TATE COHOMOLOGY AND PERIODIC LOCALIZATION OF POLYNOMIAL FUNCTORS NICHOLAS J. KUHN Abstract. In this paper, we show that Goodwillie calculus, as applied to functors from stable homotopy to itself, interacts in striking ways with chromatic aspects of the stable category. Localized at a fixed prime p, let T(n) be the telescope of a vn self map of a finite S-module of type n. The Periodicity Theorem of Hopkins and Smith implies that the Bousfield localization functor associated to T(n)* is independent of choices. Goodwillie's general theory says that to any homotopy functor F from S-modules to S-modules, there is an associated tower under F, {PdF}, such that F ! PdF is the universal arrow to a d-excisive func- tor. Our first main theorem says that PdF ! Pd-1F always admits a homotopy section after localization with respect to T(n)* (and so also after localization with respect to Morava K-theory K(n)*). Thus, after periodic localization, polynomial functors split as the product of their homogeneous factors. This theorem follows from our second main theorem which is equiva- lent to the following: for any finite group G, the Tate spectrum tG(T(n)) is weakly contractible. This strengthens and extends previous theorems of Greenlees-Sadofsky, Hovey-Sadofsky, and Mahowald-Shick. The Pe- riodicity Theorem is used in an essential way in our proof. The connection between the two theorems is via a reformulation of a result of McCarthy on dual calculus. 1.Introduction and main results Over the past twenty years, beginning with the Nilpotence and Periodic- ity Theorems of E. Devanitz, M. Hopkins, and J. Smith [DHS , HopSm , R2], there has been a steady deepening of our understanding of stable homotopy as organized by the chromatic, or periodic, point of view. During this same period, there have been many new results in homotopical algebra, many fol- lowing the conceptual model offered by T. Goodwillies's calculus of functors [G1 , G2 , G3 ]. ____________ Date: July, 2003. 2000 Mathematics Subject Classification. Primary 55P65; Secondary 55N22, 55P* *60, 55P91. This research was partially supported by a grant from the National Science F* *oundation. 1 2 KUHN Here, and in a previous paper [Ku2 ], I prove theorems illustrating a beau- tiful interaction between these two strands of homotopy theory. These re- sults say that certain homotopy functors, stratified via Goodwillie calculus, decompose into their homogeneous strata, after periodic localization. The first paper concerned a highly stuctured splitting of the important functor 1 1 . Ignoring the extra structure, one is left with an illustration of the main result here: after Bousfield localization with respect to a periodic ho- mology theory, all polynomial endofunctors of stable homotopy split into a product of their homogeneous components. We now explain our main results in more detail. The periodic homology theories we consider are K(n)*, the nth Morava K-theory at a fixed prime p and with n > 0, and the `telescopic' variants T (n)*, where T (n) denotes the telescope of a vn-self map of a finite complex of type n. A consequence of the Periodicity Theorem is that the associated Bousfield class is independent of the choice of both the complex and self map. Also, we recall that T (n)*-acyclics are K(n)*-acyclic1; thus the associated localization functors are related by LK(n) ' LK(n)LT(n). Our use of concepts from Goodwillie calculus and localization theory re- quire that we work within a good model category with homotopy category equivalent to the standard stable homotopy category. Thus we work within the category S, the category of S-modules of [EKMM ]. Goodwillie's general theory then says that a homotopy functor F : S ! S admits a universal tower of fibrations under F , .. . | | fflffl| P2F:(X): tt tttt |p2| e2 tttt fflffl| tttt jP1F4(X)4 ttt jjjj tttjjje1jjjjjt |p1| tjjjjjjtte0 fflffl| F (X) ________________//_P0F (X), such that (1) PdF is d-excisive, and (2) ed : F ! PdF is the universal natural transformation to a d-excisive functor. ____________ 1The Telescope Conjecture asserts the converse, and, these days, is consider* *ed unlikely to hold for n 2. TATE COHOMOLOGY AND PERIODIC LOCALIZATION 3 Our splitting theorem then is as follows. Theorem 1.1. Let F : S ! S be any homotopy functor. For all primes p, n 1, and d 1, the natural map pd(X) : PdF (X) ! Pd-1F (X) admits a natural homotopy section after applying LT(n). The theorem can be reformulated as follows. Let DdF (X) be the fiber of pd(X) : PdF (X) ! Pd-1F (X). Then DdF is both d-excisive and homo- geneous: Pd-1DdF ' *. The theorem is equivalent to the statement that there is a natural weak equivalence of filtered spectra Yd LT(n)PdF (X) ' LT(n)DcF (X). c=0 Example 1.2. Here is the simplest example illustrating our theorem. Let p = 2. For k 2 Z, let RPk1 be the Thom spectrum of k copies of the canonical line bundle over RP 1. [Ku1 , Ex.5.7] implies that the cofibration sequence (1.1) RP-11! RP01 ! S0 splits after K(n)-localization, for all n, even though the connecting map ffi : S0 ! RP-11is nonzero in mod 2 homology. As was, in essence, observed in a 1983 paper by J.Jones and S.Wegmann [JW ], (1.1) is the suspension of the special case X = S-1 of a natural cofibration sequence of functors (1.2) (X ^ X)hZ=2 ! P2(X) ! X. One can also construct this sequence using Goodwillie calculus: see x3. Theorem 1.1 says that (1.2) splits after applying LT(n) for all n and X, even though the connecting map ffi : X ! (X ^ X)hZ=2 is often nontrivial before localization. Remark 1.3. There are various sorts of polynomial functors studied in the lit- erature differing slightly from Goodwillie's d-excisive functors: R.McCarthy has studied d-additive functors [McC ], and his student A.Mauer-Oats [MO ] has studied an infinite family interpolating between additive and excisive. As will be explained more fully in x6, the analogue of Theorem 1.1 holds in all these generalized settings. Remark 1.4. Theorem 1.1 and Corollary 1.7 below also has consequences for using the tower {PdF (X)} to understand E*n(F (X)), where En is the usual p-complete integral height n complex oriented commutative S-algebra. 