Higher topological cyclic homology and the Segal conjecture for tori

HIGHER TOPOLOGICAL CYCLIC HO MOLOGY AND THE SEGAL CO NJECTURE F OR TORI GUNNAR CARLSSON, CHRISTOPHER L. DOUGLAS, AND BJØRN IAN DUNDAS Abstract. W e inv estigat e higher topol ogical cyclic homology as an approac h to studying c hro- matic phenomena in homotopy theory . Higher topological cyclic homology is constructed f rom the ﬁxed p oints of a version of topological Hochsc hild homology based on the n-dimensional torus, and w e prop ose it as a computationa lly tractable cousin of n-fold iterated algebraic K-theory . The ﬁxed p oints of toral topological Ho chsc hild homology are related to one another b y re- striction and F r obenius op erators. W e i n tro duce tw o additional fami l ies of op erators on ﬁxed points, the V ersch iebung, indexed on self-isogenies of the n-torus, and the diﬀerentials, indexed on n- v ectors. W e give a detailed analysis of the relations among the restriction, F r obenius, V er- sc hiebung, and diﬀerentials, producing a higher analog of the structure Hesselholt and Madsen described f or 1-dimensional topological cyclic homology . W e calculate tw o imp ortant pieces of higher topological cyclic homology , namely topological restriction homology and top ological F rob enius homology , for the sphere sp ectrum. The latter computation allows us to establish the Segal conjecture for the torus, whic h is to say to completely compute the cohomotop y t ype of the classi fying space of the torus. Contents 1. Int ro duction 2 1.1. Background and motiv ation 2 1.2. Results and future dir ections 3 2. Higher top olog ical cyclic homolo gy 5 2.1. Higher top olo gical Ho chsc hild homology 5 2.2. Restriction a nd F rob enius op erato rs 6 2.3. Cov ering homology 7 2.4. T r a nsfer maps 8 2.5. The V er schiebung 10 3. Diﬀerent ials a nd higher Witt relatio ns 11 3.1. The one-dimensio nal Witt re la tions 11 3.2. Higher diﬀerentials 12 3.3. Gcd’s and Lcm’s of matrices 14 3.4. Stable splitting of the transfer 16 3.5. Relations a mong the F rob enius, diﬀerential, and V e rschiebung 17 4. Adams op erations on covering homolog y 21 4.1. Identiﬁcation of the F r ob enius a nd restriction c a tegories 21 4.2. Actions on inverse systems 22 4.3. Examples of group actio ns on cov ering homolo gy 23 5. Calculation o f T R ( n ) for the sphere 24 5.1. The K-theo r y of ﬁnite G -sets 25 5.2. K-theor y and the equiv ariant sphere spec tr um 26 5.3. The homotopy limit over the restrictions 27 6. T F ( n ) for the sphere and the Segal conjecture for tori 28 6.1. The rank ﬁltration of the eq uiv ariant sphere sp ectrum functor 28 6.2. The cotype de c o mpo sition of the sphere and a splitting of the ra nk ﬁltration 29 6.3. The homotopy type of the ﬁxed rank- c o type co mpo ne nts of T F ( n ) 31 Appendix. Cyclic homolo gy as a ho motopy limit of F rob enius homolo g y 35 References 38 The ﬁr st author was supported in part by NSF grant DM S-0406992. The s econd author w as s upp orted i n part by an NSF Postdoctoral F ellowship. 1 1. Introduction 1.1. Bac kground and motiv ation. A casua l glance at a n y chart of ho mo topy groups o f spheres is dizzying—o ne gets the impression tha t there is some not-quite discer nible pattern. The chromatic viewp oint on stable homotopy theo ry clariﬁes ma tters: we put o n colored go ggles so that w e ca n see information only of a particular wa velength, tha t is with particular per io dicity prop erties. This organizing pr inciple is eno rmously useful b oth as a conceptual framework for co mprehending large scale phenomena in homo topy theory , and as a co mputational to ol [2 3, 24]. The traditional appro ach to chromatic phenomena pro ceeds b y what Hopkins calls “designer homotopy theory”. In this one designs, abstrac tly and w itho ut reg ard to geo metr y , sp ectra with desired chromatic prop er ties : Morav a K- theories K ( n ), Lubin-T ate sp ectr a E n , higher rea l K - theories E O n , and so on. This tact is e xtremely successful: w e input the “wa velength” we are int erested in studying, by exa mining a heigh t n formal g roup, and build colored goggles to suit this study . Even the magniﬁcent and quite g e ometric theory of top olo gical mo dular forms [19] has the feature that the chromatic infor mation is present at the outse t—the constr uc tio n of tmf mak es explicit use of the height 1 and 2 formal g roups of elliptic curves. In order to complemen t this de s igner homotopy a pproach to chromatic information, w e would like (1) a geo metric understanding of the meaning of chromatic phenomena in homotopy theory , and (2 ) a natura l construction of chromatic type n sp ectra that do esn’t b egin with the for mal g roup of height n , tha t is in which the chromatic b ehavior is an output rather than an input to the theory . Regarding the ﬁrst desiderata, vector bundles and top ologica l K-theory pr ovide an elega n t ge o metric description o f chromatic level 1 structure. Analogous geo metric frameworks for chromatic level 2 structure are b e g inning to take shap e: the Baas- Dundas -Rognes theo ry of 2-vector bundles [3, 2], and the Segal-Stolz-T eichner theor y of 2-dimensio nal conforma l ﬁeld theories [2 7, 2 8] se em particula rly promising. Regarding the second des iderata, Rognes ’ red- shift conjecture prop oses that the a lgebraic K-theory functor transfor ms appro priate chromatic type n − 1 sp ectra into c hromatic type n sp ectra. If the conjecture is corre c t, alg ebraic K- theory ﬁts the bill o f an alchemical wa y to pro duce chromatic phenomena. The red-shift conjecture naturally lea ds one to inv estigate the itera ted alg ebraic K- theory K n ( A ) of a sp ectrum A , by wa y of stepping up the chromatic “sp ectrum” . In particular, the n -fold iterated algebraic K - theory of the ring F p is a natur al ca ndidate for a basic type n spec tr um. There are t w o problems with this iterated a lgebraic K -theory approach. The ﬁrst is that it is co mputationally int ractable . It is diﬃcult enoug h to compute the eﬀect of a sing le applicatio n o f alg ebraic K - theory , much less numerous iterates all at once; indeed alr eady for tw o-fold algebraic K -theory , the computations required considerable eﬀort and ingenuit y on the par t of, for example, Ausoni a nd Rognes [1]. The second problem is that it seems plausible that des pite exhibiting chromatic level n b ehavior, itera ted algebr aic K -theory will not hav e pa rticularly plea sant universal pro per ties as a chromatic type n sp ectrum. F or instance, n -fold iterated algebr aic K -theor y might have an (a s yet tec hnically unav ailable) action by the symmetric gr oup—this ac tio n and p erha ps others would need to be equalized in or de r to appro ach a more canonical co nstruction. The pre s ent pap er is part of a progr am to give a simple, direc t construction of spectr a exhibiting higher chromatic b ehavior, ba s ed not on itera ted a lgebraic K-theor y but o n fo r ms of higher top o- logical cyclic homolo gy . The top ologic al cyc lic homology T C ( A ) of a ring sp ectrum A is built out of ﬁxed p oint sp ectra arising from a circle a ction on the top ologica l Ho chsc hild homolog y sp ectrum T H H ( A ) := A ⊗ S 1 . Mo re s peciﬁca lly , T C ( A ) is the homotopy limit of the ﬁx e d points ( A ⊗ S 1 ) G , for ﬁnite subgroups G of the circle, ov er certain restriction and F ro benius op erato rs. By w ork of Go o dwillie, McCarthy , and the third author [13, 2 1, 12], the cyclotomic trace map fro m algebr aic K-theory to top ological cyclic homo logy is a r elative equiv alence, and s o the latter is a r e asonable approximation to the for mer . Higher top olog ical cyclic homology is built fr om ﬁxed p o int sp ectra asso ciated to the sp ectrum T H H n ( A ) := A ⊗ T n —that is, we replace the circle by the n - dimensional torus, with the r esult being a higher top olog ical Ho chschild homo lo gy . In detail, higher top ological cyclic ho mology T C ( n ) ( A ) is a homoto p y limit of the ﬁxed p oints ( A ⊗ T n ) G for ﬁnite subgroups G 2 of the torus, over a collection of r estriction and F rob enius op er ators as s o ciated to s e lf- is ogenies of the torus . As yet it re ma ins a hop e that hig her topolo gical cyclic homology is an eﬀectiv e substitute for iterated algebr aic K-theory , but in a ny case we b elieve it has thr ee adv antages: ﬁrst, it is muc h more amena ble to computation; second, it is b oth co nceptually and technically m uch simpler; third, it has b etter symmetr y prop erties, co ming from isogenie s of the torus that mix the diﬀerent circle factors, and so we believe it represents a pa r ticularly natural candidate red- shift functor . In order to obtain a clea rer picture of the higher topolo gical cyclic homo logy functor, w e nee d, quite simply , to compute examples. In order to approa ch calcula tio nal techniques for hig her topolo gical cyclic homology , w e recall Hesselholt and Ma dsen’s approach to classical topo logical cyclic ho mology computations. In this case the bas ic op e rators are the res triction R p : T H H ( A ) Z /p k → T H H ( A ) Z /p k − 1 and F r ob enius F p : T H H ( A ) Z /p k → T H H ( A ) Z /p k − 1 . Instead of immediately taking the homotopy limit over the restriction and F ro be nius , Hesselholt and Madse n tr e a t the diagram of restr iction o pe rators as a prosp ectrum, and obs e rve that in addition to the F rob enius action, there are t wo additional op erato r s on this pr osp ectrum, the V e r schiebung V p : T H H ( A ) Z /p k − 1 → T H H ( A ) Z /p k and the diﬀerent ial d 1 : S 1 ∧ T H H ( A ) Z /p k → T H H ( A ) Z /p k . They prov e that these op erators satisfy the key relations F p V p = p and F p d 1 V p = d 1 . These relations, and a detailed unders tanding o f the formal structure they entail, allow Hesselholt and Madsen to do e xtensive calculations of top olo g ical cyclic homo logy , and therefore of algebr a ic K-theory [16, 17, 18]. One of our ma in tasks in the pres e n t pap er is the pro duction a nd analysis of a nalogous op erator s for hig her top olog ical cyclic homolog y . W e alrea dy hav e at o ur disp osa l restr iction R α and F rob enius F α op erators [6], indexed by self-isogenies α o f the n-torus. W e in tro duce higher V erschiebung op erators V α , a lso indexed by is o genies, a nd diﬀere n tials d v indexed by vectors v ∈ Z n . W e then establish numerous relations among these oper ators, including ana logues of the classica l relations for the pro ducts F α V β and F α d v V β . The simplest p o tent ial computation is that o f the higher top olog ical cyclic homolo gy T C ( n ) ( S ) of the sphere spectr um. This sp ectrum T C ( n ) ( S ) is the ho motopy limit o ver restr iction and F rob enius op erators on ﬁxed p oints of the higher topolo gical Ho chschild homology T H H n ( S ) of the sphere. How ever, the higher top ological Hochschild homolo gy of the s phere is ju st the sphere sp e ctrum itself, together with a new- found torus equiv ariance. W e therefore ﬁnd ourselves inv es tigating ﬁxed po int s of equiv a riant sphere sp ectra, a sub ject already of consider able classical imp or tance. The Segal conjectur e for a ﬁnite g roup G provides a description o f the homotopy t yp e of the ho mo topy G -ﬁxed po int s S hG of the sphere [7]. Said a nother way , the conjecture c o mputes the co homotopy F ( B G + , S ) = S hG of the cla ssifying space of the ﬁnite g roup. In the pro cess o f inv estigating T C ( n ) ( S ), we study a n imp or tant piece of top o logical cyclic ho- mology , namely top olo gical F r ob enius homology T F ( n ) ( S ), which is the homotopy limit only ov er the F rob enius maps. This top olog ical F ro be nius ho mology happ ens to b e homoto py equiv alent to the co homotopy sp e ctrum o f the class ifying space of the tor us. W e give a detailed and complete analysis o f the homoto p y type of the top olo gical F r ob enius homology of the sphere, a nd therefore establish the Segal conjectur e for tor i. 1.2. Results and future di rectio ns. Our ﬁrs t main theor em is the description of the relations among the restriction, F rob enius, V erschiebung, and diﬀeren tial opera tors on the ﬁxed p oints of higher top olo g ical Ho chsc hild ho mology . Theorem 1.1. Fix an o dd prime p. L et A b e a c onne ctive c ommutative ring sp e ctrum. F or α ∈ M n ( Z p ) ∩ GL n ( Q p ) an inje ctive endomorphism of Z n p , let L α := α − 1 Z n p / Z n p ⊂ T n p b e a c or- r esp onding sub gr oup of the p-adic n-torus. Denote by T α := T T n p ( A ) L α the L α -ﬁxe d p oints of the higher t op olo gic al Ho chschild homolo gy of A b ase d on the p-adic n-torus; this is a ring sp e ctrum with multiplic ation µ : T α ∧ T α → T α . 3 Ther e ar e op er ators in the stable homotopy c ate gory R α : T β α → T β (r estr iction), F α : T αβ → T β (F ro b eniu s ) , and V α : T β → T αβ (V erschiebung). Mor e over, for e ach p-adic ve ctor v ∈ Z n p , ther e is an op er ator d v : S 1 ∧ T α → T α (diﬀer ential). The r est riction and F r ob enius ar e ring maps, the diﬀer ential is a derivation, and the r estriction c ommutes with the other op er ators. These R , F , V , and d maps satisfy the fol l owing r elations. 1. µ ( V α ∧ 1) = V α µ (1 ∧ F α ) . 2. F α V β = | g c d α,β | V [lcm α,β /α ] F [lcm α,β /β ] . 3. d v F α = F α d αv ; V α d v = d αv V α . 4. F α d v V β = d b ez α gcd † α,β v V [lcm α,β /α ] F [lcm α,β /β ] + V [lcm α,β /α ] F [lcm α,β /β ] d b ez β gcd † α,β v . Her e we have chosen matr ic es g c d α,β and c oprime matric es [ α/ gcd α,β ] and [ β / g c d α,β ] such t hat α = gcd α,β [ α/ gcd α,β ] and β = g cd α,β [ β / g cd α,β ] . We have also chosen Bezout matric es b ez α and bez β such t hat [ α/ gcd α,β ] b ez α +[ β / gcd α,β ] b ez β = 1 , and c oprime matric es [lcm α,β /α ] and [lcm α,β /β ] such that α [lcm α,β /α ] = β [lcm α,β /β ] . See Theorem 3.22 for a mor e explicit and complete list of relatio ns . By taking ho motopy gro ups of the ﬁxed po in ts of higher top ologica l Ho chsc hild homolo gy , we can co ncis ely pack a ge the structure of higher to p olo gical cyclic homology in ter ms of o per ators on a pro multi-diﬀeren tial graded ring. Corollary 1. 2. Asso ciate d to a c onne ctive c ommutative ring sp e ctru m A , ther e is a pr o mult i- diﬀer ential gr ade d ring T R α q ( A ; p ) deﬁne d as fol lows. F or e ach matrix α ∈ M n ( Z p ) ∩ GL n ( Q p ) , ther e ar e gr oups T R α q ( A ; p ) := π q ( T T n p ( A ) L α ) , wher e L α = α − 1 Z n p / Z n p . A s q varies these gr oups form a gr ade d ring. F or e ach p-adic ve ct or v ∈ Z n p , ther e is a gr ade d diﬀer ential d v : T R α q ( A ; p ) → T R α q +1 ( A ; p ) ; these diﬀer entials ar e derivations, ar e line ar in the ve ctor v , and they gr ade d c ommute with one another. The c ol le ction T R α ∗ ( A ; p ) is ther efor e a multi-diﬀer ent ial gr ade d ring. As α varies these form a pr o mu lti-diﬀer ential gr ade d ring under the r estriction maps R α . Ther e i s a c ol le ction of pr o-gr ade d-ring o p er ators F α : T R αβ ∗ → T R β ∗ , and a c ol le ction of pr o- gr ade d-mo dule op er ators V α : ( F α ) ∗ T R β ∗ → T R αβ ∗ . That V α is a mo dule m ap is e qu ivalent t o F ro b enius r e cipr o city: V α ( x ) · y = V α ( x · F α ( y )) . These op er ators ar e su bje ct to the r elations 1. F α V β = | g c d α,β | V [lcm α,β /α ] F [lcm α,β /β ] . 2. d v F α = F α d αv ; V α d v = d αv V α . 