EQUIVARIANT HOMOTOPY THEORY FOR PRO-SPECTRA H. FAUSK Abstract.We extend the theory of equivariant orthogonal spectra from fi- nite groups to profinite groups, and more generally from compact Lie gro* *ups to compact Hausdorff groups. The G-homotopy theory is "pieced together" from the G=U-homotopy theories for suitable quotient groups G=U of G; a motiv* *a- tion is the way continuous group cohomology of a profinite group is buil* *t out of the cohomology of its finite quotient groups. In this category Postnikov* * tow- ers are studied from a general perspective. We introduce pro-G-spectra a* *nd construct various model structures on them. A key property of the model * *struc- tures is that pro-spectra are weakly equivalent to their Postnikov tower* *s. We give a careful discussion of two version of a model structure with "unde* *rlying weak equivalences". One of the versions only make sense for pro-spectra.* * In the end we use the theory to study homotopy fixed points of pro-G-spectr* *a. Contents 1. Introduction 2 1.1. Acknowledgements 5 2. Unstable equivariant theory 5 2.1. G-Spaces 6 2.2. Collections of subgroups of G 6 2.3. Model structures on the category of G-spaces 8 2.4. Some change of groups results for spaces 10 3. Orthogonal G-Spectra 11 3.1. JGV-spaces 11 3.2. Orthogonal R-modules 13 3.3. Fixed point and orbit spectra 13 3.4. Examples of orthogonal G-Spectra 14 3.5. The levelwise W-model structures on orthogonal G-Spectra 14 4. The stable W-model structure on orthogonal G-spectra 15 4.1. Verifying the model structure axioms 15 4.2. Fibrations 18 4.3. Positive model structures 20 4.4. Homotopy classes of maps between suspension spectra 20 4.5. The Segal-tom Dieck splitting theorem 23 4.6. Self-maps of the unit object 23 5. The W - C-model structure on orthogonal G-spectra 24 5.1. The construction of WCMR 25 ____________ Date: August 28, 2006. 1991 Mathematics Subject Classification. Primary 55P91; Secondary 18G55. Key words and phrases. Equivariant homotopy, pro-spectra, profinite groups. Support by the Institut Mittag-Leffler (Djursholm, Sweden) is gratefully ack* *nowledged. 1 2 H. FAUSK 5.2. Tensor structures on MR 26 5.3. The C-cofree model structure on MR 28 6. A digression: G-spectra for noncompact groups 28 7. Postnikov t-model structures 32 7.1. Preliminaries on t-model categories 32 7.2. The d-Postnikov t-model structure on MR 33 7.3. An example: Greenlees connective K-theory 34 7.4. Postnikov sections 34 7.5. Coefficient systems 36 7.6. Continuous G-modules 38 8. Pro-G-spectra 38 8.1. Examples of pro-G-Spectra 39 8.2. The Postnikov model structure on pro-MR 39 8.3. Tensor structures on pro- MR 42 8.4. Bredon cohomology 43 8.5. Group cohomology 44 8.6. Homotopy orbits and group homology 44 8.7. The Atiyah-Hirzebruch spectral sequence 45 9. The C-free model structure on pro- MR 46 9.1. Construction of the C-free model structure on pro- MR 46 9.2. Comparison of the free and the cofree model structures 47 10. Homotopy fixed points 49 10.1. The homotopy fixed points of a pro-spectrum 49 10.2. Homotopy orbit and homotopy fixed point spectral sequences 51 10.3. Comparison to Davis' homotopy fixed points 53 Appendix A. Compact Hausdorff Groups 55 References 56 1.Introduction This paper is devoted to explore some aspects of equivariant homotopy theory of G-equivariant orthogonal spectra when G is a profinite group. We develop the theory sufficiently to be able to construct homotopy fixed points of G-spectra * *in a natural way. A satisfactory theory of G-spectra, when G is a profinite group, requires the generality of pro-G-spectra. The results needed about model struc- tures on pro-categories are presented in two papers joint with Daniel Isaksen [* *19] [20]. Most of the theory also works for compact Hausdorff groups and discrete groups. We start out by considering model structures on G-spaces. This is needed as a starting point for the model structure on G-spectra. A set of closed subgroups * *of G is said to be a collection if it is closed under conjugation. To any collecti* *on C of subgroups of G, we construct a model structure on the category of G-spaces such that a G-map f is a weak equivalence if and only if fH , for H 2 C, is a underl* *ying weak equivalence. The collections of subgroups of G that play the most important role in this paper are the cofamilies, i.e. collections of subgroups that are closed under p* *assing EQUIVARIANT HOMOTOPY THEORY FOR PRO-SPECTRA 3 to larger subgroups. The example to keep in mind is the cofamily of open subgro* *ups in a profinite group. We present the foundation for the theory of orthogonal G-spectra, indexed on finite orthogonal G-representations, with minimal assumptions on the group G and the collection C. Most of the results extend easily from the theory develop* *ed for compact Lie groups by Mandell and May [35]. We include enough details to make our presentation readable, and provide new proofs when the generalizations to our context are not immediate. Equivariant K-theory and stable equivariant cobordism theory both extend from compact Lie groups to general compact Haus- dorff groups. A generalization of the Atiyah-Segal completion theorem is studied in [18]. Let R be a symmetric monoid in the category of orthogonal G-spectra indexed on a universe of G-representations. In Theorem 4.7 the category of R-modules, denoted MR , is given a stable model structure, such that the weak equivalences* * are maps whose H-fixed points are stable equivalences for all H in a suitable colle* *ction W. For example W might be the smallest cofamily containing all normal subgroups H of G such that G=H is a compact Lie group. A stable G-equivariant theory of spectra, for a profinite group G, is also given by Gunnar Carlsson in [5]. We would like to have a notion of "underlying equivalence" even when the triv* *ial subgroup, {1}, is not included in the collection C. We consider a more general framework. In Theorem 5.5 we show that for two reasonable collections, W and C, of subgroups of G such that W U is in C, whenever W 2 W and U 2 C, there is a model structures on MR such that the cofibrations are relative C-cell complexes and the weak equivalences are maps f such that W*(f) = colimU2CssWU*(f) is an isomorphism for every W 2 W. For example, C can be the collection of open subgroups of a profinite group G and W the collection, {1}, consisting of the t* *rivial subgroup in G. In the rest of this introduction we assume that Un(R) = 0 whenever n < 0 and U 2 W. We can then set up a good theory of Postnikov sections in MR . The Postnikov sections are used in our construction of the model structures on pro- MR . Although we are mostly interested in the usual Postnikov sections that cut off the homotopy groups at the same degree for all subgroups W 2 W, we give a general construction that allow the cutoff to take place at different degrees* * for different subgroups. In Theorem 8.4 we construct a stable model structure, called the Postnikov W - C-model structure, on pro- MR . It can be thought of as the localization of the strict model structure on pro- MR , where we invert all maps from a pro- spectrum to its levelwise Postnikov tower, regarded as a pro-spectrum. Here is * *one characterization of the weak equivalences: The class of weak equivalences in the Postnikov W -C-model structure is the class of pro-maps that are isomorphic to a levelwise map {fs}s2S such that fs becomes arbitrarily highly connected (unifor* *mly with respect to the collection W) as s increases [19, 3.2]. In Theorem 8.27 we give an Atiyah-Hirzebruch spectral sequence. It is con- structed using the Postnikov filtration of the target pro-spectrum. The spectr* *al sequence has good convergence properties because any pro-spectrum can be recov- ered from its Postnikov tower in our model structure. 4 H. FAUSK The category pro-MR inherits a tensor product from MR . This tensor structure is not closed, and it does not give a well-defined tensor product on the whole homotopy category of pro- MR with the Postnikov W - C-model structure. The Postnikov W - C-model structure on pro- MR is a stable model struc- ture. But the associated homotopy category is not an axiomatic stable homotopy category in the sense of Hovey-Palmieri-Strickland [27]. We discuss two model structures on pro - MR with two different notions of "underlying weak equivalences". Let G be a finite group and let C be the collec* *tion of all subgroups of G. There are many different, but Quillen equivalent, W - A-model structures on MR with W = {1} and A C. Two extreme model structures are the cofree model structure, with A = C, and the free model struc* *ture, with A = W = {1}. The cofibrant objects in the free model structure are retracts of relative G-free cell spectra. Now let G be a profinite group and let C be the collection of all open subgro* *ups of G. In this case the situation is more complicated. The {1}-weak equivalences are maps f such that {1}*(f) = colimU2CssU*(f) is an isomorphism. We call these maps the C-underlying weak equivalences. Let G be a nonfinite profinite group, and let C be the collection of all open subgroups of G. The Postnikov {1}-C-mod* *el structure on pro- MR is the closest we can get to a cofree model structure. It is given in Theorem 8.5. Certainly, it not sensible to have a model structure w* *ith cofibrant objects relative free G-cell complexes, because Sn ^G+ is equivalent * *to a point. In pro- MR , unlike MR , we can form an arbitrarily good approximation to the free model structure by letting the cofibrations be retracts of levelwise r* *elative G-cell complexes that become "eventually free". That is, as we move up the inve* *rse system of spectra, the stabilizer subgroups of the relative cells become smalle* *r and smaller subgroups in the collection C. The key idea is that the cofibrant repla* *cement of the constant pro-spectrum 1 S0 should be the pro-spectrum { 1 EG=N+ }, indexed by the normal subgroups N of G in C, ordered by inclusion. We use the rather technical theory of filtered model categories, developed in [19], to con* *struct the free model structure on pro- MR . This C-free model structure is given in Theorem 9.2. The C-free and C-cofree model structures on pro- MR are Quillen adjoint, via the identity maps, but there are fewer weak equivalences in the free than i* *n the cofree model structure. Thus, we actually get two different homotopy categories. We relate this to the failure of having an inner hom functor in the pro-categor* *y. Let Ho(pro- MR ) denote the homotopy category of pro- MR with the Postnikov C-model structure. Assume that X is cofibrant and that Y is fibrant in the Post- nikov C-model structure on pro- MR . Then Theorem 9.10 says that the homset of maps from X to Y in the homotopy category of the C-free model structure on pro- MR is: Ho(pro- MR ) (X ^ {EG=N+ }, Y ) while the homset in the homotopy category of the C-cofree model structure on pro- MR is: Ho (pro- MR ) (X, hocolimUF (EG=N+ , Y ), where the colimit is taken levelwise. EQUIVARIANT HOMOTOPY THEORY FOR PRO-SPECTRA 5 The Postnikov model structures are well-suited for studying homotopy fixed points. For definiteness, let G be a profinite group, let C be the collection o* *f open subgroups of G, and let R be a non-equivariant S-cell spectrum with trivial hom* *o- topy groups in negative degrees. The homotopy fixed points of a pro-G-spectrum {Yt} is defined to be the G-fixed points of a fibrant replacement in the Postni* *kov C-cofree model structure. It is equivalent, in the Postnikov model structure on R-spectra, to the pro-spectrum hocolimNF (EG=N+ , PnYt)G indexed on n and t. The spectrum associated to the homotopy fixed point pro- spectrum (take homotopy limits) turns out to be equivalent to holimt,mhocolimNF ((EG=N)(m)+, Yt)G . These expressions resemble the usual formula for homotopy fixed points. The appropriate notion of a ring spectrum in pro-MR is a monoid in pro-MR . This is more flexible than a pro-monoid. The second formula for homotopy fixed point spectra shows that if Y is a (commutative) fibrant monoid in pro- MR with the strict C-model structure, then the associated homotopy fixed point spectrum is a (commutative) monoid in MR . Under reasonable assumptions there is an iterated homotopy fixed point formul* *a. This appears to be false if one defines homotopy fixed points in the C-strict m* *odel structure on pro- MR . We obtain a homotopy fixed point spectral sequence as a special case of the Atiyah-Hirzebruch spectral sequence. The explicit formulas for the homotopy fixed points, the good convergence pro* *p- erties of the homotopy fixed point spectral sequence, and the iterated homotopy fixed point formula are all reasons for why it is convenient to work in the Pos* *tnikov C-model structure. A general theory of homotopy fixed point spectra for actions by profinite gro* *ups was first studied by Daniel Davis in his Ph.D. thesis [8]. His theory was inspi* *red by a homotopy fixed point spectral sequence for En, with an action by the ex- tended Morava stabilizer group, constructed by Ethan Devinatz and Michael Hop- kins [12]. We show that our definition of homotopy fixed point spectra agrees w* *ith Davis' when G has finite virtual cohomological dimension. Our theory applies to the example of En above, provided we follow Davis and use the "pro-spectrum K(n)-localization" of En rather than (the K(n)-local spectrum) En itself. 