4 KUHN Since it is known [H ] that K(n)*(X) = 0 if and only if E*n(X) = 0, our the- orem says that the spectral sequence associated to the tower will collapse at E1. Theorem 1.1 is deduced from a rather different result in equivariant stable homotopy theory that we now describe. If G is a finite group, let G-S denote the category of S-modules with G- action: the category of so-called `naive G-spectra'. Note that any S-module can be considered as an object in G-S by giving it trivial G-action. For Y 2 G-S, we let YhG and Y hG respectively denote associated homo- topy orbit and homotopy fixed point S-modules. There are various con- structions in the literature, more [GM ] or less [ACD , AK , Kl1, WW1 ] so- phisticated, of a natural `Norm' map N(Y ) : YhG ! Y hG satisfying the key property that N(Y ) is an equivalence if Y is a finite free G-CW spectrum. Let the Tate spectrum tG (Y ) be defined as the cofiber of N(Y ). As recently observed by J.Klein [Kl2 ], up to weak equivalence, these constructions are unique: see x2. We prove the following vanishing theorem. Theorem 1.5. For all finite groups G, primes p, and n 1, LT(n)tG (LT(n)S) ' *. This theorem will turn out to be equivalent to the following corollary. Corollary 1.6. If T (n) is the telescope of any vn-self map of a type n com- plex, then tG (T (n)) ' *. Besides implying Theorem 1.1, Theorem 1.5 also leads to the following splitting result. Corollary 1.7. For any Y 2 G-S, the fundamental cofibration sequence N(Y ) hG YhG ---! Y ! tG (Y ) splits after applying LT(n) for any n. One also immediately deduces results similar to [HSt , Cor.8.7]. Corollary 1.8. For all finite groups G, the norm map induces an isomor- phism T (n)*(BG) ~-!T (n)-*(BG). Similarly, LT(n)( 1 BG+ ) is self dual in the category of T (n)-local spectra. TATE COHOMOLOGY AND PERIODIC LOCALIZATION 5 Our two theorems are supported by three propositions. The first of these is a slight variant of results of R. McCarthy in [McC ], and establishes the connection between our two theorems. We need to recall Goodwillie's classification of homogeneous polynomial functors [G3 ]. Let d denote the dthsymmetric group. If our original functor F is finitary (terminology from [G3 ]), i.e. commutes with directed homotopy colimits, then DdF (X) is weakly equivalent to a homotopy orbit spectrum of the form (CF (d) ^ X^d)h d, where CF (d) 2 d-S is determined naturally by F . Important to us is that, even without the finitary hypothesis, there is a natural weak equivalence of the form DdF (X) ' ( dF (X))h d, where dF is a functor determined naturally by F , taking values in the category d-S. Proposition 1.9. Let F : S ! S be any homotopy functor. For all d 1, there is a homotopy pullback diagram PdF (X) ______//( dF (X))h d |pd| || fflffl| fflffl| Pd-1F (X) ____//_t d( dF (X)). This diagram is natural in both X and F . Our other two propositions together imply Theorem 1.5. The first is a new very general observation about Tate spectra. Proposition 1.10. Let R be a ring spectrum and E* a homology theory. If tZ=p(R) is E*-acyclic for all primes p, then so is tG (M) for all R-modules M and for all finite groups G. We remark that, by standard arguments, tZ=p(R) ' *, and thus is cer- tainly E*-acyclic, for all primes p such that R* is uniquely p-divisible. In particular, to apply the proposition to the pair (R, E*) = (LT(n)S, T (n)*), one need to only look at the single prime involved in the periodic theory. It is in proving our last proposition that deep results in periodic stable homotopy will be used. Proposition 1.11. For all primes p and n 1, LT(n)tZ=p(LT(n)S) ' *. At this point we need to comment on results like Theorem 1.5 in the literature. 6 KUHN The main theorem of the 1988 article by M.Mahowald and P.Shick [MS ] can be restated as (1.3) tZ=2(T (n)) ' *. A proof along their lines can presumably be done at odd primes as well. We will see that the generalization of their theorem to all primes is equivalent to Proposition 1.11, yielding one possible proof of that result. We will offer a rather different proof, using the telescopic functors of Bousfield and the author [B1 , Ku1 , B2 ]. The main theorem of the 1996 article by J.Greenlees and H.Sadofsky [GS ] reads (1.4) tG (K(n)) ' *. Their proof is elementary (in the sense that consequences of the Nilpotence Theorem are not needed), but heavily uses two special facts about K(n): it is complex oriented, and K(n)*(BZ=p) is a finitely generated K(n)*-module. Note that neither of these two facts is available when considering T (n)*. For readers interested in the simplest proof of (1.4), it is hard to imagine improving upon the clever argument given in [GS , Lemma 2.1] showing that tZ=p(K(n)) ' * , but our Proposition 1.10 offers an alternative way to proceed starting from this. The most substantial part of the main theorem of [HSa ] says that (1.5) LK(n)tG (LK(n)S) ' *. Note that, were the Telescope Conjecture true, then (1.5) and Theorem 1.5 would be equivalent; at any rate, the latter implies the former. The authors prove their theorem by starting from (1.4), and then using the Periodicity Theorem, together with the technical heart of Hopkins and D. Ravenel's proof [R2 ] that LE(n) is a smashing localization. Our proof of Theorem 1.5 bypasses the need for the Hopkins-Ravenel argument. The rest of the paper is organized as follows. In x2, we review properties of the norm map and tG , leading to a proof of Proposition 1.10. In x3, supported by the appendix, we first discuss models for LE tZ=p(LE S) for a general spectrum E, and then use telescopic functors to show that the model is contractible when E = T (n). The results of the previous two sections are combined in x4 yielding a proof of Theorem 1.5. Also in this section is a discussion of the equivalence of Theorem 1.5 and Corollary 1.6, with arguments similar in spirit to ones in [MS , HSa ]. In x5, we review what we need to about d-excisive functors, and prove Proposition 1.9 with arguments similar to those in [McC ]. In x6, we prove our splitting results, Theorem 1.1 and Corollary 1.7. As is already evident, if E is an S-module, we let LE denote Bousfield localization with respect to the associated homology theory E*. Through- out we also use the following conventions regarding functors taking values TATE COHOMOLOGY AND PERIODIC LOCALIZATION 7 in S. We write F -f!