3. F α d v V β = d b ez α gcd † α,β v V [lcm α,β /α ] F [lcm α,β /β ] + V [lcm α,β /α ] F [lcm α,β /β ] d b ez β gcd † α,β v . W e undertake computations inv olving the spher e sp ectrum. Prop ositio n 1.3. The t op olo gic al r estriction homolo gy of t he spher e, that is the homotopy limit of the ﬁxe d p oints T T n p ( S ) L α along the re striction op er ators, is T R ( n ) ( S ) ≃ Y O ⊆ Z n p B ( Z n p / O ) + Her e the pr o duct varies over the op en su b gr oups O ⊆ Z n p . Our se cond main theorem is the Segal conjecture for tori, whic h amounts to a computation o f the top olog ical F rob enius homology of the spher e. Theorem 1.4. The p -adic c ohomotopy of the classifying sp ac e of t he torus is homotopy e quivalent to the p -c ompletion of top olo gic al F r ob enius homolo gy, F ( B T n + , S p ) ≃ T F ( n ) ( S ) p and the homotopy gr oups of T F ( n ) ( S ) p ar e as fol lows: π ∗ ( T F ( n ) ( S ) p ) = Y k,c lim l  Z [ GL n ( Z p ) / Γ l,k,c ] ⊗ π ∗ (Σ ∞ S k ∧ B T k + ) /p l  4 Her e the pr o duct is over 1 ≤ k ≤ n and c is a c ol le ct ion of u nor der e d p ositive inte gers { n 1 , . . . , n k } . The limit is over l ∈ N , and the gr oup Γ l,k,c ⊂ GL n ( Z p ) is the stabilizer of a chosen sub gr oup K of C × n p l of r ank k and c otyp e c . See Theorem 6 .2 for a more co mplete description o f the pie c e s of this decomp osition. W e br ieﬂy men tion so me of the directions in whic h we in tend to ta ke this pro ject. The most impo rtant next step is explicit co mputatio ns o f the higher top olog ical cyc lic ho mology T C ( n ) ( F p ) of the Eilenberg -MacLane sp ectrum for F p . In fact, this story is a bit more subtle than we let on in the ab ove background discussio n, b ecause the mos t interesting versions of c y clic ho mo logy a re likely to come not from equalizing all restrictio n and F rob enius op erato r s (as in T C ( n ) ), but in carefully choosing subdia g rams o f o p er ators for the homotopy limit. W e touch on the kinds of choices w e hav e in mind in section 4 below, and desc r ib e another important as pec t of these sub- ho motopy limits: there ar e sys tems o f Adams op erations that survive to ac t o n the homotopy limits, and it seems likely that these op erations will play a central ro le in studying the re sulting versions of hig her top ological cyclic homology . W e rese rve, though, a mor e detailed discuss io n for ano ther o ccasio n. The computations of hig he r top ologica l cyclic homology can, in pa rt, pro ceed along lines analo- gous to the w ork of Hess e lholt and Madsen, namely using a collection o f in terrelated T a te and s pec - trum co homology sp ectral sequences, together with iterated applications o f the norm-coﬁbratio n sequence. Along the w ay in these computations , it will b e co nvenien t to for malize the structure seen in Cor ollary 1.2 into a no tion of Burns ide-Witt co mplex. Such a complex will hav e the given relations and will also hav e a c o mpatible ma p fro m the Burnside-Witt vectors for Z n p . It will b e particularly worthwhile to in vestigate an initial s uch complex , a “ de Rha m-Burnside-Witt co mplex”, and to describ e the analog of the Burnside-Witt structure for log-r ing s. W e r emark on one las t na tur al contin ua tion of this work, namely that w e hop e our ana ly sis of the cohomo topy of the classifying space o f a torus, v iewed as the ma ximal torus of a compact Lie group, can b e used a s a b o otstra p to es tablish the Seg al conjecture for all co mpact Lie g roups. 2. Higher topological cyclic homo logy 2.1. Higher top ologi cal Ho c hsc hild homology. Let A be a connective commutativ e S -a lgebra. The top o logical Ho chsc hild homo logy of A , denoted T H H ( A ), is the sp ectrum A ⊗ S 1 ; her e ⊗ is the tensor in the catego ry of commutativ e S -algebras . More concr etely , T H H ( A ) is the realiza tion of the simplicial sp ectr um whose k -th level is A ∧ ( S 1 ) k , where ( S 1 ) . is the standar d simplicial circle. In this sens e, T H H ( A ) is the “ S 1 -fold smash p ower” o f A . W e are interested not only in the homoto py t yp e but in the equiv ariant homotopy type of top o- logical Ho chschild homology . In particula r, we will b e interested in a la rge colle ction of o p er ations on the ﬁxed p oints of top ologica l Ho c hschild ho mology , including the restrictions, F rob enii, V er- schiebung, and diﬀeren tials. The restriction maps only exist on equiv ar ia nt sp ectr a that have a delicate prop erty calle d cyclotomicity . Unfortunately , T H H ( A ) is not cyclotomic and s o is not suitable for a detailed in vestigation of ﬁxed p oint str uc tur es. This trouble c an b e re ctiﬁed by constructing, as in Hesse lholt-Madsen [16], a cycloto mic s p ectr um T ( A ) that is non-equiv a riantly homotopy equiv alent to T H H ( A ). Higher top olog ical Ho chschild homolog y T H H n ( A ) is by deﬁnition the sp ectrum A ⊗ T n , where T n is the n-tor us. This sp ectrum can ag a in b e g iven very concre tely as the r ealization of a simplicial sp ectrum A ∧ ( T n ) . . As in the one-dimensional case, this sp ectrum is eq uiv ariantly misb ehav ed and requires r ectiﬁcation. O ne rectiﬁca tion is the Lo day constructio n [6], w hich pro duces a sp ectrum Λ T n ( A ) that is no n-equiv ar iantly equiv alent to T H H n ( A ), bu t which exhibits the desired hig her analogues of cyclotomicity . Mor eov er, in the one-dimensional case, Λ S 1 ( A ) recov ers the equiv a riant homotopy type T ( A ). The notation Λ X ( A ) is meant to sug gest b oth the “ X -fold smash power of A ”, that is V X A , a nd als o the o rigin of the technical asp ects o f the constr uc tio n in Lo day’s w ork. How ever, the reader who is used to the Hess elholt-Madsen notation T ( A ) mig h t b e advised to think of Λ X ( A ) as “ T X ( A )”, an eq uiv ariant topolo gical Ho chsc hild homology based on X rather than on the circle . 5 The Lo day constructio n provides a functor X 7→ Λ X ( A ) fro m space s to s pectr a, and hence ea ch Λ X ( A ) comes with an a c tion o f the entire spa c e of endomor phisms o f X . This a ction leads to a particularly r ich structure when X is a g roup G , for exa mple the n-torus T n . T o isoge nies of G (that is surjective homomor phisms with ﬁnite kernel) one can asso ciate so-ca lle d restrictio n a nd F ro benius maps relating the ﬁxe d po in t s pectr a Λ G ( A ) H , for v arying subgro ups H of G . The ho mo topy limit ov er these restriction and F ro be nius maps is a kind of “ top ological G -cyclic homology” of A . Instead of considering all iso genies o f G , we can fo cus attention on a particular class P o f iso genies and co nsider o nly the restriction and F ro b enius ma ps coming from these is ogenies. The ho motopy limit over these maps is c alled the P -cov ering ho mology of A a nd is deno ted T C P ( A ). These versions of G -cyclic homology will b e describ ed in more detail in section 2.3 and particular examples of int erest disc ussed in s ection 4.3. When the gr o up G is the p-co mplete cir cle, a nd the class P is the ﬁnite o rientation-preserving self- cov erings of the circ le , the P -covering homolog y is the or dinary top olo gical cyc lic homolog y T C ( A ) of B¨ okstedt, Hsia ng, and Mads en [5]. Notice that this class P is gene r ated by the s tandard p-fold cov er of the circ le , a nd so the only op erator s in the ho motopy limit ar e the traditiona l restriction R p (called Φ in [5]) a nd F r ob enius F p (called D in [5]). In the next section, we describ e the r estriction a nd F ro benius op era tors on the ﬁxed p oints Λ G ( A ) H for a general group G , and discuss the sp ecializ ation to our preferred group, the p-adic n-torus. 2.2. Restriction and F rob enius op erators. As b efore, co nsider a connective commutativ e S - algebra A , a nd a group G . The restriction and F rob enius are maps between ﬁxed p oints o f higher top ological Ho chsc hild homolog y Λ G ( A ): R H I : Λ G ( A ) H → Λ G/I ( A ) H/I F H I : Λ G ( A ) H → Λ G ( A ) I Here H is a ﬁnite subgroup o f G , and I is a nor mal subgr oup of H . The F ro benius ha s a concise description as the inclusion of ﬁxed points. The restriction by contrast is a more in volv ed o p er ation and we only brush aga inst its deﬁnition—techn ical details ca n b e found in [16] and [6]. Roughly sp eaking, the sp ectrum Λ G ( A ) H can be describ ed in terms of mapping s pa ces out of the union S K ⊂ H S K of ﬁxed points b y subgroups of H ; here the S are ce rtain ﬁnite H -s e ts. When we restrict the so urce of these mapping spa ces to the union S I ⊂ K ⊂ H S K of ﬁxed p oint by subgro ups c ontaining I , the res ulting sp ectrum is Λ G/I ( A ) H/I . The r estriction map has inconv eniently landed in the ﬁxed p oints of the higher top ologica l Ho ch schild homolo g y based no longer on G but on G/I . W e solve this problem by considering only situations where G/ I is itself isomorphic to G . In particular , this is tr ue provided I is the kernel of a sur jective homo morphism a : G → G with ﬁnite kernel. W e identify G/I with G by the natural isomo rphism, and ther e b y H /I with a subgro up o f G . The r estriction then ma ps be tw een ﬁxed p oints of the sing le sp ectrum Λ G ( A ). The whole situa tio n is more conv eniently and directly indexed in terms of the se lf-ho momorphisms of G , as follows. Let a, b : G → G be surjective homomorphisms with ﬁnite kernels. Let L a denote the kernel o f a . Denote b y φ a : G/L a → G the natura l is o morphism, a nd let φ ∗ a G b e G considered as a G -space with the G -action G × G a × 1 − − → G × G µ − → G . Note that φ a gives a G - isomorphism φ a : G/L a ∼ = − → φ ∗ a G . Thro ughout the pa per we will, perha ps unfor tunately , refer to bo th φ a : G/L a → G a nd φ − 1 a : G → G/L a as φ a , and also, when conv enient for notational r easons, as φ a . This is omorphism φ a gives us control ov er the G -s tructures on the v ar ious quotients G/L a ; we need a similar ly car eful v iew of the eq uiv arianc e of the ﬁx ed p oint sp ectra Λ G ( A ) L a . A pr iori, the sp ectrum Λ G ( A ) L a is a mo dule ov er S [ G/L a ] = Σ ∞ ( G/L a ) + , but we can give it an S [ G ]-mo dule structure by the comp osite S [ G ] ∧ Λ G ( A ) L a φ a ∧ 1 − − − − → S [ G/L a ] ∧ Λ G ( A ) L a µ − − − − → Λ G ( A ) L a 6 W e will often abbrevia te this comp osite action itself by µ , as we think of it as the unambiguous natural ac tio n of G o n Λ G ( A ) L a . In the description of the restriction map above, repla ce H by the kernel L ba and replace the subgroup I by L a . Note that the quotient H /I = L ba /L a is isomo rphic to L b . With th e giv en S [ G ]-mo dule structur e, the r e striction is now an S [ G ]-mo dule map: R a := R L ba L a : Λ G ( A ) L ba → Λ G ( A ) L b This is the form in which we will use the re s triction map thro ughout this pap er. In these terms, the F rob enius is a map F b := F L ba L a : Λ G ( A ) L ba → Λ G ( A ) L a The relation of the restriction and F ro be nius [6] is pleasantly s traightforw ard: R a F b = F b R a . W e brieﬂy sp ecialize the ab ov e discussion of restrictions and F rob enii to o ur case of sp ecial int erest, namely when the group is the p- a dic n-torus T n p := R n p / Z n p . The r elev ant subgroups L α ⊂ T n p arise as kernels of isogenies (surjective ﬁnite-kernel homomor phisms) α : T n p → T n p . The monoid of isogenie s o f the torus is isomorphic to the mono id M n := M n ( Z p ) ∩ GL n ( Q p ) of injective linear endomor phisms of Z n p ; the corresp ondence takes a matr ix α over Z p to the co rresp onding cov ering α/ Z n p : R n p / Z n p → R n p / Z n p . The k ernel of this map α : T n p → T n p is L α = α − 1 Z n p / Z n p ⊆ Q n p / Z n p = C × n p ∞ ⊆ T n p . F or conv e nience we now abbre viate the Lo day co nstruction of topo lo gical Ho ch schild homology a s T ( n ) := Λ T n p ( A ) and the ﬁxed p oint sp ectra as T α := Λ T n p ( A ) L α . Note well tha t the ﬁx ed p oint spectrum only depe nds on the kernel L α and not on the cov ering map α —how ever, as a T n p -equiv ar iant sp ectrum, w ith T n p -action µ ( φ − 1 α ∧ 1) descr ibed abov e, T α do es depe nd o n the particular covering, and this justiﬁes the notation. In this context we think of the isomorphism φ α : T n p /L α → T n p as “m ultiplication b y α ” , and note that it restricts to is o morphisms C × n p ∞ /L α ∼ = C × n p ∞ and L β α /L α ∼ = L β . F or a covering α ∈ M n , the r e s triction now has the fo rm of a map (really a family of ma ps), R α : T β α → T β This map dep ends, even non-equiv a riantly , on the particula r covering α and so we see ag ain that it is essential that we keep tra ck of the cov erings a nd not just the co r resp onding subgroups of the torus. The F ro benius now ha s the for m F α : T αβ → T β As it is derived fr om the inclus io n of ﬁxed p oints, the F ro benius really dep ends only o n the subg roups L α . Ho wev er , as we a re interested in the in teraction of the F ro benius , Restriction, and other op erators , it is conv enient to keep them all indexed in one place, namely on the cov erings o f the torus. 2.3. Co v ering hom ology. W e are interested in homotopy limits o ver classes of restriction and F rob enius ope rators—these limits are called covering homology and r epresent v ersions of cyclic homology . The restr ictions and F rob enii are b oth indexed on coverings of the p-adic n-tor us, but with oppo site v ariance compo sition rules. W e need an indexing diagr am enco ding this mixed- v aria nce doubling of the categ ory of cov erings, and we do this b o o k keeping using a “twisted arrow category ”, des c rib ed b elow. Let G be a gro up. Co nsider a collec tion of surjective homomor phisms a : G ։ G , and let C b e the monoid generated by these. This monoid C can b e viewed as a categor y , also de no ted C , with one ob ject. Our desired “doubling” of C is enco de d as follows. 7 Deﬁnition 2.1. Let C be a c ategory . The twiste d arr ow c ate gory A r C of C has ob jects the arrows of C ; a morphis m from d : v → y to b : w → x is a dia gram v c − − − − → w d   y b   y y a ← − − − − x in C . Tha t is, there is a morphism from d to b for every equation d = abc . W e write ( a ∗ , c ∗ ) for this morphism and no te that the comp os itio n rule rea ds ( a ∗ 0 , c 0 ∗ )( a ∗ 1 , c 1 ∗ ) = (( a 1 a 0 ) ∗ , ( c 0 c 1 ) ∗ ). Deﬁnition 2.2. Deﬁne F r ob C , resp ectively Res C , to b e the s ub ca teg ory of the t wisted a rrow category A r C with all o b jects, but only the morphisms o f the form a ∗ := ( a ∗ , id), resp ectively c ∗ := (id , c ∗ ). The basic str ucture of co vering homology [6] is encoded in a functor A r C → Sp ec. The homo- morphism a : G → G maps to the spectr um Λ G ( A ) L a . The image of the ma p c ∗ : bc → b is the restriction map R c : Λ G ( A ) L bc → Λ G ( A ) L b and the image of the map a ∗ : ab → b is the F r ob enius F a : Λ G ( A ) L ab → Λ G ( A ) L b . There ar e three impo rtant limits asso ciated to this functor , namely over the res triction subca te- gory , the F r ob enius sub categor y , and ov e r the whole twisted ar row categor y . Deﬁnition 2.3 . T R C ( A ) := holim a ∈ Res C Λ G ( A ) ker a T F C ( A ) := ho lim a ∈ F r ob C Λ G ( A ) ker a T C C ( A ) := holim a ∈A r C Λ G ( A ) ker a The last of these is called the C -cov ering ho mo logy of A. W e often restrict attention to cov erings of the p-adic n-torus, whic h as before ar e enco ded in the monoid M n of injective endomorphisms of Z n p . In this ca se we abbrev iate T R , T F , and T C as follows: T R ( n ) ( A ) := T R M n ( A ) = ho lim α ∈ Res M n T α ( A ) T F ( n ) ( A ) := T F M n ( A ) = holim α ∈ F r ob M n T α ( A ) T C ( n ) ( A ) := T C M n ( A ) = holim α ∈ Ar M n T α ( A ) Note tha t T C (1) ( A ) is not precisely ordinary top ologica l cy clic homology , b ecause T C (1) takes int o acco unt, in addition to the usual p-fold cov erings, o rientation r eversing self-cov erings of the circle. Nev ertheless we refer to T C ( n ) ( A ) as the higher to po logical cyclic ho mology of A . 