1.1. Acknowledgements. The model theoretical foundations for this paper is joint work with Daniel Isaksen. I am also grateful to him for many discussions on the foundation of this paper. I would like to thank Andrew Blumberg and Daniel Davis for their interest in the paper and their help. 2.Unstable equivariant theory We associate to a collection, W, of closed subgroups of G a model structure on the category of based G-spaces. The weak equivalences in this model structure a* *re maps f such that the H-fixed points map fH is a non-equivariant weak equivalence for each H 2 W. 6 H. FAUSK 2.1. G-Spaces. We work in the category of compactly generated weak Hausdorff spaces. Let G be a topological group. A G-space X is a topological space togeth* *er with a continuous left action by G. The stabilizer of x 2 X is {g 2 G | gx = x}* *. This is a closed subgroup of G since it is the preimage of the diagonal in X xX unde* *r the map g 7! x x gx. Let Z be any subset of X. The stabilizer of Z is the intersect* *ion of the stabilizers of the points in Z, hence a closed subgroup of G. Similarly,* * for any subgroup H of G the H-fixed points, XH = {x 2 X | hx = x for eachh 2 H}, of a G-space X is a closed subset_of X. The stabilizer of XH contains_H and is a closed subgroup of G. So XH = XH , for any subgroup H of G, where H denotes the closure of H in G. Hence, we consider closed subgroups of G only. A based G-space is a G-space together with a G-fixed basepoint. We denote the category of based G-spaces and basepoint preserving continuous G-maps by GT . Lemma 2.1. The category of based G-spaces GT is complete and cocomplete. Proof.The limits and colimits are created via the forgetful functor to spaces [* *34]. We denote the category of based G-spaces and continuous basepoint preserving maps by TG . The space of continuous maps is given a G-action by (g . f)(x) = gf(g-1x) (and topologized as the Kellyfication of the compact open topology). T* *he corresponding categories of unbased G-spaces are denoted GU and UG . Lemma 2.2. Let X and Y be two G-spaces. The action of G on TG (X, Y ) is continuous. Proof.It suffices to show that the adjoint, G x TG (X, Y ) x X ! Y , of the act* *ion map is continuous. The map is a composition of several continuous maps. The category GT is a closed symmetric tensor category, where S0 is the unit object, the smash product X ^ Y is the tensor product, and the G-space TG (X, Y* * ) is the inner hom functor. Define a functor GU ! GT by attaching a disjoint basepoint, X 7! X+ . This functor is a left adjoint to the forgetful functor GT ! GU. The morphism set GU(X, Y ) is naturally a retract of GT (X+ , Y+ ). More precisely, we have that ` GT (X+ , Y+ ) = ZGU(Z, Y ) where the sum is over all open and closed G-subsets Z of X. Let f :X+ ! Y+ be a map in GT . Then the corresponding unbased map is f|Z :Z ! Y where Z = X+ - f-1 (+). Hence, statements about based spaces often give analogous statements for unbased spaces. 2.2. Collections of subgroups of G. We are mostly concerned with cofamilies in this paper. For completeness, we consider more general collections of subgro* *ups when this is suitable. Definition 2.3. A collection W of subgroups of G is a nonempty set of closed subgroups of G such that if H 2 W, then gHg-1 2 W for any g 2 G. A collection W is a normal collection if for all H 2 W there exists a K 2 W such that K H and K is a normal subgroup of G. EQUIVARIANT HOMOTOPY THEORY FOR PRO-SPECTRA 7 Definition 2.4. A collection W of subgroups of G is a cofamily if K 2 W implies that L 2 W for all subgroups L K. A collection C of subgroups of G contained in a cofamily W is a family in W, if, for all K 2 C and H 2 W such that H K, we have that H 2 C. Let W be a collection of subgroups of G. The smallest cofamily of closed sub-_ groups of G containing W is called the cofamily closure of W and is denoted W . A cofamily is called a normal cofamily if it is the cofamily closure of a colle* *ction of normal subgroups of G. We now give some important cofamilies. Example 2.5. The collection of all subgroups U of G such that G=U is finite and discrete is a cofamily. This collection of subgroups is closed under finite int* *ersection since G=U \ V G=U x G=V . A finite index subgroup of G has only finitely many G-conjugate subgroups of G. Hence, if U is a finite index subgroup of G, then \g2GgUg-1 is a normal subgroup of G such that G= \g2G gUg-1 is a finite discrete group. Let fnt(G) be the collection of all normal subgroups U of G such that G=U is a finite discrete group. Example 2.6. Define dsc(G) to be the collection of all normal subgroups U of G such that G=U is a discrete group. This collection is closed under intersecti* *on. We call a collection that is contained in the cofamily closure of dsc(G) a disc* *rete collection of subgroups of G. Example 2.7. Let Lie(G) be the collection of all normal subgroups U of G such that G=U is a compact Lie group. This collection is closed under intersection s* *ince a closed subgroup of a compact Lie group is a compact Lie group. We call a collec* *tion that is contained in the cofamily closure of Lie(G) a Lie collection of subgrou* *ps of G. Lemma 2.8. Let G be a compact Hausdorff group, and let K be a closed subgroup of G. Then {U \ K | U 2 Lie(G)} is a subset of Lie(K), and for every H 2 Lie(K) there exists a U 2 Lie(G) such that U \ K H. Proof.Let U 2 Lie(G). The subgroup U \K is in Lie(K) since K=K \U is a closed subgroup of the compact Lie group G=U. Let H be a subgroup in Lie(K). We have that \U2Lie(G)U = 1 by Corollary A.3. Hence U \ K=H for U 2 Lie(G) is a collection of closed subgroups of the compact Lie group K=H whose intersection is the unit element. Since Lie(G) is closed under finite intersections, the descending chain property for closed sub* *groups of a compact Lie group [13, 1.25, ex. 15] gives that there exists a U 2 Lie(G) * *such that U \ K is contained in H. We order fnt(G) and Lie(G) by inclusions. We recall the following facts. Proposition 2.9. A topological group G is a profinite group precisely when G ! limU2fnt(G)G=U is a homeomorphism. A topological group G is a compact Hausdorff group precisely when G ! limU2Lie(G)G=U is a homeomorphism. 8 H. FAUSK Proof.These facts are well-known. The second claim is also proved in Proposition A.2. Even though we are mostly interested in actions by profinite groups, we find * *it natural to study actions by compact Hausdorff groups whenever possible. 2.3. Model structures on the category of G-spaces. We associate to a col- lection W of closed subgroups of G a model structure on the category of based G-spaces. Definition 2.10. Let f :X ! Y be a map in GT . The map f is said to be a W-equivalence if the underlying unbased maps fU :XU ! Y U are weak equivalences for all U 2 W. Definition 2.11. Let p: E ! B be a map in GT . We say that p is a W-fibration if the underlying unbased maps pU :EU ! BU are Serre fibrations for all U 2 W. We next define the generating cofibrations and generating acyclic cofibration* *s. We use the conventions that S-1 is the empty set and D0 is a point. Definition 2.12. Let WI be the set of maps {(G=U x Sn-1)+ ! (G=U x Dn)+ }, for n 0 and U 2 W. Let WJ be the set of maps {(G=U x Dn)+ ! (G=U x Dn x [0, 1])+ }, for n 0 and U 2 W. Recall that an object X in a category K is said to be a small object in K if ` ` a2A K(X, Ya) ! K(X, a Ya) is an isomorphism for any indexing set A and any objects Ya in K. Lemma 2.13. Let H be a closed subgroup of G, and let Y be a H-space. Then (G xH Y )+ is a small object in GT whenever Y+ is a small object in HT . Proof.The result follows from the adjunctions GT ((G xH Y )+ , Z) ~=HU(Y, Z) ~=HT (Y+ , Z|H). The restriction map GT ! HT respects arbitrary (wedge) sums. The following model structure is called the W-model structure on GT . For the definition of relative cell complexes see [25, 10.5]. Proposition 2.14. There is a proper model structure on GT with weak equiva- lences W-weak equivalences, fibrations W-fibrations, and cofibrations retracts * *of relative W-cell complexes. The set WI is a set of generating cofibrations and WJ is a set of generating acyclic cofibrations. Proof.A map p: E ! B in GT is a W-fibration if and only if it has the right lifting property with respect to all maps in WJ. A map f is a W-acyclic fibrati* *on if and only if it has the right lifting property with respect to all maps in WI. This follows from the corresponding non-equivariant result and by the fixed poi* *nt adjunction [26, 2.4]. To use the small object argument we need that GT (G=U+ ^ Sn-1, -) commutes with directed colimit of spaces obtained by adjoining cells in WI and WJ. This EQUIVARIANT HOMOTOPY THEORY FOR PRO-SPECTRA 9 follows from Lemma 2.13. The verifications of the model structure axioms follows as in [26, 2.4]. The model structure is both left and right proper. This follow* *s from the corresponding non-equivariant results since pullbacks commute with fixed po* *ints and since pushouts along closed inclusions also commute with fixed points. An alternative way to set up the model structure on GT is given in [35, III.1* *]. Let WGT , or simply WT , denote GT with the W-model structure, and let Ho(WGT ) denote its homotopy category. Proposition 2.15. Let X be a retract of a WI-cell complex, and let Y be a G-space. Then the set Ho(WT )(X, Y ) is isomorphic to the set of ordinary (base* *d) G-homotopy classes of maps from X to Y . Proof.All objects are fibrant and a retract of a WI-cell complex is cofibrant i* *n the W-model structure. The cylinder object of (a cofibrant object) X in the W-model structures is X ^ [0, 1]+ . The next result has also been proved by Bill Dwyer [14, 4.1]. Note that a G-c* *ell complex X is a WI-cell complex if and only if all its isotropy groups are in W. Corollary 2.16. Let X and Y be WI-cell complexes. If a map f :X ! Y is a W-weak equivalence, then f is a G-homotopy equivalence. To get a topological model structure on our model category we need some as- sumptions on the collection W. Definition 2.17. A collection W of subgroups of G is called an Illman collection if (G=U x G=U0)+ is a WI-cell complex for any two U and U0 in W. In particular, all Illmancollections are closed under intersections since U \* * U0 is an isotropy group of G=U x G=U0. Lemma 2.18. If W is a discrete or a Lie collection of subgroups of G and W is closed under intersection, then W is an Illmancollection of subgroups of G. ______ Proof.The_statement is clear when W is contained in dsc(G). When W is contained in Lie(G), then the claim follows from a result of Illman [29]. Lemma 2.19. Let W be an Illman collection. If X and Y are two WI-cell complexes, then X ^ Y is again (homeomorphic to) a WI-cell complex. Proof.It suffices to show that (Sn-1 x Sm-1 x G=U x G=U0 ! Dn x Dm x G=U x G=U0)+ is a relative WI-cell complex. Since W is an Illmancollection this reduces to showing that (Sn-1 x Sm-1 x Sk-1 x G=U ! Dn x Dm x Dk x G=U)+ is a relative WI-cell complex. This is so. We follow the treatment of a topological model structure given in [35, III.1]. Note that the G-fixed points of the mapping spaces in TG are the mapping spaces in GT . Let MG be a category enriched in GT . Let GM be the G-fixed category of MG . Simplicial structures are defined in [25, 9.1.1,9.1.5]. We modify the defi* *nition of a simplicial structure by model theoretically enriching MG in the model cate* *gory WT instead of the model category of simplicial sets. 10 H. FAUSK Let i: A ! X and p: E ! B be two maps in MG . Let MG (i*, p*): MG (X, E) ! MG (A, E) xMG(A,B) MG (X, B) be the G-map induced by precomposing with i and composing with p. Definition 2.20. Let MG be enriched over GT . A model structure on GM is said to be W-topological if it is topological (see [25, 9.1.2]) and the following ho* *lds: (1) There is a tensor functor X T and a cotensor functor F (T, X) in MG , for X 2 MG and T 2 TG , such that there are natural isomorphisms of based G-spaces MG (X T, Y ) ~=TG (T, MG (X, Y )) ~=MG (X, F (T, Y )), for X, Y 2 MG and T 2 TG . (2) The map MG (i*, p*) is a W-fibration whenever i is a cofibration and p is* * a fibration in GM, and if i or p in addition is a weak equivalence, then MG (i*, * *p*) is a W-equivalence. Remark 2.21. The G-fixed points of MG (i*, p*) is GM(i*, p*). So if {G} 2 W, then a W-topological model structure on GM gives a topological model structure. We prove the pushout-product axiom [39, 2.1,2.3]. Lemma 2.22. Let W be an Illman collection of subgroups of G. Assume that f :A ! B and g :X ! Y are in WI, then f g :(A ^ Y ) [A^X (B ^ X) ! B ^ Y is a W-cofibration. Moreover, if at least one of f and g is in WJ instead of WI, then f g is a W-acyclic cofibration. Proof.This reduces to our assumption on W; if U and U0 are in W, then G=U x G=U0 is a W-cell complex. See also [35, II.1.22]. Proposition 2.23. Let W be an Illmancollection of subgroups of G. Then the model structure in Proposition 2.14 is a W-topological model structure. Proof.This follows from [35, III.1.15-1.21] and Lemma 2.22. Remark 2.24. A based topological model category M has a canonical based sim- plicial model structure. In the topological model structure denote the mapping space by Map (M, N), the tensor by M X, and the cotensor by F (X, M). Here X is a based space, and M and N are objects in M. The singular simplicial set functor, sing, is right adjoint to the geometric realization functor | - |. The* * corre- sponding based simplicial mapping space is given by sing(Map (M, N)). The sim- plicial tensor and cotensor are M |K| and F (|K|, M), respectively, where K is* * a based simplicial set and M and N are objects in M. We use that |K^L| ~=|K|^|L|. A based simplicial structure gives rise to an unbased simplicial structure. W* *e get a unbased simplicial structure by forgetting the basepoint in the based simplic* *ial mapping space, and by adding a disjoint basepoint to unbased simplicial sets in the definition of the tensor and the cotensor. Hence we can apply results about (unbased) simplicial model structures to a topological model category. 