~G if f(X) : F (X) ! G(X) is a weak equivalence for all X. By a weak natural transformation f : F ! G we mean a pair of natural tranformations of the form F -g~H -h!G or F -h!H -g~G. Finally, we say that a diagram of weak natural transformations commutes if, after evaluation on any object X, the associated diagram commutes in the stable homotopy category. Acknowledgements I would like to thank various people who have helped me with this project. Randy McCarthy and Greg Arone have helped me learn about Goodwillie towers. Obviously Randy's paper [McC ] has been important to my thinking, and Greg suggested the compelling reformulation of Randy's results given in Proposition 1.9. Neil Strickland alerted me to the fact a conjecture of mine, that (1.5) was true, was already a theorem in the literature, and Hal Sadofsky similarly told me about Mahowald and Shick's theorem (1.3). Our main results have been reported on in various seminars and conferences, e.g. at the A.M.S. meetings in January, 2003, and in Oberwolfach in March, 2003. 2. Tate spectra and Proposition 1.10 2.1. Homotopy orbit and fixed point spectra. For G a fixed finite group, and Y 2 G-S, the S-modules YhG and Y hG are defined in the usual way: YhG = (EG+ ^ Y )=G, and Y hG = (Map S (EG+ , Y ))G . Both of these functors take weak equivalences and cofibration sequences in G-S to weak equivalences and cofibration sequences in S. (See [GM , Part I] for these sorts of facts.) YhG has an important additional property not shared with Y hG: it com- mutes with filtered homotopy colimits. We record the following well known facts, which are fundamental when one considers the behavior of YhG and Y hG under Bousfield localization. Lemma 2.1. If f : Y ! Z is a map in G-S that is an E*-isomorphism, then fhG : YhG ! ZhG is also an E*-isomorphism. Lemma 2.2. If Y 2 G-S is E-local, so is Y hG. 2.2. A characterization of the norm map. A recent paper by Klein [Kl2 ] exploring axioms for generalized Farrell-Tate cohomology leads to a nice characterization of norm maps, and thus Tate spectra. Proposition 2.3. Let NG (Y ), N0G(Y ) : YhG ! Y hG be natural transfor- mations such that both NG ( 1 G+ ) and N0G( 1 G+ ) are weak equivalences. 8 KUHN Then there is a unique weak natural equivalence f(Y ) : YhG -~! YhG such that the diagram NG(Y ) YhG _____//Y hG FF OO FFF |N0G(Y|) f(Y )F""FF| YhG commutes. It follows that the cofibers of NG (Y ) and N0G(Y ) are naturally weakly equivalent. We sketch the proof, using the sorts of arguments in [Kl2 ]. Call a homotopy functor H : G-S ! S homological if it preserves homo- topy pushout squares and filtered homotopy colimits. Then Klein, in the spirit of [WW2 ], observes that any homotopy functor F : G-S ! S admits a universal left approximation by a homological functor, i.e. there exists homological functor F hom, and a natural transformation F hom(Y ) ! F (Y ) satisfying the expected universal property. Applying this to the case F (Y ) = Y hG, and observing that YhG is homo- logical, shows that there is a unique weak natural transformation g : YhG ! Y hG,homyielding a commutative diagram of weak natural transformations NG(Y ) YhG J______//YOhGO JJ | JJJ | g(Y )JJ%%J| Y hG,hom. The right upward map is certainly an equivalence for Y = 1 G+ , and, by assumption, so is the top map. Thus g is a weak natural transformation between homological functors that is an equivalence when Y = 1 G+ . It follows that g is weak equivalence. Applying this same argument to N0Gyields the proposition. 2.3. Tate spectra. We refer to any natural transformation NG as in the last proposition as a norm map. The cofiber of NG (Y ) is the associated Tate spectrum, denoted tG (Y ). Both NG and tG are unique in the sense of Proposition 2.3; their existence is shown in the various references cited in the introduction. It is immediate that tG preserves weak equivalences and cofibration se- quences. From [GM , Prop. I.3.5], we deduce Lemma 2.4. If R is a (homotopy) ring spectrum with trivial G action, and M is an R-module, then tG (R) is a ring spectrum, and tG (M) is a tG (R)- module. Furthermore, RhG ! tG (R) is a map of R-algebra spectra. TATE COHOMOLOGY AND PERIODIC LOCALIZATION 9 Fix Y 2 G-S. For each subgroup H of G, Y can be regarded as being in H-S by restriction. From [GM , pp.28-29], one deduces Lemma 2.5. The assignment G=H 7- ! tH (Y ) defines a Mackey functor to the stable homotopy category. Furthermore, Y hH ! tH (Y ) is a map of Mackey functors. In x5, we will use the following familiar property of the norm map. In the literature, this explicitly appears, with a short axiomatic proof, as [AK , Prop.2.10]. Lemma 2.6. If K is a finite free G-CW complex, then for all Y 2 G-S, tG (MapS (K, Y )) ' *. 2.4. Proof of Proposition 1.10. Recall that R is a ring spectrum, and we are assuming that tZ=p(R) is E*-acyclic. We wish to show that tG (M) is also E* acyclic, for all R-modules M, and for all G. We first note that we can assume M = R. For tG (M) is a tG (R)-module, and thus the former will be E*-acyclic if the latter is. Next we show that we can reduce to the case when G is a p-group. For each prime p dividing the order of G, let Gp < G be a p-Sylow subgroup. Then we have Lemma 2.7. Given Y 2 G-S and E* a generalized homology theory, tG (Y ) will be E*-acyclic if tGp(Y ) is E*-acyclic for all p dividing the order of G. Proof.We recall that the completion of the Burnside ring A(H) is denoted bA(H). The assignment G=H 7- ! Y hH is then an bA-module Mackey functor in the sense of [MM ]. Thus so is G=H 7- ! tH (Y ), and then also G=H 7- ! E*(tH (Y )). Now [MM , Cor.4] implies the lemma. Having reduced Proposition 1.10 to the case when G is a p-group, and is thus solvable, the next lemma implies the proposition. Lemma 2.8. Let K be a normal subgroup of G, Q = G=K, R a ring spectrum, and E* a homology theory. If tK (R) and tQ (R) are both E*- acyclic, so is tG (R). Proof.For Y 2 G-S, consider the composite NK (Y )hQ hK NQ(Y hK) hK hQ hG YhG ' (YhK )hQ ------! (Y )hQ ------! (Y ) ' Y . We will know that this composite can be considered a norm map if we check that each of these maps is an equivalence when Y = 1 G+ . 