RATIONAL S1-EQUIVARIANT STABLE HOMOTOPY THEORY. J.P.C.Greenlees The author is grateful to the Nuffield Foundation for its support. Author addresses: School of Mathematics and Statistics, Hicks Building, Sheffield S3 7RH. UK. E-mail address: j.greenlees@sheffield.ac.uk 160 Part III Applications. 162 CHAPTER 14 Introduction to Part III. 14.1.General Outline The material in Part III is a collection of applications of the general theor* *y developed in Parts I and II. Accordingly, the chapters are completely independent of each ot* *her. Since most of the work has already been incorporated in the general framework, the se* *ctions are rather short, and uncluttered by technicalities. We begin with Chapter 15, which consists of five independent sections answeri* *ng the main questions motivating this study. Perhaps the most obvious problem of all, * *in the light of Haeberly's example, is to understand the behaviour of the Atiyah-Hirzebruch * *spectral sequence. We show that it collapses at E2 for all F-spaces if and only if ssT*(* *K ^ EF+) is injective. In general, the differentials encode the Adams short exact sequence,* * but there are differentials of arbitrary length. For arbitrary spaces, the main message * *is that the Atiyah-Hirzebruch spectral sequence is not a natural tool. Alternatively, we ca* *n always use a cellular decomposition of the domain to try to understand maps. In our case, * *the graded orbit category can be made quite explicit, and we can therefore do homological * *algebra over the category of additive functors on this category. We may then construct * *a spectral sequence, whose E2 term is calculable and given by homological algebra over the* * graded orbit category. It is also obviously convergent, but it does not seem a practi* *cal tool in general, because it is usually a half-plane spectral sequence. The moral is tha* *t we should not decompose spectra by Eilenberg-MacLane spectra or by cells, but rather by i* *njectives in the standard model. Another basic construction of spectra is by taking the suspension spectrum of* * a space. Equivariantly it is known that suspension spectra have certain special properti* *es, such as tom Dieck splitting, which are not enjoyed by all spectra. In particular, the i* *nclusion of the H-fixed points can be regarded as a map of T-spaces, and this has implicati* *ons for the model. However, since our model has no record of purely unstable information, w* *e cannot hope for a precise characterization of suspension spectra. The special case of K-theory is interesting because it has Bott periodicity, * *and hence Euler classes of its own. The representing spectrum also turns out to be forma* *l in the torsion model, so that the K-theory of any spectrum depends only on its homolog* *y in the torsion model. On the other hand, our analysis is really only the beginning of* * a study 163 164 14. INTRODUCTION TO PART III. melding the formalisms of the present work with the geometric information of K-* *theory: in particular it would be interesting to understand the Chern character in more* * detail and to relate our results to the work of Brylinski and collaborators [3]. After our analysis of function spectra, the rational Segal conjecture is littl* *e more than an elementary example: in the torsion model DET+ is formal, and represented by the* * natural map tF* tF*-! tF*-! tF*=OF = 2I -! Q[c; c-1]=Q[c]: We also present a more naive approach by way of comparison. In Chapter 16 we consider the well known T-equivariant cohomology theory given* * by cyclic cohomology. This is very simple rationally; more generally rational Tate* * spectra are also rather simple, but we may make certain intriguing algebraic connections. F* *inally we are able to identify the integral Tate spectrum t(KZ) of integral complex K-the* *ory KZ. This is of interest because t(KZ) is known to be H-equivariantly rational for a* *ll finite subgroups H. In fact, writing KZ for integral complex K-theory for emphasis, we* * identify the spectra t(KZ) ^ "EF and t(KZ) ^ EF+ together with the map of which t(KZ) is the fibre. The first is obtained from K-theory with suitable coefficients by i* *nflating and smashing with "EF, and the second is rational, and we can identify it as an inj* *ective Euler- torsion OF-module. The final Chapter 17 is more substantial. We turn to examples gaining their im* *portance from algebraic K-theory. B"okstedt, Hsiang and Madsen define the topological c* *yclic co- homology of a ring or a space [2]. It is obtained by performing various constru* *ctions on the topologicial Hochschild homology spectrum and can be used as a close approx* *imation to the completed algebraic K-theory of suitable rings. Hesselholt and Madsen [1* *7] identify the structure required of a T-spectrum X for one to be able to construct T C(X)* *. Spec- tra X with this structure are called cyclotomic spectra. The motivation for th* *e notion of a cyclotomic spectrum comes from the free loop space X = map(T; X) on a T-fi* *xed space X. This has the property that if we take K-fixed points we obtain the T=K* *-space map(T=K; X), and if we identify the circle T with the circle T=K by the |K|th r* *oot isomor- phism we recover X. For spectra, one also needs to worry about the indexing uni* *verse, but a cyclotomic spectrum is basically one whose geometric fixed point spectrum* * K X, regarded as a T-spectrum, is the original T-spectrum X. After the suspension sp* *ectrum of a free loop space, the principal example comes from the topological Hochschild * *homology of T HH(F ) of a functor F with smash products. Given such a cyclotomic spectru* *m X, one may construct the topological cyclic spectrum T C(X) of B"okstedt-Hsiang-Ma* *dsen [2], which is a non-equivariant spectrum. An intermediate construction of some inter* *est is the T-spectrum T R(X). Although these constructions are principally of interest pro* *finitely, it is instructive to identify the cyclotomic spectra in our model and follow the c* *onstructions through. We identify the rational cyclotomic spectra in the torsion model: they* * are the spectra X so that the function [N] : F -! torsionQ[c] - modules modelling EF+ ^* * X is constant, and so that the structure map tF* V - ! N commutes with any translati* *on of the finite subgroups. It therefore factors through tF* V -! tF*=OF V and th* *e map OF=OF V - ! N is a direct sum of copies of Q[c; c-1]=Q[c] V - ! [N](1). Fur- thermore, we may recover Goodwillie's theorem [7] that for any cyclotomic spect* *rum X 14.2. PROSPECTS AND PROBLEMS. 165 we have T C(X) = XhT: topological cyclic cohomology coincides with cyclic cohom* *ology in the rational setting. 14.2. Prospects and problems. The main theoretical problem is to show that the equivalence of the category * *of T-spectra and the derived category of the standard model can be obtained from a chain of * *equivalences arising from adjoint pairs of functors on underlying Quillen model categories. * *This would inevitably be linked with a better understanding of the meaning of the standard* * model. One of the most interesting prospective applications is that of understanding* * rational T-equivariant elliptic cohomology. Constructions have recently been given by Gr* *ojnowski [8] and by Ginzburg-Kapranov-Vasserot [6], and the cohomology of any T-space is* * a sheaf over an elliptic curve. One can ask if these theories are represented. This wou* *ld involve considering sheaves of T-spectra over an elliptic curve, and it would seem a se* *nsible first step to consider sheaves of objects of A. Both in this case and that of K-theory, there is the task of relating the gen* *eral model to the geometry of the cohomology theory: in practice this will involve concent* *ration on the Chern character, and comparison with the work of Brylinski [3]. There are a* * number of other classes of spectra which we do not understand as well as we should lik* *e, such as suspension spectra, free loop spaces, THH, ...... In the present work we have concentrated entirely on the circle group T. Alt* *hough it is unlikely to be possible to give so complete a picture as we have done for T-* *spectra, we hope to consider other small groups in due course. The continuous quaternio* *n and dihedral groups are prime candidates, both by virtue of their simplicity and th* *e prospects for applications. However, consideration of the case of Mackey functors [11] sh* *ows that it is necessary to replace the underlying algebra of OF-modules by that