2.4. Some change of groups results for spaces. We describe the usual adjoint functors related to change of groups. Let K be a closed subgroup of G. The forg* *etful functor from GT to KT is given by restricting the G-action to K. It has a left adjoint given by sending X to G+ ^K X and a right adjoint given by sending X to TK (G+ , X). Let N be a normal subgroup of G. A functor from G=N-spaces to G-spaces is induced by the quotient map G ! G=N. The N-fixed point functor is EQUIVARIANT HOMOTOPY THEORY FOR PRO-SPECTRA 11 a right adjoint functor, and the N-orbit functor is a left adjoint functor. In * *general these six functors do not behave well with respect to the model structures on t* *he categories of G-spaces, G=N-spaces, and K-spaces. We give some conditions on the collections of subgroups of G, G=N, and K that guarantee that we get Quillen adjunctions. Let WK be a collection of subgroups of K, let WG be a collection of subgroups* * of G, and let WG=N be a collection of subgroups of G=N. The forgetful functor from WG GT to WK KT is a Quillen right adjoint functor if WK is contained in WG . It is a Quillen left adjoint functor if, in addition, H \ K 2 WK for every H 2 WG . The functor from WG=N G=NT to WG GT is a right Quillen adjoint functor if {HN=N | H 2 WG } WG=N . It is a left Quillen adjoint functor if, in addition, {HN | HN=N 2 WG=N } WG . ______ ______ Example 2.25. The forgetful functor from Lie(G)GT_to_Lie(K)KT is both a left and a right Quillen adjoint functor if K_is_in_Lie(G). It is neither a left nor* * a right Quillen adjoint functor if K is not in Lie(G). _________ __Let_N be a normal subgroup of G. Then the functor from Lie(G=N)G=NT to Lie(G)GT is both a left and a right Quillen adjoint functor. 3. Orthogonal G-Spectra Equivariant orthogonal spectra for compact Lie groups was introduced by Man- dell and May in [35]. We generalize their theory to allow more general groups. * *We develop the theory with minimal assumptions on the collection of subgroups used to define cofibrations and weak equivalences. We follow Chapters 2 and 3 of the* *ir work closely. 3.1. JGV-spaces. We define universes of G-representations. Definition 3.1. A G-universe U is a countable infinite direct sum 1i=1U0 of a real G-inner product space U0 satisfying the following: (1) the one-dimension* *al trivial G-representation is contained in U0; (2) U is topologized as the union * *of all finite dimensional G-subspaces of U (each with the norm topology); and (3) the G-action on all finite dimensional G-subspaces V of U factors through a compact Lie group quotient of G. If G is a compact Hausdorff group, then the G-action on a finite dimensional G-representation factors through a compact Lie group quotient of G by Lemma A.1. This is not true in general (consider the representation Q=Z < S1). We only use the finite dimensional G-subspaces of U, so one might as well assume that U0 is a union of such. Definition 3.2. Let SV denote the one-point compactification of a finite dimen- sional G-representation V . The last assumption in Definition 3.1 is added to guarantee that spaces like * *SV have the homotopy type of a finite G-cell complex. Definition 3.3. If the G-action on U is trivial, then U is called a trivial uni- verse. If each finite dimensional orthogonal G-representations is isomorphic to* * a G-subspace of U, then U is called a complete G-universe. 12 H. FAUSK All compact Hausdorff groups have a complete universe. However, it might not be possible to find a complete universes with a countable dimension. Traditiona* *lly, the universes have been assumed to have countable dimension [37, IX.2.1]. Remark 3.4. There are alternative notions of a G-universes. We use the or- thogonal finite dimensional G-representations, that factor through a compact Lie quotient group of G, as the indexing representations. This suffices to construc* *t a sensible equivariant homotopy theory for compact Hausdorff groups with the weak equivalences determined by the cofamily closure of Lie(G). We recall some definitions from [35, II]. Definition 3.5. Let U be a universe. An indexing representation is a finite dimensional G-inner product subspace of U. If V and W are two indexing repre- sentations and V W , then the orthogonal complement of V in W is denoted by W - V . The collection of all real G-inner product spaces that are isomorphic to an indexing representation in U is denoted V(U). When U is understood, we write V instead of V(U) to make the notation simpler. Definition 3.6. Let JGV be the unbased topological category with objects V 2 V and morphisms linear isometric isomorphisms. Let GJ V denote the G-fixed category (JGV)G . Definition 3.7. A continuous G-functor X :JGV! TG is called a JGV-space. (The induced map on hom spaces is a continuous unbased G-map.) Denote the cat- egory of JGV-spaces and (enriched) natural transformations by JGVT . Let GJ VT denote the G-fixed category (JGVT )G . Definition 3.8. Let SVG:JGV! TG be the JGV-space defined by sending V to the one point compactification SV of V . The external smash product __^:J V V V V G T x JG T ! (JG x JG )T is defined to be X__^Y (V, W ) = X(V ) ^ Y (W ) for X, Y 2 JGV and V, W 2 V. The direct sum of finite dimensional real G-inner product spaces gives JGV the structure of a symmetric tensor category. A topological left Kan extension giv* *es an internal smash product on JGVT [36, 21.4, 21.6]. We give an explicit descrip* *tion of the smash product. Let W be a real N-dimensional G-representation in V(U). Choose G-representations Vn of dimension n and Vn0of dimension N - n in V(U) for n = 0, 1, . .,.N. For example let Vn = VN0-n be the trivial n-dimensional G-representation, Rn. Then we have a canonical equivalence X ^ Y (W ) ~=_Nn=0JGV(W, Vn VN0-n) ^O(Vn)xO(VN0-n)X(Vn) ^ Y (VN0-n). The inner hom from X to Y is the JGV-space V 7! JGVT (X(-), Y (V -)) given by the space of continuous natural transformation of JGVT -functors. The internal smash product and the inner hom functor give JGVT the structure of a closed symmetric tensor category [35, II.3.1,3.2]. The unit object is the func* *tor that sends the indexing representation V to S0 when V = 0, and to a point when V 6= 0. By passing to fixed points we also get a closed symmetric tensor struct* *ure on GJ VT . EQUIVARIANT HOMOTOPY THEORY FOR PRO-SPECTRA 13 3.2. Orthogonal R-modules. For the definition of monoids and modules over a monoid in tensor categories see [34, VII.3 and 4]. The functor SVGis a strong symmetric functor. Hence the JGV-space SV is naturally a symmetric monoid in GJ VT . The following definition is from [35, II.2.6]. Definition 3.9. An orthogonal G-spectrum X is a JGV-space X :JGV! TG together with a right module structure over the symmetric monoid SV in GJGVT . Denote the category of G-spectra by JGVS. Let GJ VS be the G-fixed category (JGVS)G . The smash product and inner hom functors of orthogonal spectra are the smash product and inner hom functors of SV -modules, respectively. So the category of orthogonal G-spectra JGVS is itself a closed symmetric tensor category with SV * *as the unit object [35, 3.9]. The fixed point category GJ VS inherits a closed ten* *sor structure from JGVS. Explicit formulas for the tensor and inner-hom functors are obtained from the formulas after Definition 3.8 and [35, II.3.9]. Definition 3.10. We call a monoid R in GJ VS an algebra. We say that R is a commutative algebra, or simply a ring, if it is a symmetric monoid in GJ VS. We sometimes add: orthogonal, G, and spectrum, to avoid confusion. Let R be an orthogonal algebra spectrum. Definition 3.11. An R-module is a left R-module in the category of orthogonal spectra. Let MVRdenote the category of R-modules. The category of R-modules is bicomplete. If R is a commutative monoid, then the category MR is a closed symmetric tensor category [35, III.7]. A monoid T in the category of R-modules is called an R-algebra. Any R-algebra is an S-algebra. Let T be an R-algebra. Then the category of T -modules, in the category of R-modules, is equivalent to the category of T -modules, in the category of S-modules, when T is regarded as an S-algebra. We now give a pair of adjoint functors between orthogonal G-spectra and G-spaces. The V -evaluation functor V :JGVS ! TG is given by X 7! X(V ). We abuse language and let V also denote the functor V precomposed with the forgetful functor from R-modules to orthogonal spectra. There is a left adjoint, denoted RV, of the V -evaluation functor in the categ* *ory of R-modules. The R-module RVZ, for a G-space Z, sends W 2 V(U) to (3.12) RVZ(W ) = Z ^ O(W )+ ^O(W-V )R(W - V ) when V W , and to a point otherwise [36, 4.4]. This functor is called the V -* *shift desuspension spectrum functor and is also denoted FV and 1V(when R = S) in [35]. When V = 0 we denote this functor by 1R. We have that RVZ ~= SVZ^R. 3.3. Fixed point and orbit spectra. We define fixed point and the orbit spectra. The details on adjunction functors and change of universes from [35, V] extends* * to our setting. The results on Quillen adjoint functors between the model structur* *es (constructed in later sections) require some assumptions like the ones given in Subsection 2.4. We do not make those results explicit. 14 H. FAUSK Let X be an orthogonal spectrum and let H be a subgroup of G. Then the quotient X=H is defined to be X=H(V ) = X(V )=H with structure maps X=H(V ) ^ SW ! X(V )=H ^ SW =H ~=(X(V ) ^ SW )=H ! X(V W )=H. The H-orbit spectrum is a G-spectrum with trivial H-action. Let H be a closed subgroup of G. Let X be a U-spectrum where U is a uni- verse with trivial H-action. Then the H-fixed point spectrum XH is defined by XH (V ) = (X(V ))H for V 2 V(U), and the structure map is XH (V ) ^ SW ~=(X(V ) ^ SW )H ! XH (V W ). This is a NG H-spectrum. One can also define geometric fixed point spectra as in [35, V.5]. 3.4. Examples of orthogonal G-Spectra. Let T :TG ! TG be a continuous G-functor. Then we define the corresponding JGV-space by T O SVG:V(G) ! TG . This JGV-space is given an orthogonal G-spectrum structure by letting T (SV ) ^ SW ! T (SV W ) ~=T (SV ^ SW ) be the adjoint of the map SW ! TG (SV , SV ^ SW ) T!TG (T (SV ), T (SV ^ SW )) where the first map is a G-map adjoint to the identity on SV ^ SW . We can define a G-equivariant K-theory spectrum for a compact Hausdorff group G. If X is a compact G-space, then KG (X) is the Grothendieck construction on the semiring of isomorphism classes of finitely generated real bundles on X.* * The Atiyah-Segal completion theorem generalizes to compact Hausdorff groups if we make use of a suitable completion functor [18]. Let G be a compact Hausdorff group. We define a Thom spectrum as T OG (V ) = colimUT OG=U (V ) where the limit is over U 2 Lie(G) such that V has a trivial U-action. For more detail see [18, 7]. 3.5. The levelwise W-model structures on orthogonal G-Spectra. We make some minor modifications to the discussion of model structures in [35, III* *]. Throughout this subsection we work in the category of R-modules MR for a ring R. The category of R-modules can be described as the category of continuous D-spaces for an appropriate diagram category D. The objects are the same as those of JGV, but the morphisms are more elaborate. See [36, sec.23] and [35, I* *I.4] for details. Interpreted as a continuous diagram category in GT , we give MR the projective model structure inherited from the W-model structure on GT [25, 11.3.2]. Definition 3.13. Let 1RWI denote the collection of RVi, for all i 2 W I and all indexing representations V in U. Let 1RWJ denote the collection of RVj, f* *or all j 2 WJ and all indexing representation V in U. We call the following model structure on orthogonal G-spectra the levelwise W-model structure. Proposition 3.14. The category of R-modules has a compactly generated proper model structure with levelwise W-weak equivalences and levelwise W-fibrations (as JGV-diagrams). The cofibrations are generated by 1RWI, and the acyclic cofibrations are generated by 1RWJ. If W is an Illmancollection, then the model structure is W-topological. EQUIVARIANT HOMOTOPY THEORY FOR PRO-SPECTRA 15 Proof.The proof of the first part is similar to [36, 6.5]. The adjunction betwe* *en RVand V , gives that the maps in the classes 1RWI and 1RWJ are Hurewitz cofibrations (satisfies the homotopy extension property). Hence relative W-cell complexes are Hurewitz cofibrations. Since W is an Illmancollection the cofibra* *tion hypothesis [35, III.2.6] holds. The results of Theorem 2.7 in [35, III] extends* * if we replace based G-CW complexes with based W-CW complexes. Definition 3.15. A spectrum X is called a W - -spectrum if the adjoint of the structure maps, X(V ) ! W-V X(W ) are unbased W-equivalences of spaces for all pairs V W in V(U). 