10 KUHN As there is an equivalence of S-modules with K-action ` 1 G+ ' 1 K+ , gK2Q it follows that NK ( 1 G+ ), and thus NK ( 1 G+ )hQ , is an equivalence. As there are equivalences of S-modules with Q-action NK ( 1 G+) 1 1 ( 1 G+ )hK --------~ ( G+ )hK -!~ Q+ , it follows that NQ (( 1 G+ )hK ) is an equivalence. We conclude from this discussion that if both NK (R)hQ and NQ (RhK ) are E*-isomorphisms, then NG (R) will also be an E*-isomorphism, and thus tG (R) will be E*-acyclic. By assumption, tK (R) is E*-acyclic. Thus NK (R) is an E*-isomorphism. By Lemma 2.1, NK (R)hQ is also. By assumption, tQ (R) is E*-acyclic. As tQ (RhK ) is a tQ (R)-module, we conclude that tQ (RhK ) is also E*-acyclic, so that NQ (RhK ) is an E*- isomorphism. 3. Telescopic functors and Proposition 1.11 The goal of this section is to prove that LT(n)tZ=pLT(n)S ' *. We will prove this by establishing that the localized unit map LT(n)S ! LT(n)tZ=pLT(n)S is null. In outline our argument showing this is as follows. It is well known that tZ=pS can be written as certain inverse limit of Thom spectra. Starting from this, we will show that the unit map S ! tZ=pS factors though an inverse limit of `connecting maps' associated to the Goodwillie tower of the functor 1 1 applied to spheres in negative dimensions. We warn the reader of technical complications: odd primes are less pleasant than p = 2, we use the theorems of W.H.Lin and J.Gunawardena establishing the Segal conjecture for Z=p, and a key homological calculation is deferred to an appendix. It will follow that the localized unit will factor through the inverse limit * *of the localized connecting maps. That this inverse limit is null will then be an easy consequence of constructions of Bousfield and the author [B1 , Ku1 , B2] showing that LT(n) factors through 1 . These `telescopic' constructions heavily use the Periodicity Theorem of Hopkins and Smith [HopSm ], and thus are also heavily dependent on the Nilpotence Theorem of [DHS ]. 3.1. Models for LE tZ=pLE Y and LE t pLE Y . If ff is an orthogonal real representation of a finite group G, we let S(ff) and Sffrespectively denote the associated unit sphere and one point compactified sphere. Thus S(ff) TATE COHOMOLOGY AND PERIODIC LOCALIZATION 11 has an unbased G-action while the G-action on Sffis based, and there is a cofibration sequence of based G-spaces S(ff)+ ! S0 ! Sff. Fix a prime p, and let æ denote p acting on Rp= (R) in the usual way. The action of Z=p < p on S(æ) is free, and one concludes that the infinite join S(1æ) is a model for EZ=p. This quickly leads to the following well known description of tZ=p. Lemma 3.1. (Compare with [GM , Thm.16.1].) For Y 2 G-S, there is a natural weak equivalence tZ=pY ' holim MapS (Skj, Y )hZ=p. k We need a generalization of this. Lemma 3.2. For Y 2 G-S, there is a natural weak equivalence LE tZ=pLE Y ' holim LE (MapS (Skj, Y )hZ=p). k If (p - 1)! acts invertibly on E*, e.g. if E is p-local, there is a natural weak equivalence LE t pLE Y ' holim LE (MapS (Skj, Y )h p). k These equivalences are also natural with respect to the partially ordered set of Bousfield classes , and there are commutative diagrams LE t pLE Y___~__//holimk LE (MapS (Skj, Y )h p) | | | | fflffl| fflffl| LE tZ=pLE Y _~__//_holimk LE (MapS (Skj, Y )hZ=p). Proof.By definition, LE tZ=pLE Y is the cofiber of LE NZ=p(LE Y ) : LE (LE Y )hZ=p ! LE (LE Y )hZ=p. The domain of this map can be simplified: LE YhZ=p ! LE (LE Y )hZ=p 12 KUHN is an equivalence. Meanwhile, the range of this map rewritten via the fol- lowing chain of natural weak equivalences: LE (LE Y )hZ=p~-(LE Y )hZ=p ~-!Map hZ=p S (S(1æ)+ , LE Y ) ~-!holimMap (S(kæ) , L Y )hZ=p k S + E ~-!holimL Map (S(kæ) , L Y )hZ=p k E S + E -~ holimL Map (S(kæ) , L Y ) k E S + E hZ=p -~ holimL Map (S(kæ) , Y ) . k E S + hZ=p The crucial second to last equivalence here is induced by norm maps which are equivalences since Z=p acts freely on S(kæ). Thus LE tZ=pLE Y has been identified: LE tZ=pLE Y~-!holim cofiber{LE YhZ=p ! LE MapS (S(kæ)+ , Y )hZ=p} k ~-!holim L Map (Skj, Y ) . k E S hZ=p The proof of the statements for t p are similar, noting that, under the hypothesis that (p - 1)! acts invertibly on E*, the norm maps MapS (S(kæ)+ , LE Y )h p ! MapS (S(kæ)+ , LE Y )h p will still be equivalences. For r 0, and X an S-module, we let DrX = (X^r)h r, and we recall that there are natural transformations DrX ! Dr X. Specializing to r = p, a quick check of definitions verifies the next lemma. Lemma 3.3. There is a natural weak equivalence kDp -k X ' MapS (Skj, X^p)h p, and thus there is a p-local equivalence t pS ' holim k+1DpS-k . k Define dk : S ! k+1DpS-k to be be the composite S unit--!t pS -! k+1DpS-k . As the restriction map t pS ! tZ=pS is unital, our various observations combine to yield the following proposition. Proposition 3.4. If E is p-local, LE tZ=pLE S ' * if and only if holim LE dk : LE S ! holimLE k+1DpS-k k k is null. TATE COHOMOLOGY AND PERIODIC LOCALIZATION 13 3.2. The Goodwillie tower of 1 1 . Recall that 1 Z denotes the sus- pension spectrum of a space Z, and that 1 has right adjoint 1 , where 1 X is the zerothspace of a spectrum X. Let Pr(X) denote the rth functor in the Goodwillie tower of the functor 1 1 : S ! S. Thus this Goodwillie tower has the form .. . | | fflffl| P3(X)99 sss ssss p3(X)|| e3(X)ssss fflffl| ssss ii4P2(X)4 sse2(X)iiiiiiss sss iiiiii p2(X)|| ssiiiiise1(X) fflffl| 1 1 X _________________//P1(X). This tower has the following fundamental properties. (1) If X is 0-connected, then 1 1 X ! holimrPr(X) is an equivalence. (2) The fiber of pr(X) : Pr(X) ! Pr-1(X) is naturally weakly equivalent to Dr(X). (3) There are equivalences D1X ' P1X ' X, and via the second of these, e1(X) : 1 1 X ! P1X can be identified the with evaluation map ffl(X) : 1 1 X ! X. All of these properties can be deduced from Goodwillie's general theory. For an explicit discussion of these (and more) see [AK ] or [Ku2 ]. 