of sheaves* * over spaces of subgroups, since the topology on the space of subgroups can no longer be ign* *ored. For groups of rank greater than 1, the injective dimension of the relevant algebrai* *c categories will be greater than 1. There is therefore no prospect of obtaining splittings* * for formal reasons, and models must be based on a more complete geometric understanding th* *an we have used here. 166 14. INTRODUCTION TO PART III. CHAPTER 15 Classical miscellany. The sections in this chapter are independent of each other; each answers a natu* *ral question about rational T-spectra. In Section 15.1 we give a complete analysis of the behaviour of the Atiyah-Hi* *rzebruch spectral sequence for F-spectra, generalising the study in Section 1.4. Sectio* *n 15.2 sets up a calculable spectral sequence for calculating maps between T-spectra from a* * cellular decomposition, based on the graded orbit category. Section 15.3 shows how the e* *xistence of tom Dieck splitting makes the models of suspension spectra very special. Sectio* *n 15.4 finally returns to complex T-equivariant K-theory, and identifies its place in the tors* *ion model, showing that it is formal. Finally, Section 15.5 identifies the functional dual* * DET+, giving the rational analogue of the geometric equivariant Segal conjecture. 15.1.The collapse of the Atiyah-Hirzebruch spectral sequence. The purpose of this section is to analyse the Atiyah-Hirzebruch spectral sequ* *ence for F-spectra. We suppose given an arbitrary T-equivariant cohomology theory K*T(.), and con* *sider the Atiyah-Hirzebruch spectral sequence Es;t2= HsT(X; K_tT) =) K*T(X): This may be constructed either by using the skeletal filtration of X or, in com* *plete gener- ality, by using the Postnikov filtration of K. It is conditionally convergent i* *f X is bounded below. First, let us suppose that K = ET(2n)+; we observe that, if X is free, the on* *ly relevant part of the Mackey functor K_*Tis the nonequivariant homotopy of K. Thus the sp* *ectral sequence is concentrated on the lines q = 1 and q = 2n + 2. There are therefor* *e no differentials except d2n+2: Hp(X=T) = HpT(X; K_2n+2T) -! Hp+2n+2T(X; K_1T) = Hp+2n+2(X=T): If we take the special case X = ET(2m)+, we see that the differential must be g* *iven by multiplication by cn+1 in order to give the correct answer, as calculated by th* *e Adams short exact sequence. In particular, the differential is non-zero if and only i* *f m > n. 167 168 15. CLASSICAL MISCELLANY. Theorem 15.1.1. If X is an F-spectrum then the Atiyah-Hirzebruch spectral coll* *apses at E2 if ssT*(K) is injective over OF. Conversely, if the Atiyah-Hirzebruch Spe* *ctral sequence collapses at E2 for all F-spectra X, then ssT*(K) is injective. More precisely, any differential d2i+1is zero, and the nonzero differentials d* *2n+2are all explained by the above example, in a sense to be made precise in the proof. Proof: The first observation is that if X is an F-spectrum then K*T(X) = [X; K]* **T= [X; K ^ EF+]*T. Since the spectral sequence is natural, we may suppose that bot* *h X and K ^ EF+ have homotopy in even degrees. We argue that if ssT*(K) is injective th* *en the E2 term is entirely in even total degrees, and hence the spectral sequence collaps* *es. First note that for any F-spectrum T , K*T(T ) = Hom(ssT*(T ); ssT*(K ^ EF+)): Thus, taking T = oe0H, we see that, for any finite subgroup H, the H-equivarian* *t basic homotopy groups are purely in odd degrees, since ssT*(oe0H) = Q is odd. Thus t* *he part of the graded Mackey functor K_*Tover F is entirely in odd degrees. On the othe* *r hand, since X is an F-spectrum, [X; HM]*T= [X; HM ^ EF+]*T, and HM ^ EF+ is a wedge of copies of E,Lwith one factor for each basis element of V (H) = eH M(H). Thus* *, if we let I M = HI(H) V (H) we have H*T(X; M) = Hom(ssT*(X); I M); which is entirely in odd degrees. Thus E*;*2= H*T(X; K_*T) is in even total deg* *rees as claimed. Now suppose that the Atiyah-Hirzebruch spectral sequence does not collapse, an* *d that x 2 Ep;qrsupports a non-zero differential dr(x) = y 6= 0. We shall prove that r* * = 2n + 2 for some n, and that the differential is explained by naturality and the differenti* *als described above. We may pick a representative x02 Ep;q1= [X(p)=X(p-1); K]p+qTfor x. This shows* * that x0is supported on a map oepH- ! X=X(p-1)-! p+qK. Replacing X by X=X(p-1)and suspending, we may assume that X is (-1)-connected and p = 0. We may therefore * *replace K by its connective cover K10without changing the fate of x in the spectral seq* *uence. Now, letting Knmdenote the Postnikov section of K with non-zero homotopy groups in d* *egree i with m i n, we consider the Postnikov tower of K: Kr-1r-1-! Kr-10 -! Krr # Kr-2r-2-! Kr-20 -! Kr-1r-1 # K22 -! K20 -! K33 # K11 -! K10 -! K22 # x X - - -! K00 -! K11 By hypothesis, x : X -! K00lifts to x(r): X -! Kr-20so that the composite x(r):* * X -! Kr-20-! Kr-1r-1is essential and represents y. Since X is an F-spectrum, the beh* *aviour 15.2. ORBIT CATEGORY RESOLUTIONS. 169 is unaltered if the diagram is smashed with EF+. Since HM ^ EF+ is injective, a* *ll maps at the E2-term are detected by their d-invariant, and it is thus appropriate to* * examine the effect of taking homotopy of the above diagram smashed with EF+. The basic obse* *rvation is that, ssT*(HM ^ EF+) = I M; which is in odd degrees. Furthermore, x is detected by ssT1, and we need only l* *ook at the odd graded part. This immediately shows that all odd differentials are zero, so tha* *t r = 2n + 2 for some n. Next, we note that the maps ssT*(K2s+10^ EF+) -! ssT*(K2s0^ EF+) are injective in odd degrees, whilst the maps ssT*(K2s+20^ EF+) -! ssT*(K2s+10^ EF+) are surjective in odd degrees. Furthermore, the image of the composite consists* * of elements divisible by c. We thus find the diagram ssT*(K2n0^ EF+) -! ssT*(K2n+12n+1^ EF+) = 2n+3I K_2n* *+1T % # ssT*(X)-! ssT*(K00^ EF+) = I K_0T ; and we know that some element "zof ssT2n+3(X) maps to z 2 ssT2n+3(K2n0^ EF+), a* *nd cn+1z detects x, whilst the image of z in ssT2n+3(K2n+12n+1^ EF+) detects y. We there* *fore find a map ssT*(E(2n+2)) = I2n+20-! ssT*(X) with "zas the image of the top class, and "xas the image of the bottom class. B* *y the Adams short exact sequence, this is realised by a map E(2n+2)-! X. __|_| 15.2. Orbit category resolutions. Integrally one expects the cellular decomposition to be unhelpful in global c* *alculations because one does not know the stable homotopy groups of spheres. Rationally, ev* *erything is much simpler. For finite groups, cells are Eilenberg-MacLane spectra, and h* *ence the cellular decomposition is simply another way of viewing the complete splitting * *[14]. In the present T-equivariant context, cells are not all Eilenberg-MacLane spectra, but* * one may understand the entire graded category of natural or basic cells. We shall conce* *ntrate on the graded category hSB* of basic cells (i.e. the full subcategory of the graded st* *able category with the basic cells as objects). One thus views the entire homotopy functor ss_T*(X) of X as a module over the* * graded category hSB*. By the Yoneda lemma, the case when X is a cell plays the role of* * a free object, and a resolution is form of cellular approximation. We understand maps * *of degree 0 from the discussion of Mackey functors presented in Appendix A. Referring to 2.