2.4. T ransfer m aps. The restrictio n and F rob enius op erators by no means exha us t the structure of the ﬁxe d p oints o f top ological Ho chsc hild homolog y . Our next stop is the V erschiebung op erator , which is derived from the transfer maps asso ciated to the pro jectio ns G/L b → G/L ab . Assume G is a compact Lie g roup (p-adic or otherwise), a nd consider a s b e fo re a mono id C of linear self-cov erings of G , that is of surjective homomor phisms a : G → G with ﬁnite kernel L a . F or a, b ∈ C , choos e a G -repr esentation W a nd an o pe n G -em bedding i : W × G/L b ֒ → W × G/L ab ov er the pro jection p r a : G/L b → G/L ab . The one-p oint compactiﬁcation o f this embedding, that is the Thom constructio n, is a G -map tr L a : S W ∧ ( G/L ab ) + → S W ∧ ( G/L b ) + called the L a -transfer. The tra nsfer is indep endent of the choice o f em b edding in the following sense. Choose a complete univ erse of G -representations, and for a ny G -space X , let Q G ( X ) = 8 colim W Map ∗ ( S W , S W ∧ X ) where the co limit is taken ov e r the univ erse. Then tw o diﬀerent choices of embeddings give homo to pic transfers tr a : Q G ( G/L ab ) → Q G ( G/L b ) . F or insta nce, if G is the circle and a is multiplication b y the p ositive integer n (or rather the n -th power o pe r ation, as we are writing G mult iplicatively), w e hav e an em bedding C × G ֒ → C × φ ∗ n G 1 × φ n − − − → C × G/L n where C is the co mplex plane with the usual circle action—the embedding can b e given ex plicitly by , for example, ( w, z ) 7→  nz + 1 1+ | w | w, z n  . As a res ult we hav e the des ired transfer tr n : S C ∧ ( G/L n ) + → S C ∧ G + . The prop erties of the tr ansfer we will need are the following. (Here ≃ means “homoto pic after applying Q G ”, and similarly fo r commutativit y claims.) Prop ositio n 2.4 . F or a, b : G → G surje ctive ﬁnite-kernel homomorphisms of c omp act Lie gr oups, the tr ansfers tr a of the pr oje ctions pr a : G → G/ L a and pr a : G/L b → G/L ab and the tr ansfers tr b of the pr oje ctions p r b : G → G/L b and pr b : G/L a → G/L ba satisfy the fol lowing r elations. 1. tr a tr b ≃ tr ba . If f : G → G ′ is an isomorphism of Lie gr oups and a ′ = f af − 1 , then tr a ′ ≃ f tr a f − 1 . If a is an isomorphism, then tr a ≃ id . 2. The tra nsfer is a G -mo dule map; that is the fol lowing diagr am c ommu tes: G + ∧ ( S W ∧ ( G/L b ) + ) 1 ∧ tr a ← − − − − G + ∧ ( S W ∧ ( G/L ab ) + ) µ   y µ   y S W ∧ ( G/L b ) + tr a ← − − − − S W ∧ ( G/L ab ) + Her e µ abbr eviates the c omp osite G + ∧ ( G/L b ) + φ b ∧ 1 − − − → ( G/L b ) + ∧ ( G/L b ) + µ − → ( G/L b ) + . 3. F ro benius recipr o city . T he tr ansfer is a c omo dule map: the diagr am S W ∧ ( G/L b ) + ∧ ( G/L ab ) + S W ∧ ( G/L ab ) + ∧ ( G/L ab ) + tr a ∧ 1 o o S W ∧ ( G/L b ) + ∧ ( G/L b ) + 1 ∧ ( pr a ) + O O S W ∧ ( G/L b ) + 1 ∧ ∆ + O O S W ∧ ( G/L ab ) + tr a o o 1 ∧ ∆ + O O c ommut es, wher e ∆ is the diagonal. 4. The double coset formula. Assu me G is c onne cte d, and c onsider a c ommuting diagr am of self-c overings G ˜ b ← − − − − G a   y ˜ a   y G b ← − − − − G with L ˜ a ∩ L ˜ b = { 1 } . Th en G/L ˜ b pr ˜ b ← − − − − ` L ˜ b \ L a ˜ b /L ˜ a G pr a   y pr ˜ a   y G/L a ˜ b = G/ L b ˜ a pr b ← − − − − G/L ˜ a is c artesian and tr b pr a ≃ | L ˜ b \ L a ˜ b /L ˜ a | · pr ˜ a tr ˜ b . These prop erties ar e standa rd. The s e cond and third pro per ties follow from the na turality o f the transfer, whic h along wit h the ﬁrst co mp os itio n prop erty a pp ea r s already in Kahn and Pr iddy’s original paper [20]; the double c o set for m ula predates ev en the deﬁnition of the transfer. Lest these 9 formulas see m askew, the reader s hould keep in mind that, despite the notation, tr a is the tr a nsfer for the pro jection map pr a and not for the map a itse lf. 2.5. The V e rsc hi ebung. Our ﬁrst new op erator on the ﬁxed p oints of top olog ical Ho chsc hild homology , the V er schiebung, is induced by transfer maps for pro jections ass o ciated to cov erings. As be fo re let G b e a compa ct Lie gr oup, W a r eal G -r epresentation, and a a ﬁnite se lf-cov ering of G . Re c a ll that the Lo day construction Λ G ( A ) is, a priori, a Γ-space, and so asso ciates to a ﬁnite set a spa c e, or more generally to a s implicial set a simplicial space, therefore by r ealization a spa ce. (Henceforth we g enerally do no t distinguish b et ween s pa ces and their singular c omplexes or b etw een simplicial spaces a nd their rea lizations.) An immediate consequence of the pro of of the fundamental coﬁbration sequence [6] is that the stabilization ma p Λ G ( A )( K ) L a → Map ∗ ( S W , Λ G ( A )( S W ∧ K )) L a ∼ = Map ∗ ( S W ∧ ( G/L a ) + , Λ G ( A )( S W ∧ K )) G is a weak equiv alence. Here K is a test input to the Γ-spa ce Λ G ( A ). (In proving this equiv alence, the fundamental coﬁbra tion sequence provides the induction step with resp ect to the order of L a . Alternatively one ca n appro ach this stabilizatio n problem via equiv ar iant obstruction theo ry as in B¨ okstedt-Hsiang- Ma dsen [5 ]). Hence the transfer y ields a map Λ G ( A )( K ) L b ∼ − − − − → Map ∗ ( S W ∧ ( G/L b ) + , Λ G ( A )( S W ∧ K )) G tr ∗ a   y Λ G ( A )( K ) L ab ∼ − − − − → Map ∗ ( S W ∧ ( G/L ab ) + , Λ G ( A )( S W ∧ K )) G As K v arie s these ﬁt tog ether into a map in the stable homotopy ca tegory; this map is the V er- schiebung V a : Λ G ( A ) L b → Λ G ( A ) L ab . By construction the V erschiebung co mm utes with restriction o per ators, that is R a V b = V b R a , a nd is v ar iously related to the F rob enius a s follows: Prop ositio n 2.5. L et A b e a c ommut ative c onne ctive S -algebr a, G a c omp act c onne cte d Lie gr oup, and a and b ﬁnite self-c overings of G . L et T a := Λ G ( A ) L a . 1. F ro benius r ecipro city . V b is a mo dule map in the sens e that the fol lowing diagr am c ommutes: T b ∧ T ab V a ∧ 1 / / 1 ∧ F a   T ab ∧ T ab µ   T b ∧ T b µ   T b V a / / T ab In p articular V a F a = V a (1) · . 2. The do uble coset formula. F a V b = | L ˜ b \ L a ˜ b /L ˜ a | · V ˜ b F ˜ a wher e ˜ a and ˜ b ar e self-c overings such that a ˜ b = b ˜ a and L ˜ a ∩ L ˜ b = 1 . 3. F a µ ( a + ∧ 1) = µ (1 ∧ F a ) : G + ∧ T ab → T b and V a µ = µ ( a + ∧ V a ) : G + ∧ T b → T ab . 4. If G ˜ b ← − − − − G a   y ˜ a   y G b ← − − − − G is a c artesian s qu ar e of ﬁ nite self-c overings, then F a µ (1 ∧ V b ) γ = µ (1 ∧ F a V b ) + F a V b µ 10 wher e γ : G + ∧ T ˜ a ` G + ∧ T ˜ a G + ∧ T ˜ a → G + ∧ T ˜ a is t he map induc e d by ( a + ∧ 1) + ( b + ∧ 1) and the maps in the pushout ar e ˜ b + ∧ 1 and ˜ a + ∧ 1 . Otherwise said, t he diagr am G + ∧ T ˜ a a + ∧ 1 % % L L L L L L L L L L µ (1 ∧ F a V b ) + + V V V V V V V V V V V V V V V V V V V V V V V V V V G + ∧ T ˜ a ˜ b + ∧ 1 9 9 r r r r r r r r r r ˜ a + ∧ 1 % % L L L L L L L L L L G + ∧ T ˜ a F a µ (1 ∧ V b ) / / T ˜ b G + ∧ T ˜ a b + ∧ 1 9 9 r r r r r r r r r r F a V b µ 3 3 h h h h h h h h h h h h h h h h h h h h h h h h h h c ommut es up to homotopy. F rob enius re c ipro city and the double cose t form ula follow from the cor resp onding prop erties of the transfer. Pr op erty 3 simply recor ds the exp ected equiv ar iance pr op erties of the F r ob enius and V erschiebung, and pro per t y 4 is a combination of the tw o parts of prop erty 3 . 3. Differentials and higher Witt rela tions 3.1. The one-dim ensional Witt relations. The ﬁxed p oints T ( A ) C p k of o rdinary top olo gical Ho ch schild homo logy are related by the oper ators restr iction R : T ( A ) C p k → T ( A ) C p k − 1 and F rob e- nius F : T ( A ) C p k → T ( A ) C p k − 1 . By deﬁnition to po logical cyclic homolog y T C ( A ) is the ho motopy limit o f the collec tio n { T ( A ) C p k } over these tw o ser ies of op erator s. W e ca n break this homotopy limit into tw o stages by considering either o nly the Res triction o r only the F rob enius op era tors. The homotopy limit ov er the restric tio n maps is called T R ( A ); top ologica l cyclic homolog y T C ( A ) ca n then be o btained as the homotopy equalizer of T R ( A ) F ⇒ id T R ( A ). Alternately , the ho motopy limit ov er the F r ob enius maps is called T F ( A ); top ologica l cyclic homo logy ca n then b e re c ov e r ed a s the homotopy equa lizer of T F ( A ) R ⇒ id T F ( A ). In o rder to compute the homotopy g roups of top ological cy clic homolo gy , it is useful to explo it t wo additional op era tio ns on the homo to p y groups of the ﬁxed p oint spectra . The ﬁrst of these we hav e already discusse d, the V erschiebung V : π ∗ ( T ( A ) C p k − 1 ) → π ∗ ( T ( A ) C p k ). As the V er schiebung arises from a transfer map, it is o nly well deﬁned a t the level of homotopy . The second ope r ation is a diﬀer en tial d : π ∗ ( T ( A ) C p k ) → π ∗ +1 ( T ( A ) C p k ); the diﬀerential is, roughly sp eaking , multiplication by the fundamental class of S 1 on the S 1 -sp ectrum T ( A ) C p k . The co lle ction R , F , V , d sa tisﬁes c e rtain fundamental r e lations: RF = F R , RV = V R , R d = dR , F V = p , V F = V (1) · , F dV = d . Moreover d is a diﬀerential and a gra ded deriv ation. Hess elholt and Ma dsen [18] prop os ed viewing the groups π ∗ ( T ( A ) C p k ) as a pro system with resp ect to the maps R , and viewing F , V , and d a s oper ators on this prosystem. They formaliz ed this structure in the notion of a Witt co mplex, and constructed an initia l ob ject in the categ ory of Witt co mplexes. This universal Witt complex is called the de Rham-Witt complex, in part b ecause it receives a canonical sur jectiv e map fro m the de Rham complex on Witt v ectors. By studying the structure of the de Rham-Witt complexes of r ings and also of log-ring s, Hesselholt a nd Madsen were able to do extensive calculations o f top ological cyclic homolo gy; they applied these ca lc ula tions with g reat success to, among other problems, the a nalysis o f the algebr aic K-theory of lo cal ﬁelds [17]. In the following sectio ns we beg in the inv e stigation o f similar structures on higher top ologica l Ho ch schild homology , fo cusing on establishing the basic rela tions among the higher restriction, F rob enius, V erschiebung, a nd diﬀerentials. Though we do not cla im to have determined all p os sible relations among these oper ators, we have established higher a nalogs of all the classical r elations—in particular, we describ e a n int riguing splitting that o c curs in the fundamental higher F dV relation. 11 3.2. Higher diﬀerentials. W e now in tro duce the diﬀeren tials in our higher a nalog of the Witt structure. These maps a r e derived from the T n p -action on T ( n ) ( A ) L α by means of a stable splitting of the torus, and as such are dep endent on the matrix α and not—as the F rob enius and V e r schiebung maps are—o nly on the gr oup L α . The transfer map for the pro jectio n T 1 → ∗ is a stable map S 1 → T 1 + which pro duces a stable splitting T 1 + ≃ S 0 ∨ S 1 . In par ticular the s table homotopy of the torus is π S ∗ ( T 1 + ) ∼ = π S ∗ ( S 0 ) ⊕ π S ∗ ( S 1 ). The transfer ma y b e given explicitly by the pro jection σ : S C = S 2 → S 2 /S 1 ∼ = S 1 ∧ T 1 + sending a point z ∈ S C to | z | 1+ | z | ∧ z | z | ∈ R / Z ∧ T 1 + . By smashing this map with itself k times, we ge t a map σ : S k C → S k ∧ T k + . Concretely , this k -fold pr o duct of the transfer is the Thom construction on the em b edding R k × T k ֒ → C k of a tubular neighbor ho o d of the standard inclusio n T k ⊆ C k . Altogether we get a stable splitting T k + ≃ _ T ⊆{ 1 ,...,k } S | T | ∼ = k _ j =0  S j  ∨ ( k j ) W e will often wr ite maps b etw een T k + ’s a s matr ices in terms of this last basis. Lemma 3.1. With r esp e ct to t he stable splitting T n + ≃ W k j =0  S j  ∨ ( k j ) the multiplic ation µ : T 2 + → T 1 + is homotopic to the map  1 0 0 0 0 1 1 η  : S 0 S 1 S 1 S 2 → S 0 S 1 Pr o of. It is enough to verify that the S 2 → S 1 comp onent is given by η , a nd this follows from the diagram S 2 C σ ∧ 1 − − − − → S 1 ∧ T 1 + ∧ S C 1 ∧ µ − − − − → S 1 ∧ S C 1 ∧ σ   y 1 ∧ σ   y S 1 ∧ T 1 + ∧ S 1 ∧ T 1 + (1 ∧ µ ) τ − − − − → S 1 ∧ S 1 ∧ T 1 + The top µ r efers to the natura l action of T 1 on S C . The square is s trictly commutativ e, and the top horizo ntal compo site is homotopic to η [1 5].  R emark 3.2 . F or α ∈ M n Z p , we hav e an induced stable map α + : ( T n p ) + → ( T n p ) + . F rom the ab ov e description of the multiplication µ we can describ e α + in terms of the stable splitting. If n = 1 and α = a ∈ Z p then the map α + is g iven b y [ 1 0 0 a ]. If n = 2 and α =  a b c d  then α + is g iven b y     1 0 0 0 0 a b abη 0 c d cdη 0 0 0 a d − bc     : S 0 S 1 S 1 S 2 → S 0 S 1 S 1 S 2 F or g eneral α = [ a ij ] ∈ M k × n Z p the stable map α + is given by the matrix indexed by the s ubsets of { 1 , . . . , k } a nd { 1 , . . . , n } with S - T -entry M S,T =   X f : T ։ S sign( f ) Y j ∈ T a f ( j ) j   η | T |−| S | : S | T | p → S | S | p where the sum is over all surjective f : T ։ S and the sign is with resp ect to the order ing given by the fact that we are consider ing s e ts of na tur al num b ers . 12 If p is an o dd prime, multiplication by η is nullhomotopic, and these formulae simplify consider - ably . In particular we have that if α ∈ M n Z p , then S n C p σ − − − − → ( S n ∧ T n p + ) p det( α )   y 1 ∧ α +   y S n C p σ − − − − → ( S n ∧ T n p + ) p commutes up to stable homotopy . Deﬁnition 3. 3. Let X b e a T n p -sp ectrum a nd ℓ ∈ M k × n Z p . Deﬁne the ℓ th diﬀerential as the stable map d ℓ : S k ∧ X → X given b y the co mpo site S k C ∧ X σ ∧ 1 − − − − → S k ∧ T k p + ∧ X 1 ∧ ℓ + ∧ 1 − − − − − → S k ∧ T n p + ∧ X µ − − − − → S k ∧ X Prop ositio n 3.4. If ℓ ′ ∈ M n × k ′ Z p and ℓ ′′ ∈ M n × k ′′ Z p , then d ℓ ′ d ℓ ′′ ≃ d ℓ wher e ℓ is t he n × ( k ′ + k ′′ ) matrix obtaine d by placing the c olumns of ℓ ′ b efor e those of ℓ ′′ . F or p an o dd prime, ℓ ∈ M n × k Z p , and γ ∈ M k Z p , we have d ℓγ ≃ de t( γ ) · d ℓ In p articular, if ℓ ∈ M n Z p , then d ℓ ≃ de t( ℓ ) · d 1 . If ℓ ∈ M n × k Z p has r ank less than k , then d ℓ ≃ 0 . Pr o of. The ﬁr st statement fo llows from the commutativit y of T k ′ p × T k ′′ p ℓ ′ × ℓ ′′ − − − − → T n p × T n p ∼ =   y µ   y T k p ℓ − − − − → T n p The second statement follo ws fro m the obser v ation in remar k 3 .2 that if γ ∈ M n Z p and η = 0 then (1 ∧ γ + ) σ ≃ σ det( γ ). Finally , if ℓ has less than maximal r ank, then ℓ = ℓ ′ γ where γ ∈ M k Z p has zero determinant.  Note that the equiv alence d ℓγ ≃ det γ · d ℓ implicitly depends on a p -completion; of course, this completion is unnecessar y if the matr ix γ is integral. R emark 3.5 . The ﬁrs t part of this prop os ition says that all diﬀerentials ca n b e descr ibed a s com- po sites of “one-dimensiona l” diﬀer e n tials, that is by comp osites o f d ℓ ’s with ℓ : Z p → Z n p . F or p = 2 the seco nd part of the pr o po sition bec o mes a bit mor e complicated: d ℓγ ≃ de t( γ ) · d ℓ + X ∅6 = T ( { 1 ,...,k } M { 1 ,...,k } ,T d ℓi T where i T is the matrix of the inclusio n of T in { 1 , . . . , k } , and M S,T , a s in rema r k 3.2, consists of pro ducts of minors and powers o f η . As an example, let n = 1 and ℓ = 1. Then d 2 1 ≃ d [1 , 1] = d [1 , 0] · [ 1 1 0 0 ] . This diﬀerential is homotopic to 0 · d [1 , 0] + 1 · 1 · d 1 η + 0 · 0 · d 0 η = d 1 η , r ecov ering Hesselholt’s formula d 2 ≃ dη for the one-dimens io nal case. Lemma 3.6. L et ℓ 1 , ℓ 2 ∈ M n × 1 Z p . Then d ℓ 1 + ℓ 2 ≃ d ℓ 1 + d ℓ 2 . 