3.3. Telescopic functors. Bousfield and the author have deduced the the following consequence of the Periodicity Theorem. Theorem 3.5. There exists a functor n : Spaces ! S-modules and a natural weak equivalence n 1 X ' LT(n)X. With the result stated at the level of homotopy categories, and with K(n) replacing T (n), this is the main theorem of [Ku1 ]. However the sorts of constructions given there, and in [B1 ] (for n = 1), yield the theorem as stated: see [B2 ]. This has the following immediate corollary [Ku1 , Ku2 ]. 14 KUHN Corollary 3.6. There is a natural factorization by weak S-module maps LT(n) 1 1 X ''n(X)oo77oo OOOLT(n)ffl(X)OO oooo OOO ooo OO''O LT(n)X _________________________LT(n)X. To use this, we recall an observation about reduced homotopy functors, functors F : S ! S such that F (X) is contractible whenever X is. Good- willie observes that then there is an induced weak natural transformation F (X) -! F ( X). The naturality is with respect to both X and F . For example, if F = Dr, this natural transformation agrees with the one discussed previously. In particular, we can apply this construction to both the domain and range of the natural transformation LT(n)Pp ! LT(n)P1, evaluated on -k X for all k 0. Recalling that LT(n) commutes with suspension and P1(X) ' X, we obtain maps holim kLT(n)Pp( -k X) ! LT(n)X. k Theorem 3.7. holim kLT(n)Pp( -k X) ! LT(n)X admits a homotopy sec- k tion. Proof.A section is given by holim k(LT(n)ep( -k X) O jn( -k X)). k 3.4. Specialization to odd spheres. Standard homology calculations as in [CLM , BMMS ] imply the next lemma. Lemma 3.8. Localized at an odd prime p, DrSk ' * for odd k 2 Z, and for 2 r p - 1. Thus (for all primes p) the natural map holim kPp-1(S-k ) ! S k is a p-local equivalence. Continuing the cofibration sequence DpX ! Pp(X) ! Pp-1(X) one step to the right defines a natural transformation ffi(X) : Pp-1(X) ! DpX. Localized at p, define ffik : S ! k+1DpS-k to be the composite kffi(S-k)k+1 -k S -~ kPp-1(S-k ) ------! DpS . Proposition 3.9. holim LT(n)ffik : LT(n)S ! holimLT(n) k+1DpS-k is k k null. TATE COHOMOLOGY AND PERIODIC LOCALIZATION 15 Proof.Localized at p, there is a cofibration sequence holimLT(n) kPpS-k ! LT(n)S ! holimLT(n) k+1DpS-k . k k Theorem 3.7 says that the first map has a section. Thus the second map is null. 3.5. Proof of Proposition 1.11. A comparison of Proposition 3.4 with Proposition 3.9 shows that we will have proved Proposition 1.11 once we check the following lemma. Lemma 3.10. holim dk : S ! holim k+1DpS-k factors through k k holimffik : S ! holim k+1DpS-k . k k Proof.W.H.Lin's theorem [L ], when p = 2, and J.Gunawardena's theorem [Gun , AGM ], when p is odd, can be stated in the following way: holim dk : S ! holim k+1DpS-k k k is p-adic completion. It follows that we need to check that holim ffik 2 k ß0(holim k+1DpS-k ) ' Zp is a topological generator. As topological gen- k erators of Zp are detected mod p, the next lemma, whose proof is deferred to the appendix, completes our argument. Lemma 3.11. ffi(S-1 ) : Pp-1(S-1 ) ! DpS-1 is nonzero in mod p homol- ogy. 4. The proofs of Theorem 1.5 and Corollary 1.6 We begin this section by noting how Proposition 1.11 and Proposition 1.10 together imply Theorem 1.5. Proposition 1.11 can be restated as saying that tZ=pLT(n)S is T (n)*-acyclic. Recalling that the localization of a ring spectrum (e.g. S) is again a ring spectrum, Proposition 1.10 can then be applied to the pair (R, E*) = (LT(n)S, T (n)*), to conclude that tG LT(n)S is T (n)*-acyclic for all G. This is a restatement of Theorem 1.5. Now we turn to showing how Corollary 1.6 can be deduced from Theo- rem 1.5, and vice versa. We need to review some of the fine points of the Periodicity Theorem. (A good reference for this is [R2 ].) We fix a prime p, and work with p-local spectra. A finite spectrum F is of type n if K(n)*(F ) 6= 0, but K(i)*(F ) = 0 for i < n. Let Cn = {finiteF | F has type at leastn}. Then every F 2 Cn admits a vn self map: a map f : dF ! F such that K(n)*(f) is an isomorphism, but K(i)*(f) = 0 for all i 6= n. If n > 0, then d will necessarily be positive. In all cases, f is unique and natural up to iteration. Thus there is a well defined functor from Cn to spectra sending F to v-1nF , the telescope of any vn self map of F . We note that vn preserves both cofibration sequences and retracts. 16 KUHN The Thick Subcategory Theorem says that any thick subcategory of the category of p-local spectra, i.e. any collection of p-local finite spectra clos* *ed under cofibration sequences and retracts, is Cn for some n 0. We recall that LT(n) denotes Lv-1nFfor any F of type n. From the facts stated above, it is easily verified that this is independent of choice of F , a* *nd that for all F 2 Cn, LT(n)(F ) = v-1nF . Finally we note that if F has type n and F 0has type i 6= n, then v-1nF ^ v-1iF 0' *. Lemma 4.1. Fix a finite group G. The following conditions are equivalent. (1) tG (LT(n)S) is T (n)*-acyclic. (2) For all F 2 Cn, tG (v-1nF ) ' *. (3) For all type n complexes F , tG (v-1nF ) ' *. (4) There exists a type n complex F such that tG (v-1nF ) ' *. Note that statement (1) is the conclusion of Theorem 1.5 and (3) is the conclusion of Corollary 1.6. Clearly (2) implies (3), which in turn implies (4). To see that (4) implies (2), note that the collection of F 2 Cn such that tG (v-1nF ) ' * forms a thick subcategory contained in Cn. Such a thick subcategory will equal all of Cn if it contains any type n finite. (This type of reasoning appears in [MS ].) Now suppose (1) holds. Since v-1nF ' LT(n)F , it is an LT(n)S-module, and we see that tG (v-1nF ) is (v-1nF )*-acyclic for all finite F of type n. It* * is easy to find a type n finite F that is a ring spectrum; thus so is R = v-1nF . But then tG (R) will be an R*-acyclic R-module, and thus contractible, i.e. statement (4) holds. It remains to show that (2) implies (1). We reason as in [HSa ]. Define finite spectra F (0), . .,.F (n) by first setting F (0) = S, and then recursively defining F (i + 1) to be the cofiber of a vi self map of F (i). Ravenel [R1 ] observes that if f : dX ! X is a self map with cofiber C and telescope T , then = . Applying this n times leads to an equality of Bousfield classes n-1` = . i=0 Smashing this with tG (LT(n)S), and noting that tG (LT(n)S) ^ F (n) ' tG (LT(n)F (n)) ' tG (v-1nF (n)), TATE COHOMOLOGY AND PERIODIC LOCALIZATION 17 leads to n-1` = . i=0 Smashing this with T (n), and noting that T (n) ^ v-1iF (i) ' * if i < n, leads to = . If (2) holds, then the right side of this last equation is the Bousfield class * *of a contactible spectrum. Thus so is the left, i.e. (1) holds. 