4.The stable W-model structure on orthogonal G-spectra We define stable equivalences between orthogonal G-spectra [35, III.3.2]. Definition 4.1. The n-th homotopy group of an orthogonal G-spectrum X at a subgroup H of G is ssHn(X) = colimVssHn( V X(V )) for n 0, and n ssH-n(X) = colimV RnssH0( V -R X(V )) for n 0, where the colimit is over indexing representations in U. A map f :X * *! Y of orthogonal G-spectra is a stable W-equivalence if ssHn(f) is an isomorphism for all H 2 W and all n 2 Z. We show in Theorem 4.7 below that we can Bousfield localize the levelwise model structure on the category of orthogonal G-spectra with respect to the sta* *ble W-equivalences. We follow our program of giving model structures to the category of orthogonal G-spectra with minimal assumptions on the collection of subgroups used. We need to strengthen the notion of an Illmancollection of subgroups of G. The ext* *ra condition added is needed in the proof of Corollary 4.4 and Proposition 4.5. Definition 4.2. Given a G-universe U. We say that a collection C is a U-Illman collection if C is an Illmancollection (see Def. 2.17), and if in addition the * *following holds: whenever H is the stabilizer of and element in the universe U, then H \ U is in C for all U 2 C. If U is a trivial G-universe, then a collection is U-Illmanif it is Illman. I* *f U is a complete universe, then C is a U-Illmancollection if C is a family in the cof* *amily closure of Lie(G) (see Def. 2.4). 4.1. Verifying the model structure axioms. Lemma 4.3. Let X, Y , and Z be based G-spaces. Let W be an Illmancollection. If Z is a W-cell complex and f :X ! Y is a W-equivalence, then TG (Z, X) ! TG (Z, Y ) is a W-equivalence. Proof.The higher lim spectral sequence shows that is enough to prove the result when K is Sn+^ G=L+ for L 2 W. An adjunction gives that ssHk(TG (Sn+^ G=L+ , X)) ~=[Sn+^ Sk ^ (G=L x G=H)+ , X]G , 16 H. FAUSK where the square brackets are G-homotopy classes of maps (see Prop. 2.15). Since Sn+^ Sk is a based CW-complex and (G=L x G=H)+ is a based W-cell complex by the definition of an Illmancollection in 2.17, the result follows. Corollary 4.4. Let W be a U-Illman collection of subgroups of G. A levelwise W-equivalence of G-spectra is a stable W-equivalence. Proof.We have assumed that any finite dimensional G-representation V in the universe U is a G=U-representation for a compact Lie group quotient G=U of G. Since the collection W is U-Illman, it then follows that SV is a (finite) W-cell complex. Lemma 4.3 gives that V f(V 0) is a W-equivalence for all V, V 02 V. Each ssU*, for U 2 W, is a homology theory on the homotopy category of orthog- onal G-spectra with the levelwise W-model structure. This follows by Corollary 4.4 since the H-fixed point functor commutes with wedges and with pushout along a closed inclusion (all Hurewicz cofibrations are closed inclusions since our s* *paces are weak Hausdorff). We get a stable W-model structure on orthogonal G-spectra by Bousfield localizing the levelwiseLW-model category of orthogonal spectra wi* *th respect to the homology theory h = U2W ssU*. We give a more precise description of this stable W-model structure, and de- termine the h-local objects. We follow [35, III.4]. A set of generating cofibra* *tions is 1RWI. We give a set of generating acyclic cofibrations. Let ~V,W : RV WSW ! RVS0 be the adjoint of the map SW ! ( RVS0)(V W ) ~=O(V W )+ ^O(W) R(W ) given by sending an element w in SW to e ^ i(W )(w) where e is the identity ma* *p in O(V W ), and i: S0 ! R is the unit map. Let kV,W be the map from RV WSW to the mapping cone, M~V,W, of ~V,W. Let WK be the union of 1RWJ and the set of maps of the form i kV,W for i 2 WI and indexing representations V, W in U. The box is the pushout-product map. The set WK of maps in MR is a set of generating acyclic cofibrations. Note that if W is a U-Illmancollection of subgroups of G, A is a based W-cell complex, and V is an indexing representation in U, then A ^ SV is again a based W-cell complex by Lemma 2.19. The next result, together with Corollary 4.4, show that a map between - W-spectra is a levelwise W-equivalence if and only if it is a W-equivalence. This fundamental result is an extension of [35, III.9* *]. Proposition 4.5. Assume that W is a U-Illmancollection of subgroups of G. Let f :X ! Y be a map of W - - G-spectra. If f*: ssH*(X) ! ssH*(Y ) is an isomorphism for any H 2 W, then for all indexing representations V U f(V )*: ssH*(X(V )) ! ssH*(Y (V )) is an isomorphism for all H 2 W. So f is a level W-equivalence. Proof.Let Z be the homotopy fiber of f. It is again an - G-spectrum. We want to show that ssH*(Z) = 0 for all H 2 W, implies that ssH*(Z(V )) = 0 for any indexing representations V and any H 2 W. Fix an indexing representation V and a normal subgroup N 2 Lie(G), such that N acts trivially on V . With EQUIVARIANT HOMOTOPY THEORY FOR PRO-SPECTRA 17 these choices ( V Z(V ))H = V (Z(V )H ) for all H N. Hence ssH*+|V(|Z(V )) * *is isomorphic to ssH*( V Z(V )) for all H N in W. Since Z is an - G-spectrum, an easy argument gives that ssH*(Z(V )) = 0 for all H 2 W such that H N [35, III.9.1]. We now prove the result for subgroups H in W that are not necessarily contain* *ed in N. Fix a subgroup H 2 W. Assume by induction that ssK*(Z(V )) = 0 for all subgroups K 2 W such that K H. If L is an orbit type in V , then H \ gLg-1 is in W for all g 2 G since C is a U-Illmancollection. The argument given in [3* *5, III.9] implies that ssH*(Z(V )) = 0. We now justify that we can make the induct* *ive argument. The quotient group H=H \ N is isomorphic to H . N=N, which is a subgroup of the compact Lie group G=N. Hence the partially ordered set of closed subgroups of H containing H \ N satisfies the descending chain property. We have that ssK*(Z(V )) = 0 for all K H \ N in W. We start the induction with the subgroup H \ N which by assumption is in W. For more details see [35, III.9]. As in [35, III.4.8] and [36, 9.5], the fact that the W-model structure on G -* * T is W-topological gives the following characterization of the maps that satisfy * *the right lifting property with respect to WK. Proposition 4.6. A map p: E ! B satisfies the right lifting property with respe* *ct to WK if and only if p is a levelwise fibration and the obvious map from E(V ) * *to the pullback of the diagram W E(V W ) | | fflffl| B(V )_____// W B(V W ) is an unbased W-equivalence of spaces for all V, W 2 V(U). Proof.The only modification of the proof in [35, III.4.8] is that we now have a topological model category such that if i is a W-cofibration, and p is a W-fibr* *ation then (MR )G (i*, p*) is a W-fibration of spaces. Theorem 4.7. Let W be a U-Illmancollection of subgroups of G. Let R be a ring. The category of R-modules is a compactly generated proper W-topological tensor model category such that the weak equivalences are the stable W-equivalences, t* *he cofibrations are retracts of relative 1RWI-cell complexes, and the acyclic cof* *ibra- tions are retracts of relative WK-cell complexes. Proof.The proof is almost identical to the proofs in [35, III.4],[35, III.7.4],* * and [36, 9]. Note that the proofs uses a few lemmas, given in [35], that are not explici* *t in this paper. More details of the tensor structure are given in Lemma 5.9. We sometimes denote MR together with the W-model structure by WMR . Lemma 4.8. Let H be in W. The stable homotopy group ssHn is corepresented by RR-nG=H+ , for n 0, and by R G=H+ ^ Sn, for n 0, in the category of R-modules. The homotopy group ssHn is a homology theory which satisfies the colimit axiom. The colimit axiom says that colimassH*(Xa) ! ssH*(X) is an isomorphism, where the colimit is over all finite subcomplexes Xa of the cell complex X. 18 H. FAUSK 4.2. Fibrations. We summarize the description of the fibrations in the stable W-model structure. Proposition 4.9. A map f :X ! Y is a fibration if and only if the map X(V ) ! Y (V ) is a W-fibration and the obvious map from X(V ) to the pullback of the diagram W X(V W ) | | fflffl| Y (V )____// W Y (V W ) is a unbased W-equivalence of spaces, for all V, W 2 V. The fibrant spectra are exactly the W - -spectra. A map f :X ! Y is an acyclic fibration if and only if f is a levelwise acyclic fibration. A natural fibrant replacement functor in MR is given by sending an R-module X to the R-module (4.10) V 7! colimW W X(V W ) where the colimit is over indexing representations in U. In particular, a natu* *ral fibrant replacement of the suspension spectrum RVZ (see 3.12) is the spectrum that sends V 0to colimW W+V O(V V 0 W ) ^O(V 0 W)R(V 0 W ) ^ Z. The next Lemma (and the claim after it) follows as in [35, 3.6, 3.11]. Lemma 4.11. Let f :X ! Y be a map of spectra and let V 2 V. Then X ^ SV ! Y ^ SV is a stable W-equivalence if and only if f is a stable W-equivalence. More generally, X ^ A ! Y ^ A is a W-equivalence for all W-cell complexes A and W-equivalences X ! Y . Definition 4.12. Let W be a collection of subgroups of G, and let K be a subgro* *up of G. The intersection K\W is defined to be the collection of all subgroups H 2* * W such that H K. If W is a U-Illmancollection of subgroups of G and K 2 W, then K \ W is a U|K-Illmancollection of subgroups of G. Lemma 4.13. Let W be a U-Illmancollection of subgroups of G and let K 2 W. Let Y be a fibrant object in W - GMR . Then Y regarded as a K-spectrum is fibrant in (K \ W) - K MR . Proof.This follows from the explicit description of fibrant objects in Proposit* *ion 4.9. (Alternatively, check that G^K - is left Quillen adjoint to the forgetful * *functor from G-spectra to K-spectra.) Lemma 4.13 need not remain true when the subgroup K is not in W. For applications in Section 10 we give some assumptions that guarantee that the res* *ult remains true even when K 62 W. Lemma 4.14. Let W be a U-Illmancollection of subgroups of a compact Hausdorff group G. Let f :X ! Y be a fibration in WMR . Assume that both X and Y are W - S-cell complexes. Let K be any closed subgroup of G, and let W0 be a EQUIVARIANT HOMOTOPY THEORY FOR PRO-SPECTRA 19 U|K-Illmancollection of subgroups of K such that W0W W. Then f, regarded as a map of K-spectra, is a fibration in the W0-model structure on KMR (with the universe U|K, and R regarded as a K-spectrum). Note that X and Y are required to be W -S-cell complexes not just W -R-cell complexes. This holds if they are W - R-cell complexes and R is a W - S-cell complex. Proof.Let f :X ! Y be a W-fibration between W-cofibrant objects in WGMR . Since W0 is U|K-Illmanas a K-collection of map, it suffices, by Proposition 4.9, to show that for any L 2 W0 the map f(V )L :X(V )L ! Y (V )L is a fibration, for V 2 V, and to show that the map from X(V )L to the pullback of the diagram (4.15) Y (V )L | | fflffl| ( W X(V W ))L_____//( W Y (V W ))L is an equivalence of spaces, for V, W 2 V. We need that maps from a compact space C to the L-fixed points of X(V ) and Y (V ) factor through the UL-fixed points of X(V ) and Y (V ) for some U 2 W. Since X and Y are W - S-cell complex. A map from a compact space C into a W - S-cell complex factors through a finite sub cell complex. Hence it suffices* * to verify the claim for individual cells. Let Z be a W-cell complex space. Then a map from a compact space C into V 0Z(V ) factors through ( V 0(V )Z)U for some U 2 W. We prove this claim. Note that O(W ) and G=H are W-cell complex for every indexing representation W and every H 2 W, respectively0[29]. Recall, from 3.12, that V 0(V ) is the s* *pace Z ^ O(V )+ ^O(V -V 0)SV -V , for V V 0, and a point otherwise. A map from a compact space C into the quotient of a W-cell complex Z0divided out by a compact group action, lies in the quotient of a compact subset of Z0. The claim follows. Hence a map from a compact space C into the L-fixed points of X(V ) and Y (V ) factor through X(V )UL and Y (V )UL , respectively, for some U 2 W. We are now ready to prove the Lemma. Let (4.16) Dn+ _______//_X(V )L j || f(V|)L| fflffl| fflffl| (Dn x I)+ ____//_Y (V )L be a diagram of based spaces. There exists a U 2 W such that the map from j to f(V )L factors through f(V )UL . Since f(V )UL is a fibration we get a lift * *in the diagram 4.16. Hence f(V )L is a fibration. The proof of the second claim is similar. We note that a map from a compact space C to ( W X(V W ))L, composed with the inclusion into W X(V W ), is adjoint to0a based map from C+ ^ SW to X(V W ). Hence it factors through X(V W )U , for some U0 2 W. By choosing a smaller U U0 such that U acts trivially on W , the map from C factors through ( W X(V W ))UL . Hence to check that the map from X(V )L to the pullback of 4.15 is a weak equivalence, it suff* *ices to check this on all UL-fixed points for U 2 W. This follows by our assumptions. 20 H. FAUSK Lemma 4.17. Let W be a U-Illmancollection of subgroups of a compact Hausdorff group G. Let f :X ! Y be a (co)-n-equivalence in MR between fibrant objects X and Y in the W-model structure on GMR , that are also W - S-cell complexes. Let K be any closed subgroup of G, and let W0 be a U|K-Illman collection of subgroups of K such that W0W W. Then f regarded as a map of K-spectra is a (co)-n-equivalence in the W0-model structure on KMR . Proof.Both X and Y are fibrant, so X(V ) ! Y (V ) is a W-(co)-n-equivalence, for every indexing representation V (by a modificati* *on of the proof of 4.14). Since X and Y are cofibrant we get, as in the proof of L* *emma 4.14, that X(V )L ! Y (V )L is a (co)-n-equivalence for any L 2 W0. 