* *1.4, we see that composition in hSB* is usually zero for dimensional reasons. In fact, ther* *e are no maps of degree 2 between any pair of objects, and the only case with maps of degree * *more than 1 170 15. CLASSICAL MISCELLANY. is [oe0T; oe0T]T*= Q(QF[c-1]). For maps of degree 1 we have [oe0T; oe0T]T1= QF,* * [oe0T; oe0H]T1= Q, [oe0H; oe0T]T1= 0 and [oe0H; oe0H]T1= Q; for maps of degree 0 we have [oe0T; oe* *0T]T0= Q, [oe0T; oe0H]T0= 0, [oe0H; oe0T]T0= Q and [oe0H; oe0H]T0= Q. It therefore remains to deal with the* * composite of a degree 1 morphism (which must be of form xTT: oe1T-! oe0T, xHH: oe1H-! oe0Hor a* * multiple of the transfer trTH: oe1T-! oe0H) and a degree 0 morphism (which is either a m* *ultiple of the relevant identity or a multiple of the projection prHT: oe0H-! oe0T). We no* *w deal with these remaining cases. Lemma 15.2.1. The composites of the x's and y's are as follows. (i) xTTprHT= 0 and prHTxHH= 0, (ii) trTHprHT 6= 0 and prHTtrTHcorresponds to the inclusion of the Hth factor i* *n QF = [oe0T; oe0T]T1. Proof: Part (i) is clear since the composites lie in the zero group. The first * *fact in Part (ii) follows from the explicit geometric construction of the transfer. The second fa* *ct in Part (ii) follows by construction of the isomorphism in tom Dieck splitting [18, V.1* *1]. __|_| It is thus natural to take oH = trTHprHTas the basic generator of [oe0H; oe0H]* *T1and ffiH = prHTtrTHas a standard basis element of [oe0T; oe0T]T1. Now, suppose given any T-spectrum X, we consider [.; X]T*as a contravariant fu* *nctor on the graded orbit category. As such, we may form a projective resolution, and we* * may realise it. In fact we may construct a map P0 -! X, which is surjective in graded equiv* *ariant homotopy for all subgroups of T, and in which P0 is a wedge of cells. Now let X* *1 be the cofibre of this, and iterate to form the diagram X = X0 -! X1 -! X2 -! . . . " " " P0 P1 P2 By construction, all the maps Xs -! Xs+1 are zero in H-equivariant homotopy for* * all subgroups H, and so holim!Xs is contractible by the Whitehead theorem. s It is convenient to form the dual diagram with Xp-1 = fibre(X -! Xp), so that * *X-1 = * and X ' holim!Xp: * = X-1 -! X0 -! X1 -! . . . " " " -1P0 -1P1 -1P2 Replacing the maps by inclusions, we view this as a filtration of X with subquo* *tients Xp=Xp-1 = Pp. We may now construct a spectral sequence by applying [.; Y ]*Tto* * the diagram. It has Ep;q1= [Pp; Y ]p+qT, and Dp;q1= [Xp; Y ]p+qT. The spectral sequ* *ence lies in the right half-plane, and the differentials are cohomological, so that dr : Ep;qr-!* * Ep+r;q-r+1r, and it is evidently conditionally convergent. Finally, we see by construction t* *hat the se- quence . .-.! ss_T*(-2P2) -! ss_T*(-1P1) -! ss_T*(P0) -! ss_T*(X) -! 0 15.3. SUSPENSION SPECTRA. 171 is exact. Hence we may identify the E2 term as an Ext group, and the spectral s* *equence takes the form Ep;q2= Extp;qhSB(ss_T*(X); ss_T*(Y )) =) [X; Y ]p+qT: * The fact that the category of Mackey functors is one dimensional gives a vani* *shing line if X is bounded below. Indeed, since cells are also (-1)-connected we may ensure P* *0 is (-1)- connected, and hence that X1 is 0-connected. This formality proves that if the * *resolution is dimensionally minimal, Xp is (p - 1)-connected. However, if we use basic cel* *ls and the fact that the category of Mackey functors is of projective dimension 1, we may * *ensure that X2 is 2-connected. By iteration we see that X2sand P2sare (3s - 1)-connected; s* *imilarly X2s+1and P2s+1are 3s-connected. Thus the map X2s-1-! X is (3s - 2)-connected, a* *nd the map X2s -! X is (3s - 1)-connected. Unfortunately this only seems to be use* *ful if the homotopy of the spectrum Y is bounded above. Thus if YHq= 0 for q -1 and a* *ll H, then the nonzero entries are in the first quadrant and lie on or above the line* * q = (p - 1)=2. 15.3.Suspension spectra. In this section we suppose given a based T-space Z, and we identify the place* * of its suspension spectrum in our classification. We follow our usual convention of o* *mitting notation for the suspension spectrum functor, and using the notation T for Lewi* *s-May fixed point functor. Our basic tool is tom Dieck splitting, which states that the Lewis-May fixed * *points of the suspension spectrum of Z is _ TZ = Z _ ET=K+ ^T=K ZK ; K furthermore this is natural, and applies to stable retracts of spaces. The cruc* *ial simplifica- tion for spaces is that there is a map ZT -! Z of T-spectra, and hence a diagram "EF ^ ZT - ! EF+ ^ ZT '# # E"F ^ Z - ! EF+ ^ Z: The structure map "EF ^ Z -! EF+ ^ Z thus factors through the corresponding map for ZT, which we understand completely, since ZT is rationally a wedge of spher* *es. On the other hand, by naturality of tom Dieck splitting, we find ssT*(Z ^ E) = ss*(ET=H+ ^T=H ZH ) = H*(ET=H+ ^T=H ZH ); which we may certainly regard as computable. Summary 15.3.1. The algebraic model of the suspension spectrum of a space Z is M M tF*H*(ZT) -! 2IH*(ZT) = 2 H*(ET=H+^T=HZT) -! 2 H*(ET=H+^T=HZH ): H H 172 15. CLASSICAL MISCELLANY. The first map in the diplayed composite necessarily has zero e-invariant, and * *is simply induced by the quotient tF*-! 2I. However the second map is induced by the incl* *usion XT -! X, and may have non-zero d and e invariant. Again we may be satisfied th* *at the d-invariant is given by a homology calculation. For the e invariant, since* * the above discussion applies to retracts of spaces, we may assume that XT and EF+ ^ X are* * of pure parity, and then identify the e invariant with the Borel homology of the cofibr* *es XH =XT as in [1]. There remains the question of whether this characterizes suspension spectra. M* *ore pre- cisely, we cannot distinguish T-fixed suspensions so that we are asking if ever* *y model of the above sort is the model of a spectrum n1 X for some T-space X and some inte* *ger n. We do not have available the option of identifying the suspension spectrum * *functor, since there is no algebraic model of rational T-spaces. One might hope to reali* *ze finitely generated examples by explicit construction, but one would expect a certain amo* *unt of suspension to be necessary to achieve stability in each case. To obtain a globa* *l realization one would need a bound on these suspensions; since the model contains no data r* *elevant to the achievment of stability, there is no ready way to do this. 15.4. K-theory revisited. Let us consider the structure of the T-spectrum K representing rational equiva* *riant K-theory. We know that KH* = R(H)[fi; fi-1] for all subgroups H, where R(H) is* * the rationalization of the complex representation ring and fi 2 K-2 is the Bott ele* *ment. Now R = R(T) = Q[z; z-1], and R(H) = Q[z]=(zn - 1) when H is of order n. The restri* *ction maps are implicit in the notation here, and the induction maps to T are zero (h* *olomorphic induction maps are not part of the structure). Now, by Bott periodicity we have* * K-theory Euler classes c(H) of degree -2 for each finite subgroup H, obtained by applyin* *g the Thom isomorphism to the image of e(V (H)) in K-theory. We may apply Bott periodicity* * to obtain the usual K-theory Euler class (H) 2 R(T). In other words if H is of order n we* * have (H) = 1 - zn and c(H) = fi(H): Furthermore Y (H) = d; d|n where d is the dth cyclotomic polynomial. Let S be the multiplicative set gener* *ated by the Euler classes 1 - zn, and T be generated by the cyclotomic polynomials d; i* *n practice the geometry localizes so as to invert S, which is algebraically the same as in* *verting T , and the latter is easier to understand. Let F = S-1R = T -1R, so that ssT*(K ^ "EF) ~=F [fi; fi-1] and ssT*(K ^ EF+) ~=(F=R)[fi; fi-1]: Since the cyclotomic polynomials are coprime, an element of F=R can be written * *uniquely as a sum of terms fd(z)=d(z)n for n 1 where fd(z) 2 R is not divisible by d(z)* *. Hence M F=R = R[1=n]=R: n We should relate this to our geometric decompositions. 15.4. K-THEORY REVISITED. 173 Lemma 15.4.1. If K(H) denotes the part of K in T-Spec=H as usual then ssT*(K(H)) = (R[1=n]=R)[fi; fi-1] and cH acts as multiplication by fin times an automorphism. Remark 15.4.2. Our map cH is defined up to a non-zero rational number, whilst* * the K-theory Euler class is defined absolutely. The multiple is therefore well defi* *ned up to a non-zero scalar, but its exact value is not relevant to us at present. Crabb co* *nsiders studies this value in greater detail [5]. Proof: The only part requiring proof is that the action of cH is as claimed. W* *e shall show that cH acts as c(V (H)) times an automorphism on K ^ E. To do this, we* * let V = V (H), and unravel definitions. We have a K-theory Thom isomorphism t : K ^ SV -'! K ^ S|V,|and the F-spectrum Thom isomorphism o : SV ^ E -'! S|V |^ E; we need to know they are compat* *ible in the sense that the composites K ^ S0 ^ E -! K ^ SV ^ E -! K ^ S|V |^ E are equal. Now, we observe that both are maps of K-module spectra, and hence i* *t is sufficient to show both induce the same map S0^ E -! K ^ S|V |^ E. Maps o* *f this form are classified by K|VT|(E) ~=lim K|V(|E(2n)). Indeed the maps are cl* *assified n by what they induce in homotopy: ~= 1 1 [E ^ K; S|V |^ E ^ K]K;T0-! HomR(R=n ; R=n ): We know the K-theory Thom isomorphism gives multiplication by n. We may express* * the action of cH in its n-adic expansion as multiplication by x0+ x1n + x22n+ . .,.