13 Pr o of. The diﬀeren tial d ℓ 1 + ℓ 2 and the sum d ℓ 1 + d ℓ 2 are the upp er and lo wer compo sites of the diagram ( S 1 ∨ S 1 ) ∧ X / / ( T 1 + ∨ T 1 + ) ∧ X ℓ 1 + ∨ ℓ 2 + / / ( T n + ∨ T n + ) ∧ X / / fold   X ∨ X   S 1 ∧ X / / O O T 1 + ∧ X pinch O O ( ℓ 1 + ℓ 2 ) + / / T n + ∧ X / / X That { fold ◦ ( ℓ 1 + ∨ ℓ 2 + ) ◦ pinch } : T 1 + → T n + is stably ho motopic to ( ℓ 1 + ℓ 2 ) + is check ed by ex plicit computation in homotop y and depends essen tially on the source having only cells in dimensions zero and one.  3.3. Gcd’s and Lcm’s of matrices. In order to calculate the r elations among the F rob enius, the diﬀerential, and the V erschiebung, it is conv enien t to develop some technology descr ibing how v ario us matric es (which index the F , d , and V op erator s) int eract. W e do so in a bit of (we hop e ent ertaining) generality . Deﬁnition 3.7. Let f : A → B b e an injection o f ab elian groups. The volume | f | o f f is deﬁned to be the cardinality o f the cokernel o f f . The adjoi nt f † of f is the unique lifting f † in 0 / / A f / / | f |   B / / | f |   f †       B / f ( A ) / / 0   0 0 / / A f / / B / / B / f ( A ) / / 0 Note that if A = B = Z n , then the volume is the absolute v alue of the determinant, and the adjoint is given by plus o r minus the adjoint of a matrix in elementary linear algebr a. In Z p there are more units, and so p otentially a greater distance b etw een our adjo int and the class ical a djoin t deﬁned by explicit for mulae in volving minors. In the following, Λ will b e a principa l ideal do main, and Q a ﬁnitely genera ted free Λ-mo dule. Deﬁnition 3.8 . Deﬁne M Λ ( Q ) ⊆ E nd Λ ( Q ) to b e the submono id consis ting of the injective endo- morphisms of Q . In the situation Λ = Z p and Q = Z n p we simply write M n := M Z p ( Z n p ). Given f , g ∈ M Λ ( Q ), we say that f and g are c oprime if f + g : Q ⊕ Q → Q is surjective. The gr e atest c ommon divisor of f and g is a ma p gcd( f , g ) : Q → Q for which there are coprime ¯ f and ¯ g such that the comp osite Q ⊕ Q ¯ f + ¯ g − − − → Q gcd( f ,g ) − − − − − → Q is equal to Q ⊕ Q f + g − − − → Q ; mor e sp eciﬁcally , the greatest common divisor is the equiv a lence class in M Λ ( Q ) / Aut Λ ( Q ) c o nsisting of those d ∈ M Λ ( Q ) such that f + g = d ( ¯ f + ¯ g ) for some coprime ¯ f and ¯ g . Alternately , the gr eatest common diviso r can b e v iewed as the collection of all ma ps Q → Q that factor into an is omorphism Q ∼ = ( f + g )( Q ⊕ Q ) follow ed by the inclusion ( f + g )( Q ⊕ Q ) ⊆ Q . Note that such factorizations exist b ecause ( f + g )( Q ⊕ Q ) is a free mo dule of r ank equal to the ra nk of Q —this follows b ecause f is injective and submo dules of the free mo dule Q over the P ID Λ are free. Note that | gcd( f , g ) | = 1 and g cd( f , g ) = Aut Λ ( Q ) are just other wa ys o f saying that f and g are coprime. A simple diagra m chase gives the following lemma. Lemma 3.9. If f , g ∈ M Λ ( Q ) then | f || g | | Q/ ( f ( Q ) ∩ g ( Q )) | = | g c d( f , g ) | . Deﬁnition 3.10. F or ea ch coprime pair f , g ∈ M Λ ( Q ), cho ose a section (b ez f ⊕ be z g ) : Q → Q ⊕ Q of f + g . The maps b ez f and b ez g such that f b ez f + g b ez g = 1 are the analogs of class ic al Bezout nu mbers. In this coprime case, choose the representativ e 1 for the gr eatest co mmon divis or o f f and g . If f and g ar e not coprime cho ose a representativ e in M Λ ( Q ) for the gre atest common divis or and call it gcd( f , g ), by abuse of no ta tion. Let ¯ f , ¯ g ∈ M Λ ( Q ) b e the copr ime pair given uniquely 14 by f + g = gcd( f , g )( ¯ f + ¯ g ) and let b ez f ⊕ be z g be equal to b ez ¯ f ⊕ be z ¯ g , s o that f b ez f + g b ez g = gcd( f , g ). Note that the diagr am Q g   Q ˜ f o o ˜ g   Q Q f o o is cartesia n if and only if the diagram Q ¯ g   Q ˜ f o o ˜ g   Q Q ¯ f o o is cartesia n. When these diagrams are cartes ia n, denote the comp osite g ˜ f = f ˜ g by lcm( f , g ), giving the pleasant equation | f || g | = | g cd( f , g ) || lcm( f , g ) | . F or ea ch pair f and g , we in fact choose a sp eciﬁc pullba ck of f and g , ther efore sp eciﬁc maps ˜ f and ˜ g ; this in turn pins down a particular map lcm( f , g ). R emark 3.1 1 . W e summarize the ab ov e discussion and introduce some notation clarifying the rela- tionship of { ¯ f , ¯ g , ˜ f , ˜ g } to { f , g } . Giv en natural num be r s m and n , there are n umbers h m i := m/ g cd( m, n ) = lcm( m, n ) /n h n i := n/ g cd( m, n ) = lc m( m, n ) /m In our noncommutativ e gener alization, there is not one notion here but tw o, accor ding to whether we gener alize the expression “ m/ gcd( m, n )” or “lcm( m, n ) /n ”. The genera lizations a re denoted, resp ectively ¯ f and ˜ f . W e sometimes use the expression [ f / gcd( f , g )] to me an ¯ f , and the expres- sion [lcm( f , g ) /g ] to me an ˜ f . T o reiterate then, the endomo r phisms [ f / gcd( f , g )] and [ g / gcd( f , g )] are deﬁned to be c o prime endomorphisms for which ther e is a mor phisms gcd( f , g ) such that f = gcd( f , g )[ f / gcd( f , g )] and g = gcd( f , g )[ g / gcd( f , g )]. Similarly , t he endomo rphisms [lcm( f , g ) /g ] and [lcm( f , g ) / f ] a re deﬁned to b e coprime endomorphisms s uch that f [lcm( f , g ) /f ] = g [lcm( f , g ) /g ]; that co mmon pro duct is called lcm( f , g ). Notice that with the ab ov e deﬁnitions of Be z out endo- morphisms, we have the relation [ f / gcd( f , g )] b ez f +[ g / g c d ( f , g )] b ez g = 1. W e sometimes wr ite gcd( f , g ) as g cd f ,g and similarly lcm( f , g ) as lcm f ,g . Example 3.12 . Prop ositions 2.4.4 a nd 2.5.2 describ e relations for tr ansfer maps a nd the V e r schiebung. When the group G in ques tio n is the torus T n p , the order of the double cose t | L ˜ b \ L a ˜ b /L ˜ a | app ear ing in those formulae is precis ely the volume | gcd( a, b ) | . Here L a = a − 1 Z n p / Z n p ⊂ T n p is the kernel o f the covering asso cia ted to the ma trix a . Let K b e a ﬁeld containing Λ, and set b Λ = K / Λ. This quotient b Λ should b e thought o f as the classifying space of Λ; for ex ample, for the inclusion Z ⊂ R , the classifying space is b Z = R / Z , the circle. (T otar o has a lso considere d this notio n o f classifying spaces o f rings.) When as b efore Q is a ﬁnitely g e ne r ated fre e Λ-mo dule, let b Q = b Λ ⊗ Λ Q . When f ∈ M Λ ( Q ) there is an induced surjection bf : bQ → b Q with kernel l f ∼ = Q/f ( Q ) and an is omorphism φ f : bQ /l f → bQ induced by m ultiplication by f , a s illustrated by the diagram 0 − − − − → Q − − − − → K ⊗ Λ Q − − − − → bQ − − − − → 0 f   y f   y ∼ = bf   y 0 − − − − → Q − − − − → K ⊗ Λ Q − − − − → bQ − − − − → 0 (Here the middle map is a n is omorphism by Cramer ’s rule , s ince det ( f ) is inv ertible in K ). As an example, let Λ = Z ⊆ R = K , Q = Z and let f b e multiplication by m ∈ Z . Then bf : bQ → bQ is the m -fold cov ering o f the circle bQ = R / Z . Lemma 3.13. L et f , g ∈ M Λ ( Q ) . Ther e is an h ∈ M Λ ( Q ) su ch that g = hf if and only if l f ⊆ l g ⊆ bQ . In p articular, ther e is an h ∈ Aut Λ ( Q ) such t hat g = hf if and only if l f = l g . 15 Pr o of. This follows by a diag ram chase inv olving 0 − − − − → Q − − − − → K ⊗ Λ Q − − − − → bQ − − − − → 0 g x   g x   ∼ = bg x   0 − − − − → Q − − − − → K ⊗ Λ Q − − − − → bQ − − − − → 0 f   y f   y ∼ = bf   y 0 − − − − → Q − − − − → K ⊗ Λ Q − − − − → bQ − − − − → 0  Lemma 3.14. If f : Λ k → Λ n is an inje ction, then ther e ar e γ ∈ GL n (Λ) , γ ′ ∈ GL k (Λ) s u ch that γ ′ f γ is r epr esente d by a diagonal k × k -matrix fol lowe d by the standar d inclu s ion Λ k ⊆ Λ n . Pr o of. The fundamen tal theor em for ﬁnitely generated mo dules over a PID gives us an isomo rphism Λ n /f (Λ k ) ∼ = Λ n − k ⊕ L k i =1 Λ /λ i Λ, for some λ i ∈ Λ. Lifting this isomor phism gives the result.  If in addition Λ is lo cal n umber ring with maxima l ideal genera ted by π , then the dia gonal matr ix in questio n can b e chosen to have p owers of π on the dia gonal. 3.4. Stable s plitting of the transfer. In the last section we analyz ed the interactions of endo- morphisms of mo dules over a PID. W e a re of course mos t interested in the case of the P ID Λ = Z p and the mo dule Q = Z n p . Asso c ia ted to a matrix α ∈ M n = M n ( Z p ) ∩ GL n ( Q p ), there is a self- cov ering of the p-co mplete n- torus. The V erschiebung map for α is by deﬁnition the V er schiebung map for that induced self-cov ering. This V erschiebung was deﬁned in section 2.5 in terms of the transfer map for the pro jection a sso ciated to the cov ering. The hig her diﬀerentials in tro duced in section 3.2 were built using the stable splitting of the n-torus. In orde r to determine how the diﬀerentials and the V er s chiebung interact, we must ther e - fore desc r ib e the relations hip of the stable splitting of the n-torus to th e transfers ass o ciated to pro jections of the tor us—giving such a description is the purpo se of this section. Cho ose once and for all an e q uiv aria nt transfer map tr p : ( T /L p ) + → T + . W e will need the following fact proved in Hesse lho lt [15]. Lemma 3.15. With r esp e ct t o the stable e quivalenc e T + ≃ S 0 ∨ S 1 the c omp osite T + φ p + − − − − → ( T /L p ) + tr p − − − − → T + is given by t he matrix τ p =  p ( p − 1) η 0 1  F or α, β ∈ M n we ana lyze the trans fer maps tr β αβ : ( T n /L αβ ) + → ( T n /L β ) + as follows. F a ctor α = γ ′ δ γ wher e γ ′ , γ ∈ GL n ( Z p ) and δ is a diago nal matrix with dia gonal en tries p ow ers of p . The diagram T n /L β φ γ β ← − − − − T n pro j .   y pro j .   y T n /L αβ φ γ β ← − − − − T n /L δ is a pullback. (In fact the horizon tal maps are isomorphisms. Note the absence of γ ′ from the diagram—it was a bsorb ed in an implicit equality in the low er righ t corner b etw een T n /L δ and 16 T n /L γ ′ δ . There is a seco nd implicit equality in the upp er right corner b etw een T n and T n /L γ − 1 .) The transfers tr β αβ and tr δ are therefore related by the diagr am ( T n /L β ) + φ γ β + ← − − − − ( T n ) + tr β αβ x   tr δ x   ( T n /L αβ ) + φ γ β + ← − − − − ( T n /L δ ) + Moreov er, the diag onal transfer tr δ : ( T n p /L δ ) + → ( T n p ) + can b e descr ib ed explicitly as the p - completion of smashes of powers o f tr p . Note that if η acts trivially , and we view p + as g iven by the matrix  1 0 0 p  : S 0 S 1 → S 0 S 1 , the comp os ite tr p φ p + is given by the adjoint ( p + ) † =  p 0 0 1  . This contin ues to b e true more generally : if α ∈ M n then α + can be view ed as a 2 n × 2 n -matrix thr ough the n -fold smash of the stable equiv alence T + ≃ S 0 ∨ S 1 , a nd (if η acts triv ially) the transfer is given in this basis by ( α + ) † . W e will in the future o cca sionally write ( α + ) † for tr α φ α + even if η do es not act trivially . Corollary 3.1 6. L et δ ∈ M n Z b e a diagonal matrix with p -p ower entries. Then, under the stable e quivalenc e T + ≃ S 0 ∨ S 1 the tr ansfer ( δ + ) † : ( T n p ) + → ( T n p ) + is given by t he matrix τ δ = O 1 ≤ j ≤ n  δ j j ( δ j j − 1 ) η 0 1  If α ∈ M n then S 1 ∨ · · · ∨ S 1 inc / / ( T n p ) + ( α + ) † / / ( T n p ) + pr oj / / S 1 ∨ · · · ∨ S 1 is given by α † . If η acts trivial ly then S 1 ∨ · · · ∨ S 1 inc − − − − → ( T n p ) + inc   y ( α + ) †   y ( T n p ) + ( α † ) + − − − − → ( T n p ) + c ommut es in the st able homotopy c ate gory. 3.5. Relations among the F rob enius, diﬀe rential, and V ersc hiebung. In the one-dimensio nal case, the V erschiebung is r elated to the F rob enius and diﬀerential by the rela tions F V = p , V F = V (1) · , and F dV = d . In Pr op osition 2 .5, w e established multi-dimensional a nalogs of the ﬁrst tw o relations, namely F α V β = | g c d α,β | V ˜ β F ˜ α and V α F α = V α (1) · ; (here we hav e made use of the o bserv atio n in example 3.1 2). It remains only to analy ze the comp osite F α d ℓ V β . Lemma 3.17. L et α, β ∈ M n and let ℓ ∈ M n × k Z p . Then F α d ℓ V α : S k ∧ T ( n ) ( A ) L β → T ( n ) ( A ) L β is homotopic to the c omp osite S k ∧ T β σ ∧ 1 / / ( T k p ) + ∧ T β ℓ + ∧ 1 / / ( T n p ) + ∧ T β ( α + ) † ∧ 1 / / ( T n p ) + ∧ T β φ β ∧ 1 / / ( T n p /L β ) + ∧ T β µ / / T β wher e ( α + ) † := tr α φ α + , the map tr α : ( T n p /L α ) + → ( T n p ) + is the tr ansfer, and as b efor e T β is shorthand for T ( n ) ( A ) L β . 17 Pr o of. If we show the stable commutativit y of ( T n p ) + ∧ T β φ β + ∧ 1 / / ( T n p /L β ) + ∧ T β µ / / T β ( T n p /L α ) + ∧ T β tr α ∧ 1 O O ( T n p ) + ∧ T β φ α + ∧ 1 o o φ αβ + ∧ 1 / / 1 ∧ V α   ( T n p /L αβ ) + ∧ T β tr α ∧ 1 O O 1 ∧ V α   ( T n p ) + ∧ T αβ φ αβ + ∧ 1 / / ( T n p /L αβ ) + ∧ T αβ µ / / T αβ F α O O then we are done: a moment’s reﬂection shows that the tw o maps in question can b e obtained by precomp osing with S k ∧ T β σ ∧ 1 / / ( T k p ) + ∧ T β ℓ + ∧ 1 / / ( T n p ) + ∧ T β . The squa re obviously c ommu tes. The upp er left p entagon comm utes b eca use T n p pro j   φ β / / T n p /L β pro j   T n p /L α T n p φ α o o φ αβ / / T n p /L αβ is a pullbac k—here of course the horizontal maps are isomorphisms . The right p entagon commut es by arguments ana logous to those in the pro o f of the one-dimensional F dV rela tion, sp eciﬁcally by dualizing the V erschiebung and F rob enius a nd ex pr essing the p entagon as a co mpo site of transfer pullback s quares for pr o ducts of to ri, a la diagram 1.5 .2 of Hesselho lt [15].  Corollary 3.18. In the situation of the lemma, if α ∈ GL n Z p then F α d ℓ V α = d α − 1 ℓ Pr o of. W e need only understand ( α + ) † := tr α φ α + . Here φ α is simply α − 1 : T n p → T n p /L α = T n p and tr α is the identit y .  This for m ula may seem surpris ing, given that F α : T L αβ → T L β and V α are the identit y o f sp ectra when α is an isomo rphism, but follows from the fact that they ar e not equiv ariant, and d ℓ is sensitive to the T n p -action. Lemma 3 .17 can more elegantly b e rephras ed in terms of a less useful v ariant of the diﬀer e n tial: for a stable map v : ( T k p ) + → ( T n p ) + let D v be the co mpo site ( T k p ) + ∧ X v ∧ 1 / / ( T n p ) + ∧ X µ / / X , so that d ℓ = D ( ℓ + ) ( σ ∧ 1). Then we get that F α D v V α = D ( α + ) † v Corollary 3.19. L et α ∈ M n Z . Assume mu lt iplic ation by η is nul lhomotopic on T L β (for example if it is an Eilenb er g-MacL ane sp e ctrum or if p is o dd). If ℓ ∈ M n × 1 Z p , then F α d ℓ V α = d α † ℓ and F α d 1 V α = | α | det( α ) · d 1 Pr o of. The ﬁr st par t follows by Corollar y 3 .16 and Lemma 3.17. F or the second par t, note that, if η = 0, the “ n -dimensiona l part” of ( α + ) † is a co lumn of zeros except for the ﬁrst en try which is the unit | α | / det( α ).  F or diﬀerentials d ℓ , with ℓ an n × k matrix for 1 < k < n , we see that F α d ℓ V α is giv en b y int ermediate matrices of minors. No te a ls o that the num b er | α | det( α ) ∈ Z p app earing in the ab ove formula is inv ertible. 18 Lemma 3.20. L et ℓ : Z k p → Z n p and α ∈ M n . Then 1. d ℓ F α = F α d αℓ 2. V α d ℓ = d αℓ V α Pr o of. The ﬁr st equation ho lds b ecause the diagram ( T n p ) + ∧ T αβ pro j ∧ 1 / / φ β + ∧ 1   ( T n p /L α ) + ∧ T αβ φ β + ∧ 1   ( T n p /L β ) + ∧ T αβ pro j ∧ 1 / / 1 ∧ inc   ( T n p /L αβ ) + ∧ T αβ µ / / T αβ inc   ( T n p /L β ) + ∧ T β µ / / T β commutes, a nd the tw o sides o f the equation can be obtained by precompo sing with S k ∧ X L αβ σ ∧ 1 − − − − → T k p ∧ X L αβ ℓ + ∧ 1 − − − − → T n p ∧ X L αβ The second eq ua tion is similar.  Piecing tog ether our accumulated understanding of the interactions among these o p er ations, we can establish the ﬁnal relation: Theorem 3.21. L et p b e an o dd prime, and let α, β ∈ M n , ℓ : Z p → Z n p . Cho ose a r epr esentative D of gcd α,β and c oprime ¯ α, ¯ β ∈ M n with α = D ¯ α and β = D ¯ β . F urt hermor e, cho ose a splitting ( s, t ) to ¯ α + ¯ β , so t hat ¯ αs + ¯ β t = 1 . Then F α d ℓ V β = d sD † ℓ F ¯ α V ¯ β + F ¯ α V ¯ β d tD † ℓ Pr o of. F ro m Corolla ry 3 .19 and Lemma 3.20, we have F α d ℓ V β = F ¯ α F D d ℓ V D V ¯ β = F ¯ α d D † ℓ V ¯ β = F ¯ α d ¯ αsD † ℓ + ¯ β tD † ℓ V ¯ β = F ¯ α ( d ¯ αsD † ℓ + d ¯ β tD † ℓ ) V ¯ β = F ¯ α d ¯ αsD † ℓ V ¯ β + F ¯ α d ¯ β tD † ℓ V ¯ β = d sD † ℓ F ¯ α V ¯ β + F ¯ α V ¯ β d tD † ℓ  Note that by Prop os ition 2.5.2, F ¯ α V ¯ β = V ˜ β F ˜ α , where ˜ α and ˜ β ar e coprime ma trices with α ˜ β = β ˜ α . The right hand side o f this F dV rela tio n could thereby b e r ewritten in terms of dV F and V F d . W e summarize the s tructure o f the ﬁx e d p oint sp ectra T ( n ) ( A ) L α in the following omnibus the- orem. W e state the results in the homotop y catego r y , though those r elations only in volving the restriction and F ro benius may b e lifted to sp ectra. Theorem 3.22 . Fix an o dd prime p. L et A b e a c onne ctive c ommutative ring sp e ctrum. F or α ∈ M n ( Z p ) ∩ GL n ( Q p ) an inje ctive endomorphism of Z n p , let L α := α − 1 Z n p / Z n p ⊂ T n p b e the c orr esp onding sub gr oup of the p-adic n-torus. Denote by T α := T T n p ( A ) L α the ﬁx e d p oints of the n-dimensional top olo gic al Ho chschild homolo gy of A ; t his is a ring sp e ctrum with multiplic ation denote d µ : T α ∧ T α → T α . Ther e ar e op er ators in the stable homotopy c ate gory R α : T β α → T β (r estr iction), F α : T αβ → T β (F ro b eniu s ) , and V α : T β → T αβ (V erschiebung). Mor e over, for e ach p-adic ve ctor v ∈ Z n p , ther e is an op er ator d v : S 1 ∧ T α → T α (diﬀer ential). These R , F , V , and d maps satisfy the fol lowing r elations. 19 1. R α µ = µR α ; F α µ = µF α ; d v (1 ∧ µ ) = µ ( d v ∧ 1) + µ (1 ∧ d v )( τ ∧ 1) ; if α ∈ GL n Z p then F α = id and V α = id . 2. R α R β = R αβ ; F α F β = F β α ; V α V β = V αβ ; d v + w = d v + d w ; d v (1 ∧ d w ) = d w τ ( d v ∧ 1) τ . 3. R α F β = F β R α ; R α V β = V β R α ; R α d v = d v R α . 4. µ ( V α ∧ 1) = V α µ (1 ∧ F α ) ; F α V β = | g c d α,β | V [lcm α,β /α ] F [lcm α,β /β ] . 