4.3. Positive model structures. We give some brief remarks about other model categories of spectra. Prespectra are defined by replacing the category V(U), * *in Definitions 3.6 and 3.9, by a smaller category consisting of the indexing repre* *sen- tations and the inclusions. There is a stable W-model structure on the category of pre-spectra. This model category is Quillen equivalent to the stable W-model structure on G-orthogonal spectra [35, III.4.16]. We can also consider model structures on the category of algebras. We need to remove some of the cofibrant and acyclic cofibrant generators to make sure t* *he free symmetric algebra construction takes acyclic cofibrant generators to stable W-equivalences. Let R+WI and R+WJ consist of all V -desuspensions of ele- ments in WI and WJ by indexing representations V in U such that V G 6= 0. The positive levelwise W-model structure on the category of orthogonal spectra is the model structure obtained by replacing 1RWI and 1RWJ by R+WI and R+WJ, respectively. The positive stable W-model structure on orthogonal spec- tra is obtained by replacing WK by the set WK+ consisting of the union of R+J and the maps i kV,W with i 2 I and V G 6= 0. The discussion of the positive mod* *el structure goes through as in [35, III.5]. Proposition 4.18. Let R be a commutative monoid in the category of G-orthogonal spectra. Then there is a compactly generated W-topological model structure on t* *he category of R-algebras such that the fibrations and weak equivalences are creat* *ed in the underlying positive W-model category of orthogonal G-spectra. The same applies to the category of commutative R-algebras. 4.4. Homotopy classes of maps between suspension spectra. We first give a concrete description of the set of morphisms between suspension spectra in the W-stable homotopy category on MR . We then prove some results about vanishing of the negative stable stems; they are used in Section 7. Recall that WT denote GT with the W-model structure. Lemma 4.19. Let X and Y be two based G-spaces. Then there is a natural isomorphism Ho(WMR )( 1RX, 1RY ) ~=Ho(WT )(X, colimW W (R(W ) ^ Yc)), where Yc is a cofibrant replacement of Y . EQUIVARIANT HOMOTOPY THEORY FOR PRO-SPECTRA 21 Proof.Recall the description of 1Rin 3.12. The functors 1Rand 0 are a Quillen adjoint pair. The result follows by replacing X and Y by cofibrant objects, Xc * *and Yc, in WT , and then replace 1RYc by a fibrant object as in 4.10. Corollary 4.20. Let X and Y be two based G-spaces. Then there is a natural isomorphism Ho(WMS)( 1 X, 1 Y ) ~=Ho(WT )(X, colimW W SW Y ). In particular, if X is a finite W-cell complex, then Ho(WMS)( 1 X, 1 Y ) ~=colimWHo(WT )(X ^ SW , Y ^ SW ). We next show that the negative stable stems are zero. In what follows homotopy means usual homotopy (a path in the space of maps). Lemma 4.21. Let V be a finite dimensional real G-representation, with G-action factoring through a Lie group quotient of G. Let X be a based G-space, and let n > 0 be an integer. Then any based G-map SV ! SV ^ X ^ Sn is G-null-homotopic. Proof.Assume the action on SV factors through the compact Lie group quotient G=K. The problem reduces to show that SV ! SV ^ XK ^ Sn is G=K-null homotopic for all n > 0. Hence we can assume that G is a compact Lie group. By Illman's triangulation theorem SV is a finite G-cell complex [29]. We choose* * a G-CW structure on SV . Let (G=HixDni)+ be a cell of SV . We take the Hi-fixed points of SV and compare the real manifold dimensions, denoted dim, of the fixed points of SV and the cells in SV . This gives that ni= dim(V Hi) - dim(NG Hi=Hi* *). To prove the Lemma it suffices to show that any given map f :SV ! SV ^ X ^ Sn extends over the cone SV ^ I of SV . There is a sequence SV = Y-1 ! Y0 ! Y1 ! . .!.YN = SV ^ I where Yn+1 is obtained from Yn by a pushout W G=Hi +^ Sni ______//_Yn | | | | W fflffl| fflffl| G=Hi +^ Dni+1 ____//_Yn+1 where the wedge sum is over all i such that ni= n, and N satisfies ni N for all i. Hence it suffices to show that any map _G=Hi + ^ Sni ! SV ^ X ^ Sn is G-null homotopic for all i. This is equivalent to showing that Sni ! SV Hi^XHi^Sn is n* *ull homotopic, which is true because ni= dim(V Hi) - dim(NG Hi=Hi) < dim(V Hi) + n. Lemma 4.22. Let U be any G-universe, and let W be a U-Illmancollection of subgroups of G. Then we have that Ho(WMS)( 1 G=H+ , 1 G=K+ ^ Sn) = 0, for all H, K 2 W and n > 0. 22 H. FAUSK Proof.Since W is a U-Illmancollection of subgroups of G, SV ^ G=H+ is a finite W-cell complex by Lemma 2.19 and compactness of SV ^ G=H+ . Corollary 4.20 gives that the group Ho(WMS)( 1 G=H+ , 1 G=K+ ^ Sn) is isomorphic to colimV 2V(U)Ho(WT )(SV ^ G=H+ , SV ^ G=K+ ^ Sn). It suffices to show that any map SV ^ G=H+ ! SV ^ G=K+ ^ Sn is G-null homotopic. This is equivalent to show that SV ! SV ^ (G=K)+ ^ Sn is H-null homotopic. This follows from Lemma 4.21. This Lemma allow us to form W-CW-complex approximation. Lemma 4.23. Let W be a U-Illman collection of subgroups of G. Let T be an S-module such that TjH = 0, for j < n and H 2 W. Then there is a cell complex, T 0, built out of cells of the form 1Rk0Sk-1^G=H+ ! 1Rk0Dk^G=H+ , for k-k0 n and H 2 W, and a W-weak equivalence T 0! T . Proof.The approximation can be constructed as a W-CW-complex using Lemma 4.22. Lemma 4.24. Let W be a U-Illman collection of subgroups of G. Let R and T be two S-modules. If RHi= 0, for i < m and H 2 W, and TjH = 0, for j < n and H 2 W, then (R ^ T )Hk= 0 for k < m + n and H 2 W. Proof.We can replace R and T by WI-cell complexes made of cells in dimension greater or equal to m and n, respectively by Lemma 4.23. The spectrum analogue of Lemma 2.19 gives that R ^ T is again a W-cell complex made out of cells in dimension greater or equal to m + n. The result now follows from Lemma 4.22. Proposition 4.25. Let W be a U-Illmancollection of subgroups of G. Let R be a ring spectrum such that RHn= 0 for all n < 0 and H 2 W. Then we have that Ho(WMR )( 1RG=H+ , 1RG=K+ ^ Sn) = 0, for all H, K 2 W and n > 0. Proof.The group Ho(WMR )( 1RG=H+ , 1RG=K+ ^ Sn) is isomorphic to Ho(WMS)( 1 G=H+ , 1 G=K+ ^ R ^ Sn). The result now follows from Lemma 4.21 and Proposition 4.24. (Let T be 1 G=K+ ^ Sn.) Lemma 4.26. Let W be a U-Illmancollection of subgroups of G. Let R be a ring such that RHn= 0, whenever n < 0 and H 2 W. Let T be an R-module such that TjH = 0, for j < n and H 2 W. Then there is a cell complex, T 0, built out of cells of the form 1RSk-1 ^ G=H+ ! 1RDk ^ G=H+ , for k n, and a W-weak equivalence T 0! T . Proof.This follows from Lemma 4.25 and the proof of Lemma 4.23. If the universe U is trivial and K is a not subconjugated to H in G, then the* *re are no nontrivial maps from 1 G=H+ ^ Sn to 1 G=K+ ^ Sm . We take advantage of this to strengthen Lemma 4.25. EQUIVARIANT HOMOTOPY THEORY FOR PRO-SPECTRA 23 Proposition 4.27. Let U be a trivial universe. Let W be an Illmancollection of subgroups of G. Let R be ring spectrum such that RHn= 0 for all n < 0 and H 2 W. Then, for each H, K in W, we have that Ho(WMR )( 1RG=H+ , 1RG=K+ ^ Sn) = 0, whenever n > 0 or H is not subconjugated to K. Proof.If n > 0, then the result follows from 4.25. If H is not subconjugated to* * K, then Ho(WMR )( 1RG=H+ , 1RG=K+ ^ Sn) ~=colimm ss0 m (G=K+ ^ (R ^ Sn)(Rm ))H . This is 0 since G=KH+is the basepoint. 4.5. The Segal-tom Dieck splitting theorem. We consider homotopy groups of suspension spectra. Let G be a_compact_Hausdorff group, let U be a complete G-universe, and let MS have the Lie(G)-model structure. Proposition 4.28. If Y is a G-space, then there is an isomorphism of abelian groups L H ss*(EWG H+ ^WGH Ad(WGH)Y H) ! ssG*( 1SY ) ______ where the sum is over all G-conjugacy classes of subgroups H in Lie(G). Proof.We have an isomorphism colimN2Lie(G)ss*(colimV 2UN( V SV Y N)G ) ! ssG*( 1SY ) and UN is G=N-complete. The result follows from the splitting theorem for com- pact Lie groups [33, V.9.1]. If U is a complete G-universe, then U restricted_to K is again a complete K-universe [17, Sec. 3]. So for any K 2 Lie(G), the stable K-homotopy groups of a G-space Y calculated in the G-homotopy category are isomorphic to those calculated in the K-homotopy_category. Hence the calculation of the n-th homo- topy group at K 2 Lie(G)of a G-space Y reduces to Proposition 4.28 (with G replaced by K). 4.6. Self-maps of the unit object. The additive tensor category, Ho(WMR ), is naturally enriched in the category of modules over the ring, Ho(WMR )( 1RS0, 1RS0), of self map of the unit object, 1RS0, in the homotopy category. Let us denote * *this ring by BRW, and denote BS0Wsimply by BW . The ring BRW depends on G, W, R, and the G-universe U. If G 2 W, then we can identify BRW with ssG0(R). If A is an R-algebra, then BAW is an BRW-algebra. Since all algebras of ortho* *g- onal spectra are S-algebras, it is important to understand the ring BW . If G is a compact Lie group, the universe is complete, and W is the collectio* *n of all closed subgroups of G, then BW is naturally isomorphic to the Burnside rin* *g, A(G), of G [37, XVII.2.1]. ______ Lemma 4.29. Let U be a complete G-universe and let W be the collection Lie(G). Then the self-maps of S0, in the homotopy category of WM, is naturally isomorph* *ic to colimU2Lie(G)A(G=U), 24 H. FAUSK where A(G=U) ~=Ho(WM)(G=U+ , S0) is the Burnside ring of the Lie group G=U and the maps in the colimit are induced by the quotient maps G=U+ ! G=V+ , for V < U in Lie(G). In general, it is difficult to determine BW . For example, when G is a finite* * group and W is a family, then the proof of the Segal conjecture gives that the ring B* *W is isomorphic to the Burnside ring A(G) of G completed at the augmentation ideal \H2W ker(A(G) ! A(H)), where the maps A(G) ! A(H) are the restriction maps [37, XX.2.5]. We give an elementary observation which shows that different collections W might give rise to isomorphic rings BW . Lemma 4.30. Let N be a normal subgroup of a finite group G, and let WN be the family of all subgroups contained in N. If X 2 GT has a trivial G-action and Y 2 GT , then Ho ({N}MR )( 1 X, 1 Y ) ! Ho(WN MR )( 1 X, 1 Y ) is an isomorphism. In particular, B{N} is isomorphic to BWN . Proof.Let X0 be a cell complex_replacement_of X built out of cells with trivial G-actions. The space EWN is Lie(G)-equivalent to E(G=N). This is a {N}-cell complex. Hence X0^ EG=N+ ! X0 is a cofibrant replacement of X both in the {N} and and in the WN -model categories. A WN -fibrant replacement Y 0of Y is also a {N}-fibrant replacement. Remark 4.31. If X does not have trivial G-action, then the homotopy classes [ 1 X, 1 Y ] in Lemma 4.30 are typically different for the collections {N} and* * WN , respectively. 5.The W - C-model structure on orthogonal G-spectra Let R be a ring and let C be a U-Illman collection of subgroups of G. We define K-equivalences in the C-model structure on the category of R-modules, MR , for K not necessarily in C. Then we construct a model structure with weak equivalences detected by a collection W of subgroups of G that is not necessari* *ly contained in C. We start by briefly describing the W - C-model structure on MR in the case when W is contained in C. Let C be a U-Illmancollection of subgroups of G. Let H 2 C. Then ssH* is a corepresented homology theory that satisfies the colimit axiom by Lemma 4.8. The direct sum L h = K2W,n2Z ssKn is also a homology theory that satisfies the colimit axiom. The h-equivalences are closed under pushout along cofibrations in CMR . We can now (left) Bousfield localize CMR with respect to the homology theory h [4] [25, 13.2.1]. Hence for any subcollection W in C there is a model structure on G-spectra such that the cofibrations are retracts of relative C-cell complexes and the weak equivalence* *s are maps f such that ssHn(f) is an isomorphism, for all H 2 W and all n 2 Z. EQUIVARIANT HOMOTOPY THEORY FOR PRO-SPECTRA 25 5.1. The construction of WCMR . We define homotopy groups in the C-model category with respect to subgroups not necessarily in C. Definition 5.1. Let C be a U-Illman collection_of subgroups of G. Let K be a subgroup of G such that the closure UK 2 C, for all U 2_C._ The n-th stable homotopy group at K is defined to be K*(X) = colimU2C ssUK*(X). The colimit is over the category with objects subgroups U of G that are in C * *and with morphisms containment of subgroups. The colimit is directed since C is an Illmancollection. If K 2 C, then K*and ssK*are canonically isomorphic functors. Definition 5.2. Let C and W be two collections of_subgroups_of G. Then the product collection CW has elements the closure, UH , of the product subgroup UH in G, for all U 2 C and all H 2 W. Example 5.3. The collection W = {1} satisfies CW C for any collection C. If C is a cofamily, then CW C for any collection W. Definition 5.4. Let W be a collection of subgroups of G such that CW C. Then we say that a map f between orthogonal spectra is a W-equivalence if Kn(f) is an isomorphism for all K 2 W and all integers n. Directed colimts of abelian groups respect direct sums and exact sequences. So K* is a homology theory which satisfies the colimit axiom by Lemma 4.8. The direct sum L h = K2W,n2Z Kn is again a homology theory which satisfies the colimit axiom. Hence we can Bous- field localize with respect to h. Theorem 5.5. Let C be a U-Illmancollection of subgroups of G, containing the subgroup G, and let W be any collection of subgroups of G such that CW C. Then there is a cofibrantly generated proper simplicial model structure on MR s* *uch that the weak equivalences are W-equivalences and the cofibrations are retracts* * of relative C-cell complexes. Proof.There exists a set K of relative C-G-CW complexes with sources C-G-CW complexes such that a map p has the right lifting property with respect to all h-acyclic cofibrations with cofibrant source, if and only if p is a fibration a* *nd p has the right lifting property with respect to K. To find such a set of maps K we use the cardinality argument of Bousfield.QWe need to take into account both the cardinality of G and the cardinality of V R(V ), where the product is over indexing representations in the universe U. The class of h-equivalences is clos* *ed under pushout along C-cofibrations. Hence we can apply [25, 13.2.1] to conclude that if p has the right lifting property with respect to the maps in the set K, then it has the right lifting property with respect to all h*-acyclic cofibrati* *ons. Hence there is a cofibrantly generated left proper model structure on MR with t* *he specified class of cofibrations and weak equivalence [25, 4.1.1]. It remains to* * show that the model structure is right proper and simplicial. We show that the model structure is right proper. The U-fixed point functor, for U 2 C, takes pull-back squares to pull-back squares, and fibrations to fibr* *ations. Hence the claim follows from properness of the model category of orthogonal spe* *ctra [36, 9.2]. 