* *where xi2 R. It suffices to prove that the F-spectrum Euler class is multiplication b* *y n mod 2nfor a non-zero scalar , i.e. that x0 = 0 and x1 = . First, we know that the map oe0H-! oe0H^ K induces the permutation module map Q = eH A(H) -! eH R(H) = R=n. Now, we need to understand something of the map E(2n)-! K ^ E(2n), and we can infer enough by considering the diagram 0 Q Q k k k ~= T 2k ssT2k+1(E(2k-2))-! ssT2k+1(E(2k))-! ss2k+1(oeH) # # # ssT2k+1(E(2k-2)^ K)-! ssT2k+1(E(2k)^ K)-! ssT2k+1(oe2kH^ K) k k k R=kn R=k+1n R=n: This shows that the image gk of the generator of ssT2k+1(E(2k)) in ssT2k+1(E* *(2k)) = R=k+1nis k modulo knfor a non-zero rational number k. 174 15. CLASSICAL MISCELLANY. Now consider the diagram Q Q k k ~= T kV (0) ssT2k+1(E(2k))-! ss2k+1( E ) ~=# #~= ckH T kV ssT2k+1(E) -! ss2k+1( E) # # ssT2k+1(E ^ K)-! ssT2k+1(kVE ^ K) k k R=1n R=1n: We see that ckHtakes the image of gk 2 R=k+1nto the image of 1 2 R=n. Now gk is mapped to k=k+1nmodulo elements annihilated by kn. We conclude from the case k * *= 1 that x1 = 1 as required. The general case shows that k = k1. __|_| Corollary 15.4.3. The Q[cH ]-module ssT*(K(H)) is injective. Indeed, if Linis * *the space of Laurent polynomials in z with poles of order at most i at a primitive nth ro* *ot of unity, then multiplication by n gives an isomorphism Li+1n=Lin-! Lin=Li-1n. Thus ssT*(* *K(H)) ~= I(H) (L1n=L0n)[fi; fi-1], and hence K(H) ' E ^ S0[ (L1n=L0n)[fi; fi-1] ]: __|_| Accordingly K is characterized by tF* ssT*(K ^ "EF), ssT*(K ^ EF+) and the hom* *omor- phism between them. To make sense of the following statement, note that (F=R)[f* *i; fi-1] is a module over OF, since it is F-finite; more explicitly, if x 2 R[1=n]=R and H * *is of order n, then cH x = nxfi as mentioned above. Of course, multiplication by -1nis not * *defined on F=R, but it makes sense for F . Theorem 15.4.4. The T-spectrum K is the unique T-spectrum for which tF* ssT*(K* * ^ "EF) -! ssT*(K ^ EF+) is the map ^qK: tF* F [fi; fi-1] -! (F=R)[fi; fi-1] __ described as follows._ For x 2 OF of degree -2k we have ^qK(x ffil) = xffil+k,* * and ^qK(c-kH ffil) = -knffil-k. Proof: The value of ^qK(x ffil) is immediate from our method of calculating ss* *T*(K ^ EF+). For ^qK(c-kH ffil) we apply 6.1.2, using the compatibility statement in 15.4.1* * to relate it to our present naming of elements. Consider the diagram K -1^e K ^ S-kV (H) -'! K ^ S-2k -'! K # # # # K ^ "EF -' K ^ "EF ^ S-kV (H) -'! K ^ "EF ^ S-2k -'! K ^ "EF # r # # # K ^ EF+ - K ^ EF+ ^ S-kV (H) -'! K ^ EF+ ^ S-2k -'! K ^ EF+: 15.5. THE GEOMETRIC EQUIVARIANT RATIONAL SEGAL CONJECTURE FOR T. 175 The first horizontal is induced by e : S-kV (H)-! S0, the second is the K-theor* *y Thom class, and the third is multiplication by the integer Bott class. By 6.1.2 the * *relevant map is induced by r. Applying ssT*we see by definition that the composite from right to left in th* *e first row is multiplication by (kV (H)), and hence this is also true in the second row. * *Since the bottom right hand vertical induces projection, we identify the second vertical * *r in the lower ladder as stated. __|_| ___ It is tempting to rewrite the description of ^qKabove as ^qK(x ffil) = xffil* *+k for all x 2 tF*, but this makes no sense, since F is not an OF-module. This suggests we* * should perform some algebraic completion to F . This can be achieved geometrically by * *replacing K with F (EF+; K). This is a reasonable thing to do since the fibre of the comp* *letion map is the F-contractible spectrum F (E"F; K), which is determined by its homotopy * *groups. 15.5. The geometric equivariant rational Segal conjecture for T. In this section we aim to analyse the functional dual DET+ = F (ET+; S0); the* * title is something of a misnomer since neither Segal nor anyone else has made a conje* *cture about DET+. It completes the description given in [9, 12] of the integral funct* *ional dual. It would be possible to give an entirely algebraic treatment since we have a mo* *del for function spectra, but we shall first present a more direct treatment so as to a* *void relying on the more complicated bits of algebra. Since ET+ = E<1>, we should really discuss theWmore generalQquestion of ident* *ifying DE. Indeed, we should also consider DEF+ ' D( HE) ' H DE. Since the initial stages of the analysis are easier to understand for EF+, we begin with * *that. We already know EF+ ^ DEF+ ' EF+, and that ssT*(DEF+) = tF*; accordingly we h* *ave the cofibre sequence DEF+ -! "EF ^ S0[tF*] -a! EF+; where a induces projection onto the positive dimensional part. In the algebrai* *c model DEF+ is thus described by the element ^a2 Hom OF(tF*tF*; 2I) given by smashing * *a with DEF+ and looking in ssT*. For definiteness we emphasize that the second copy of* * tF*is the new one. Lemma 15.5.1. The map ^a: tF* tF*-! 2I is given by ^a(x y) = a*(xy). Note that ^arealizes Tate duality between negative and positive parts of tF*. Proof: Given an OF-map : tF*-! M, so that (1) = m the value of on OF tF* follows, and if the components of m are uniquely divisible by the relevant cH t* *hen is determined. The result follows provided x is of positive degree. The real conte* *nt of the lemma is that the formula is valid also when a*(x) = 0. To begin with we note how a nontrivial map "EF -! 2k+1EF+ is detected. As mot* *iva- tion, we observe that the obvious example is the quotient of "EF = S1V (F)by Sk* *V (F). If it were possible to simply desuspend by smashing with a putative spectrum S(-k-1)V* * (F), then the map would be detected in homotopy. Since the spectrum SV (F)is not inverti* *ble we must be slightly less direct by concentrating on a single subgroup H at a time,* * and using 176 15. CLASSICAL MISCELLANY. oekV (H)instead of SkV (F). This part of the analysis is given in the proof of * *15.5.2 below. __|_| The following identifies DE, and the special case H = 1 gives DET+. Proposition 15.5.2.There is a cofibre sequence DE -! "EF ^ S0[ Q[cH ; c-1H] ] -b! E and the extension is determined by the fact that ^b2 Hom OF(tF* Q[cH ; c-1H]; 2I(H)) = Hom Q[cH](Q[cH ; c-1H] Q[cH ; c-1H]; * *2I(H)) is given by ^b(x y) = b*(xy), which represents a perfect duality of Q[cH ; c-1* *H] Proof: We smash the standard cofibre sequence EF+ -! S0 -! "EF with DE; the terms are identified with those in the statement by the following lemma. Lemma 15.5.3. (i)There is an equivalence EF+ ^ DE ' E: (ii)There is an isomorphism ssT*(TDE) ~=Q[cH ; c-1H]: Remark: Part (i) of 15.5.3 with H = 1 corrects statements in the rational analy* *sis of [9]. More precisely, the space EF+ should be replaced by EG+ in 1.6, Theorem B, 4.8,* * and on the right hand side of 4.5 and page 359 line -3. The correction is discussed in* * more detail in [12]. Proof of 15.5.3: (i) The equivalence EF+ ^DEW' E follows since EF+ ^DE is a retract of EF+ ^ DEF+ ' EF+. Indeed EF+ ' H E, and the idempotent for * *all subgroups K 6= H annihilates DE. Hence EF+ ^ DE is a retract of E, and* * by homotopy groups it is an equivalence. (ii) The identification of TDE is immediate from 2.4.1. __|_| It remains to show that ^b(xy) = b*(xy); this follows when x is of positive de* *gree exactly as in 15.5.1. Now suppose x = c-kHfor k 0; the verification that ^b(c-kH y) = * *b*(c-kHy) in this case will also complete the proof of 15.5.1. We take the cofibre sequence in the statement and smash it with oenV (H). Sinc* *e oenV (H) is formed from S0 and various basic cells with isotropy H, we have oenV (H)^ "E* *F ' "EF; by the Thom isomorphism E ^ oenV (H)' E ^ S2n, and also oenV (H)^ DE ' D(E ^ oe-nV (H)) ' D(E ^ S-2n): Thus the cofibre sequence becomes S2n^ DE -! "EF ^ S0[ Q[cH ; c-1H] ] 1^b-!2n+1E: Because the final term is 2n-connected and the first has ssT*only in degrees 2* *n it follows that ssT*(1 ^ b) is surjective for all n. Taking n -k - 1 establishes the requ* *ired formula for ^b. __|_| 15.5. THE GEOMETRIC EQUIVARIANT RATIONAL SEGAL CONJECTURE FOR T. 177 Now let us turn to the general question of what can be said about DX = F (X; * *S0) if we already understand X in the standard model. Thus we suppose that X has mo* *del B = (N -! tF* V ), and we let Q -! tF* H denote the torsion model of DX. First we recall that S0 has model (OF -! tF*), this has torsion part I, and t* *he natural fibrant model is as the fibre of the map e(Q) -! f(2I). We have seen in Section* * 9.3 that if S is the torsion part of X, the torsion part of DX is Rf Hom(S; I): The vertex is described by the fibre sequence H -! V *-! E-1Hom(tF* V; 2I): However the real aim here is to give a description of the complete model. We sh* *ow that the obvious cofibre sequence DX -! DX ^ "EF -! DX ^ EF+ is also natural from the algebraic point of view. Lemma 15.5.4. If B = (N -! tF* V ), then the functional dual of B is describe* *d by i j R^ RHom (B; S0) = fibre e(V *) -! R^ f(Hom(N; 2I)) : Proof: We use the injective resolution S0 -! e(Q) -! f(2I), and deduce that i j RHom (B; S0) = fibre Hom (B; e(Q)) -! Hom (B; 2f(I)) : Now Hom (B; e(Q)) is the injective e(V *), whilst Hom (B; 2f(I)) = f(Hom(N; 2* *I)). Applying ^ we obtain the result. __|_| We can say a little more about the term R^ f(Hom(N; 2I)) in the description F* *or an arbitrary OF-module M we proved in Example 8.5.8 that there is a fibre seque* *nce R^f(M) -! e(E-1M) -! f(E-1M=M). 