5. Polynomial functors and Tate cohomology In this section we sketch a proof of Proposition 1.9. As I hope will be clear, this proposition is just a variant of [McC , Prop.4], and our proof uses precisely the same ideas that McCarthy does. 5.1. Review of Goodwillie calculus. In the series of papers [G1 , G2 , G3 ], Tom Goodwillie has developed his theory of polynomial resolutions of homotopy functors. We need to summarize some aspects of Goodwillie's work as they apply to functors from S-modules to S-modules. Throughout we cite the version of [G3 ] of June, 2002. In [G2 ], Goodwillie begins by defining and studying the total homotopy fiber of a cubical diagram. For example the total homotopy fiber of a square X0 _____//_X1 | | | | fflffl| fflffl| X2 ____//_X12 is the homotopy fiber of the evident map from X0 to the homotopy pullback of the square with X0 omitted. A cubical diagram is then homotopy cartesian if its total fiber is weakly contractible. Dual constructions similarly define total homotopy cofibers and homotopy cocartesion cubes. We note that in a stable model category like S, a cubical diagram is homotopy cartesian exactly when it is homotopy cocartesion. A cubical diagram is strongly homotopy cocartesion if each of its 2 dimen- sional faces is homotopy cocartesion. A functor is then said to be d-excisive if it takes strongly homotopy cocartesion (d + 1)-cubical diagrams to homo- topy cartesian cubical diagrams. In [G3 ], given a functor F , Goodwillie proves the existence of a tower {PdF } under F so that F ! PdF is the universal arrow to a d-excisive functor, up to weak equivalence. For functors with range in a stable model category, Goodwillie [G3 ] gives a description of how DdF (X), the fiber of PdF (X) ! Pd-1F (X), can be computed by means of cross effects. We describe how this goes in our setting. 18 KUHN Let F : S ! S be a functor. Let d = {1, 2, . .,.d}. In [G3 , x3], crdF , the dth cross effect of F , is defined to the the functor of d variables given as t* *he total homotopy fiber ` (crdF )(X1, . .,.Xd) = TotFib F ( Xi). T d i2d-T A d-variable homotopy functor H : Sd ! S is reduced if H(X1, . .,.Xd) is contractible whenever any of the Xi are. Given such a functor, its multi- linearization L(H) : Sd ! S is defined by the formula (5.1) L(H)(X1, . .,.Xd) = hocolimn n1+...+ndH( n1X1, . .,. ndXd). i!1 This will be 1-excisive in each variable. Now define dF : S ! d-S by the formula dF (X) = L(crdF )(X, . .,.X). Then [G3 , Theorems 3.5, 6.1] says that there is a natural weak equivalence (5.2) DdF (X) ' ( dF )(X)h d. We need to explain some of the ideas behind this formula. Firstly, d(F ) ! d(PdF ) is always an equivalence, and it follows that one can assume the original functor F is d-excisive. If F is d-excisive then crdF is already 1-excisive in each variable [G3 , Prop.3.3], and so dF (X) can be identified with (crdF )(X, . .,.X). In this case, the natural map DdF (X) ! PdF (X) identifies with the natural transformation ffd(X) : ( dF )(X)h d ! F (X) defined to be the composite `d ( dF )(X)h d ! F ( X)h d ! F (X). i=1 W d Here the second map is induced by the fold map i=1X ! X. Goodwillie proves (5.2) by verifying that crd(ffd) is an equivalence, so that Dd(ffd) is an equivalence. Enroute to this, he shows that there is a natural equivariant weak equivalence crd( dF ) ' d+ ^ crdF. 5.2. Dual constructions. In [McC ], McCarthy investigates `dual calculus'. In this spirit, replacing wedges by products, fibers by cofibers, etc., leads to constructions dual to the above. In particular, given F : S ! S, we define crdF : Sd ! S by the formula Y (crdF )(X1, . .,.Xd) = TotCofibF ( Xi), T d i2T TATE COHOMOLOGY AND PERIODIC LOCALIZATION 19 and then we define dF : S ! d-S by dF (X) = L(crdF )(X, . .,.X). Because both the domain and range of F is a stable model category, one sees that each of the natural transormations crdF ! crdF and dF ! dF are weak equivalences. If F is d-excisive then dF (X) can be identified with (crdF )(X, . .,.X). In this case, we define the weak natural transformation ffd(X) : F (X) ! ( dF )(X)h d to be the zig-zag composite F (X) ! F (Xd)h d ! ( dF )(X)h d -~ ( dF )(X)h d. Here the first map is induced by the diagonal X ! Xd. Arguments dual to Goodwillie's show that the next lemma holds. Lemma 5.1. (Compare with [McC , Lemmas 3.7,3.8].) Let F : S ! S be d-excisive. (1) crd(ffd), and thus Dd(ffd), is an equivalence. (2) There is a natural equivariant weak equivalence crd( dF ) ' MapS ( + , crdF ). 5.3. Proof of Proposition 1.9. Proposition 1.9 is a formal consequence of Lemma 5.1. First of all, we observe the following. Lemma 5.2. (Compare with [McC , proof of Prop.4].) Let F be d-excisive. Then t d( dF ) is (d - 1)-excisive. Thus the cofibration sequence Dd(( dF )h d) ! Pd( dF )h d ! Pd-1(( dF )h d) identifies with the norm sequence ( dF )h d ! ( dF )h d ! t d( dF ). Proof.For the first statement, we check that crd(t d( dF )) ' *: crd(t d( dF )) ' t d(crd( dF )) ' t d(MapS ( + , crdF )) ' *. Here we have used Lemma 5.1(2) and Lemma 2.6. As ( dF )h d is d-excisive and homogeneous, the second statement fol- lows. 20 KUHN Now we turn to the proof of Proposition 1.9. We can assume that F is d-excisive. Assuming this, the last lemma implies that the weak natu- ral tranformation ffd(X) : F (X) ! ( dF (X))h d induces a commutative diagram of weak natural transformations DdF (X) _____//_( dF (X))h d | | | | fflffl| fflffl| PdF (X) ______//( dF (X))h d | | | | fflffl| fflffl| Pd-1F (X) ____//_t d( dF (X)). In this diagram each of the vertical columns is a homotopy fibration sequence of S-modules. The top map is a weak equivalence thanks to Lemma 5.1(1). Thus the bottom square is a homotopy pullback diagram. 