26 H. FAUSK We show that the model structure is simplicial. This is where we use the as- sumption that G is contained in C. The tensor and cotensor functors are given by 1R|K|^X and F ( 1R|K|, X), respectively, for a simplicial set K and an R-module X. The simplicial hom functor is given by singGMR (X, Y ). It is clear that the pushout-product map applied to a simplicial cofibration and a C-cofibration in * *MR is again a C-cofibration. If the simplicial cofibration is acyclic, then the pu* *shout- product map is in fact a C-acyclic cofibration. This follows since MR , with the C-model structure, is a C-simplicial model structure. It suffices to show that* * if X2 ! Y2 is a W - C-acyclic cofibration with C-cofibrant source, then the map from the the pushout of 1RSn-1 ^ X2_____// 1RDn ^ X2 | | fflffl| 1RSn-1 ^ Y2 to 1RDn ^ Y2 is again a W - C-acyclic cofibration [39, 2.3]. This is the case * *since our weak equivalences are given by a homology theory in the homotopy category of the tensor C-model structure on MR (see Theorem 4.7). This model structure is called the W - C-model structure on MR . The W-model structure is the W - W-model structure. We sometimes denote MR together with the W - C-model structure by WCMR . Proposition 5.6. Let C1 C2 be two U-Illmancollections of subgroups of G and let W be a collection of subgroups of G such that C1W C1 and C2W C2. Then the identity functors WC1MR ! WC2MR and WC2MR ! WC1MR are left and right Quillen adjoint functors, respectively. Hence a Quillen equivalence. Given two U-Illmancollections C1 and C2 such that C1W C1 and C2W C2. Then the union of the two collections C = C1 [ C2 is also a U-Illmancollection such that CW C. The identity functors are left Quillen equivalences from the C1 - W-model structure on MR , and from the C2 - W-model structure on MR to the C1 [ C2 - W-model structure on MR . Remark 5.7. One can also construct a W - C-model structure on the category of based G-spaces, GT . This is obtained by localizing with respect to the clas* *s of W-homotopy equivalences. 5.2. Tensor structures on MR . The category MR is a closed symmetric tensor category. We follow [39, 2] when considering the interaction of model structur* *es and tensor structures. A model structure is said to be tensorial if the follow* *ing pushout-product axiom is valid. Definition 5.8. The pushout-product axiom [39, 2.1]: Let f1: X1 ! Y2 and f2: X2 ! Y2 be cofibrations. Then the map from the pushout, P, to Y1 Y2 in the EQUIVARIANT HOMOTOPY THEORY FOR PRO-SPECTRA 27 diagram f1 1 X1 X2 _____//Y1 X2 :: 1 f2|| ||:::: fflffl| fflffl|1:f2: X1 Y2________//PL :: VVVVV LLL :: VVVVVV LLL :: f 1VVVVV LLL:: 1 VVV**AEAE:&&L Y1 Y2, is again a pushout. If, in addition, one of the maps f1 or f2 is a weak equival* *ence, then P ! Y1 Y2 is also a weak equivalence. The monoid axiom [39, 2.2]: Any acyclic cofibrations tensored with arbitrary object in M is a weak equivalence. Moreover, arbitrary pushouts and transfinite composition of such maps are weak equivalences. Lemma 5.9. The W-levelwise model structure on MR satisfies the pushout- product axiom and the monoid axiom. Proof.Since 1R is a left adjoint functor, the verification of the pushout-prod* *uct axiom for MR reduces to WT , which is Lemma 2.22. It suffices to check the monoid axiom [39, 2.2]. The acyclic cofibrant genera* *tors are of the form RV(G=U x Dn)+ ! RV(G=U x Dn x [0, 1])+ . This is a deformation retract. So its smash product with any spectrum X is again a deformation retrac* *t. Hence a W-equivalence. Pushout along a deformation retract is again a deformati* *on retract. The class of W-equivalences is closed under transfinite composition. H* *ence the class of acyclic cofibrations tensor arbitrary objects in MR are contained * *in W, and pushout and transfinite compositions of such maps is again in W. Lemma 5.10. Let M, be a closed tensor category. If M is a cofibrantly generat* *ed model structure that satisfies the pushout-product axiom, then a localized model structure on M, with the same cofibrations and a larger class of weak equivalen* *ces, W0, also satisfies the pushout-product axiom provided the following holds: (1) the sources of the cofibrant generators of M are cofibrant; (2) the class of cofibrations in W0 is closed under pushouts; and (3) if X is cofibrant and f is a cofibration in W0, then X f is in W0. Proof.The first part of the pushout-product axiom is immediate since the cofibr* *a- tions are unchanged in the localized model structure. Let f1: X1 ! Y2 be a cofibration in W0. Let f2: X2 ! Y2 be a cofibrant generator. Consider the diagram f1 1 X1 X2 _____//Y1 X2 :: 1 f2|| ||:::: fflffl| fflffl|1:f2: X1 Y2________//PL :: VVVVV LLL :: VVVVVV LLL :: f 1VVVVV LLL:: 1 VVV**AEAE:&&L Y1 Y2, 28 H. FAUSK where P is the pushout. The first and third conditions give that f1 1X2 and f1 1Y2 are cofibrations in W0. The second condition gives that X1 Y2 ! P is in W0. The two out of three axiom now gives that P ! Y1 Y2 is in W0. Proposition 5.11. The tensor (closed) category MR , with the W-C-model struc- ture, satisfies the pushout-product axiom. Hence it is a tensor model category. Proof.This follows from Lemmas 5.9 and 5.10. The third condition in the Lemma reduces to cell level considerations. See the proof of Theorem 5.5. Remark 5.12. In fact, Remark 5.7 and Proposition 5.11 give that the W - C-model structure on MR is a W - C-topological model structure. Proposition 5.13. Let U be a complete G-universe. Let W be a U-IllmanLie col- lection of subgroups of G. Then the dualizable objects in WMR (with the W-model structure) are precisely retracts of V(U)-desuspensions of finite W-cell comple* *xes. Proof.The proof in [37, XVI 7.4] goes through with modifications to allow for general R-modules instead of S0-modules. 5.3. The C-cofree model structure on MR . The W - C-model structure on MR is of particular interest when W = {1}. Definition 5.14. We say that f is a C-underlying equivalence if {1}(f) = colimU2CssU (f) is an equivalence. The name, C-underlying equivalence, is justified by the next lemma. Lemma 5.15._Assume_that G is a compact Hausdorff group and let C be the col- lection Lie(G). Let R be a non-equivariant ring spectrum. Then a map f :X ! Y between cofibrant objects (retracts of C-cell complexes) is a C-underlying equi* *va- lence if and only if f is a non-equivariant weak equivalence. Proof.This follows as in the proof of Lemma 4.14. Theorem 5.16. Assume that U is a trivial G-universe. Let C be an Illmancol- lection of subgroups of G such that G 2 C. Then there is a cofibrantly generat* *ed proper simplicial tensor model structure on MR such that the weak equivalence are C-underlying equivalences and the cofibrations are retracts of relative C-c* *ell complexes. Proof.This is a special case of Theorem 5.5 and Proposition 5.11. We refer to this model structure as the C-cofree model structure on MR . Remark 5.17. We require the universe to be trivial as part of the definition of the C-cofree model structure. When {1} is in C there is no loss of generality in making this assumption. 6.A digression: G-spectra for noncompact groups In this section we consider an example of a model structure on orthogonal G- spectra where the homotopy theory is "pieced together" from the genuine homotopy theory of the compact Lie subgroups of G. This example is inspired by conversat* *ions with Wolfgang L"uck. This section plays no role later in the paper. The model structure we construct below in Proposition 6.5 is in many ways opposite to the model structure (to be discussed) in Theorem 8.4: Compact Lie EQUIVARIANT HOMOTOPY THEORY FOR PRO-SPECTRA 29 subgroups versus compact Lie quotient groups, ind-spectra versus pro-spectra, p* *ro- universes versus ind-universes. The difficulties here lies in dealing with inv* *erse systems of universes for the finite subgroups of G. Let G be a topological group, and let X be a trivial G-universe. Let R be a symmetric monoid in the category of orthogonal G-spectra indexed on X . Let M denote the category of R-modules indexed on X . Definition 6.1. Let FG denote the family of compact Lie subgroups of G. If G is a discrete group or a profinite group then FG is the family of finite subgroups of G. The results in this section remains true if we replace FG with * *any collection of subfamilies such that whenever J < G and H 2 F0J, then H 2 F0G. By Proposition 3.14 there is a cofibrantly generated model structure on M such that the cofibrations are retracts of relative 1RFG I-cell complexes and * *the weak equivalences are levelwise FG -equivalences. We would like to stabilize M with respect to H-representations for all compact Lie subgroups H of G. An H-representation might not be a retract of a G-representation restricted to H (there might not be any nontrivial H-representations of this form). Our approach is to localize M with respect to stable H-homotopy groups de- fined using a complete H-universe, one universe for each H in FG . Definition 6.2. An FG -universe consists of an H-universe UH , for each H 2 FG , such that whenever H K, then UK is a subuniverse of UH |K. For any subgroups H K L the three resulting inclusions of universes are required to be compatible. We say that the FG -universe, {UH }, is complete if UH is a complete H-univer* *se for each H 2 FG . Lemma 6.3. There exists a complete FG -universe. Proof.Choose a complete H-universe U0Hfor each H 2 FG . Let UH be defined to be L K H (U0K|H) where the sum is over all K 2 FG that contains H. Let H be a compact Lie group. Then there is a stable model structure on orthogonal H-spectra, indexed on a trivial H-universe, that is Quillen equivale* *nt to the "genuine" model structure on orthogonal H-spectra indexed on a complete H-universe. This is proved in [35, V.1.7] (note that the condition V V0, used there, is not necessary). The point of view of doing "genuine" stable equivaria* *nt homotopy theory in the category of spectra indexed on a trivial universe has be* *en advocated by Morten Brun. Let H be a compact Lie group and let V and V0be collections of H-representati* *ons containing the trivial H-representations. Typically, V is the collection V(U) * *of all H-representations that are isomorphic to some indexing representation in an H-universe U. There is a change of indexing functor 0 V IVV0:JGVS ! JG S defined in [35, V.1.2]. The functor IVV0is an equivalence of categories with IV* *0Vas the inverse functor. The functor IVV0is a strong symmetric tensor functor. Thes* *e, and other, claims are proved in [35, V.1.5]. 