178 15. CLASSICAL MISCELLANY. CHAPTER 16 Cyclic and Tate cohomology. This is a short chapter, but fits naturally between its neighbours. The first s* *ection identifies rational cyclic cohomology, the second gives an algebraic description of the Ta* *te construc- tion on rational spectra, and the third gives a description of the Tate spectru* *m of integral complex equivariant K-theory. 16.1. Cyclic cohomology. In this section we consider periodic cyclic cohomology. We begin by observing* * that the rationalisation of the integral cyclic cohomology is the cyclic cohomology of t* *he rationals, so that the two possible interpretations coincide. It was proved in [14] that the representing spectrum for cyclic cohomology wi* *th coef- ficients in an abelian group A is the Tate spectrum t(HA) = F (ET+; HA) ^ "ET. * * The following lemma is special to bounded cohomology theories. Lemma 16.1.1. For any Mackey functor A the rationalization of t(HA) is t(H(A * * Q)). Proof: The essential point is that F (ET+; HA) = holim F (ET(n)+; HA), and that* * the n maps induced in [X; .]T by those of the inverse system are ultimately isomorphi* *sms for each finite X. This inverse limit therefore commutes with direct limit under de* *gree zero selfmaps of HA. Thus t(HA) ^ S0Q = holim!( F (ET+; HA) ^ "ET; m! ) m ' F (ET+; holim!(HA; m!)) ^ "ET m __ ' F (ET+; HA Q) ^ "ET= t(H(A Q)): |_| Henceforth we suppose A is rational. Lemma 16.1.2. Provided A is rational, t(HA) is F-contractible. We therefore h* *ave an equivalence t(HA) ' "EF ^ S0[ssT*(t(HA))]: 179 180 16. CYCLIC AND TATE COHOMOLOGY. Proof: Indeed t(HA)|H = t(HA|H ), and the rational Tate cohomology of any finit* *e group is 0. __|_| For any abelian group A we have ssT*(t(HA)) = ^HC* A = A[c; c-1]. 16.2.Rational Tate spectra. In this section we discuss the Tate construction of [14], which generalizes th* *e periodic cyclic cohomology discussed in the previous section. Recall that the Tate const* *ruction on a T-spectrum is defined by t(X) = F (ET+; X) ^ "ET. This simplifies considerabl* *y in the rational case, and it seems worth giving a complete description of the Tate con* *struction in the category of rational T-spectra. We begin with the warning that if X is integral, the map t(X) -! t(X ^ S0Q) ne* *ed not be a rational equivalence, so that Lemma 16.1.1 above is special to suitabl* *y bounded theories like HA. An example is given by complex K-theory, since t(KZ)|H is non* *-trivial and rational for all non-trivial finite subgroups H [10, 14, 15]; the following* * lemma shows this is false for t(KQ). We revert to our global assumption that all spectra are rational. Lemma 16.2.1. The natural map t(X) = F (ET+; X) ^ "ET-! F (ET+; X) ^ "EF is an equivariant equivalence. Thus t(X) is an F-contractible spectrum determin* *ed by its homotopy groups. Proof: We give two proofs. Firstly, the Tate construction commutes with restri* *ction: t(X)|H = t(X|H ). The lemma follows from the fact that the Tate construction is* * trivial on rational spectra for finite groups. One way of seeing this is to use the fact t* *hat if H is finite and e 2 A(H) is the idempotent with support 1 then EH+ = eS0 and "EH = (1 - e)S* *0. For the second proof, we compare the cofibre sequence ET+ - ! S0 -! E"T with EF+ -! S0 -! "EF. We see that the lemma is equivalent to showing that the natur* *al map f : F (ET+; X) ^ ET+ -! F (ET+; X) ^ EF+ is an equivalence. However, the co* *fibre of f is a wedge of terms F (ET+; X) ^ E with H 6= 1; this is contractible, a* *s one sees from the fact that F (ET+; X) ^ oe0H' * by using cofibre sequences and passing * *to direct limits. __|_| Proposition 16.2.2.If X is a rational T-spectrum with associated module M = ss* *T*(X^ ET+) over Q[c1], then t(X) is the F-contractible spectrum with homotopy groups ^H0(c(M) H^-1(M) 1) (c1) where ^H*(c1)denotes local Tate cohomology in the sense of [10]. 16.3. THE INTEGRAL T-EQUIVARIANT TATE SPECTRUM FOR COMPLEX K-THEORY. 181 Remark 16.2.3. There are two methods for calculating the local Tate cohomolog* *y; since (c1) is principal both are very simple. The second description simplifies furth* *er because M is torsion. In fact, the local Tate cohomology is only non-zero in codegrees 0 * *and -1, and for these cases we have ^H-i(c(M) = (L(c1)M)[1=c ] = limi(M; c ); 1) i 1 1 where L(c1)*denotes the left derived functors of completion at (c1). Note that * *this is only likely to be equal to (c1)-adic completion when M is finitely generated. Howeve* *r, since M is torsion, when it is finitely generated it is already complete; the Tate cohomol* *ogy therefore vanishes, as we know it must for geometric reasons. Proof: Observe F (ET+; X) ' F (ET+; X ^ ET+), so that if M = ssT*(X ^ ET+), the* *re is an exact sequence 0 -! Ext(2I; M) -! [ET+; X]T*-! Hom(I; M) -! 0: This is precisely parallel to the algebraic situation. We may split X into eve* *n and odd parts, and thus the exact sequence splits. Therefore F (ET+; X) is modelled by * *the com- plex Hom(P K(c1); M) where P K(c1) is a complex of projectives approximating th* *e stable Koszul complex Q[c1]-! Q[c1; c-11]; the homology of this complex calculates the* * left de- rived functors of c1-completion [13]. In particular, when X is even, [ET+; X]T** *is L(c1)0M in even degrees and L(c1)1M in odd degrees. Now conclude that there is a split exact sequence 0 -! Ext(2I; M)[1=c] -! t(X)T*-! Hom(I; M)[1=c] -! 0: Therefore, if T T(c1)(M) is the complex of the second avatar in the notation of* * [10], t(X) is modelled by the corresponding torsion free model, e(T T(c1)(M)). Thus, if X is * *even, t(X)T* is ^H0(c1)(M) in even degrees and H^-1(c1)(M) in odd degrees. __|_| 16.3. The integral T-equivariant Tate spectrum for complex K-theory. In this section we apply the general theory to identify the Tate spectrum of * *complex equivariant K-theory KZ integrally. However, we note that t(KZ) is not rational* *, and its rationalization is not t(KQ), so this is not an application of the previous sec* *tion. Before we state the theorem, recall that the representation ring R(T) = Z[z; * *z-1], and that the Euler class of the representation zn is 1-zn. In particular, we let O * *= 1-z and find R(T)^(O)= Z[[O]]; indeed, z = 1 - O is invertible in Z[O]=(On), so that Z[O] -!* * Z[O; z-1] = Z[z; z-1] induces an isomorphism of (O)-completions. We write Z((O)) for the lo* *calization Z[[O]][O-1], and S for the mulitiplicative set generated by the Euler classes. * *Note that if n 2 the Euler class 1 - zn is a multiple of O. However, although the multiplie* *r is a unit in Q((O)), it is not a unit in Z((O)) Q. 182 16. CYCLIC AND TATE COHOMOLOGY. Theorem 16.3.1. The Tate spectrum t(KZ) is F-equivalent to a rational spectrum, and is thus determined by the homotopy type of T KZ and its rational type. Ther* *e is an equivalence of KZ-module spectra T KZ ' KS-1Z((O)): The rational spectrum t(KZ) ^ S0Q is classified in the torsion model by M tF* S-1Z((O))[fi; fi-1] -! S-1Z((O))=Z((O))[fi; fi-1] = Z((O))=1|H|[fi; * *fi-1]; H6=1 where n is the nth cyclotomic polynomial, and the structure map is described as* * in 15.4.4. Proof: First note that t(KZ) = F (ET+; KZ)^E"T, so that we may calculate its co* *efficient ring from the Atiyah-Segal completion theorem. First, equivariant K-theory has * *coefficients R(T)[fi; fi-1], with R(T) = Z[z; z-1], and the K-theory Euler class of zn is 1 * *- zn. By the Atiyah-Segal completion theorem, ssT*(F (ET+; KZ)) = R(T)^(O)[fi; fi-1]. Consider the cofibre sequence t(KZ) -! t(KZ) ^ "EF -! t(KZ) ^ EF+: We shall identify t(KZ) ^ "EF, t(KZ) ^ EF+, and the map between them in turn. Firstly, since O is an Euler class, ssT*(t(KZ) ^ "EF) = S-1Z((O))[fi; fi-1]; where S is the multiplicative set generated by 1 - zn for n 1. Now S-1Z((O)) i* *s flat over Z, and hence the coefficients of T t(KZ) are the same as those of K-theory with* * coefficients in S-1Z((O)). Lemma 16.3.2. There is an equivalence T KZ ' KS-1Z((O)) of non-equivariant KZ-module spectra. Proof: First note that T t(KZ) is a module over KZ. Now let MS-1Z((O)) be a no* *n- equivariant Moore spectrum, and construct a map f : MS-1Z((O)) -! t(KZ) inducing an isomorphism in ssT0. Now form the composite KS-1Z((O)) = K ^ MS-1Z((O)) -! K ^ T t(KZ) -! T t(KZ) in which the first map is obtained from f by applying K ^ T (.) to f, and the s* *econd uses the module structure. By construction this induces an isomorphism in homotopy, * *and is therefore an equivalence. __|_| Next, we claim that t(KZ) ^ EF+ is rational. This is immediate from the fact * *that, t(KZ)|H is rational for all finite subgroups H [10,W14, 15]. Indeed, we know th* *at t(KZ) ^ T=H+ is induced from t(KZ)|H . Rationally EF+ ' HE, so t(KZ) ^ EF+ has a corresponding splitting. The summand for E<1> = ET+ is trivial, so we may choos* *e H 6= 1 and consider t(KZ) ^ E. From the identification of cH , we find it has homo* *topy groups Z((O))=1nin each even degree, and in particular it is injective. 