation map M x Sec(M, K) -! K. Recall from Propositions 13.7.4 and 13.7.5 that we can commute suspension spectrum functors past smash products and base change functors. Lemma 16.5.9. With these definitions of t and k, o ' -V 1 t: S ~= -V 1 SV -! -V 1 r!S ' r!CM and , ' -VM 1Mk :r*r!CM ' -VM 1Mr*r!S -! -VM 1MSVM' SM . Proof. By [59, III.5.2], the dual of t is the projection ffi :M+ -! S0. This means that the following diagram is stably homotopy commutative. SV ^ M+ _t^id//_T ^ M+ id^ffi|| |ev| fflffl| fflffl| SV ^ S0 _________SV Here ffi :M+ = r!r*S0 -! S0 is the counit of the space level adjunction (r!, r** *), and we can identify 1 ffi with the counit oe : 1 M+ ~=r!r*S -! S of the spectr* *um level adjunction (r!, r*). Applying -V 1 to the diagram and passing to adjoin* *ts, the right vertical arrow becomes ffM :r!CM ' -V 1 T -! D(M+ ) = Dr!r*S, by the proof of Lemma 16.2.7. Comparing the resulting diagram with the diagram that defines o, we conclude that o ' -V 1 t. For the identification of ,, we consider the composite equivalence r*r!CM ' Dr*r!SM in the diagram that defines , to be an identification of the dual of r** *r!SM . 16.5. PROOF OF THE WIRTHM"ULLER ISOMORPHISM FOR MANIFOLDS 225 To identify the dual of i modulo that identification, we observe that the follo* *wing diagram is commutative. r*r!S ^M S0M___________k^id__________//_SVM^M S0M id^i|| |||| fflffl| || r*r!S ^M r*r!S0M~=_//_r*(r!S ^ r!S0M)r*ev//_r*SV Indeed, recalling that k = " O r*e"vand rewriting the diagram in more familiar notation, it becomes M x T __idxe"v//_M x Sec(M, M x SV ) idxi|| |"| fflffl| fflffl| M x (T ^ M+ )___idxev__//_M x SV , and both composites send (m, x) to (m, ev(x ^ m)). Applying -V 1 to the first diagram and comparing with the definition of ,, we conclude that , ' -VM 1Mk. The following space level result will imply Proposition 16.5.4. Proposition 16.5.10. Let K be a G-space over M. There is a natural map uK :r*r!(K ^M S ) -! VMK in Ho GKM which satisfies the following properties. (i)When K = S0M, uK ' k :r*r!S -! SVM. (ii)For a based G-space T , the following diagram commutes in Ho GKM . r*r!(TM ^M K ^M S ) __'__//TM ^M r*r!(K ^M S ) uTM ^M K|| |id^uK| fflffl| ~= fflffl| VM(TM ^M K) ____________//TM ^M VMK Here the top equivalence is given by (11.4.2) and (11.4.5) in Ho GKM . (iii)The following diagram commutes in Ho GK*. r!(K ^M S ) ^ SV_____id^t____//r!(K ^M S ) ^ r!S ' || |'| fflffl| fflffl| r!( VMK ^M S )oo_r!(uK_^idr!(r*r!(K)^M_S ) ^M S ) Here the right vertical equivalence is given by (11.4.5) in Ho GKM . The le* *ft vertical equivalence is the composite r!(K ^M S ) ^ SV ' r!(K ^M S ^M SVM) ' r!(K ^M SVM^M S ), where the second equivalence is obtained by moving the copy of So from SVM~= So ^M S and amalgamating it with the displayed copy of S . 226 16. THE WIRTHM"ULLER AND ADAMS ISOMORPHISMS Proof of Proposition 16.5.4. Let X = 1MK. Define ,X to be the map -VM 1MuK -V r*r!(X ^M CM ) ' -VM 1Mr*r!(K ^M S )_________//_ M 1M VMK ~=X. Using Proposition 16.5.10(ii), we see that , VMX can be identified with VM,X ,* * which in turn can be identified with 1MuK . To show that the diagram (16.5.6) commut* *es, it suffices to show that the diagram obtained from it by applying VM commutes. We have just identified the lower horizontal arrow of the resulting diagram in * *space level terms. Similarly, the definition (16.5.3) of its upper horizontal arrow, * *together with Lemma 16.5.9 and Proposition 16.5.10(i), identifies its upper horizontal a* *rrow, with Y serving as a dummy variable. More explicitly, using the projection formu* *la (11.4.5), we see that the diagram can be rewritten as id^uS0M r*r!(r*Y ^M S )_'__//r*Y ^M r*r!S____//_r*Y ^M SVM r*r!(`^id)|| |`^id| fflffl| fflffl| r*r!