5. d v F α = F α d αv ; V α d v = d αv V α ; F α d v V β = d b ez α gcd † α,β v V [lcm α,β /α ] F [lcm α,β /β ] + V [lcm α,β /α ] F [lcm α,β /β ] d b ez β gcd † α,β v . Notation: I n the ab ove, we have a bbr eviate d f or example 1 ∧ V as V and R ∧ R as R and the like, when no c onfusion is p ossible. In item (1), τ is a twist, and the e quations r e c or d the facts t hat R and F ar e ring maps, d v is a derivation, and F α and V α only d ep end on the gr oup L α . The terms in items (4) and (5) ar e deﬁne d as fol lows. Cho ose matric es g c d α,β and c oprime [ α/ gcd α,β ] and [ β / gcd α,β ] su ch that α = gcd α,β [ α/ gcd α,β ] and β = gcd α,β [ β / g cd α,β ] . N ex t cho ose “Bezout” matric es b ez α and b ez β such that [ α/ gcd α,β ] b ez α +[ β / gcd α,β ] b ez β = 1 . Final ly c ho ose c oprime matric es [lcm α,β /α ] and [lcm α,β /β ] such that α [lcm α,β /α ] = β [lcm α,β /β ] ; t hat c ommon pr o duct is by deﬁnition lcm α,β . A t the risk of rep etition, w e follow the lead of Hesselho lt and Madsen in taking homotopy gr oups and r epack a ging some of this data int o a particula r kind of pro diﬀerential ring . In the following p is aga in an o dd pr ime. Corollary 3.23. Asso ciate d to a c onne ct ive c ommutative ring sp e ctru m A , ther e is a pr o mult i- diﬀer ential gr ade d ring denote d T R α q ( A ; p ) and deﬁne d as fol lows. F or e ach matrix α ∈ M n ( Z p ) ∩ GL n ( Q p ) , we have a gr oup T R α q ( A ; p ) := π q ( T T n p ( A ) L α ) ; her e L α = α − 1 Z n p / Z n p . As q varies these gr oups form a gr ade d ring. F or e ach p-adic ve ctor v ∈ Z n p , ther e is a gr ade d diﬀer en t ial d v : T R α q ( A ; p ) → T R α q +1 ( A ; p ) ; these diﬀer entials ar e derivations, ar e line ar in the ve ctor v , and they gr ade d c ommut e with one anothe r. The c ol le ction T R α ∗ ( A ; p ) is ther efor e a multi-diﬀer ential gr ade d ring. As α varies these form a pr o multi-diﬀer ential gr ade d ring under t he r est riction maps R α . Ther e i s a c ol le ction of pr o-gr ade d-ring o p er ators F α : T R αβ ∗ → T R β ∗ , and a c ol le ction of pr o- gr ade d-mo dule op er ators V α : ( F α ) ∗ T R β ∗ → T R αβ ∗ . Both F α and V α dep end only on the gr ou p L α . Her e ( F α ) ∗ T R β ∗ denotes T R β ∗ with the T R αβ ∗ -mo dule stru ctur e determine d by pr e c omp osition with F α . Note t hat the fact t hat V α is a mo dule map is e quivalent to F r ob enius r e cipr o city: V α ( x ) · y = V α ( x · F α ( y )) . These op er ators ar e su bje ct to the r elations 1. F α V β = | g c d α,β | V [lcm α,β /α ] F [lcm α,β /β ] 2. d v F α = F α d αv ; V α d v = d αv V α 3. F α d v V β = d b ez α gcd † α,β v V [lcm α,β /α ] F [lcm α,β /β ] + V [lcm α,β /α ] F [lcm α,β /β ] d b ez β gcd † α,β v It may be that this list of relations, together with a few discussed ea rlier concerning higher diﬀerentials, is in a useful sense complete. In this case, it is clear how to utilize and abstract the structure seen in the coro llary: add a map (compatible with the ope r ators) fr om the Burnside-Witt vectors [1 1, 14] int o the deg ree zer o par t of the pro mu lti-diﬀerential gr aded ring—we would call 20 the resulting structure a Burnside - Witt complex . (Compare Hesselholt-Ma dsen [18].) It would be worth studying the initial such complex, which serves the r o le o f a higher analog of the de Rham- Witt co mplex. In future work we will make pr ecise and inv es tigate these notions of Bur nside-Witt complex and “ de Rham-Bur nside-Witt” complex. 4. Adams opera tions o n covering ho mology Recall that in the end w e ar e int erested in covering homolo gies T C J ( A ), that is in versions of higher topolo gical cyclic homology—these are ho mo topy limits of the ﬁxed po in ts T ( n ) ( A ) L α under collections of res triction and F rob enius op era tors a sso ciated to s ubmonoids J o f the self-is o genies o f the p-a dic n-torus. There are g r oups of Ada ms op er a tions acting on the ﬁxed p oints T ( n ) ( A ) L α , and for appropr iate choices of submonoids J , these oper ations survive the homotopy limit to pro duce int eresting actions on the cov e ring homology T C J ( A ). Indeed, these actions will play a cr ucial role in our future in vestigation of chromatic phenomena a rising from higher to p olo gical cy c lic homology . 4.1. Iden tiﬁcation of the F rob enius and restri ctio n categories. W e b egin by explicitly iden- tifying the structure of the indexing catego r ies for F rob enius and res triction o per ators, in the case of higher top olog ical Ho chschild homology based on the p-adic n-torus. Recall from se c tion 2.3 that M n := M n ( Z p ) ∩ GL n ( Q p ) a nd F rob M n and Res M n are the sub categ ories of the t wisted arr ow category A r M n consisting resp ectively of the maps α ∗ and α ∗ . Lemma 4 .1. Two obje cts in F rob M n ar e c onne cte d by at m ost one morphism; similarly for Re s M n . The gr oup GL n ( Z p ) acts b oth on the left and the right of b oth F r ob M n and Res M n . Pr o of. Note that since the ma trices in M n are in vertible o ver Q p we hav e ca nc e llation: f g h = f g ′ h ⇒ g = g ′ . Hence, in both R es M n and F r ob M n there is at most one arrow betw een tw o ob jects. If a ∗ : a b → b ∈ F rob M n and g ∈ GL n ( Z p ) w e set g · a ∗ = ( g ag − 1 ) ∗ : g ab → g b and a ∗ · g = a ∗ : abg → bg . The other a ction is similar.  Notice that wher eas there is at most one map b et ween tw o ob jects in Res M n or F rob M n , in the “amalga mation” A r M n there can b e many . Lemma 4.2. F or α ∈ M n , t he assignment α 7→ L α = α − 1 Z n p / Z n p deﬁnes an e quivalenc e of c ate gories F rob M n ≃ GL n ( Z p ) \ F rob M n ∼ = S ub op T n p fr om F r ob M n to t he opp osite of the c ate gory S ub T n p of ﬁnite sub gr oups of T n p = R n p / Z n p and inclu- sions, and induc es an isomorphism b etwe en GL n ( Z p ) \ F rob M n and S ub op T n p . Dual ly, the assignment β 7→ β Z n p deﬁnes an e quivalenc e Res M n ≃ R es M n /GL n ( Z p ) ∼ = Op Z n p b etwe en Res M n and the c ate gory O p Z n p of op en sub gr oups of Z n p and inclusions, and induc es an isomorphi sm b etwe en the orbit c ate gory Res M n /GL n ( Z p ) and O p Z n p . Pr o of. Since all ﬁnite s ubg roups of T n p are of the form L α for some α , Lemma 3 .13 is just another wa y o f stating that α 7→ L α induces an isomorphism b etw e en GL n ( Z p ) \ F rob M n and S ub op T n p . Cho ose a s ection σ of the pro jection p : M n → GL n ( Z p ) \M n . F or every β ∈ M n there is a unique γ β ∈ GL n ( Z p ) such that σ ( p ( β )) = γ β β . Let Σ : GL n ( Z p ) \ F rob M n → F rob M n be deﬁned by Σ( p ( β )) = σ ( p ( β )) = γ β β and Σ( p ( β ) → p ( αβ )) = ( γ β β → γ αβ αβ = ( γ αβ αγ − 1 β ) γ β β ), giving a section to the pro jection to the orbit categor y , with γ as the natura l isomo rphism esta blishing the equiv alence. The last statemen t is dual. The analog of Lemma 3.1 3 for Re s M n is the sta temen t that tw o op en subgr oups α Z n p and β Z n p of Z n p are equa l if and only if β − 1 α ∈ GL n ( Z p ), and mor e gene r ally α Z n p ⊆ β Z n p if and o nly if β − 1 α ∈ M n .  21 Henceforth we abbrev iate S ub op T n p by S ub . In o ur la ter computation of top ologica l F r ob enius ho- mology , we will need the following r esult concer ning the orbit struc tur e of this ca tegory of subgroups. Corollary 4.3. L et K ∈ S ub . The orbit of K un der t he GL n ( Z p ) -action on S ub c onsists of t he sub gr oups that ar e abstr actly gr oup-isomorphic to K . The st abilizer of K has ﬁnite index in GL n ( Z p ) , and if K ⊆ L p l id n for some l , t hen the en tir e orbit wil l c onsist of sub gr oups of L p l id n . Pr o of. Clearly , all elements in the orbit of K are abstractly iso morphic to K . Conv er sely , a ssume K ′ ∈ S ub a nd given a group isomor phis m x : K ∼ = K ′ . Choo s e matrices α, α ′ ∈ M n such that K = L α and K ′ = L α ′ and co nsider the diag r am 0 / / Z n p γ     / / α − 1 Z n p   / / / / K / / x ∼ =   0 0 / / Z n p   / / ( α ′ ) − 1 Z n p / / / / K ′ / / 0 By Nak ay ama’s lemma the dotted middle vertical map exists a nd is a n isomorphism, and so the dotted left vertical map exists, and is repr esented by a matrix, say γ . This mea ns that K ′ = L αγ − 1 = γ · K . Lemma 3.1 3 implies that L α ⊆ L p l id n if and o nly if p l α − 1 ∈ M n , and so certainly such an l exists. F urthermore, since p l γ α − 1 = γ · p l α − 1 we a ls o have that L αγ − 1 ⊆ L p l id n . The stabilizer has ﬁnite index b ecause there are only ﬁnitely many subgroups of L p l id n .  R emark 4.4 . Occasio nally we will be exclusively interested in homoto p y limits over sub categories of F r ob M n , and for thos e pur po ses we can av oid the p -adic in tegers , a s follows. Let F r ob M n Z be the full sub category o f F rob M n with ob jects inte gr al matrices with determinants a p ow er of p . The inclusion F rob M n Z ⊆ F r ob M n is an equiv alence of ca tegories, and the a ssignment α 7→ L α induces an isomor phism betw een GL n ( Z ) \ F rob M n Z and the categor y S ub op C × n p ∞ of ﬁnite subgroups of C × n p ∞ . 4.2. Actions on in verse systems. In this section, w e descr ibe actions by gro ups I ⊆ GL n ( Z p ) on the c overing homolo gy T C J ( A ), for submonoids J ⊆ M n of isogenies of the to rus. Prop ositio n 4.5. L et I ⊆ GL n ( Z p ) b e a sub gr oup and J ⊆ M n a submonoid su ch that if γ ∈ I and β ∈ J then γ β γ − 1 ∈ J . In this c ase t he gr oup I acts on T C J ( A ) := holim β ∈A r J T ( n ) ( A ) L β by sending γ ∈ I to the op er ator R γ F γ − 1 . Pr o of. Observe that if γ ∈ I and α, β ∈ J then T γ αβ γ − 1 F γ − 1 − − − − → = T γ αβ R γ − − − − → T αβ F γ αγ − 1   y F α   y F α   y T γ β γ − 1 F γ − 1 − − − − → = T γ β R γ − − − − → T β and T γ αβ γ − 1 F γ − 1 − − − − → = T γ αβ R γ − − − − → T αβ R γ β γ − 1   y R γ β γ − 1   y R β   y T γ αγ − 1 F γ − 1 − − − − → = T γ α R γ − − − − → T α commute.  22 The q uestion is ho w to c ho ose appr opriate I and J . The intersection b etw ee n I and J should be kept as small a s p o ssible, since any element in the intersection will a lready be in the indexing category of the homo to p y limit, a nd s o the action will b e tr ivial. The minimal J of interest is the free submonoid generated b y p , whic h consists of the ma trices p k id n for k ∈ N . An y α ∈ M n commutes with these matrices, so in this ca se we ma y let I = GL n ( Z p ): Corollary 4.6. L et ∆ ⊆ M n b e the fr e e submonoid gener ate d by p · id n . The gr oup GL n ( Z p ) acts on the “top olo gic al diagonal cyclic homolo gy” T C ∆ ( A ) = holim p k id n ∈A r ∆ T ( n ) ( A ) L p k id n . R emark 4 .7 . When β ∈ GL n Z p , the restrictio n map R β is an isomorphism o f T n p -sp ectra, which can b e rather co nfusing. In k eeping track o f the action, the rea der may ﬁnd c onsolation in the fact that the diagra m T n p × T n p /L αβ 1 × φ β   T n p /L αβ × T n p /L αβ φ αβ × 1 o o µ / / φ β × φ β   T n p /L αβ φ β   T n p × T n p /L α T n p /L α × T n p /L α φ α × 1 o o µ / / T n p /L α commutes, where the solid vertical maps keep tr ack of the r eindexing implement ed by the restric tio n map, while the horizo n tal maps keep track of the actions. At r o ot, the restriction map for a n inv er tible β is equa l to the functorial action by β : T n p → T n p , namely the map T β : T T n p ( A ) → T T n p ( A ), and it is this actio n tha t survives to T C ∆ ( A ). Note that the target T T n p ( A ) of the map T β has its T n p -action twisted by φ β , so that T β remains equiv ariant. 4.3. Examples of group actions on co v ering homol ogy. An interesting cla ss of examples arises as follows. Let B be a Z p -algebra that is free and ﬁnitely gener ated of rank n as a Z p -mo dule. Cho ose a basis for B and let M B = B ∩ ( B ⊗ Q ) ∗ , that is the elemen ts of B that are inv ertible o ver Q . Using the basis, the mono id M B is natura lly identiﬁed a s a submonoid of M n . W e may then consider T C B ( A ) := T C M B ( A ) = holim β ∈A r M B T ( n ) ( A ) L β Example 4.8 . Consider the 2-adic Gaussia n in tegers B = Z 2 [ i ]. Up to multiplication by units of B , any elemen t in M B is a pow er of 1 + i ∈ M B . Note that π 0 T R B ( A ) = W B ( π 0 A ), where W B ( π 0 A ) is the Burnisde-Witt v ectors for π 0 A o ver the module B ∼ = Z × 2 2 [6]. In this example of the 2 - adic Gaussian integers, T C B ( A ) is th e ho mo topy limit of T R B ( A ) F 1+ i ⇒ id T R B ( A ). As a result, we hav e π − 1 T C B ( A ) = W B ( π 0 A ) F 1+ i ; this la st express ion denotes the F ro be nius c o inv ar i- ants o f the Burnside-Witt r ing. The rea de r should contrast this example with the diago nal cyclic homology T C ∆ ( A ) of Corollary 4.6. Both examples come from ﬁxed p o int s of the 2-adic 2-toral top ological Hochsc hild ho mology , and in both examples the index ing categor y is “linear” in the sense that it is gener a ted by a single pair of F r ob enius and Restriction o per ators. Ho wev er, in the diagonal cyclic case, π − 1 T C ∆ ( A ) = W Z × 2 2 ( π 0 A ) F 2 · id 2 . Note that in the Ga us sian in tegers example, F 2 = F 2 i = ( F 1+ i ) 2 —the tw o co equalizer s, with resp ect to F 1+ i and F 2 · id 2 , are not in general the same. The reader is in vited to make these calculations explicit in the case of A = Z , using the equiv a lence W Z × 2 2 ( Z ) ∼ = K 0 (almost ﬁnite Z × 2 2 -sets). Other imaginary quadr atic order s provide similarly interesting examples. W e g et an action of the g roup Aut Z p -algebra ( B ) o f Z p -algebra automorphisms o f B as follows. Using the chosen ba sis o f B , view Aut Z p -algebra ( B ) as a submo no id of GL n ( Z p ); an automorphism g corresp onds to a ma trix denoted x g . As ab ov e, identify M B with the corre s po nding submonoid of M n . The automorphism g ∈ Aut Z p -algebra ( B ) acts on the submono id M B by g ( β ) = x g β x − 1 g , 23 where β ∈ M B ⊂ M n . The diagra m T ( n ) ( A ) L g ( αβ γ ) F x g R x − 1 g − − − − − − → T ( n ) ( A ) L αβ γ F gγ R gα   y F γ R α   y T ( n ) ( A ) L g ( β ) F x g R x − 1 g − − − − − − → T ( n ) ( A ) L β commutes (b ecaus e the diag ram B b 7→ gb − − − − → B β ·   y g ( β ) ·   y B b 7→ gb − − − − → B commutes). Hence we hav e a natural transforma tio n A r M B β 7→ T ( n ) ( A ) L β * * V V V V V V V V V V g   RF g x  y y y y y y y y y y y y y y Spec A r M B β 7→ T ( n ) ( A ) L β 4 4 h h h h h h h h h h where R F g := R x g F x − 1 g . The automorphism group Aut Z p -algebra ( B ) therefor e acts on T C B ( A ). Example 4 .9 . . 1. Let G b e a group o f order n and B = Z p [ G ]. In this case G acts on T C B ( A ). 2. Let O K be the r ing of integers in the deg ree n extensio n K of Q p . The Galois gr oup G = Gal ( K / Q p ) acts on T C O K ( A ). F or instance, if n = 2 and p 6 = 1 mod 4, then we hav e “complex co njugation” on T C O K ( A ). 3. Let O b e a ma ximal o rder in a division alge br a of ra nk n 2 ov er Q p . The algebra automo r- phisms o f O —a nd in particula r the units O ∗ —act on T C O ( A ). The reader who may wonder at the cur ious a pp ea r ance o f the n 2 -fold itera ted top ologica l Ho chschild homolo gy in this example is in vited to compare with the origin of n 2 -fold ab elian v a rieties in the chromatic level n version o f Behrens and Lawson’s top olo g ical a utomorphic forms [4]. W e will give a detailed study of this example in particular in future work. 5. Calcul a tion o f T R ( n ) f or the sphere Recall that T R ( n ) ( A ) is the homotopy limit ov er the res triction op erators acting o n ﬁxed p oints of higher top olog ical Ho chsc hild homolog y T ( n ) ( A ). This “top olog ical restrictio n homology” is an impo rtant w ay-station en route to computations of top olo gical cyclic ho mology . In this section we calculate top olo gical restriction homolo gy for the sphere sp ectrum: Prop ositio n 5.1. Ther e is an e quivalenc e T R ( n ) ( S ) ≃ Y O ⊆ Z n p B ( Z n p / O ) + wher e the pr o duct varies over the op en sub gr oups O ⊆ Z n p . Our computation of T R ( n ) ( S ), as w ell as o ur computation of T F ( n ) ( S ) in section 6, will b e based on a comparison of the ﬁxed p oints o f top olog ical Ho chsc hild homo logy with the K - theo ry of equiv aria nt ﬁnite sets. Below we give a detailed account of the homotopy type and the functorial prop erties of these K - theory sp ectra . Then we rela te the K -theor y of equiv ar iant s e ts to the ﬁxed po int s o f the equiv aria nt spher e spec trum. Finally w e no te that the topo logical Ho chsc hild homo logy of the sphere is the equiv ariant sphere sp ectrum, and thereby co mplete the description of T R ( n ) ( S ). 