16.3. THE INTEGRAL T-EQUIVARIANT TATE SPECTRUM FOR COMPLEX K-THEORY. 183 Finally, the map t(KZ)E"F -! t(KZ) ^ EF+ factors through the rationalization t(KZ) ^ "EF - ! t(KZ) ^ "EF ^ S0Q, and the resulting map t(KZ) ^ "EF ^ S0Q -! t(KZ) ^ EF+ is classified by its d-invariant since the codomain is injective. * *The map from tF* S-1Z((O))[fi; fi-1] is the analogue of that in Theorem 15.4.4. __|_| 184 16. CYCLIC AND TATE COHOMOLOGY. CHAPTER 17 Cyclotomic spectra and topological cyclic cohomology. In this chapter, we study various T-spectra arising from algebraic K-theory. Va* *rious con- structions are used to define suitable targets for trace maps from algebraic K-* *theory, and the most sophisticated takes B"okstedt's Topological Hochschild homology of a r* *ing, and forms the associated topological cyclic spectrum in the sense of B"okstedt-Hsia* *ng-Madsen [2]. Madsen has recently given a very helpful general survey [20]. The topological cyclic construction can be applied to any T-spectrum with app* *ropriate extra structure, and we begin in Section 17.1 by identifying the extra structur* *e involved in specifying such a `cyclotomic' spectrum. In the following section, we illus* *trate this by considering the basic examples: free loop spaces on a T-fixed space, and top* *ological Hochschild homology of a functor with smash products. Finally, in Section 17.3 * *we analyse the topological cyclic construction on rational cyclotomic spectra. 17.1.Cyclotomic spectra. We must begin by recalling the definition of a cyclotomic spectrum. The basic* * idea is that it is a spectrum X with the property analogous to that of the free loop sp* *ace Z, on a T-fixed based space Z, namely that for any finite subgroup K the fixed point se* *t (Z)K is equivalent to the original space Z. The analogue should be that any fixed point* * spectrum K X is equivalent to X again. Of course K X is really a T=K-spectrum, so we m* *ust begin by explaining exactly how we interpret it as a T-spectrum indexed on the origin* *al universe. In addition, we want to avoid redundant structure, so we simply require that th* *e resulting equivalences are transitive. __ We wish to consider the group T and all its quotients T = T=K by finite subgr* *oups ~= * *__ K. We want transitive systems of structure, so we first let ae = aeK : T -! * *T be the isomorphism given by taking_the |K|th root. If we index our T-spectra on a com* *plete universe U, we index our T-spectra on the complete universe UK . However we wan* *t these universes to be comparable, so we say that a complete T-universe U is cyclotomi* *c if it is ~= * K __ provided with isomorphisms U -! aeK U . Identifying T and T via aeK , this als* *o specifies isomorphisms UL -! ae*K=L(U L)K=L. We require that these are transitive in the * *sense that if L K then the composite U -! ae*LUL-! ae*L(ae*K=LUL)K=L = ae*KUK 185 186 17. CYCLOTOMIC SPECTRA AND TOPOLOGICAL CYCLIC COHOMOLOGY. is the isomorphismLfor K. One such cyclotomic universe is the direct sum U = n2* *ZUn where Un = i2Nzn with the isomorphisms suggested by the indexing. Suppose then that X is a T-spectrum indexed_on the cyclotomic universe U . Th* *us, for any finite subgroup K, K X is a T-spectrum indexed on UK ; by pullback alo* *ng the isomorphism aeK we obtain a T-spectrum ae*KK X indexed on ae*KUK, which may be * *viewed as a T-spectrum ae!KK X indexed on U by using the cyclotomic structure of the u* *niverse. A cyclotomic structure on X consists of a transitive system of T-equivalences rK : ae!KK X -'! X: By transitivity, the essential pieces of the structure come from the cases that* * K is of prime order. Although this structure is really designed to capture profinite information, t* *here is enough residue rationally to make it worthwhile identifying the cyclotomic obje* *cts in the algebraic model of rational T-spectra. The essential idea is that in a cyclotom* *ic spectrum all finite subgroups behave in an analogous way, differing only in the multipli* *city with which information occurs. There is no significant constraint on total fixed points T * *X. The first step of our analysis was to split F-spectra into the parts over different subgr* *oups, so it is easy to describe the cyclotomic structure in these terms. A spectrum X is cyclo* *tomic if we have specified equivalences X(C1) ' X(C2) ' X(C3) ' : :.:This uniformity itself* * imposes constraints on the assembly map of a T-spectrum. Let us now describe the algebraic model for cyclotomic spectra more precisely.* * It is useful to bear in mind the torsion model rather than the standard model. Definition 17.1.1.The ring of cyclotomic operations is the polynomial ring Q[c* *0]on a single generator c0 of degree -2. The standard injective I0 is defined by the e* *xact sequence 0 -! Q[c0]-! Q[c0; c-10] -! 2I0 -! 0: The cyclotomic torsion category Cthas obj* *ects (2I0 V -! T0) where V is a graded vector space and T0 is a torsion Q[c0]-module* *. The morphisms are given by commutative squares as usual. __|_| Lemma 17.1.2. The category Ctis abelian and 2 dimensional. Hence we may form t* *he derived category DCt. Proof: The proof is precisely analogous to that for the torsion model category * *At. __|_| Again, it is convenient to have a 1-dimensional model; the analogue of the sta* *ndard model is considerably simplified in the present context. Definition 17.1.3.The standard cyclotomic category C has objects Q[c0]-maps N0* * -! 2I0 V with N0 a torsion module. The morphisms are given by commutative squares * *as usual. Lemma 17.1.4. The cyclotomic category C is abelian and 1-dimensional. Hence we* * may form the derived category DC . Furthermore passage to fibre dgC -! dgCt and pas* *sage to cofibre dgC -! dgCt induce inverse equivalences of derived categories, so that * *DC ' DCt. 17.1. CYCLOTOMIC SPECTRA. 187 Proof: The proof is similar to the case of the standard model, but with the sim* *plification that the cofibre functor arrives in the correct category before passing to homo* *logy. __|_| Now define a functor : Ct-! At as follows. For an object we define (2I0 V -s0!T0) to be the composite M LH s0M (tF* V -! 2I V = 2I0 V0 -! T0): H H Here the first map is induced by the quotient tF*-! tF*=OF = 2I, and the second* * is the direct sum of countably many copies of s0 made into a OF-module in the obvious * *way. The functor is obviously exact and hence induces a functor : DCt -! DAt : We may now state a precise theorem. Theorem 17.1.5. A T-spectrum admits the structure of a cyclotomic spectrum if* * and only if it corresponds to an object of DAt equivalent to one in the image of . Note that the condition in the theorem gives a rather satisfactory characteri* *zation of cyclotomic spectra. It essentially says that a cyclotomic spectrum is one that* * has two properties. Firstly, the structure map factors through that for its geometric * *fixed point spectrum (as happens for suspension spectra) and secondly, that all finite subg* *roups behave alike. If a spectrum admits a cyclotomic structure then a structure is imposed by ch* *oosing particular equivalences between the idempotent pieces of the torsion part of th* *e model. Note that for a spectrum X with torsion model tF* V -! T admitting a cyclotomic structure the corresponding cyclotomic spectrum is simply 2I0 V - ! T0 where T* *0 = e1T = ssT*(ET+ ^ X) and the map is obtained by factoring s through the projecti* *on and applying the idempotent e1. Proof: We have explained how to put a cyclotomic structure