5.4. Polynomial functor variants. McCarthy and his student Mauer- Oats [MO ] have explored various different notions of what it might mean to say a functor F : A ! B is polynomial of degree at most d, with d-excisive and d-additive as two special cases. In these variants B should surely be a reasonable model category, but A can often be a category with much less structure. As a hint of why this might be true, note that the definition of cross effects only uses the existence of finite coproducts in A. If B is any stable model category admitting norm maps, and A is also appropriately stable, then the evident analogue of Proposition 1.9 still holds. The discussion above goes through with one little change: the formula (5.1) for the (multi)linearization process L needs to be adjusted to reflect the notion of degree 1 functor at hand. Note that our proof of Proposition 1.9 didn't use this formula (nor did McCarthy's arguments in [McC ]). Of relevance to the next section, we note that these variants of L are still homotopy colimits, and thus preserve E*-isomorphisms. 6.Localization and the proofs of Theorem 1.1 and Corollary 1.7 In this section, we show how our vanishing Tate cohomology result, The- orem 1.5, leads to the splitting results Theorem 1.1 and Corollary 1.7. To simplify notation, we let L = LT(n). Proof of Corollary 1.7.Let Y be an S-module with G action. We wish to show that the norm sequence N(Y ) hG YhG ---! Y ! tG (Y ) splits after applying L. Thus we need to construct a left homotopy inverse to L(N(Y )). TATE COHOMOLOGY AND PERIODIC LOCALIZATION 21 The localization map Y ! LY induces a commutative diagram N(Y ) YhG ______________//_Y hG | | | | fflffl|N(LY ) fflffl| (LY )hG ___________//(LY )hG . Applying L to this yields the diagram L(N(Y )) L(YhG )______________//_L(Y hG) |o| |L(''hG)| fflffl|L(N(LY )) fflffl| L((LY )hG )____~_____//_L((LY )hG ). Here the left vertical map is an equivalence, as homology isomorphisms are preserved by taking homotopy orbits (Lemma 2.1). The lower map, L(N(LY )), is an equivalence by Theorem 1.5: its cofiber, L(tG (LY )), is a module over L(tG (LS)), and is thus contractible. Our desired left homotopy inverse is now obtained by composing the right vertical map of the diagram with the inverses of the two indicated equiva- lences. Proof of Theorem 1.1.We are given a functor F : S ! S and wish to prove that DdF (X) ! PdF (X) ! Pd-1F (X) splits after applying L. Thus we need to construct a left homotopy inverse to LDdF (X) ! LPdF (X). We need a lemma that plays the role that Lemma 2.1 played in the pre- vious proof. Call a natural transformation F ! G an E*-isomorphism, if F (X) ! G(X) is an E*-isomorphism for all X. Formula (5.2) says that Dd is the composition of constructions each of which preserve E*-isomorphisms, and thus we have Lemma 6.1. If F ! G is an E*-isomorphism, then so is DdF ! DdG. Remark 6.2. This lemma holds for the variants on the notion of d-excisive, as discussed above in x5.4. Armed with this lemma, Theorem 1.1 is proved as follows. The localization natural transformation F ! LF , together with Proposi- tion 1.9, induce a commutative diagram DdF _____//Dd(LF )_~__//_ d(LF )h d | | | | | | fflffl| fflffl| fflffl| PdF _____//_Pd(LF_)___// d(LF )h d. 22 KUHN Applying L to this, gives the diagram LDdF _~__//_LDd(LF )_~__//L( d(LF )h d) | | | | | o| fflffl| fflffl| fflffl| LPdF _____//_LPd(LF )____//L( d(LF )h d). Here the top left natural transformation is an equivalence by the lemma just stated. The right vertical natural transformation is an equivalence by Theorem 1.5, as its cofiber, L(t d( d(LF )), is an L(tG (LS))-module, when evaluated on any X. (Though not necessarily local, due to the hocolimit construction L, d(LF )(X) is nevertheless an LS-module.) Our desired left homotopy inverse is now obtained by composing the nat- ural transformation along the bottom of this diagram with the inverses of the three indicated equivalences. Appendix A. Proof of Lemma 3.11 We begin with some needed notation. Recall that Pr denotes the rth Goodwillie approximation to the functor 1 1 . We let ffi(X) : Pr-1(X) ! DrX denote the connecting map for the cofibration sequence DrX ! Pr(X) ! Pr-1(X). Given any reduced homotopy functor F : S ! S, we let (X) : F (X) ! F ( X) denote the canonical natural map. Fixing a prime p, all homology will be with Z=p coefficients. The Steenrod operations act on H*(X) as operations lowering dimensions. To unify the `even' prime and odd prime cases, we let P1 = Sq2, when p = 2. Thus, for all primes p, P1 lowers degree by 2p - 2. The goal of this appendix is to prove Lemma 3.11, which we restate more precisely. Lemma A.1. ffi* : H-1(Pp-1(S-1 )) ! H-1( Dp(S-1 )) is an isomorphism of one dimensional Z=p-modules. Recall that H*(DrX) is a known functor of H*(X), both additively, and as a module over the Steenrod algebra. Furthermore, the behavior of * : H*( DrX) ! H*(Dr X) is known. See [CLM , BMMS ]. TATE COHOMOLOGY AND PERIODIC LOCALIZATION 23 Naturality implies that there is a commutative diagram: ffi* -1 H-1(Pp-1(S-1 )) ________________//_H-1( Dp(S )) o|| o|| fflffl| ffi* fflffl| H-1(Pp-1(O-1HZ))O _____________//_H-1( Dp(O-1HZ))O |P1*| o|P1*| | ffi* | H2p-3(Pp-1( -1HZ)) ____________//H2p-3( Dp( -1HZ)) ||* o||* fflffl| -2ffi* fflffl| H2p-3( -2Pp-1( HZ)) __________//_H2p-3( -1Dp( HZ)), where the top vertical maps are induced by the inclusion S-1 ! -1HZ. The top square is a square of homology groups of lowest degree. That the indicated maps are isomorphisms, all between one dimensional vector spaces, is an easy consequence of facts from [CLM , BMMS ]. For example, the middle right map is an isomorphism due to the Nishida relation P1*fiQ1x = fiQ0x 2 H-2(Dp( -1HZ)), for x 2 H-1( -1HZ). Using this diagram, to show that the top map is nonzero, and thus an isomorphism, it suffices to show that the lower left map and the bottom map are each isomorphisms. We state each of these as a separate lemma (one in dual form). Lemma A.2. * : H2p-3(Pp-1( -1HZ)) ! H2p-3( -2Pp-1( HZ)) is an isomorphism of one dimensional Z=p-modules. Proof.When p = 2, is an equivalence, and so * is