24 5.1. The K-theory of ﬁnite G -sets. F or G a ﬁnite group, let S ets G denote the categ ory whos e ob jects a re ﬁnite G -sets and whose morphisms ar e G -equiv ar iant isomorphisms. W e b egin b y re - viewing the structure of the K -theory sp ectra K ( S ets G ) as describ ed in Segal [25] or Ca r lsson [9]. Note that any transitive G -set X is isomorphic to one of the fo rm G/K , where K is a s ubg roup of G . The conjuga c y class o f K is determined by X , and conv ersely determines X up to iso morphism. An y ﬁnite G -set X has a decomp osition X ∼ = ` s i =1 G/K i , and s and the collection of conjugacy classes of subgro ups (with multiplicities) a re deter mined uniquely by X and in turn deter mine X up to is o morphism of G -s ets. It follows that there is an equiv alence o f categor ie s S ets G ∼ = Y [ K ] S [ K ] where the pro duct is over the conjuga cy classes of subgro ups of G , and where S [ K ] denotes the full sub categ ory of S e ts G on G -sets X for which the stabilizer of every x ∈ X is in the conjugacy class [ K ]. E ach S [ K ] is a symmetric monoidal sub category of S ets G . The automorphism gro up of the o b ject ( G/K ) n is the wrea th pro duct Σ n ≀ W G ( K ), where W G ( K ) denotes the quotient gro up N G ( K ) /K , with N G ( K ) the normalizer of K in G . It is now an easy cons equence of the Bar ratt- Priddy-Quillen Theorem [22] that the sp ectrum a s so ciated to the symmetric monoidal category S [ K ] is the susp ension sp ectrum Σ ∞ B W G ( K ) + . Ther e is a cor resp onding deco mpo sition of sp ectra K ( S ets G ) ∼ = _ [ K ] B W G ( K ) + where aga in the wedge sum is over the conjugacy classe s of subgroups o f G . W e now re strict our attent ion to ﬁnite a be lia n G , in which case the K -theor y is simply K ( S ets G ) ≃ _ K ⊆ G B ( G/K ) + W e will need to understand the functor ia l b ehavior of this decompo sition under g r oup ho momor- phisms. Let i : G ′ ֒ → G be the inclus io n of a subg roup. A quotient G/K , r egarded as a G ′ -set by restriction of the action a long i , decompos es a s G/K ∼ = ` G/K · G ′ G ′ /K ∩ G ′ . This deco mp os ition is reﬂected in K -theor y as follows. The induced map i ∗ : K ( S e ts G ) → K ( S ets G ′ ) sends the K -theor y factor K ( S [ K ]) corr esp onding to the subg roup K to the factor K ( S [ K ∩ G ′ ]) cor resp onding to the subgroup K ∩ G ′ . Mor eov er, the map on that facto r is the transfer B ( G/K ) + → B ( G ′ /K ∩ G ′ ) + asso ciated to the inclusion G ′ /K ∩ G ′ ֒ → G/K . The K - theory of ﬁnite se ts is functorial not only for inclusio ns of groups , but also for surjective homomorphisms of g roups. If f : G ։ G ′ is a sur jection with kernel H there is a “ restriction map” f ! : S ets G → S ets G ′ sending the ﬁnite G -s et X to its ﬁxed p oints X H . In particular f ! sends the G -set G/K to the empty set if H 6⊆ K and to G ′ /f ( K ) if H ⊆ K . Under the equiv alence ab ov e, this ma p f ! corres p onds to the pr o jection W K ⊆ G B ( G/K ) + → W J ⊆ G ′ B ( G ′ /J ) + induced by the isomorphism G/K ∼ = G ′ /f ( K ) if H ⊆ K , a nd the trivial map if not. F or those a s p erplexed a s the authors by the class ical conﬂict o f terminolog y , we oﬀer the following dictionary betw een the languag e of the Burnside ring and that o f the Witt s tr ucture on topo lo gical Ho ch schild homology . Here i : H ⊆ G is the inclusion of a subgr oup, a nd f : G → G/H is the pro jection. G -sets T op ologi cal Ho chsc hild Homology Restriction to action by subgroup F rob enius = inclusio n of ﬁxed po in ts i ∗ : S e ts G → S e ts H F : T ( n ) ( A ) G → T ( n ) ( A ) H Inducing up a ction X 7→ G × H X V erschiebung = tra nsfer i ∗ : S e ts H → S e ts G V : T ( n ) ( A ) H → T ( n ) ( A ) G T aking ﬁxed p oints X 7→ X H Restriction f ! : S ets G → S ets G/H R : T T n p ( A ) G → T T n p /H ( A ) G/H 25 5.2. K-theory and the equiv arian t sphere s p ectrum. When G is a ﬁnite g roup, there is an equiv alence b etw een the K -theor y of the category of ﬁnite G -sets and the G -ﬁxed points of the equiv aria n t sphere sp ectrum—this is the equiv aria n t Barratt-P riddy-Quillen theo rem. The top ological restric tio n homolo g y T R ( n ) ( S ) is a ho motopy limit of G -ﬁxed po in ts of the sphere fo r v arying G . T o ex press T R ( n ) ( S ) in terms of K -theo r y , w e need an equiv alenc e betw een the ﬁxed po int s of the sphere and the K -theory of ﬁnite sets that intert wines the restrictio n o p er ation on the spher e w ith a ﬁxed p oint op er a tion on sets. In this section w e constr uct an explicit c hain o f equiv alences that has the necess ary functoriality pr op erties: S G ≃ − → N T G ≃ ← − N B G ≃ − → N C G ≃ ← − N D G ≃ − → K ( S ets G ) The intermediate sp ectra N T G , N B G , N C G , and N D G are describ ed be low. W e use Γ- spaces as our mo del of sp ectra, and g enerally follow the conﬁgur ation space viewpo in t Segal develop ed in his pro of of the non-equiv aria n t Ba r ratt-Pr iddy-Quillen theor em [26]. Let S G denote the Γ-spa ce taking a ﬁnite based set F to the space ho colim V Map G ( S V , F ∧ S V )—this is the G -ﬁxed p oint sp ectrum of the sphere. As before S ets G is the symmetric monoidal catego r y o f ﬁnite G -sets, and let ¯ K ( S ets G ) denote the as s o ciated Γ-category; that is, the ca tegory ¯ K ( S ets G )( F ) has ob jects the equiv ariantly ( F \∗ )-lab eled ﬁnite G -sets, with morphisms the equiv aria n t label- preserving maps. Reca ll that the K -theo ry spectr um K ( S e ts G ) is the Γ-space deﬁned by the levelwise nerve of ¯ K ( S ets G ). The Γ- category ¯ K ( S ets G ) enco des abstract ﬁnite G -sets. The topolo gical Γ-categ ory C G will enco de conﬁgura tions o f ﬁnite G -sets in G -repre s ent ations, and the to po logical Γ-catego ry T G will enco de the s e lf- ma ps of G -s pheres determined by the Thom construction on the normal bundles of those conﬁgurations of G -se ts. The top olog ical Γ-categ o ries B G and D G will pr ovide technical bridges b etw een the T ho m categor y , the co nﬁguration catego ry , and the abstra c t G -set catego r y . W e pr o ceed directly to the deﬁnitions . Fix a complete universe U o f orthog onal G -r e pr esentations. The ob jects o f the ca tegory D G ( F ) cons is t of triples ( X, V , φ ), wher e X is a n equiv a riantly ( F \ ∗ )- lab eled ﬁnite G -set, V is a subrepresentation of U , and φ : X → V is an equiv ariant embedding. A mor phism of D G ( F ) from ( X, V , φ ) to ( Y , W, ψ ) is an inclusion V ⊂ W of subrepresentations of U and a la b el-pr eserving equiv ariant isomo rphism γ : X → Y such that ψγ = φ . The map D G → ¯ K ( S ets G ) of top olo gical Γ-categor ies takes D G ( F ) to ¯ K ( S ets G )( F ) by sending the triple ( X, V , φ ) to the lab eled G -set X . The resulting map N D G → K ( S ets G ) of Γ - spaces is a levelwise equiv alence: the homotopy ﬁb er of D G ( F ) → ¯ K ( S ets G )( F ) is a co n tractible equiv a riant embedding category , and the e quiv alence follows by topo logical Quillen Theorem A. Next we deﬁne the conﬁg uration category C G . The ob jects of C G ( F ) are pairs ( C, V ), where V is a subrepresentation of U and C is an equiv ar iantly ( F \∗ )-lab eled ﬁnite G -s ubset of V . A morphism of C G ( F ) from ( C, V ) to ( D, W ) is an inclusion V ⊂ W of subrepr e s ent ations taking C la b el-pr eservingly isomorphically o nt o D . The equiv a lence C G ( F ) ← D G ( F ) takes the triple ( X, V , φ ) to the pair ( φ ( X ) , V ). The categ ory B G is an enla rgement of C G that includes balls a r ound the co nﬁgurations. The ob jects of B G ( F ) are pairs ( A, V ), where again V is a subrepresentation of U , and A is an equiv a riant collec tion o f disjoin t op en metric ba lls in V whose comp onents are e q uiv ariantly ( F \∗ )-la be led. A morphism from ( A, V ) to ( B , W ) is a n inclusion V ⊂ W of subrepresentations taking A into B by a prop er map and inducing a lab el-pres e r ving is o morphism o f the set of centers of the A -balls to the set of centers of the B -balls. The eq uiv alence B G ( F ) → C G ( F ) takes the pair ( A, V ) to the pair ( A cent , V ), where A cent denotes the set o f centers of the balls of A . The top olo gical Γ-ca tegory T G is deﬁned as follows. The ob jects o f T G ( F ) cons ist of pa irs ( V , f ), where V is a subrepr esentation of U , and f : S V → F ∧ S V is an equiv ariant map. A morphism from ( V , f ) to ( W , g ) is an inclusion V ⊂ W of subrepres ent ations together with a distinguished equiv aria nt homotopy class of eq uiv ariant homotopies relative to f from g to Σ W − V f . The map T G ( F ) ← B G ( F ) takes a pa ir ( A, V ), wher e A is a lab eled co llection of ba lls in V , to the pair ( V , f ), where f is the Pontry agin-Thom constructio n on the op en inclusion A ⊂ V , mo diﬁed so that each ball of A ma ps by a standa rd linear isomorphis m to V a nd so that a ba ll of A lab eled by p ∈ F ma ps to the p -comp onent of the targe t F ∧ S V ; a morphism of B G ( F ) from ( A, V ) to ( B , W ) maps to the 26 morphism of T G ( F ) given b y the inc lus ion V ⊂ W to g ether with the homotopy class of a standar d linear homotopy from the Pont ryagin-Thom co ns truction on B ⊂ W to the ( W − V )-susp ension of the Pontry agin-Thom co nstruction on A ⊂ V . Unlik e the equiv alences b etw een N B G , N C G , N D G and K ( S e ts G ), the map N T G ← N B G is not a levelwise equiv a lence, but it is a stable eq uiv alence of Γ-spaces . The last map S G → N T G do es not come from a map of Γ-ca teg ories, but is a levelwise eq uiv alence. Chose the standard model for the homotopy co limit of a functor X from a dia gram I to spa ces, namely ho colim X = colim  a i → j X ( i ) ⊗ N ( j / I ) op → → a i X ( i ) ⊗ N ( i /I ) op  The functor in question is X S G ( F ) ( V ) = Map G ( S V , F ∧ S V ), deﬁned on the diagra m of subrepre - sentations of U . The map S G ( F ) → N T G ( F ) sends the simplex { ( V , f ) ⊗ ( V ⊂ V 1 ⊂ · · · ⊂ V k ) } o f Map G ( S V , F ∧ S V ) ⊗ N ( V /I ) op to the simplex { ( V , f ) → ( V 1 , Σ V 1 − V f ) → · · · → ( V k , Σ V k − V f ) } o f N T G ( F )—in the latter simplex the morphisms are given b y the homotopy classes of the identities. All the ab ov e co nstructions a re functorial with resp ect to changing the gr oup G . That is, f or i : H → G an inclusion o f ﬁnite gro ups, and for f : G → G ′ a s urjection o f ﬁnite gr o ups, ea ch of the equiv alences commutes w ith the ma ps i ∗ , i ∗ , and f ! . 5.3. The hom otop y lim it o v er the restrictions. W e now have all the ingre die nts to express the r estriction sy stem of ﬁxed p oints o f top ologica l Ho chschild homology in terms o f a system of K - theory functors o f categor ie s of G -sets, and thereby to co mpute top olog ical restriction ho mology . Prop ositio n 5. 2. L et G b e a ﬁnite gr oup and S a fr e e G -sp ac e. Ther e is a we ak e quivalenc e T S ( S ) G ≃ S G . When f : G ։ G ′ is a surje ction the fol lowing diagr am c ommutes: T S ( S ) G ≃ / / R   S G ≃ / / _ _ _ f !   K ( S ets G ) f !   T S ( S ) G ′ ≃ / / S G ′ ≃ / / _ _ _ K ( S ets G ′ ) When i : G ֒ → G ′ is an inje ction, the diagr am c ommutes with ( R , f ! ) r eplac e d by ( F , i ∗ ) or by ( V , i ∗ ) . Pr o of. It rema ins only to e stablish the ﬁrs t equiv a lence, and the compatibility of that e q uiv alence with f ! , i ∗ , and i ∗ . Without loss of gener ality we may a ssume S is ﬁnite. In this case a coﬁnality arg umen t shows that T S ( S ) is equiv alent to ho colim V Map ∗ ( S V , S V ), wher e the homo topy colimit o ccurs over G - representations. The equiv a lence T S ( S ) G ≃ − → S G follows a nd this map ob viously resp ects the re- striction, the inclusion of ﬁxed p oints, and the transfer .  Corollary 5.3. Ther e ar e e quivalenc es T R ( n ) ( S ) ≃ ho lim O ⊆ Z n p K ( S ets Z n p / O ) ≃ Y O ⊆ Z n p B ( Z n p / O ) + wher e the O ’s vary over the op en sub gr oups of Z n p . Pr o of. The ﬁrs t equiv alence is immediate from P rop osition 5.2, using Lemma 4.2 to express the restriction homotopy limit in terms of o p en s ubg roups of Z n p . In section 5 .1, we noted that K ( S ets G ) ≃ W K ⊆ G B ( G/K ) + for ﬁnite abelian G . Moreov er w e observ ed that for a surjection f : G → G ′ , the restrictio n ma p f ! : K ( S e ts G ) → K ( S ets G ′ ) corr esp onds to the pro jection W K ⊆ G B ( G/K ) + → W J ⊆ G ′ B ( G ′ /J ) + that induces an is o morphism b etw ee n the B ( G/K ) and B ( G ′ /f ( K )) factors whenever ker f ⊂ K . The second equiv a lence in the coro llary follows.  Notice that the homotopy limit here is deceivingly complica ted: it is enough to take the ho motopy limit over the ﬁnal sub catego ry · · · ⊆ p k +1 Z n p ⊆ p k Z n p ⊆ · · · ⊆ Z n p corres p onding to the r estrictions in T R a long the inclusio ns L p k id n ⊆ L p k +1 id n . 27 6. T F ( n ) f or the sphere and the Segal conjecture f or tori In this section, we calcula te the “top ologica l F rob enius homo logy” T F ( n ) ( S ) of the spher e sp ec- trum; recall that this is the homotopy limit ov er the F r ob enius op erators acting on the ﬁxed points of higher top olog ical Ho chsc hild homolog y . On the one hand, this calcula tion is closely related to the problem of under standing the higher top ologica l cyclic homo logy of the sphere—indeed in the Appendix w e prove that top ologica l cyclic homolo gy is a ho motopy limit of restriction op erator s on top olo gical F ro be nius homology , and we use our computatio n of the F rob enius homolog y of the sphere to in vestigate the diagonal cy clic homology of the sphere. On the other hand, it is not diﬃcult to see that the function sp ectrum F ( B T n + , S p ) is homoto py equiv alent to T F ( n ) ( S ) p . Our ev aluation of top ologica l F rob enius homolo g y therefore gives a precise descr iption of the homotopy t yp e of F ( B T n + , S p ), that is of the p -adic cohomotopy o f the classifying space of the torus. The description of F ( B G + , S p ) for ﬁnite G is known as the Se gal c onje ctur e for G , a nd was carried out in the 1980 s. Partial results for G compact Lie were o bta ined, but they focused on the analysis of π 0 of the function sp ectrum. In pa rticular, a complete a na lysis o f the case o f a torus w as not obtained. W e give such an a na lysis b elow. W e hop e the res ult ca n moreover b e used to complete our understa nding of the compact Lie group version of the Segal conjecture. W e b egin by relating the co homotopy sp ectrum of the classifying space of the torus to top olog ical F rob enius ho mology , and by summarizing our co mputation of the latter. Prop ositio n 6.1. Ther e is an e quivalenc e F ( B T n + , S p ) ≃ T F ( n ) ( S ) p Pr o of. The cohomotopy of the classifying space of the tor us is equiv alent to a homotopy limit of cohomotopy sp ectra of c la ssifying spaces of ﬁnite g r oups, whic h sp ectra are in turn r elated to ﬁxed po int s of equiv ar iant sphere s pec tr a and thereby to F rob enius homolog y: F ( B T n + , S p ) ≃ holim F ( B G + , S p ) ≃ holim S G p ≃ ho lim T T n p ( S ) G p ≃ (holim T T n p ( S ) G ) p = T F ( n ) ( S ) p Here all the homotop y limits may occur ov er either the ﬁnite subgroups of T n p or o ver the ﬁnal sub c ategory o f diagona l subgroups C × n p l . The ﬁrst e q uiv alence is establishe d by direct calculation [8], and the second is the Segal co njectur e for ﬁnite gr oups [7]. The third equiv ale nce is Prop ositio n 5.2 and the four th is immediate.  Theorem 6.2. The homotopy gr oups o f the hi gher top olo gic al F r ob enius homolo gy of the spher e sp e ctrum ar e as fol lows: π ∗ ( T F ( n ) ( S ) p ) = Y k,α lim l  Z [ GL n ( Z p ) / Γ l,k,α ] ⊗ π ∗ (Σ ∞ S k ∧ B T k + ) /p l  Her e t he pr o duct is over 1 ≤ k ≤ n and α is a c ol le ction of unor der e d p ositive inte gers { n 1 , . . . , n k } . The limit is over l ∈ N , and the gr oup Γ l,k,α ⊂ GL n ( Z p ) is determine d as fol lows. Consider sub gr oups K of C × n p l such that the minimal numb er of gener ators of the qu otient gr oup C × n p l /K is exactly k (we say that K has r ank k ), and the c ol le ction o f ex p onents of p in the st andar d cyclic p-gr oup de c omp osition of C × n p l /K is { l − n 1 , . . . , l − n k } (we say that K has c otyp e α ). The gr oup GL n ( Z p ) acts on the s et of s u b gr oups K of C × n p l with r ank k and c otyp e α , and Γ l,k,α is the stabilizer of any chosen K under this GL n ( Z p ) -action. The conceptual orig in o f this decomp os ition o f the homotopy o f T F ( n ) ( S ) and a more detailed description of the terms inv olved in the decomp osition are given in the fo llowing sections . 