on an object in the * *image of . Any imprecision will be eliminated in the course of the proof in the reverse * *direction. Suppose then that X is a cyclotomic spectrum with cyclotomic structure maps r* *K : ae!KK X -'! X as required. We already know from Section 10.2 the effect of pass* *age to geometric fixed points. Indeed, by Theorem 10.2.6, if M is the model of X then * *eM is the model_for K X where e is the idempotent supported_on_the_subgroups containing * *K. Here O __Fis identified with eOF by letting a subgroup H of T correspond to its inve* *rse image in T. The effect_of ae!Ksimply results from identifying subgroups of T with those * *of the same order in T. __ __ * * __ Define nK : F -! F by letting n(H ) be the subgroup of T with the same order * *as H , ~= __ and consider the induced ring isomorphism n*K: OF -! O__F. 188 17. CYCLOTOMIC SPECTRA AND TOPOLOGICAL CYCLIC COHOMOLOGY. __ Lemma 17.1.6. The functor ae!K: T-Spec -! T-Spec corresponds to pullback alo* *ng n*Kin the usual sense that the diagram __ ae!K T-Spec -! T-Spec '# #' __ nK# DA -! DA commutes up to natural isomorphism, and similarly for torsion models. __|_| It is then clear (for example by using the cyclotomic structure for K itself) * *that the part eK M of the model over any subgroup K will be the same as the piece e1M over th* *e trivial subgroup. This also forces the map to factor as specified. If M is an object of the torsion model with zero differential, we see that the* * structure map must be zero on any element of form 1 v; otherwise it would have nonzero i* *mage in eH T for some H, and hence for all finite subgroups H. This contradicts F-fi* *niteness of T . Since this argument passes to an injective resolution, it applies to all di* *fferential graded objects. __|_| 17.2. Free loop spaces. For a based space Z, we intend to identify the place of the free loop space Z * *in the present scheme. In particular we may consider K*T(Z), which is a conjectural ap* *proxima- tion to Ell*(Z). We restrict attention to the case that Z = Y is a suspension. Here, Carlsson a* *nd Cohen [4] prove the splitting _ ET+ ^T Y = (ECn)+ ^Cn Y ^n: n Hence, rationally we have M ssT*(ET+ ^ Y ) = {H*(Y )n }Cn n with trivial H*(BT) action. This leaves us to describe a map M 2I0 H*(Y ) -! 2 {H*(Y )}Cn: n This necessarily has zero d-invariant. Indeed, this is obvious if H*Y has even * *parity. In the general case we see that the map is induced by ET+ ^TY -! ET+ ^TY ; by duality * *it is sufficient to consider cohomology, and the domain has torsion free cohomolog* *y whilst the codomain has torsion cohomology. Now, exactlyLas in the case of suspension spec* *tra, the e invariant is the element of Ext(4I0 H*(Y ); n 2H*(Y )) corresponding to the ex* *tension obtained by applying homology to ET+ ^T Y -! ET+ ^T (Y )=Y -! ET+ ^T 2Y: 17.2. FREE LOOP SPACES. 189 Since Q[c1]acts trivially on H*(Y ), one might hope the extension is always obt* *ained by tensoring a universal extension M 0 -! Q -! E -! 2I0 -! 0 n with 2H*(Y ). There is another important example of cyclotomic spectra. Example 17.2.1. Topological Hochschild Homology: Suppose that F is a functor with smash products in the sense of B"okstedt. O* *ne may define a cyclotomic spectrum T HH(F ), which comes with a spectral sequence HH*(F (S0)*) =) T HH(F )* for calculating its homotopy groups. One may then hope to calculate ssT*(ET+ ^T* *T HH(F )) using the skeletal filtration of ET+. It is always the case that T T HH(F ) ' S0, and so the structure map of the c* *yclotomic spectrum T HH(F ) takes the form 2I0 -! 2T HH(F )hT*: By definition, we always have a map from the identity functor to F and hence a * *cyclotomic map S0 = T HH(I) -! T HH(F ). Since this is an equivalence of geometric fixed p* *oints, and the structure map for S0 has zero d-invariant we deduce that the structure * *map for an arbitrary functor F has zero d-invariant. It would be interesting to unders* *tand its e-invariant more precisely. One case of particular interest is when the FSP arises from a ring R. In thi* *s case ssT*(ET+ ^ T HH(R)) = HC*(R), which can be calculated by the algebraic Loday-Qu* *illen double complex. It remains to identify the torsion model structure map, but we * *can obtain information by naturality from the unit Q -! R. In fact we have the diagram "EF ^ DEF+ - ! EF+ '# #' "EF ^ DEF+ ^ T HH(Q) - ! EF+ ^ T HH(Q) '# # "EF ^ DEF+ ^ T HH(R) - ! EF+ ^ T HH(R) in which we understand the top row precisely as the structure map of the sphere* *. Thus we only need to understand the algebraic map 2I0 = 2HC*(Q) -! 2HC*(R). This is in fact either zero or injective: this follows from the Tate spectral sequence * *by naturality. Indeed t(T HH(R)) is a module over t(T HH(Q)), and hence over the ring spectrum* * t(HQ), whose coefficients are Q[c0; c-10]: thus the behaviour is completely determined* * by the image of the unit. If R is augmented, then of course the map is injective. This special case is not too far from the general case, because any rational * *FSP arises from a simplicial ring, and ssT*(ET+ ^T HH(Ro)) can be calculated algebraically* *, since there is a T-map T HH(Ro) -! HH(Ro) which is a non-equviariant rational equivalence. * * __|_| 190 17. CYCLOTOMIC SPECTRA AND TOPOLOGICAL CYCLIC COHOMOLOGY. 17.3. Topological cyclic cohomology of cyclotomic spectra. In the first instance, the topological cyclic cohomology of a ring is designed* * to be the target of a refined trace map from the algebraic K-theory. Hesselholt and Mads* *en have shown that the cyclotomic trace is very close to being an isomorphism in many c* *ases [17]. The construction of the topological cyclic cohomology in this case begin* *s with the topological Hoschschild homology, and the definition of a cyclotomic spectrum a* *bstracts precisely what is required to make the construction. Goodwillie has identified the topolical cyclic cohomology of the topological H* *oschschild homology of a rational functor with smash products [7], and we generalize this * *to an arbitrary cyclotomic spectrum. This is not a deep result, but it demonstrates t* *he character of the topological cyclic cohomology and illustrates the adequacy of the presen* *t theory. The author is grateful to L. Hesselholt for many helpful discussions. We must begin by describing the construction. For any T-spectrum X, if L K we* * have an inclusion of the Lewis-May fixed points FLK : K X -! LX; the letter F is cho* *sen because it corresponds to the Frobenius map in algebraic K-theory. To avoid co* *nfusion, the reader should ignore for the duration of the present section the fact that * *FLK induces the restriction map from K-equivariant to L-equivariant homotopy groups. The cy* *clotomic structure supplies a second set of maps RKL: K X -! LX defined as follows. Firs* *t we let L* be the subgroup of K with order |K=L|. Now consider*the inclusion*X -! X ^E"* *[6 L*]; applying Lewis-May L*-fixed points we obtain a map L X -! L X. Applying ae!L*a* *nd the cyclotomic structure we obtain * ! L* rL* ae!L*L X -! aeL* X -! X; finally we apply L-fixed points and obtain the required map *L* L ! L* L ae!K=LK X = ae!K=LK=L X = (aeL* X) -! X: Again, the letter R is chosen because the induced map is the restriction map in* * algebraic K-theory. To simplify notation, we index F and R simply by the order of the quotient K=L* *. Thus we find F1 = R1 = 1, FrFs = Frsand RrRs = Rrs. It turns out that the Frobenius * *and restriction maps also commute. The most familiar version of the topological cyclic cohomology construction is* * simply to take the the homotopy inverse limit of the system of non-equivariant fixed poin* *t spectra under the restriction and Frobenius maps: T C0(X) = holim(holim(K X; R)); F ) = holim(holim(K X; F ); R): It may help later motivation to view this as the homotopy fixed point object of* * an `action of a category'. It turns out that the intermediate object T R0(X) = holim (K X; R); K has significance of its own, so we prefer the first description T C0(X) = holim* *(T R0(X); F ). Furthermore, we note that the above construction shows that the map RKL: ae!KK * *X -! 17.3. TOPOLOGICAL CYCLIC COHOMOLOGY OF CYCLOTOMIC SPECTRA. 