30 H. FAUSK Lemma 6.4. For each compact Lie subgroup H of G the functor ssH*(IV(UH)V(X)-) is a homology theory on M with the levelwise FG -X -model structure, that satis* *fies the colimit axiom. See the discussion after Corollary 4.4. Note that any family of subgroups of H is an UH -Illman collection. (See Definition 4.2). Proof.This follows since IVV0respects homotopy colimits and weak equivalences since V(X ) V(U) [35, V.1.6]. We localize the stable FG - X -model category with respect to the homology theory given by L h = H ssH*(IV(UH)V(X)-), where the sum is over all H 2 FG . Proposition 6.5. Given an FG -universe {UH }. Then there is a cofibrantly gen- erated proper stable model structure on M such that the cofibrations are retrac* *ts of relative 1RFG I-cell complexes and the weak equivalences are the h*-equivalenc* *es. If FG is an Illman collection of subgroups of G (see Def. 2.17), then the model structure satisfies the pushout-product axiom. This model structure is called the stable {UH }-model structure on M. Proof.See the proof of Theorem 5.5. The argument given there shows that the model structure is proper. If FG is an Illman collection, then the model struct* *ure is tensorial by Lemma 5.10 and [35, III.3.11]. The cofibrant replacement of 1 S0 in this model structure (regardless of {UH* * }) is given by 1 (EFG )+ , where EFG is an FG -cell complex such that (EFG )H is contractible whenever H 2 FG , and empty otherwise [35, IV.6]. Lemma 6.6. Assume G is a discrete group. If X is a G-cell complex, then X ^ (EFG )+ is a cofibrant replacement of X. Proof.Note that G=J+ ^ G=H+ is an FG -cell complex, whenever H 2 FG and J is an arbitrary subgroup of G. The collapse map (EFG )+ ! S0 induces an FG -equivalence X ^ (EFG )+ ! X Lemma 6.7. If G has no compact Lie subgroups besides {1} (e.g. torsion-free dis- crete groups), then the stable {UH }-model structure M is the stable model stru* *cture with underlying weak equivalences. Lemma 6.8. If G is a compact Lie group, then the stable {UH }-model structure on M is Quillen equivalent to the {all} - UG -model structure on M. Let J be a subgroup of G. Let R be a monoid in the category of orthogonal G-spectra. Let M denote the category of R-modules in the category of orthogonal G-spectra indexed on X , and let M0 denote the category of R|J-modules in the category of orthogonal J-spectra indexed on X |J. Let {UH } be an FG -universe, and set {UH }H2FJ be the FJ-universe. Note that the condition in the next Lemma is trivially satisfied if G is a di* *screte group. EQUIVARIANT HOMOTOPY THEORY FOR PRO-SPECTRA 31 Lemma 6.9. Assume that G=K+ has the structure of a J - FJ-cell complex for any K 2 FG . Give M the stable FG - {UH }-model structure, and give M0 the FJ - {UH }H2FJ -model structure. Then the functor FJ(G+ , -): MJ ! MG is right Quillen adjoint to the restriction functor MG ! MJ. Proof.The restriction functor from G-spectra to J-spectra respects weak equiva- lences by the definition of weak equivalences. Since G=K is a J - FJ-cell compl* *ex for all K 2 FG , by our assumption, the relative G - FG -cell complexes are also relative J - FJ-cell complexes. Lemma 6.10. Assume G is a discrete group. Give M the stable FG -{UH }-model structure, and give M0 the stable FJ - {UH }H2FJ -model structure. Then the functor G+ ^J - :MJ ! MG is left Quillen adjoint to the restriction functor MG ! MJ. Proof.Since G+ ^J J=K+ ~=G=K+ , and the functor G+ ^J - respects change of universe functors and colimits, it follows that G+ ^J - respects cofibrations. Let f :X ! Y be a J -FJ-equivalence. We observe that G+ ^JX is isomorphic to W HgJ2H\G=J H+ ^gJg-1\H gX as an orthogonal H-spectrum. Hence the map G+ ^Jf is an FG -weak equivalence if each H+ ^gJg-1\H g(f) is a H-equivalence for H 2 FG . This follows from [35, V.1.7,V.2.3] since g(f) is a K-equivalence for every K gJg-1 \ H, because K 2 FG and K gJg-1 implies that K 2 FgJg-1. Lemma 6.11. Assume G is a discrete group. Let X be in M0and let Y be in M. Then [G+ ^J X, Y ]G ~=[X, (Y |J)]J, where the first hom-group is in the homotopy category of the {UH }-model struc- ture on R-modules, and the second hom-group is in the homotopy category of the {UH }H2FH -model structure on R|H-modules. In particular, if X and Y are G-spectra and H 2 FG , then [G=H+ ^ X, Y ]G ~=[X, Y ]H . Remark 6.12. A better understanding of the fibrations, or at least the fibrant objects would be useful. They are completely understood when G is a compact Lie group [35, III.4.7,4.12]. Calculations in the stable {UH }-homotopy theory redu* *ces to calculations in the stable homotopy categories for the compact Lie subgroups* * of G (via a spectral sequence). This follows from Lemma 6.10 using a cell filtrati* *on of a cofibrant replacement of the source by a FG -cell complex. 32 H. FAUSK 7. Postnikov t-model structures We modify the construction of the W-C-model structure on MR by considering the n - W-equivalences, for all n, instead of W-equivalences. This is used when we give model structures to the category of pro-spectra, pro- MR , in Section 8* *.2. The homotopy category of a stable model category is a triangulated category [26, 7.1]. We consider t-structures on this triangulated category together with a lift of the t-structure to the model category itself. The relationship betwe* *en n-equivalences and t-structures is given below in Definition 7.4 and Proposition 7.10. 7.1. Preliminaries on t-model categories. We recall the terminology of a t- structure [3, 1.3.1] and of a t-model structure [20, 4.1]. Definition 7.1. A homologically graded t-structure on a triangulated category D, with shift functor , consists of two full subcategories D 0 and D 0 of D, subjected to the following three axioms: (1) D 0 is closed under , and D 0 is closed under -1; (2) for every object X in D, there is a distinguished triangle X0! X ! X00! X0 such that X02 D 0 and X002 -1D 0; and (3) D(X, Y ) = 0, whenever X 2 D 0 and Y 2 -1D 0. For convenience we also assume that D 0 and D 0 are closed under isomorphisms in D. Definition 7.2. Let D n = nD 0, and let D n = nD 0. Remark 7.3. A homologically graded t-structure (D 0, D 0) corresponds to a co- homologically graded t-structure (D 0, D 0) as follows: D n = D -n and D n = D -n . Definition 7.4. The class of n-equivalences in D, denoted Wn, consists of all maps f :X ! Y such that there is a triangle F ! X f!Y ! F with F 2 D n . The class of co-n-equivalences in D, denoted coWn, consists of all maps f such that there is triangle X f!Y ! C ! X with C 2 D n . If D is the homotopy category of a stable model category K, then a map f in K* * is called a (co-)n-equivalence if the corresponding map f in the homotopy category, D, is a (co-)n-equivalence. We use the same symbols Wn and coWn for the classes of n-equivalences and co-n-equivalence in K and D, respectively. Definition 7.5. A t-model category is a proper simplicial stable model category K equipped with a t-structure on its homotopy category together with a functori* *al factorization of maps in K as an n-equivalence followed by a co-n-equivalence in K. T-model categories are discussed in detail in [20]. They give rise to interes* *ting model structures on pro-categories. EQUIVARIANT HOMOTOPY THEORY FOR PRO-SPECTRA 33 7.2. The d-Postnikov t-model structure on MR . We construct t-model struc- tures on MR making use of Postnikov sections. In Section 8 we use this t-model structure to produce model structures on the category of pro-spectra. We allow Postnikov sections where the cut-off degree of ssH*depends on H. Construction 7.6. Assume that D is the homotopy category of a proper simplicial stable model category M. Let D be the homotopy category of M. Let D 0 be a strictly full subcategory of D that is closed under . Define D n to be nD 0. Define Wn as in Definition 7.4, and lift Wn to M. Let C denote the class of cofibrations in M, and define Cn = Wn \ C and Fn = injCn. Let D n+1 be the full subcategory of D with objects isomorphic to hofib(g) for all g 2 Fn. If th* *ere is a functorial factorization of any map in M as a map in Cn followed by a map in Fn, then D 0, D 0 is a t-structure on D. Hence the model category M, the factorization, and the t-structure on D is a t-model structure on M [20, 4.12]. Let C and W be collections of subgroups of G such that CW C. Let R be a ring, and let D be the homotopy category of WCMR . Definition 7.7. A class function on W is a function d: W ! Z [ {-1, 1} such that d(H) = d(gHg-1), for all H 2 W and g 2 G. Definition 7.8. Let d be a class function. Define a full subcategory of D by Dd 0= {X | Ui(X) = 0 fori < d(U), U 2 W}. Let Dd 1be the full subcategory of D given by Construction 7.6. If H is in the closure of a sequence Ha and ssHan(X) = 0, then ssHn(X) = 0, for a G-space X. Hence it suffices to consider continuous class functions, whe* *re Z [ {-1, 1} has the topology given by letting the open sets be {n N}, for N 2 Z [ {-1, 1}, and W has the Hausdorff topology. The next result is needed to get a t-model structure on MR . Lemma 7.9. Any map in MR factors functorially as a map in Cn followed by a map in Fn. Moreover, there is a canonical map from the n-th factorization to the (n - 1)-th factorization. Proof.The proof is similar to the proof of Theorem 5.5. See also [16, Appendix]. Lemma 7.10. Let d be a class function on W. The W - C-model structure on MR , the two classes Dd 0and Dd 0, together with the factorization in Lemma 7.9 is a t-model structure. Proof.This follows from Theorem 5.5 and [20, 4.12]. This t-model structure is called the d-Postnikov t-model structure on WCMR . We call the 0-Postnikov t-model category simply the Postnikov t-model category. A map f of spectra is an n-equivalence with respect to the d-Postnikov t- structure if and only if Um(f) is an isomorphism for m < d(U) + n and Ud(U)+n* *(f) is surjective for all U 2 W. 34 H. FAUSK 7.3. An example: Greenlees connective K-theory. To show that there is some merit to the generality of d-Postnikov t-structures, we recover Greenlees * *equi- variant connective K-theory as the d-connective cover of equivariant K-theory for a suitable class function d. Let G be a compact Lie group, and let W = C be the class of all closed subgroups of G. Note that {1} is an open closed point i* *n C. (In fact, any family gives an open subset of C by Montgomery and Zippin's theor* *em [38].) Let Pn denote the n-th Postnikov section functor, and let Cn denote the * *n-th connective cover functor. Lemma 7.11. Let G be a compact Lie group. Let d be the class function such that d({1}) = 0 and d(H) = -1 for all H 6= {1}. Then X n = F (EG+ , PnX) is a functorial truncation functor for the d-Postnikov t-model structure on WMR* * . The n-th d-connective cover is given by the homotopy pullback of the left most square in the diagram (7.12) X n _____________//X_________//F (EG+ , Pn+1X) | | || | | || fflffl| fflffl| || F (EG+ , CnX)____//_F (EG+ ,_X)__//F (EG+ , Pn+1X). In particular, (KG ) 0 is Greenlees' equivariant connective K-theory [22, 3.1]. Proof.Axiom 1 of a t-structure is satisfied since F (EG+ , Pn+1X) ~=F (EG+ , Pn+1X). We combine the verification of axioms 2 and 3 of a t-structure. Let X n denote the homotopy fiber of the natural transformation X ! F (EG+ , Pn+1X). Since X ! F (EG+ , X) is a non-equivariant equivalence we conclude, using Diagram 7.12, that X n and CnX are non-equivariant equivalent. Hence X n 2 D n for all X 2 D. If Y 2 D n and X 2 D, then D(Y, F (EG+ , Pn+1X)) = 0 since Y ^ EG+ is in CnD. This example can also be extended to arbitrary compact Hausdorff groups [18]. 7.4. Postnikov sections. Suppose d is a constant function and R has trivial W-homotopy groups in negative degrees. Then there is a useful description of the full subcategory D 0 of the homotopy category D of WCMR . Definition 7.13. We say that a spectrum R is W-connective if Un(R) = 0 for all n < 0 and all U 2 W. In other words R is W-connective if R 2 D 0 for the Postnikov t-structure on WCMR . Proposition 7.14. Let R be a W-connective ring. Then there is a t-structure on the homotopy category D of WCMR defined by the two full subcategories of D: D 0 = {X | Ui(X) = 0 wheneveri < 0, U 2 W} and D 0 = {X | Ui(X) = 0 wheneveri > 0, U 2 W}. EQUIVARIANT HOMOTOPY THEORY FOR PRO-SPECTRA 35 Proof.Recall that the full subcategory D 1 of D has objects Y such that D(X, Y * *) = 0 for all X 2 D 0 [3, 1.3.4]. Proposition 4.25 gives that G=H+ ^ Sn 2 D 0, for * *all n 0 and all H 2 C. This gives that D 1 {Y | Ui(Y ) = 0 wheneverU 2 W, i 0}. The converse inclusion is proved in two steps. We first prove it when W C. Assume that X 2 D 0 and that Ui(Y ) = 0, whenever U 2 W and i 0. By Lemma 4.23 there is a C-cell complex approximation X0 ! X such that X0 is a cell complex built from cells in non-negative dimensions, and X0 ! X is a C-isomorphism in non-negative degrees, hence a W-equivalence. So we get that D(X, Y ) = 0. Since this is true for all X 2 D 0 we conclude that Y 2 D 1. We now consider the general case. Assume that X 2 D 0 and that Ui(Y ) = 0, whenever U 2 W and i 0. By the first part of the proof there are C-truncations k :X0 ! X and l :Y ! Y 0such that: (1) ssHi(X0) = 0, for all H 2 C and i < 0, ssHi(k) is an isomorphism, for all H 2 C and i 0, (2) ssHi(Y 0) = 0, f* *or all H 2 C and i 0, and ssHi(l) is an isomorphism, for all H 2 C and i < 0. Our assumptions on X and Y give that X0 ! X and Y ! Y 0are W-equivalences. Hence D(X, Y ) = D(X0, Y 0) vanish. Since X 2 D 0 was arbitrary we conclude that Y 2 D 1. The result follows. Hence a map g is a co-n-equivalence in the (0-)Postnikov t-structure if and only if Wm(g) is an isomorphism for m > n and Un(g) is injective for U 2 W. When the universe is trivial we can give a similar description of the t-model structure for more general functions d. We say that a class function d: W ! Z [ {-1, 1} is increasing if d(H) d(K) whenever H K. Proposition 7.15. Assume the G-universe U is trivial, and let R be a W-connecti* *ve ring. Let d be an increasing class function. Then there is a t-structure on the* * ho- motopy category D of WCMR defined by the two full subcategories of D: Dd 0= {X | Ui(X) = 0 wheneveri < d(U), U 2 W} and Dd 0= {X | Ui(X) = 0 wheneveri > d(U), U 2 W}. Proof.This follows from 4.27, the proof of Proposition 7.14, and [20, 4.12]. Definition 7.16. Let X be a spectrum in MR . The n-th Postnikov section of X is a spectrum PnX together with a map pnX :X ! PnX such that Um(PnX) = 0, for m > n and all U 2 W, and m (pnX): Um(X) ! Um(PnX) is an isomorphism, for all m n and all U 2 W. A Postnikov system of X consists of Postnikov factorization pn :X ! PnX, for every n 2 Z, together with maps rnX :PnX ! Pn-1X, for all n 2 Z, such that rnX O pnX = pn-1X. Dually, one defines the n-th connected cover CnX ! X of X. The n-th connected cover satisfies Uk(CnX) = 0, for k n, and Uk(CnX) ! Uk(X) is an isomorphism, for k > n. Definition 7.17. A functorial Postnikov system on MR consists of functors Pn, for each n 2 Z, and natural transformation pn :1 ! Pn and, rn :Pn ! Pn-1 such that pn(X) and rn(X), for n 2 Z, is a Postnikov system for any spectrum X. Proposition 7.18. Let R be a W-connective ring. Then the category WCMR has a functorial Postnikov system. 36 H. FAUSK Proof.This follows from Lemma 7.9. Remark 7.19. Classically, one also requires that the maps rnX are fibrations for every n and X. We can construct a functorial Postnikov tower with this property, if we restricted ourself to the full subcategory D n for some n [20, Sect.7]. 