(X ^M S ) _________1______________//_X ^M SVM M uK Consider the dummy variable Y levelwise. We see from the case K = S0M of Proposition 16.5.10(ii) that, at level V , the top row is the map ur*Y (V.)Ther* *efore the diagram commutes levelwise by the naturality of u. To prove that the composite (16.5.5) is the identity map, we apply V to it.* * To abbreviate notation, write Y = r!(X ^M CM ) and consider the following diagram. V(id^o) V V Y____________________________________//UUU (Y ^ r!CM ) UUUUU Vo lll || UUUYUU lllll '| || UUUUU llll' | || Vr!(,X^M id) **U uull fflffl| V Yoo____________ V r!(r*Y ^MSCM ) Y ^ r!S SSS | ' || SS'SSSSS '| fflffl| SSS)) fflffl| r!(X ^M S ) oo__________r!(,X^M_id)_________r!(r*Y_^M S ) We must prove that the triangle at the upper left commutes. The arrows marked ' are given by (11.4.5) and the evident equivalence VMCM ' 1 S . The upper triangle commutes by the definition of oY in terms of o, the bottom trapezoid commutes by naturality, and the triangle at the right commutes by inspection of projection formula isomorphisms. Thus it suffices to prove that traversal of* * the perimeter gives a commutative diagram, and this will hold for X if it holds for* * VMX. In that case, we see from Lemma 16.5.9, Proposition 16.5.10(ii), and a diagram chase that the perimeter agrees with the diagram that is obtained by applying t* *he suspension spectrum functor to the diagram in Proposition 16.5.10(iii). The proof of Proposition 16.5.10 is based on the following construction of a natural map wK :r*r!K -! "K^M So that will give rise to the required map uK . Here K" is a suitably "fattened u* *p" version of K. 16.5. PROOF OF THE WIRTHM"ULLER ISOMORPHISM FOR MANIFOLDS 227 Construction 16.5.11. As in [77, 11.5], identify the tangent bundle of M with the normal bundle of the diagonal embedding M -! M x M. Let U be a tubular neighborhood of the diagonal. Let pr1and pr2be the projections M x M -! M and let ssi:U -! M be their restrictions to U. For an ex-space (K, pK , sK ) ov* *er M, consider the following diagram of retracts, where is the diagonal and ' is* * the inclusion of U in M x M. Note that ssi= priO ' and define "K= (ss1)!ss*2K. pr2 M _______//UUUUUUUUUU'//_GGM_x_M______//_NMF | UUUUUUUUUUUUU|ss1GGG|NNpr1NN | FFrFF | |UUUUUU##_____|______N''_ r | ""// | | M ______________M ___________* | | | | | | | fflffl| fflffl||* fflffl| | fflffl|| K ______//ss2K__|_E_//_M xMK_____|_____//KD | EEE| MMMM | DD | || || E"fflffl|"|| M&&fflffl||| !!fflffl|D | | "K | r*r!K________//r!K | | | | | | | fflffl| fflffl|| fflffl| | fflffl|| M _______//UUUUU|UUU_G//_M x_M___|N___//_MF | UUUUUUUUUUU|GGG NNN | FF | UUUUUUUfflffl|UUU##G NN'fflffl|' F"fflffl|"F M _______________M __________//_* The floor and ceiling of the diagram are identical. The back wall is formed by * *pulling K back along the maps of base spaces and then the front wall is obtained from t* *he back wall by applying lower shriek functors. Here we have used the canonical isomorphism pr1!(M x K) = pr1!pr*2K ~=r*r!K associated to the pullback square on the right side of the floor. Since the ssi are homotopy equivalences, the ma* *ps (16.5.12) K -! ss*2K -! "K at the left are equivalences when K is q-cofibrant and q-fibrant, and we denote* * the displayed composite equivalence as ~. To get a better feeling for the spaces in the diagram, we make the following schematic picture. ___________________________|_O_______ fflfflO _________f|fflfl__________ fflffl ____________ff|flfl____________ fflfflO ________________fff|lfl__________| fflO ____________________ffl|____|fflfflU fflfflO____________________fffl|fl______|fflffl|Um fflfflO____________________ffflf|l___|| fflffl______________________fflff|l_____________|| | O | | | | | O | | | | | O | | | | | | | * | | | O | |ss2K|Um | K | | O | | | | || O | | || | | ffl```` `|`|` ` ` ``_`|`_`_ffflfl|`_____________ | ffl______|||___________|___ffflfl___________________jj | fflfflffl |_____________|__ffflfl_______________||fflffljj* *jjj || fflffl__________________||_fflffl_________fflffl|fflfflffl* *jjjjjj | ffl______________________|_fflffl______fflffl|fflfflfflfflj* *jjjjj |ffl_______________________|fflfflfflfflfflffljjjjjM ____________________________ffl||ffl________fflfflfflfflfflff* *ljjjjjjjj ____________________________fflfflfflffljjjjjM The cube represents M x K with M in the horizontal direction and K in the other two directions. It is sitting over M x M with vertical projection and we think * *of the top face as being the image of the section. We can view ss*2K as the subspa* *ce of 228 16. THE WIRTHM"ULLER AND ADAMS ISOMORPHISMS M x K sitting over the neighborhood U of the diagonal in M x M. Passing to the front face of the diagram, the fibers of "Kover points m in M are the slices ss* **2K|Um of ss*2K as displayed with basepoint obtained by identifying the fiber Um = ss-* *11(m) over m in the bundle U to a single point. We therefore think of "Kas a fattening of the space K that sits over the diagonal in M x M. For a sub G-space A of a G-space K over M, not necessarily sectioned, passage to fiberwise quotients gives an ex-G-space K=M A over M with total space K [A M. Given two such pairs (K, A) and (L, B), we obtain a product pair by setting (K, A) xM (L, B) = (K xM L, K xM B [ A xM L). Its fiberwise quotient is the ex-G-space (K=M A) ^M (L=M B) over M. The pair (M x M, M x M - U) is a model for the Thom complex T o, and we can identify T o with the quotient space (M x M)=(M x M - U). More relevantly for us, the fiberwise quotient (M x M)=M (M x M - U) is a model for So. View M xK = pr*2K, M xM, and U as G-spaces over M via projection to the first factor and embed U in M x K by sending (m, n) to (m, sK (n)). We have the diagonal map M x K -! (M x K, U) xM (M x M, M x M - U) of G-spaces over M that sends (m, k) to ((m, k), (m, pK (k))) for m 2 M and k 2* * K. It induces the top map in the following diagram in GKM . M x K R____//_((M x K)=M U) ^M So RR OO | RRRRR | | RRRR | fflffl| R(( | r*r!K_____wK____//"K^M So Here (M x K)=M U is obtained from M x K by identifying all points of the form (m, sK (n)) such that (m, n) 2 Um to a single basepoint in the fiber over m. * *It therefore contains K" as a subspace, and this gives the right vertical inclusio* *n. The image of the top arrow lands in the image of the right vertical arrow since* * if (m, pK (k)) is not in Um , then (m, k) maps to the basepoint in So and therefor* *e to the basepoint in (M x K)=M U ^M So. This gives the diagonal arrow. Note that r*r!K = M x (K=sK (M)) and the left vertical arrow is the obvious quotient map. Since the diagonal arrow maps (m, x) with x 2 sK (M) to the base point of the fiber over m in "K^M So , it is constant on the fibers of the left vertical arr* *ow. It therefore factors through a map wK . Explicitly, wK is specified by ( (16.5.13) wK (m, [x]) = [m, x] ^ [m, pK (x)]if (m, pK (x)) 2 Um , * otherwise, where m 2 M, x 2 K, and the square brackets denote equivalence classes. Proof of Proposition 16.5.10. Here we are working in homotopy cate- gories, and we may assume that K is qf-fibrant and qf-cofibrant. Let L = K^M S . We define the map uK in Ho GKM by the natural zig-zag ~ r*r!L_wL__//"L^M Sooo~^M_id__L ^M So _____// VMK of arrows in GKM , where ~ is induced by the isomorphism ~= S ^M So _twist//_So ^M_S___//SVM 16.5. PROOF OF THE WIRTHM"ULLER ISOMORPHISM FOR MANIFOLDS 229 and ~: L -! "Lis an equivalence as in (kappak). Proof of (i) We must show that uS0M' k in Ho GKM . Using our zig-zag definition of uK , we see that it suffices to show that the composite ~-1 o~^id M x T __k__//M x SV____//_S ^M S_____//fS^M So is homotopic to the map wS defined in (16.5.13). As noted in the proof of Lemma 