6.1. The rank ﬁltration of the equiv ariant s phere sp ectrum functor. W e expre ss to p olo gical F rob enius homology as a homoto p y limit of equiv ariant sphere spec tr a, and then de s crib e a rank ﬁltration of these sp ectra. By deﬁnition T F ( n ) ( S ) is the homotopy limit holim F r o b M n T ( n ) ( S ) L α . In section 4.1, we s aw that the F rob enius indexing category is equiv alent to the catego ry S u b := S ub op T n p of ﬁnite subgroups 28 of T n p . By Prop osition 5.2, the ﬁxed po ints o f top ologica l Ho chsc hild homolog y are equiv alent to the ﬁxed p oints of the equiv ariant sphere sp ectrum. In par ticular, top olog ical F rob enius homolog y can b e e x pressed as T F ( n ) ( S ) p ≃ holim G ∈ S ub ( S G ) p . Thr oughout this se ction we will abbreviate the functor in this homo topy limit by Φ : S ub → Sp ec ; that is Φ( G ) = ( S G ) p and Φ( i : H ֒ → G ) = i ∗ . Moreov er, in lig ht of the results of section 5.2 and by abuse of no tation, w e will not distinguish betw een the eq uiv ariant spher e sp ectrum S G and the K -theor y of G -sets K ( S ets G )—indeed mo s t of our a nalysis will occur in the w orld of G -sets—and we w ill genera lly let p-completion b e implicit. The rank ﬁltration of the equiv ariant spher e spectrum functor Φ is obtained b y ﬁltering the symmetric monoidal category S ets G . F or any transitive G -set X ∼ = G/K , we deﬁne the r ank o f X to be the minimal n umber o f genera tors r equired to generate G/K . W e then deﬁne C G [ k ] to b e the full s ub categ ory of S ets G of those G -sets all of whose o rbits have rank less than or equal to k . These sub catego r ies hav e the following prop erties. • C G [0] is equiv ale n t to the catego ry of ﬁnite sets with trivial G -action, hence is equiv alent to the sphere sp ectrum. C G [ n ] is all of S ets G . • C G [ k ] is a symmetric monoidal subcateg ory of S ets G , who se asso ciated spectr um w e will denote by S G [ k ]. W e have an increasing seque nce of sp ectr a S G [0] ⊆ S G [1] ⊆ . . . ⊆ S G [ n ] = S G • The sub categor ies C G [ k ] are preser ved under the ma p Φ( i ), whe r e i : H ֒ → G , and so they will cre ate their own s p ectr um-v alued diag rams Φ[ k ], with Φ[ k ]( G ) := K ( C G [ k ]). W e now have a ﬁltration o f holim S ub Φ by holim S ub Φ[0] ⊆ holim S ub Φ[1] ⊆ . . . ⊆ holim S ub Φ[ n ] = holim S ub Φ The relative terms in this ﬁltration a re the sp ectra holim S ub Φ[ k ] / Φ[ k − 1], in view o f the following general result ab o ut sp ectrum homotopy limits. Prop ositio n 6.3. Le t C denote a smal l c ate gory, and supp ose that we have two sp e ctrum-value d functors F and G on C , to gether with a natur al tr ansformation ϕ : F → G . L et C ( ϕ ) denote the functor with C ( ϕ )( x ) e qual to the mapping c one of F ( x ) → G ( x ) . Ther e is a c oﬁbr ation se quenc e of sp e ctr a holim C F → ho lim C G → holim C C ( ϕ ) . Our next ta sk is to ev a luate the sub quotients ho lim S ub Φ[ k ] / Φ[ k − 1] of the rank ﬁltration. T o - ward that end, we express the quotients S G [ k ] / S G [ k − 1 ] themselves a s K -theo r y sp ectra of symmetric monoidal categorie s. Deﬁne the symmetric mono idal c ategory C G h k i to be the categ ory of ﬁnite G -sets a ll o f whose orbits hav e rank equal to k . Given an inclusion of a belia n g roups i : H ֒ → G , deﬁne a functor i ∗ : C G h k i → C H h k i as follows. F or a ﬁnite G -s et X , a ll of whose or bits hav e ra nk k , decompo s e X a s X = X ′ ` X ′′ , where X ′ is the unio n o f all H -or bits of X that have rank k . The functor i ∗ takes X to X ′ . Altog e ther the construction C G h k i gives a contrav a riant functor I from the category of ﬁnite subgro ups of T n to the categor y o f small ca tegories. Prop ositio n 6.4. The quotient S G [ k ] / S G [ k − 1 ] is the K -t he ory sp e ctru m asso ciate d to the sym- metric monoidal c ate gory C G h k i of ﬁ n ite G -set s whose orbits al l have r ank k . F or i : H ֒ → G an inclusion of ab elian gr oups, the map K ( I ( i ∗ )) : S G [ k ] / S G [ k − 1 ] → S H [ k ] / S H [ k − 1] is induc e d by the natur al r estriction map i ∗ : S G [ k ] → S H [ k ] . F rom now o n, we will write S G h k i for the sub quotient S G [ k ] / S G [ k − 1]. 6.2. The cot yp e de comp ositi on of the sphere and a splitting of the rank ﬁl tration. W e can simplify the homotopy limit in top olo gical F rob enius ho mology a s follows. The full s ubca tegory D n of S u b on the dia gonal subgr oups C × n p l of T n p is ﬁnal, and s o T F ( n ) ( S ) ≃ holim l ∈D n Φ | D n . Restricting to the s ubca tegory D n will a llow us to pro duce a deco mpo sition of the sub quotients of the rank ﬁltr a - tion, which in turn will for ce the r ank ﬁltration to split: ho lim S ub Φ ≃ W n k =0 holim S ub Φ[ k ] / Φ[ k − 1]. 29 F or any subgroup K ⊆ C × n p l ⊂ T n , w e will let the t yp e of K deno te the collection of exp onents of p o ccurring in the deco mpo sition of C × n p l /K into a dire c t sum o f ﬁnite cyclic p -gr oups. That is , if the type of K is the set { e 1 , e 2 , . . . , e t } , then we hav e C × n p l /K ∼ = t M i =1 C p e i W e observe that (i) 1 ≤ e i ≤ l and (ii) t ≤ n . By the c otyp e o f K , we mea n the collection { l − e 1 , l − e 2 , . . . , l − e t } , and we denote it by ct l ( K ), to emphasize that it dep ends on l . W e will need the following res ult regarding this inv a riant. Prop ositio n 6.5. L et K ⊆ C × n p l , and supp ose that rk ( C × n p l /K ) = r k ( C × n p l − 1 /K ∩ C × n p l − 1 ) Then ct l ( K ) = c t l − 1 ( K ∩ C × n p l − 1 ) Pr o of. W e ﬁrst reinterpret ct l ( K ). Since K is a subgr oup of C × n p l , it can b e de c o mpo sed as K ∼ = ( C p l ) s ⊕ n − s M i =1 C p f i where 0 ≤ f i ≤ l − 1. The num b ers s and the collectio n of num b ers f i are unique, up to a p ossible reorder ing o f the f i ’s. The corresp onding decomp ositio n for K ∩ C × n p l − 1 is o f the form K ∩ C × n p l − 1 ∼ = C s + t p l ⊕ n − s − t M i =1 C p f i where t is the num b er of v a lues of i for which f i = l − 1. It is cle ar from the deﬁnitions that the rank of C × n p l /K is n − s , and that the ra nk of C × n p l − 1 /K ∩ C × n p l − 1 is n − s − t . Since we are ass uming that rk ( C × n p l /K ) = rk ( C × n p l − 1 /K ∩ C × n p l − 1 ), it fo llows that in our case t = 0, s o in fact we hav e f i ≤ l − 2 for all i . Finally , we now have ct l ( C × n p l /K ) = { f 1 , f 2 , . . . , f n − s } = ct l − 1 ( C × n p l − 1 /K ∩ C × n p l − 1 ) as requir ed.  W e next consider the symmetr ic mono idal ca teg ory C C × n p l h k i whose ob jects are the ﬁnite C × n p l -sets all of whose orbits hav e ra nk k . Note that k ≤ n . Any orbit in a ﬁnite C × n p l -set has the form C × n p l /K , where K is a subgroup. By the c otyp e of the o rbit, we will mean ct l ( K ). Any ﬁnite C × n p l -set X whose or bits all hav e rank k has a canonica l decompo sition X ∼ = a α X α Here X α denotes the union of all orbits whose cotype is equal to α . F urther, α rang es over unor de r ed families { n 1 , n 2 , . . . , n k } , where the n i ’s are p ositive integers. It follows that the symmetric monoidal category C C × n p l h k i has a dec o mpo sition C C × n p l h k i ∼ = Y α C C × n p l h k , α i where C C × n p l h k , α i is the symmetric monoidal catego ry of ﬁnite C × n p l -sets all of whose orbits hav e rank k a nd a ll o f whose cotypes ar e α . W e will wr ite S C × n p l h k , α i for the corr esp onding sp ectrum; 30 we now hav e a decomp os itio n of sp ectra S C × n p l h k i ≃ _ α S C × n p l h k , α i Prop ositio n 6.6. The r est riction maps S C × n p l h k i → S C × n p l − 1 h k i r esp e ct the c otyp e de c omp osition. This result is a consequence o f Prop osition 6.5: as long a s the r anks sta y constant, so do the t yp es. W e can deﬁne functor s Φ[ k , α ] on D n by Φ[ k , α ]( C × n p l ) = S C × n p l h k , α i . O bserve tha t for an y ﬁxed l , the set of p oss ible cotypes for C × n p l -orbits is ﬁnite, s ince the integers n i inv o lved must be le s s than or equal to l . Co nsequently , we hav e a decomp os ition of Φ[ k ] / Φ[ k − 1 ] into a pr o duct o f functors Φ[ k , α ]. Corollary 6.7. Ther e ar e e quivalenc es holim S ub Φ[ k ] / holim S ub Φ[ k − 1] ≃ holim S ub Φ[ k ] / Φ[ k − 1] ≃ holim D n Φ[ k ] / Φ[ k − 1] ≃ Y α holim D n Φ[ k , α ] W e can b o otstr ap this decomp ositio n in to a splitting of the r ank ﬁltration: Theorem 6.8. The pr oje ction holim l ∈D n Φ[ k ]( l ) → holim l ∈D n Φ[ k ]( l ) / Φ[ k − 1]( l ) splits in the homotopy c ate gory. (Her e Φ[ k ]( l ) is shorthand for Φ[ k ]( C × n p l ) .) Mor e p r e cisely, the diagr am holim l Q α Φ[ k , α ]( l ) / / ≃ * * U U U U U U U U U U U U U U U U U holim l Φ[ k ]( l )   Q α holim l Φ[ k , α ]( l ) ∼ = O O holim l Φ[ k ]( l ) / Φ[ k − 1 ]( l ) c ommut es, wher e the horizontal map is induc e d by inclusion of c ate gories, t he diago nal map is induc e d fr om the e qu ivalenc e of Cor ol lary 6. 7 , the left vertic al map is t he c anonic al isomorphism, and the right vertic al map is the pr oje ction. Pr o of. When l is ﬁxed, the upp er tria ngle is exactly the identiﬁcation of the ﬁltra tion quotien t Φ[ k ]( l ) / Φ[ k − 1]( l ) with the ﬁnite pro duct ov er the cotypes. Consider ing the homoto p y limit over l , Corolla ry 6 .7 shows that the dia gonal map bec o mes an equiv alence.  Corollary 6.9. Ther e is an e qu ivalenc e T F ( n ) ( S ) p ≃ holim S ub Φ ≃ Y k,α holim D n Φ[ k , α ] Pr o of. The ﬁrst eq uiv alence was ment ioned at the b eginning of section 6.1. The second is a com- bination of Theorem 6.8, which splits the rank ﬁltration in to its sub quotients, and Co rollary 6.7, which deco mp os es the sub quo tients by cotype.  6.3. The homotopy ty p e of the ﬁxed rank-cot yp e comp onents of T F ( n ) . The next stage is to ev alua te the homo topy o f the individual rank-co type compo nent s holim D n Φ[ k , α ]. Before directly confronting the r ank-cotype computation, we re cord a few necessar y genera lities ab out group a ctions on sp ectra. Consider a gro up G , a subgroup K ⊆ G , and a basep oint-preserving action of K o n a based space X . The based space G + ∧ K X is the q uotient o f G + ∧ X by the equiv alence relatio n g k ∧ x ≃ g ∧ k x . W e extend this notion to sp ectra , for a naive action of K on a sp ectrum X , by letting G + ∧ K X b e the sp ectrum as so ciated to the presp ectrum G + ∧ K X ( − ). This constructio n has the following univ ersal pro p er t y . 31 Prop ositio n 6.10. Given a sp e ctrum Y with G -a ction, a sub gr oup K ⊆ G , a sp e ctru m X with K - action, and a K -e quivariant map ψ : X → Y , t her e is a unique G -e quivariant extension ψ : G + ∧ K X → Y that agr e es with ψ on X = K + ∧ K X ⊆ G + ∧ K X . The non- e quiv aria n t homotopy type of G + ∧ K X can b e describ ed as follows. Lemma 6.11. L et K b e a sub gr oup of G , and let X b e a sp e ctru m with a K - action. L et { g α } α ∈ A denote a set of c oset r epr esentatives for G/K , so that t he c osets ar e given by { [ g α ] } α ∈ A . The map θ : G/ K + ∧ X → G + ∧ K X , deﬁne d levelwise by [ g α ] ∧ x 7→ g α ∧ K x , is an e quivalenc e of sp e ctr a. The naturality pro per ties of this construction a re as follows. Supp os e we hav e a group G , sub- groups K ′ ⊆ K ⊆ G , g r oup actio ns of K and K ′ on spectr a X and X ′ resp ectively , and an equiv aria nt map f : X ′ → X . There is a naturally deﬁned ma p η : G ∧ K ′ X ′ → G ∧ K X induced by the levelwise ma p η ( g ∧ K ′ x ′ ) = g ∧ K f ( x ′ ). Choo se families of coset representatives { g α } α ∈ A and { g ′ β } β ∈ B for G/K and G/K ′ resp ectively . The coset [ g ′ β ] determines a K -cos et, w hich can b e written in the fo rm [ g α ] for a unique v a lue of α . This mea ns that there is a distinguished ele ment k β ∈ K so that g ′ β = g α k β . Lemma 6.12. The diagr am G/K ′ + ∧ X ′ θ − − − − → G ∧ K ′ X ′ λ   y η   y G/K + ∧ X θ − − − − → G ∧ K X c ommut es, wher e λ is the map induc e d by the levelwise e quation λ ([ g ′ β ] ∧ x ′ ) = [ g α ] ∧ k β f ( x ′ ) —her e α is chosen so that the K -c oset [ g ′ β ] is e qual to [ g α ] . W e now int ro duce some ter minology concerning the colle c tion o f g roups o f a ﬁxed cotype. F or each k a nd α , let M [ k , α ]( l ) denote the based set o btained by a djoining a disjo int basep oint to the set of subg roups K ⊆ C × n p l such that C × n p l /K has r ank k a nd K has cotype α . W e note that there are maps θ ( l , l ′ ) : M [ k , α ]( l ) → M [ k , α ]( l ′ ) whenever l ≥ l ′ , deﬁned by θ ( l, l ′ )( K ) = K ∩ C × n p l ′ when C × n p l ′ /K ∩ C × n p l ′ has rank k and K ∩ C × n p l ′ has co type α , and θ ( l , l ′ )( K ) = ∗ o ther wise. W e no w describ e this set M [ k , α ]( l ) a s a quotient of GL n ( Z p ). Lemma 6.13. The gr oup GL n ( Z p ) acts on the b ase d set M [ k , α ]( l ) . The action is tr ansitive on non-b asep oint elements, and the stabilizer of e ach non-b asep oint element is a ﬁnite index su b gr oup. Ther efor e, M [ k , α ]( l ) c an b e describ e d as ( GL n ( Z p ) / Γ l ) + , wher e Γ l is the stabilizer of an element in M [ k , α ]( l ) . Pr o of. Since ﬁxing r ank and cotype ﬁxes the a bstract isomorphism type of th e subg roup, this is Corollar y 4 .3.  W e pr o ceed to analyze K ( C C × n p l h k , α i ), wher e as b efore C C × n p l h k , α i is the symmetric mo no idal category o f ﬁnite C × n p l -sets all o f who se o r bits have r ank k and all of whose cotypes are α . Cho ose any subgroup K ⊆ C × n p l for which the orbit C × n p l /K ha s ra nk k a nd such that K has co type α . Let Γ denote the stabilizer of K under the action o f GL n ( Z p ) on M [ k , α ]( l ). Let C C × n p l h K i ⊆ C C × n p l h k , α i denote the symmetric monoidal sub category o n those C × n p l -sets all of whos e points hav e K as their stabilizer. On the o ne hand, the group Γ clea rly acts on this category C C × n p l h K i via its action o n the quotient g roup C × n p l /K , and co ns equently acts o n the cor resp onding sp ectrum. O n the other hand, the full group GL n ( Z p ) a cts on the catego ry C C × n p l h k , α i b ecause it acts on the group C × n p l . 32 The inclusion C C × n p l h K i ⊆ C C × n p l h k , α i is clear ly Γ-equiv ariant, so P rop osition 6 .10 yields a map o f sp ectra ρ : GL n ( Z p ) + ∧ Γ K ( C C × n p l h K i ) → K ( C C × n p l h k , α i ) . Prop ositio n 6.14 . The map ρ : GL n ( Z p ) + ∧ Γ K ( C C × n p l h K i ) → K ( C C × n p l h k , α i ) is an e qu ivalenc e. Pr o of. Using Lemma 6.11 note that the map ρ ha s source GL n ( Z p ) + ∧ Γ K ( C C × n p l h K i ) ≃ ( GL n ( Z p ) / Γ) + ∧ K ( C C × n p l h K i ) ≃ _ ¯ g ∈ GL n ( Z p ) / Γ K ( C C × n p l h K i ) The target o f the map ρ is K  C C × n p l h k , α i  ≃ K   Y L ∈M [ k,α ] ( l ) C C × n p l h L i   By Lemma 6.1 3, we hav e a preferr ed bijection ( GL n ( Z p ) / Γ) + ¯ g 7→ ¯ g · K − − − − − → M [ k , α ]( l ) Under these equiv alences the ma p ρ takes the ¯ g -summand in the wedge to the ( ¯ g · K )- th fac to r in the K -theory pro duct—the map is induced by the isomorphism C C × n p l h K i ∼ = C C × n p l h ¯ g · K i given by ¯ g . Because Γ is of ﬁnite index the ma p from the wedge to the pr o duct is a n equiv a lence.  