an isomorphism. When p is odd, the situation is more complicated, and we proceed as follows. We have a commutative diagram * -2 H2p-3(Pp-1( -1HZ)) __________//_H2p-3( Pp-1( HZ)) | o|| | fflffl| || H2p-3(P2( -1HZ)) o| | | | | | fflffl| * fflffl| H2p-3( -1HZ) ________~________//_H2p-3( -1HZ) with indicated isomorphisms. Thus, to show the top map is an isomorphism, we need to check that the lower left map is an isomorphism. Equivalently, we need to check that ffi* : H2p-3( -1HZ) ! H2p-3( D2 -1HZ) 24 KUHN is zero. The map ffi : -1HZ ! D2 -1HZ factors through : 2D2 -2HZ ! D2 -1HZ, and this map is zero on H2p-3: the range is one dimensional, spanned by the suspension of a *-decomposable of the form x * y, with x 2 H-1( -1HZ) and y 2 H2p-3( -1HZ). But nonzero *-decomposables are never in the image of * : H*( D2(X)) ! H*(D2( X)). With our final lemma, we have reached the heart of the matter. Lemma A.3. ffi* : H2p-1( Dp( HZ)) ! H2p-1(Pp-1( HZ)) is an iso- morphism of one dimensional Z=p-modules. Proof.Since HZ is 0-connected, the Goodwillie tower Pr( HZ) converges strongly to 1 1 ( HZ) = 1 S1. Thus the associated 2nd quadrant spectral sequence converges strongly to H*(S1). For this to happen, P1(x) must be in the image of ffi*, where x 2 H1(Pp-1( HZ)) is a nonzero element, for otherwise P1(x) 6= 0 2 H2p-1(S1). Thus ffi* is nonzero, and is thus an isomorphism. Remark A.4. In work in progress, the author is studying the spectral se- quence converging to H*( 1 X) with E-r,*+r1= H*(DrX). The sort of argument just given generalizes to show that the first interesting differential is dp-1 : H*-1(DpX) ! H*(X). This differential is determined by H*(X) as a module over the Steenrod algebra, and has image imposing the unstable condition on H*(X). References [AGM] J. F. Adams, J. H. Gunawardena, and H. Miller, The Segal conjecture for e* *lemen- tary abelian p-groups, Topology 24 (1985), 435-460. [ACD] A. Adem, R. L. Cohen, and W. G. Dwyer, Generalized Tate homology, homotopy fixed points and the transfer, Algebraic topology (Evanston, 1988), A.M.S.Co* *nt.Math. 96 (1989), 1-13. [AK]S. T. Ahearn and N. J. Kuhn, Product and other fine structure in polynomial* * reso- lutions of mapping spaces, Alg. Geom. Topol. 2 (2002), 591-647. [B1]A. K. Bousfield, Uniqueness of infinite deloopings for K-theoretic spaces, * *Pacific J. Math. 129 (1987), 1-31. [B2]A. K. Bousfield, On the telescopic homotopy theory of spaces, Trans. A. M. * *S. 353(2001), 2391-2426. [BMMS] R. R. Bruner, J. P. May, J. McClure, and M. Steinberger H1 Ring Spectra * *and their Applications, Springer L. N. Math. 1176, 1986. [CLM] F. R. Cohen, T. Lada, and J. P. May, The Homology of Iterated Loop Spaces, Springer L. N. Math. 533, 1976. [DHS]E. S. Devinatz, M. J. Hopkins, and J. H. Smith, Nilpotence and stable homo* *topy theory. I, Ann. Math. 128 (1988), 207-241. [EKMM] A.D.Elmendorf, I.Kriz, M.A.Mandell, J.P.May, Rings, modules, and algebr* *as in stable homotopy theory, A.M.S. Math. Surveys and Monographs 47, 1997. [G1]T. G. Goodwillie, Calculus I: the first derivative of pseudoisotopy, K-theo* *ry 4 (1990), 1-27. [G2]T. G. Goodwillie, Calculus II: analytic functors, K-theory 5 (1992), 295-33* *2. TATE COHOMOLOGY AND PERIODIC LOCALIZATION 25 [G3]T. G. Goodwillie, Calculus III: the Taylor series of a homotopy functor, pr* *eprint, Brown University, June, 2002. [GM] J. P. C. Greenlees and J. P. May, Generalized Tate cohomology, Memoirs A.M* *.S. 113 (1995), no. 543. [GS]J. P. C. Greenlees and H. Sadofsky, The Tate spectrum of vn-periodic comple* *x ori- ented theories, Math.Zeit. 222 (1996), 391-405. [Gun]J. H. C. Gunawardena, Segal's conjecture for cyclic groups of (odd) prime * *order, J. T. Knight Prize Essay, Cambridge, 1980. [HopSm]M. J. Hopkins and J. H. Smith, Nilpotence and stable homotopy theory. II* *, Ann. Math. 148 (1998), 1-49. [H]M. Hovey, Cohomological Bousfield classes, J. Pure Appl. Alg. 103 (1995), 45* *-59. [HSa]M. Hovey and H. Sadofsky, Tate cohomology lowers chromatic Bousfield class* *es, Proc. A. M. S. 124 (1996), 3579-3585. [HSt]M. Hovey and N. P. Strickland, Morava K-theories and localisation, Memoirs* * A.M.S. 139 (1999), no. 666. [JW]J. D. S. Jones and S. A. Wegmann, Limits of stable homotopy and cohomotopy groups, Math. Proc. Camb. Phil. Soc. 94 (1983), 473-482. [Kl1]J. R. Klein, The dualizing spectrum of a topological group, Math. Ann. 319* * (2001), 421-456. [Kl2]J. R. Klein, Axioms for generalized Farrell-Tate cohomology, J. Pure Appl.* * Algebra 172 (2002), 225-238. [Ku1]N. J. Kuhn, Morava K-theories and infinite loop spaces, Algebraic topology* * (Arcata, CA, 1986), Springer Lecture Notes in Math. 1370 (1989), 243-257. [Ku2]N. J. Kuhn, Localization of Andr'e-Quillen-Goodwillie towers, and the peri* *odic ho- mology of infinite loopspaces, preprint, 2003. [L]W. H. Lin, On conjectures of Mahowald, Segal and Sullivan, Math. Proc. Camb.* * Phil. Soc. 87 (1980), 449-458. [MS]M. Mahowald and P. Shick, Root invariants and periodicity in stable homotop* *y the- ory, Bull. London Math. Soc. 20 (1988), 262-266. [MO] A. J. Mauer-Oats, Goodwillie Calculi, thesis, University of Illinois, 2002. [MM] J. P. May and J. E. McClure, A reduction of the Segal conjecture, Current * *trends in Algebraic Topology (London, Ontario, 1981), Canadian Math. Soc. Conference P* *roc. 2, Part 2 (1982), 209-222. [McC]R. McCarthy, Dual calculus for functors to spectra, Homotopy methods in al* *gebraic topology (Boulder, 1999), A.M.S. Contemp. Math. Series 271 (2001), 183-215. [R1]D. C. Ravenel, Localization with respect to certain periodic homology theor* *ies, Amer. J. Math. 106 (1984), 351-414. [R2]D. C. Ravenel, Nilpotence and Periodicity in Stable Homotopy Theory, Annals* * of Mathematics Study 128, 1992. [WW1] M. Weiss and B. Williams, Automorphisms of manifolds and algebraic K-theo* *ry II, J. Pure Appl. Algebra 62 (1989), 47-107. [WW2] M. Weiss and B. Williams, Assembly, Novikov conjectures, index theorems a* *nd rigidity, Vol. 2 (Oberwolfach, 1993), London Math. Soc. Lecture Note Ser. 22* *7 (1995), 332-352. Department of Mathematics, University of Virginia, Charlottesville, VA 22903 E-mail address: njk4x@virginia.edu