16.5.9, k(m, [v]) = (m, ev([v] ^ m)), where v 2 S and brackets denote equivalence classes. Recall that the map evdepends on a choice of a tubular nei* *gh- borhood of M in x M (as in the proof of Lemma 16.2.7). We use the obvious choice {(v, m) | v 2 , m 2 M, and (p (x), m) 2 U}. Under our identification of the normal bundle of : M ! M x M and thus of its tubular neighborhood U with o, this tubular neighborhood is identified with M x V ~= o. When not in the section, we can view points [m, n] 2 So = (M xM)=M (M xM -U) as vectors (m, n) in the tangent space Um ~=om Vm ~=V of M at m. We then have that (m, ev([v] ^ m)) = (m, (p (v), m) + v). To identify this point in the image of ~, let u 2 m be such that (p (v), m)+v = (m, p (v))* *+u in V and note that u depends continuously on m and v. Since ~(u ^ [m, p (v)]) = (m, (m, p (v)) + u), the composite displayed above is given by ( (~ ^M id)~-1k(m, [v]) = [m, u] ^ [m, p (v)]if (m, p (v)) 2 Um , * otherwise. A linear homotopy in the fibers of shows that this map is homotopic to wS . Proof of (ii) Inspection of the construction of w gives the following naturality diagram for based G-spaces T and ex-G-spaces K over M. r*r!(TM ^M K)___'____//TM ^M r*r!K . w || |id^w| fflffl| ' fflffl| (TM^^M K ) ^M So____//_TM ^M K" ^M So Here, using r*r!~= (ss1)!ss*2, the bottom equivalence is the following applicat* *ion of the projection formula. (ss1)!ss*2(r*T ^M'K)(ss1)!(ss*2r*T ^M ss*2K) ' (ss1)!(ss*1r*T ^M ss*2K) ' r*T ^M (ss1)!ss*2K This use of the projection formula is compatible with its use for r! to obtain * *the equivalence of the top row. Analogous naturality diagrams for the other two maps in the definition of uK give the conclusion. Proof of (iii) Again let L = K ^M S . Expanding the diagram in the statement of (iii) in terms of the definition of uK , we must prove that the following di* *agram commutes in Ho GK*, where the equivalences here are the vertical arrows of the 230 16. THE WIRTHM"ULLER AND ADAMS ISOMORPHISMS diagram in (iii). r!( VMKO^MOS ) oo____'________r!L_^ SV_______id^t______//r!L ^ r!S r!~|| |'| | |fflffl r!(L ^M So ^M S )__r!(~^id)//_r!("L^M So ^M Sr)!(wL^idr!(r*r!L)^MoSo)_ We chase the diagram starting in r!(L^M So ^M S ) and mapping to r!("L^M So ^M S ). Let x = k ^ w 2 L = K ^M S , u 2 So, and v 2 S be points in fibers over a given m 2 M. Using square brackets to denote passage to quotient spaces (the lower shriek functors), we see that r!(~ ^ id) sends [x ^ u ^ v] to [[m, x] ^ u* * ^ v]. The definitions of ~ and of the top left equivalence (which is the left vertical eq* *uivalence in the diagram of the statement) are arranged in such a way that the composite * *of ~ and the inverse of the equivalence sends [x^u^v] to [x^(u+v)]. Let t(u+v) = [z], z 2 S , and let n = p (z). Chasing [x ^ u ^ v] around the top of the diagram, w* *hen we do not arrive at the basepoint we arrive at the point [[n, x] ^ [n, m] ^ z],* * where [n, m] is an element of U ~=o. We can identify the target space with r!("L) ^ SV using the identification of So ^ S with M x SV and the projection formula. Then our two maps are homotopic by a homotopy h that can be written in the form h([x ^ u ^ v], s) = [m + s[m, n], x] ^ (u + v + 2s[n, m]). Here m + s[m, n] denotes a point on the path from m to n in M that is the image under the exponential map of the line segment from 0 to [m, n] in the tangent space at m. The Thom map takes u + v in SV (which in M x SV is based at m) to the point z in S based at n. In SV , we have u + v = [m, n] + z. Since [n, m] = -[m, n] under the identification of o with U (as in [77, 11.5]), we se* *e that the homotopy ends at the composite around the top of the diagram, and it clearly begins at r!(~ ^ id). Bibliography [1]Hans Joachim Baues. Algebraic homotopy. 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