7.5. Coefficient systems. In this subsection we describe the Eilenberg-Mac Lane objects in the Postnikov t-structure. Let C be a U-Illmancollection. Let W be a collection such that WC C. Let R be a W-connective ring spectrum. Definition 7.20. The heart of a t-structure (D 0, D 0) on a triangulated catego* *ry D is the full subcategory D 0 \ D 0 of D consisting of objects that are isomorp* *hic to object both in D 0 and in D 0. The heart of a t-structure is an abelian category [3, 1.3.6]. Definition 7.21. An R-module X is said to be an Eilenberg-Mac Lane spec- trum if Un(X) = 0, for all n 6= 0 and all U 2 W. Lemma 7.22. Let C and W be collections of subgroup of G such that CW C, and let R be a W-connective ring spectrum. If d: W ! Z is the 0-function, then the heart of the homotopy category of WCMR is the full subcategory consisting of the Eilenberg-Mac Lane spectra. Proof.This follows from Proposition 7.14. We give a more algebraic description of the heart in terms of coefficient sys* *tems when W C. Let D denote the homotopy category of WCMR . Definition 7.23. The orbit category, O, is the full subcategory of D with objec* *ts 1RG=H+ , for H 2 W. This definition makes sense since W is contained in C. The orbit category de- pends on G, W, C, and the G-universe U. Definition 7.24. A W -R-coefficient system is a contravariant additive func- tor from Oop to the category of abelian groups. Denote the category of W - R-coefficient systems by G. This is an abelian category. An object Y in D naturally represents a coefficient system given by 1RG=H+ 7! D( 1RG=H+ , Y ). We make use of Lemma 4.8 in the next definition. Definition 7.25. Let X be an R-module spectrum. The n-th homotopy coefficient system of X, ssWn(X), is the coefficient system naturally represented by X ^ RR* *nS0 when n is positive, and by X ^ 1RS-n when n is negative. Lemma 7.26. There is a natural isomorphism G(ssW0( 1RG=H+ ), M) ~=M(G=H) for any H 2 W. Proof.This is an immediate consequence of the Yoneda Lemma. Proposition 7.27. Let R be a W-connected ring spectrum. The functor ssW0 in- duces a natural equivalence from the full subcategory of Eilenberg-Mac Lane spe* *ctra in the homotopy category of WCMR , to the category of W -R-coefficient systems. EQUIVARIANT HOMOTOPY THEORY FOR PRO-SPECTRA 37 Proof.We need to show that for every coefficient system M, there is a spectrum HM such that ssW0(HM) is isomorphic to M as a coefficient system, and further- more, that ssW0 induces an isomorphism D(HM, HN) ! GW (M, N) of abelian groups. We construct a functor, H, from G to the homotopy category of spectra. The natural isomorphism in Lemma 7.26 gives a surjective map of coefficient systems L L f(M): H2W M(G=H+) ss0(G=H+ ) ! M. This construction is natural in M. Let CM be the kernel of f(M) and repeat the construction with CM in place of M. We get an exact sequence (7.28)L L L L K2W CM (G=K+)ssW0(G=K+ ) ! H2W M(G=H+)ssW0(G=H+ ) ! M ! 0. This sequence is natural in M. We have that H2W M(G=H+) ssW0(G=H+ ) is naturally isomorphic to W W ssW0( H2W M(G=H+) G=H+ ) and W W GW (ssW0(G=K+ ), ssW0( H2W M(G=H+) G=H+ )) is naturally isomorphic to W W D(G=K+ , H2W M(G=H+) G=H+ ). Hence there is a map W W W W h(M): K2W CM (G=K+)G=K+ ! H2W M(G=H+) G=H+ , unique up to homotopy, so ssW0(h(M)) is isomorphic to the leftmost map in the exact sequence 7.28. Proposition 4.25 says that ssWn(G=K+ ) = 0, for all n < 0 and K 2 W. So ssn(hocofib(h(M))) = 0, for n < 0, and there is a natural isomorphism ssW0(hocofib(h(M))) ~=M. Now define HM to be the zeroth Postnikov section, P0hocofib(h(M)), of the homotopy cofiber of h(M). We get that HM is an Eilenberg-Mac Lane spectrum and there is a natural isomorphism ssW0(HM) ~=M. Conversely, let X be an Eilenberg-Mac Lane spectrum. Then there is a natural equivalence HssW0(X) ! X in D. This proves the first part. The map hocofib(h(M)) ! HM induces an isomorphism [HM, HN] ! [hocofib(h(M)), HN]. We then get an exact sequence W W W W 0 ! [HM, HN] ! [ H2W M(G=H+) G=H+ , HN] ! [ K2W C(G=K+)G=K+ , HN], where the rightmost map is induced by h(M) and the leftmost map is injective. Applying ssW0 gives an isomorphism between the last map and the map L L G( H2W,M(G=H+) ssW0(G=H+ ), N) ! G( K2W,C(G=K+) ssW0(G=K+ ), N). The kernel of this map is G(M, N), so ssW0 induces an isomorphism [HM, HN] ! G(M, N) 38 H. FAUSK of abelian groups. Remark 7.29. When d is not a constant class function, then the homotopy groups of the objects in the heart need not be concentrated in one degree. For exam- ple the heart of the t-structure in Lemma 7.11 consists of spectra of the form F (EG+ , HM), where M is an Eilenberg-Mac Lane spectrum. The heart of the Postnikov t-structure on D is not well understood for general functions d and n* *on- connective ring spectra R. 7.6. Continuous G-modules. When W 6 C it is harder to describe the full subcategory of Eilenberg-Mac Lane spectra as a category of coefficient systems.* * We give_a_description_of the heart of the Postnikov t-structure on the homotopy ca* *tegory of fnt(G)-free model structure on MR when G is a compact Hausdorff group. Let R0 denote the (continuous) G-ring colimUssU0(R). We have that {1}0(X) ~= colimUssU0(X) is a continuous R0 - G-module. Proposition 7.30. The heart of D is equivalent to the category of continuous R0 - G-modules and continuous G-homomorphisms between them. Proof.Let M be a continuous R0 - G-module. For any m 2 M let st(m) denote the stabilizer, {g 2 G | gm = m}, of m. We get a canonical surjective map L h: R0[G=st(m)] ! M where the sum is over all elements m in M. The map R[G=st(m)] ! M, corre- sponding to the summand m, is given by sending the element (r, g) to r.gm. This* * is a G-map since g0r . g0gm = g0(rgm), for g02 G. Repeating this construction with the kernel of h gives a canonical right exact sequence of continuous R0-G-modul* *es L f L (7.31) R0[G=U0] ! R0[G=U] ! M ! 0. We want to realize this sequence on spectra level. We have that {1}0(R ^ G=U+ ) ~=R0[G=U] ______ {1} as R0 - G-modules, for all U 2 fnt(G). The map f is realized as 0 applied to a map h(M): _ R ^ G=U0+! _R ^ G=U+ . Let Z be the homotopy cofiber of h(M). The right exact sequence in 7.31 is naturally isomorphic to {1}0applied * *to the sequence W W R ^ G=U0+! R ^ G=U+ ! Z. This follows from Proposition 4.25. We define HM to be the 0th Postnikov sectio* *n, P0(Z), of Z. Proposition 4.25 gives that {1}n(HM) = 0 when n 6= 0, and there is a natural isomorphism {1}0(HM) ~=M of continuous R0 - G-modules, for any continuous R0- G-module M, and there is a natural equivalence H {1}0(X) ! X in D, for an object X in the heart. It remains to show that {1}0is a full and * *faithful functor. The same argument as in the proof of Proposition 7.27 applies. 8. Pro-G-spectra In this section we use the W -C-Postnikov t-model structure on MR , discussed in Section 7, to give a model structure on the pro-category, pro- MR . For term* *i- nology and general properties of pro-categories see for example [20, 31]. We re* *call the following. EQUIVARIANT HOMOTOPY THEORY FOR PRO-SPECTRA 39 Definition 8.1. Let M be a collection of maps in C. A levelwise map g = {gs}s2S in pro- C is a levelwise M-map if each gs belongs to M. A pro-map f is an essentially levelwise M-map if f is isomorphic, in the arrow category of pro-C, to a levelwise M-map. A map in pro- C is a special M-map if it is isomorphic to a cofinite cofiltered levelwise map f = {fs}s2S with the property that for e* *ach s 2 S, the map Msf :Xs ! limt m. Hence the map from the second expression to 10.19 is an equivalence. Since (EG=N+ )n only has cells in dimension less than or equal to n we get th* *at F ( 1 (EG=N+ )(n), Y ) ! F ( 1 (EG=N+ )(n), Pm Y ) is an equivalence when n < m. Hence the map from the first expression to 10.19 * *is an equivalence. Hence, the spectrum associated to our definition of the homotopy fixed point pro-spectrum agrees with Davis' definition when G is a profinite group with fin* *ite virtual cohomological dimension. Corollary 10.20. If Y is a strict fibrant (commutative) R-algebra in pro- MR , then Y hGK, for all K, are also (commutative) R-algebras in pro- MR . Proof.By Proposition 10.18 we get that Y hGK is equivalent to {hocolimU(hocolimNF ( 1 (EG=N+ )(n), Yb))UK }b,m. The result follows since the pro-category is cocomplete by [20, 11.4], directed* * colim- its of algebras are created in the underlying category of modules, and fixed po* *ints preserves algebras. EQUIVARIANT HOMOTOPY THEORY FOR PRO-SPECTRA 55 Appendix A. Compact Hausdorff Groups In this appendix we recall some well known properties of compact Lie groups. We show that the relationship between compact Lie groups and compact Hausdorff groups are analogue to the relationship between finite groups and profinite gro* *ups. We give a point set topological remark. Since we work in the category of weak Hausdorff spaces closed subgroups of compact spaces are again compact. We first note that if G is a compact Hausdorff group, then the finite dimension* *al G-representations are all obtained from G=N-representations via a suitable quo- tient map G ! G=N where G=N is a compact Lie group quotient of G. Lemma A.1. Let V be a finite dimensional G-representation. Then the G-action on V factors through some compact Lie group quotient G=N of G. Proof.A G-representation V is a group homomorphism ae: G ! GL (V ). The action factor through the image ae(G). Since G is a compact group ae(G), with the subspace topology from GL (V ), is a closed subgroup of the Lie group GL (V ). Hence ae(G) is itself a Lie group. Again, since G is compact Hausdorff* *, the subspace topology on ae(G) agrees with the quotient topology from ae. Hence we have a homeomorphism G= kerae ~=ae(G); and G= kerae is a compact Lie group. Recall from Example 2.7 that Lie(G) denotes the collection of closed normal subgroups, N, of G such that G=N is a compact Lie group. We consider the inverse system of quotients G=N such that G=N is a compact Lie group. If G=N and G=K are compact Lie groups, then G=N \ K is again a compact Lie group, since it is a closed subgroup of G=N x G=K. Hence the inverse system is a filtered inverse system. In the next theorem it is essential that we work in the category of weak Haus* *dorff compactly generated topological spaces. Proposition A.2. Let X be a topological space with a (not necessarily continuou* *s) G-action. Then the G-action on X is continuous_if_and only if the action by G on X=N is continuous for all subgroups N 2 Lie(G)and the canonical map ae: X ! limNX=N, where the limit is over all N 2 Lie(G), is a homeomorphism. Proof.Assume that ae is a homeomorphism. Then the G-action on X is continuous since the G-action on limNX=N is continuous. We now assume that the G-action on X is continuous. We first show that ae: X ! limNX=N is a bijection. The Peter-Weyl theorem for compact Hausdorff groups implies that there are enough finite dimensional real G-representations to distinguish any t* *wo given elements in G [1, 3.39]. Hence \N N is {1}, and ae is injective. Now let * *{NxN } be an element in limNX=N. Since G is a compact group and since the G-action on X is continuous we get, for every N 2 Lie(G), that NxN is a compact subset of X. We get that GxN = GxV for all N and V in Lie(G). We denote this compact set K. Since \N N = 1 and NxN \ V xV N \ V xN\V , we conclude that the intersection of the closed sets NxN , for N 2 Lie(G), is a point. Call this poi* *nt x. We then have that ae(x) = {NxN }. So ae is surjective. 56 H. FAUSK We need to show that ae is a closed map. This amounts to showing that for any closed set A of X, and for any N 2 Lie(G) we have that N . A is a closed subset of X. When A is a compact (hence closed) subset of X this follows since N . A is the image of N x A under the continuous group action on X. Since we use the compactly generated topology the subset N . A of X is closed if for all compact subsets K of X the subset (N . A) \ K is closed in X. This is true since (N . A) \ K = (N . (A \ (N . K))) \ K and N . K is a compact subset of X. Hence ae is a homeomorphism. Corollary A.3. Any compact Hausdorff group G is an inverse limit of compact Lie groups. Proof.This follows from Theorem A.2 by letting X be G. Corollary A.4. The category of GT is a retract of the category pro- GLie(G)T . Proof.A G-space X is sent to the pro- GLie(G)-space {X=N}. The retract map is given by taking the inverse limit. By Theorem A.2 the composite is isomorphic to the identity map on GT . Let X and Y be two G-spaces. We have that GT (X, Y ) ! limNcolimN GT (X=N, Y=V ) is a bijection (but not necessarily a homeomorphism). Remark A.5. Let G be a profinite group. We observe that in the category of sets, X is a continuous G-set if and only if colimN XN ! X is a bijection. On the other hand, in the category of compactly generated space* *s, X is a continuous G-space if and only if X ! limNX=N is a continuous G-space. It is worth mentioning that the category of pro-compact Lie groups is equival* *ent to the category of compact Hausdorff groups. This follows since a closed subgro* *up of a compact Lie group is again a compact Lie group. 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Shipley, Algebras and modules in monoidal model categorie* *s. Proc. London Math. Soc. (3) 80 (2000), no. 2, 491-511. E-mail address: fausk@math.uio.no Department of Mathematics, University of Oslo, 1053 Blindern, 0316 Oslo, Norw* *ay