191 ae!LLX is a map of T-spectra so T R(X) = holim (ae!KK X; R) K is a T-spectrum with underlying spectrum T R0(X). However, we warn that the ide* *ntifica- tion of all terms with T R(X) means that the Frobenius maps are not maps of T-s* *pectra. We shall identify the relevant equivariance below. For non-profinite work, Goodwillie points out that the diagram given by the r* *estriction and Frobenius maps should be augmented by adding in the circle action; we may n* *ow think of an action by a topological category. Since R also commutes with the Frobeniu* *s, passing to limits under R, we obtain a diagram with a copy of T R(X) for each finite su* *bgroup, and Frobenius maps relating them; the quotient category acts on T R(X). For the* * present we view all objects as the same and hence we think of having an action of the m* *onoid M occurring in a split exact sequence 1 -! T -! M -! Z>0 -! 1; in fact if w; z 2 * *T then (wFr)(zFs) = wzrFrs. This leads to the definition T C(X) = T R(X)hM ' (T R(X)hT)hF: The following result may simply be regarded as evidence that the definition is * *a reasonable one: rationally, the topological cyclic construction is a complicated way of do* *ing something familiar. Theorem 17.3.1. (Goodwillie ) For any rational cyclotomic spectrum X we have * *an equivalence of rational spectra T C(X) ' XhT; so that the topological cyclic cohomology agrees with the Borel cohomology. Goodwillie proves this in the case that X = T HH(F ) for a rational functor F* * with smash products [7, 14.2]. Proof: The first step is to note that homotopy fixed points commute with homoto* *py inverse limits, and that the homotopy fixed point spectrum of a non-equivariantly contr* *actible spectrum is contractible. Thus T R(X)hT = (holim ae!KK X)hT K = holim ((ae!KK X)hT) K = holim ((ET+ ^ ae!KK X)hT) K This shows that it is really only necessary to understand X(1) = ET+ ^ X. Of co* *urse the end result is simply a non-equivariant rational spectrum, so it is only necessa* *ry to calculate homotopy groups. The main result of Part I is that the T-free spectrum ET+ ^ ae!K X is determi* *ned by its homotopy groups as modules over Q[c1]. Lemma 17.3.2. If X is a cyclotomic spectrum with ssT*(ET+ ^ X) = T0 then M ssT*(ET+ ^ ae!KK X) = T0 LK 192 17. CYCLOTOMIC SPECTRA AND TOPOLOGICAL CYCLIC COHOMOLOGY. and hence _ ET+ ^ ae!KK X = X(1): LK Proof: This is immediate from 17.1.6 together with our exact identification of * *Lewis-May fixed points in Theorem 12.2.2, (or more directly from 12.3.2). __|_| The relevant inverse system thus has Kth term given as a wedge of copies of th* *e spectrum X(1) indexed by the subgroups of K. It will perhaps be clearest if we think of * *this as the set of functions from the finite set [ K] of subgroups of K to X(1). The advan* *tage is that it permits a helpful notation for maps: any function f : A -! B of finite * *sets induces f* : X(1)B -! X(1)A. If f is an inclusion the map f* is simply projection. Lemma 17.3.3. When L K, the restriction map RKL induces the projection corre- sponding to the function KL: [ L] -! [ K] defined by requiring KL(H) to have or* *der |H| . |K=L|, in the sense that the diagram 1^RKL !L ET+ ^ ae!KK X -! ET+ ^ aeL X '# #' W (KL)* W HK X(1) -! HL X(1) commutes. Proof:*Recall*that L* denotes*the subgroup of K with order |K=L|. The effect of* * the map L X -! L (X ^ "E[6 L*]) = L X follows from our account of the Lewis-May fixed points. Now we just need to rename subgroups using ae!L*17.1.6, and apply L-fix* *ed points as described in 12.3.2. __|_| Corollary 17.3.4. We have an equivalence Y __ T R(X)hT ' X(1)hT: |_| H Notice that the above argument could also be used to identify the T-spectrum T* * R(X) exactly in the algebraic model. Indeed the maps ae!KK X -! ae!LLX are all iden* *tified exactly, and we can form the homotopy inverse limit in the algebraic model. Ho* *wever, since inverse limits do not preserve F-free objects, the answer is not very att* *ractive. Our present purpose requires much less; indeed, since T R(X)hT is just a rational s* *pectrum, and it remains only to understand the action of F on homotopy groups. Lemma 17.3.5. When L K the Frobenius map FLKinduces the projection correspond- ing to the inclusion iKL: [ L] -! [ K] in the sense that the diagram 1^FKL L ET+ ^ K X -! ET+ ^ X '# #' W (iKL)*W HK X(1) -! HL X(1) 17.3. TOPOLOGICAL CYCLIC COHOMOLOGY OF CYCLOTOMIC SPECTRA. 193 commutes. Proof: The first necessity is to understand the statement. We begin with a map * *K X -! LX, which we may view as a map of T spectra indexed on UK . Once we have smashed with ET+ the universe may be replaced by a complete one, and we obtain a map in* * the category for which we have a model. To understand the map we factor K X -! LX as K X -! infK X -! LX. If we view these as maps of T=L spectra, the second map is the counit of the K=L f* *ixed point adjunction, completely understood by 12.2.2 and the contents of Section 11.3. * *The first map has the property that it is a nonequivariant equivalence. The result now fo* *llows from our description of the adjunction. __|_| It remains only to index the terms so that the relevant structure is visible,* * and to verify that the circle action does not get in the way. We begin by replacing subgroups by their orders, and defining a category as f* *ollows. The object set Z>0 x Z>0 consists of integer points in the strictly positive or* *thant. There are morphisms (OEs; aet) : (m; n) -! (ms; nt) for s; t 2 Z>0. Next, consider t* *he diagram D of divisors defined by D(m; n) = {d | d dividesmn} and OEs : D(m; n) -! D(ms;* * n) is inclusion, aet: D(m; n) -! D(m; nt) is the multiplication by t. Finally, for an* * object Y we consider the contravariant functor Y D defined by taking functions from D into * *Y ; thus on objects, Y D(m; n) = Y D(m;n). The connection with the restriction and Frobeniu* *s diagram is immediate from 17.3.5 and 17.3.3. The maps will be clearest if we replace D(* *m; n) by the set of rational numbers i=jwhere i divides m and j divides n. It is easy to che* *ck that these fractions are in bijective correspondence to divisors of mn: if d divides mn th* *e relevant fraction is d=n. With this indexing, both R and F simply drop irrelevant coordi* *nates. Now, passing to limits under restriction maps we obtain Y D(m) := lim Y D(m; n), whi* *ch simply n consists of sequences (yi=j) with i dividing m. The map Fs : Y D(ms) -! Y D(m) * *again simply drops coordinates with numerator dividing ms but not m. In other words, * *if we now identify Y D(m) with Y D(1) by dividing the coordinate indexes by m we find Fs * *is the shift map specified by mutiplying indices by s and ignoring fractions with an integer* * numerator bigger than 1. The system consists of surjections, so lim1(Y D(1); F ) = 0, and* * evidently m the only compatible families are those with all coordinates equal: lim (Y D(1);* * F ) = Y . m This description suggests that we should have a means for discussing subgroups * *of T with fractional orders, which suggests we should be considering the solenoid S := li* *m(T; s) which is the inverse limit of copies of the circle under the power maps s . We must now check that the fact we have taken homotopy T-fixed points between* * the R and F stages does not invalidate the above procedure. The time has come to be* * precise about the equivariance of the Frobenius maps. First, note that although we hav* *e the behaviour Fsz = zsFs in the monoid M, so that Fs is identified with the map s * *: ET+ -! *sET+, we expect the reverse type of behaviour for the objects acted upon. Lemma 17.3.6. The Frobenius map induces a map of T-spectra along s, in the s* *ense that Fs : *sT R(X) -! T R(X) is a map of naive T-spectra. 194 17. CYCLOTOMIC SPECTRA AND TOPOLOGICAL CYCLIC COHOMOLOGY. Proof: We must remember that the map Fs arose from the inclusions K X -! LX, which is a map of T-spectra. However, when we have applied ae!in the appropriat* *e way, we must insert the power map s to retrieve the equivariance. __|_| The relevant map T R(X)hT -! T R(X)hT is then obtained by passage to fixed poi* *nts from F ( s; Fs) : *sF (ET+; T R(X)) = F ( *sET+; *sT R(X)) -! F (ET+; T R(X)): The relevant untwisting result is as follows. Lemma 17.3.7. The sth power map s : ET+ -! *sET+ is a stable rational equiva- lence. __|_| Q * Let Y = n X(1), and consider the map Fs : sY -! Y of T-spectra. the commutat* *ive diagram F(1;Fs) F (ET+; *sY ) -! F (ET+; Y ) F ( s; 1) "' "= F( s;Fs) F ( *sET+; *sY )-! F (ET+; Y ) * * Q hT has an equivalence in its left hand vertical. Hence we can untwist the action o* *n n X(1) . Corollary 17.3.8. Rationally,Qwe may identify the system of copies of T R(X)hT* * under the Frobenius map with n>0X(1)hT and with the Frobenius Fs acting via multipli* *cation by s shifts. __|_| The theorem now follows. __|_|