Prop ositio n 6 .14 establishes the ho motopy type of the v a lue s Φ[ k , α ]( l ) = K ( C C × n p l h k , α i ) of the functor Φ[ k , α ]. W e now describ e the maps b etw ee n the v arious sp ectra Φ[ k , α ]( l ). W e begin by choosing for each l a subgroup K l ⊆ C × n p l , with K l ∈ M [ k , α ]( l ), and K l ∩ C × n p l = K l ′ whenever l ≥ l ′ . F o r example, decomp ose C × n p l as C × n − k p l × C × k p l , and select a subgr oup E ⊆ C × k p l whose family of cyclic facto r s is α . Then the groups K l ′ = C × n − k p l ′ × E form an appro priate family of subgroups. W e deﬁne Γ l to b e the stabilize r of K l under the GL n ( Z p )-action. It is clea r that Γ l +1 ⊆ Γ l . W e now deﬁne τ l : C C × n p l h K l i → C C × n p l − 1 h K l − 1 i to b e the functor o btained by restricting C × n p l /K l -sets along the inclusion C × n p l − 1 /K l − 1 ֒ → C × n p l /K l . This functor is clear ly K l -equiv ar iant, and a n applicatio n o f the universal pro p er t y in P rop osition 6.10 yields a map o f sp ectra t l : GL n ( Z p ) ∧ Γ l K ( C C × n p l h K l i ) → GL n ( Z p ) ∧ Γ l − 1 K ( C C × n p l − 1 h K l − 1 i ) Prop ositio n 6.15. The functor Φ[ k , α ] is natur al ly e quivalent t o t he functor Ψ given by Ψ( C × n p l ) = GL n ( Z p ) ∧ Γ l K ( C C × n p l h K l i ) , and taking C × n p l − 1 ֒ → C × n p l to the map t l : Ψ( C × n p l ) → Ψ ( C × n p l − 1 ) . In other wor ds, the diagr am GL n ( Z p ) ∧ Γ l K ( C C × n p l h K l i ) ρ − − − − → ≃ K ( C C × n p l h k , α i ) Φ[ k , α ]( l ) t l   y Φ[ k,α ](“ l − 1 ֒ → l ”)   y GL n ( Z p ) ∧ Γ l − 1 K ( C C × n p l − 1 h K l − 1 i ) ρ − − − − → ≃ K ( C C × n p l − 1 h k , α i ) Φ[ k , α ]( l − 1) c ommut es. Next we would like to desc r ibe the ho motopy groups of holim D n Φ[ k , α ] in terms o f the homotopy limit of the factors K ( C C × n p l h K l i ). The g eneral situation we ar e in is a s follows. Consider a proﬁnite group G and a sequence o f ﬁnite index s ubgroups Γ l ⊆ G , l ≥ 0 , w ith Γ l +1 ⊆ Γ l . Let · · · → X l → X l − 1 → · · · → X 0 be a n in verse system of sp ectra, and supp ose for a ll l the spectr um X l has a Γ l -action such that the map X l → X l − 1 is equiv a riant. Prop os ition 6.10 provides ma ps of s pec tr a · · · → G + ∧ Γ l X l → G + ∧ Γ l − 1 X l − 1 → · · · → G + ∧ Γ 0 X 0 33 and by Lemma 6.11 a nd Lemma 6 .12 this inv erse system is equiv alent to the s ystem · · · → G/ Γ l + ∧ X l → G/ Γ l − 1 + ∧ X l − 1 → · · · → G/ Γ 0 + ∧ X 0 W e hav e e quiv alences holim l G/ Γ l + ∧ X l ≃ ho lim l,k G/ Γ l + ∧ X k ≃ ho lim l G/ Γ l + ∧ holim k X k Let X denote ho lim k X k . W e now need only understa nd the homotopy gr o ups o f homotopy limits of inv erse systems of the for m G/ Γ l + ∧ X . Prop ositio n 6.16 . If π t ( X ) is a ﬁnitely gener ate d Z p -mo dule, t hen π t holim l ( G/ Γ l + ∧ X ) ∼ = lim l ( Z [ G/ Γ l ] ⊗ π t ( X ) /p l ) Pr o of. The s ystems π t ( G/ Γ l + ∧ X ) are Mittag -Leﬄer, as the maps a r e surjective, and th us the cor - resp onding lim 1 terms v anish. Th us π t holim l G/ Γ l + ∧ X ∼ = lim l Z [ G/ Γ l ] ⊗ π t ( X ). B ecause Z [ G/ Γ l ] is free and ﬁnitely generated, provided π t ( X ) is a ﬁnitely g enerated Z p -mo dule, it is mor eov er the case that lim l Z [ G/ Γ l ] ⊗ π t ( X ) ∼ = lim l Z [ G/ Γ l ] ⊗ lim k π t ( X ) /p k ∼ = lim l Z [ G/ Γ l ] ⊗ π t ( X ) /p l .  W e now fo cus on our particula r situatio n: take G to b e GL n ( Z p ), a nd Γ l to b e, as before, the stabilizer of an elemen t o f M [ k , α ]( l ). Let X l be K ( C C × n p l h K l i ) p , and let the maps in the inv erse system b e restr ictions alo ng the inclusions C × n p l /K l ֒ → C × n p l +1 /K l +1 . F rom the deﬁnitions we hav e an equiv a lence K ( C C × n p l h K l i ) ≃ Σ ∞  B C p l − n 1 × B C p l − n 2 × · · · × B C p l − n k  + where α is the family { n 1 , n 2 , . . . n k } , and the maps in the in verse system a re the transfers obta ined from the cov ering s paces B C p l − n 1 × B C p l − n 2 × · · · × B C p l − n k → B C p l +1 − n 1 × B C p l +1 − n 2 × · · · × B C p l +1 − n k F or each fa ctor B C p l − n i , we hav e the S 1 -transfer Σ ∞ S 1 ∧ B T + → Σ ∞ ( B C p l − n i ) + . W e may smash k such ma ps tog e ther to obtain a ma p Σ ∞ S k ∧ B T k + → Σ ∞ ( B C p l − n 1 × B C p l − n 2 × · · · × B C p l − n k ) + . Standard compa tibilit y formulae for the transfer show that these maps ﬁt together to yield a map Σ ∞ S k ∧ B T k + → holim l Σ ∞  B C p l − n 1 × B C p l − n 2 × · · · × B C p l − n k  + After p -completion, this map b ecomes an equiv alence , as ca n b e check ed by homo logical calculations of the transfer maps. The homoto p y gr oups of (Σ ∞ S k ∧ B T k + ) p are evidently ﬁnitely g enerated Z p - mo dules. The following is now a conse quence of Pr op osition 6.16 a nd Corolla ry 6.9. Theorem 6. 1 7. The homotopy gr oups of the ﬁxe d r ank-c otyp e pie c e o f the t op olo gic al F r ob enius homolo gy of the spher e ar e π ∗ (holim D n Φ[ k , α ]) ∼ = lim l  Z [ GL n ( Z p ) / Γ l ] ⊗ π ∗ (Σ ∞ S k ∧ B T k + ) /p l  In c oncr ete terms, this me ans that for every fr e e gener ator in π t (Σ ∞ S k ∧ B T k + ) , ther e is a su mmand of the form lim i,j ( Z /p i )[ GL n ( Z p ) / Γ j ] in π t (holim D n Φ[ k , α ]) , and for every ﬁnite cyclic summand C in π t (Σ ∞ S k ∧ B T k + ) , ther e is a summand of t he form C ⊗ lim l Z [ GL n ( Z p ) / Γ l ] . The homotopy gr oups of the t op olo gic al F r ob enius homolo gy T F ( n ) ( S ) p , and t her efor e the c oho- motopy gr oups of the classifyi ng sp ac e of the torus, ar e given as t he pr o duct of t hese terms as k and α vary over the r ank and c otyp e. 34 Appendix. Cyclic homology as a homotopy l imit of Fr obenius homol ogy Ordinary top olog ical c yclic ho mology can b e expresse d in ter m of topo logical F r o be nius homology: T C ( A ) = holim  T F ( A ) R ⇒ id T F ( A )  Note that this homotopy limit is equal to the homoto p y limit of the diagr am with a sing le o b ject T F ( A ) and with mo rphisms the self-maps R k : T F ( A ) → T F ( A ) for 0 ≤ k < ∞ . The purp os e of this app endix is tw o-fold. Fir st, w e show that higher top ologic al cy clic homo logy is the homotopy equalizer of the action of all restriction maps on hig her top olo g ical F ro be nius homology: Prop ositio n A.1. Ther e is an e quivalenc e T C ( n ) ( A ) ≃ ho lim M n T F ( n ) ( A ) wher e M n is, as b efor e, the monoid of iso genies of the t orus, and wher e the action of the element s m ∈ M n on T F ( n ) ( A ) is induc e d by t he r est riction op er ators R m . The pro p os ition shows that the computation in sec tio n 6 of T F ( n ) ( S ) is an essential ingr edient in any investigation o f higher top olo gical cyclic homolo gy . The second purp os e o f this app endix is t o illustrate the p ow er of express ing cyclic homolog y in ter ms of F r ob enius homo logy by applying the ca lculation o f T F ( n ) ( S ) to desc rib e the ﬁltration quotients o f the rank ﬁltration o f the diagonal cyclic ho mology of the sphere : Theorem A. 2. Th e r estriction maps c orr esp onding to diago nal iso genies (t hose of t he form p l · id n ) pr eserve the r ank ﬁltr ation of the F r ob enius homolo gy of the spher e T F ( n ) ( S ) . The r ank ﬁltr ation ther efor e desc ends to diagonal cyclic homolo gy T C ∆ ( S ) , and the homotopy gr oups of the ﬁltr ation quotients of t he diagonal cyclic homolo gy of the spher e ar e π ∗ (( T C ∆ ( S )[ k ] /T C ∆ ( S )[ k − 1 ]) p ) ∼ = Y α ∈O k lim l  Z [ GL n ( Z p ) / Γ l,k,α ] ⊗ π ∗ (Σ ∞ S k ∧ B T k + ) /p l  Her e O k is the set of unor der e d c ol le ctions of p ositive inte gers { e 1 , e 2 , . . . , e k } su ch that at le ast one e i is e qual t o 1; the gr oup Γ l,k,α is, as b efor e, the st abilizer of a sub gr oup of C × n p l of ra nk k and c otyp e α . The attaching map o f the ﬁltratio n of diag onal cyclic homology is no n-trivial ev en in the one- dimensional cas e, and it remains a n op en problem to determine the a tta ching maps of the ﬁltration for higher diagonal cyclic homology . W e b egin b y developing a slig ht gener alization o f Pr o po sition A.1. Recall that the twisted arr ow category A r M of a monoid M , where M is vie wed a s a category with one ob ject µ , has ob jects the elements µ m − → µ of the monoid, and morphis ms ( m 4 ∗ , m 2 ∗ ) : m 1 → m 3 given b y diagra ms µ m 2 − − − − → µ m 1   y m 3   y µ m 4 ← − − − − µ T op olog ical cyclic homolo gy is the homotop y limit (of th e ﬁxed po ints of to p olo gical Ho chsc hild homology) o ver A r M n for the monoid M n of isogenies. T o po logical F rob enius homology is the homotopy limit ov er the sub categ ory with m 2 = id, a nd similarly top ological restriction homolog y for the subcateg ory with m 4 = id. F o r an y submonoid K ⊂ M n , there is a cons truction inter- mediate b etw een to po logical cyclic and top ological F rob enius homolo gy . Le t A r M n [ K ] deno te the sub c ategory of A r M n whose morphisms have m 2 ∈ K , a nd deﬁne the K -r e la tive top ologica l cyclic homology by T C ( n ) K ( A ) := holim m ∈A r M n [ K ] T m ( A ) As b efore T m ( A ) deno tes the ﬁxed p oints of T ( A ) by the kernel of the iso geny m . Prop ositio n A.1 is a sp ecial case of the following result. 35 Theorem A.3. The monoid K ⊂ M n acts on T F ( n ) ( A ) and t her e is an e quivalenc e T C ( n ) K ( A ) ≃ holim K T F ( n ) ( A ) Pr o of. Let Ψ deno te the functor fro m A r M n to sp ectra with Ψ( m ) = T m ( A ), that is the functor whose homotopy limit g ives top olog ical cy clic homolog y . W e henceforth a bbr eviate M n as M . W e beg in by reformulating the relative a rrow category as a ca tegorical semi-dir ect pro duct: Lemma A.4. Ther e is an isomorphi sm A r M [ K ] ∼ = K ⋉ F rob M . Before describing t he isomor phism we r e call th e deﬁnition of this semi-direct pro duct. Given a category C a nd a functor F : C op → Cat, the semi-direct pro duct C ⋉ F ha s ob jects pair s ( c, x ), wher e c ∈ C and x ∈ F ( c ). A morphism of C ⋉ F from ( c, x ) to ( c ′ , x ′ ) is a pair ( f : c → c ′ , φ : x → F ( f )( x ′ )). The co mp os itio n of ( f , φ ) : ( c, x ) → ( c ′ , x ′ ) and ( g , ψ ) : ( c ′ , x ′ ) → ( c ′′ , x ′′ ) is ( g f , F ( f )( ψ ) ◦ φ ). This categoric al se mi- direct pro duct is a slig h t v ariant o f Thomaso n’s Grothendieck c o nstruction [29]. In the lemma , F r ob M is viewed as a functor fro m K op to Cat taking the single o b ject µ to F r ob M and taking the mor phism k to the r ight action functor k : F rob M → F rob M given on ob jects by k · ( µ m − → µ ) = µ mk − − → µ , and on morphisms by sending (id ∗ , m 4 ∗ ) : m 1 → m 3 to (id ∗ , m 4 ∗ ) : m 1 k → m 3 k . The isomor phis m in the lemma is given by the functor K ⋉ F r ob M → A r M [ K ] taking the pair ( µ, µ m − → µ ) to the a rrow µ m − → µ and taking the mo r phism ( µ k − → µ, (id ∗ , m 4 ∗ ) : m 1 → m 3 k ) : ( µ, µ m 1 − − → µ ) → ( µ, µ m 3 − − → µ ) to the morphism ( k ∗ , m 4 ∗ ) : m 1 → m 3 . W e now need only dec o mpo se the ho motopy limit ov er the semi-dire c t pro duct: Lemma A.5. Ther e is a natur al e quivalenc e holim K ⋉ F r ob M Ψ ≃ holim K (holim F r o b M Ψ) . Here the functor Ψ on the right hand side is implicitly restricted to the subca tegory F r ob M of K ⋉ F r ob M . This equiv a lence ha s nothing to do with th e particular categories in question a nd o ccurs for any catego rical semi-direct pr o duct—the r esult is usually called the Thomaso n theor em for homotopy limits and is discussed for example in Chach´ o lski-Scherer [1 0].  W e use the theorem to express dia gonal cyclic homo logy in ter ms of F rob enius homo lo gy . Corollary A.6. Ther e is an e quivalenc e T C ∆ ( A ) ≃ holim ∆ T F ( n ) ( A ) . Pr o of. The chain of equiv alences is the following T C ∆ ( A ) = ho lim A r ∆ T m ( A ) ≃ holim ∆ (holim F r o b ∆ T m ( A )) ≃ holim ∆ ( holim F r o b M T m ( A )) = holim ∆ T F ( n ) ( A ) . The ﬁrst equiv alence follows from the theorem, a nd the seco nd b ecaus e F rob ∆ is ﬁnal in F r ob M .  Because the mo noid ∆ is gener ated b y p · id n , we can reexpr ess this homoto p y limit as holim ∆ T F ( n ) ( A ) ≃ holim  T F ( n ) ( A ) φ ⇒ id T F ( n ) ( A )  where φ is the ac tion of p · id n on T F ( n ) ( A ) induced by the restriction op erator for that matrix. Now we sp ecialize to studying th e dia g onal cyclic homology of the spher e, and show that the op erator φ pr eserves the ra nk ﬁltra tion of T F ( n ) ( S ). Recall that the rank ﬁltra tion on the equiv a riant sphere sp ectr um, describ ed in sectio n 6 .1, induces a ﬁltra tion of topo logical F rob enius homology , and we denote this ﬁltration by T F ( n ) ( S )[ k ]. Prop ositio n A. 7 . The r estriction op er ator φ on T F ( n ) ( S ) pr eserves the r ank ﬁ lt r ation T F ( n ) ( S )[ k ] . This prop osition follows by direc tly chec king that the ﬁltered equiv aria n t sphere spectrum functors Φ[ k ]( G ) := K ( C G [ k ]) des crib ed in section 6.1, namely the functors whose homotopy limits ar e T F ( n ) ( S )[ k ], extend from functors on F r ob M to functor s on A r M [∆]. 36 Because the r estriction r e spe c ts the ﬁltration, we hav e a n induced ﬁltra tion, denoted T C ∆ ( S )[ k ], on dia gonal cyclic homolo gy . The ﬁltration quotients here can b e desc r ib e d as a homoto p y equa lizer on the ﬁltr ation quotients of F rob enius homolog y: T C ∆ ( S )[ k ] /T C ∆ ( S )[ k − 1] ≃ holim  T F ( n ) ( S )[ k ] /T F ( n ) ( S )[ k − 1 ] φ ⇒ id T F ( n ) ( S )[ k ] /T F ( n ) ( S )[ k − 1 ]  It remains only to determine the actio n of φ on the ﬁltration quotients of F rob enius homolog y , and for this we utilize the co t yp e decomp ositio n of sectio n 6.2. Recall the decomp ositio n T F ( n ) ( S )[ k ] /T F ( n ) ( S )[ k − 1] ≃ Y α holim l Φ[ k , α ] where Φ[ k , α ] is the rank k , cotype α par t of the equiv ariant sphere sp ectrum functor, and wher e α v ar ies ov er collections of unor dered p ositive in tegers { e 1 , e 2 , . . . , e k } . Let P k denote the s e t of such co llections, a nd deﬁne a map θ : P k → P k by θ ( { e 1 , e 2 , . . . , e k } ) = { e 1 + 1 , e 2 + 1 , . . . , e k + 1 } . Roughly speaking , the operator φ shifts the factors in the cotype decomposition along the map “ θ − 1 ”: Prop ositio n A. 8. The op er ator φ : Q α holim l Φ[ k , α ] → Q α holim l Φ[ k , α ] m aps the c otyp e factor holim l Φ[ k , θ ( α )] by a homotopy e quivalenc e to t he c otyp e factor holim l Φ[ k , α ] , and for any α 0 c ontaining some e i = 1 , the op er ator φ maps the factor holim l Φ[ k , α 0 ] to t he b asep oint. Mor e sp e ciﬁc al ly, ther e is an identiﬁc ation Y α ∈ P k holim l Φ[ k , α ] ≃ Y α 0 ∈O k F (( { θ j ( α 0 ) } j ≥ 0 ) + , ho lim l Φ[ k , α 0 ]) wher e O k is the set of c ol le ctions { e 1 , e 2 , . . . , e k } with at le ast one e i e qual t o 1, and the right hand side of this e quivalenc e is a pr o du ct of function sp e ctr a with discr ete sour c es. With r esp e ct to this identiﬁc ation, the op er ator φ is e qual to Q α 0 ∈O k F ( θ, holim l Φ[ k , α 0 ]) . Pr o of. The iden tiﬁcation is seen a s follo ws. F or α = { e 1 , e 2 , . . . , e k } , choo se an l > max i ( e i + 1 ). The restriction asso cia ted to the inclus io n C × n p l ֒ → C × n p l +1 induces a bijection from M [ k , θ ( α )]( l + 1 ) to M [ k , α ]( l ), where, as in section 6.3, the expression M [ k , α ]( l ) deno tes the result of a djoining a disjoint basep oint to the set of subgroups K ⊆ C × n p l with C × n p l /K of r ank k and K o f cotype α . This bijection yields a distinguished equiv alence o f holim l Φ[ k , θ ( α )] and holim l Φ[ k , α ] and the identiﬁcation in the prop osition follows. That th e op erato r φ co rresp onds to F ( θ, − ) in this ident iﬁcation is seen by directly tracing its action throug h the given equiv a le nces.  The homotopy equaliz er o f φ and the ident ity evidently has a single factor for e a ch primitive co t yp e α = { e 1 , e 2 , . . . , e k } , that is for those co t yp es with so me e i = 1, and the computatio n of the rank- cotype comp onents from section 6 .3 yields our desired result: Corollary A. 9 . The homotopy gr oups of the ﬁltr ation quotients of t he ra nk ﬁltr ation of the diagonal cyclic homolo gy of the spher e ar e as fol lows: π ∗ (( T C ∆ ( S )[ k ] /T C ∆ ( S )[ k − 1 ]) p ) ∼ = Y α ∈O k lim l  Z [ GL n ( Z p ) / Γ l,k,α ] ⊗ π ∗ (Σ ∞ S k ∧ B T k + ) /p l  . Ackno wledgments W e would like to express o ur gratitude to the Institut Mittag-Le ﬄer a nd to Stanfo r d Universit y for their supp ort and ho spitality at v arious stages of this pro ject. 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Dep ar tment of Ma thema tics, St anford Univ ersity, P alo Al to, CA 94305 , USA E-mail addr ess : gunnar@m ath.stanfor d.edu Dep ar tment of Ma thema tics, St anford Univ ersity, P alo Al to, CA 94305 , USA E-mail addr ess : cdouglas @math.stanf ord.edu Dep ar tment of Ma thema tics, University of Bergen, N-500 8 Bergen, Nor w a y E-mail addr ess : dundas@m ath.uib.no 38

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