TOPOLOGICAL CYCLIC HOMOLOGY VIA THE NORM

Transcription

1 TOPOLOGICAL CYCLIC HOMOLOGY VIA THE NORM VIGLEIK ANGELTVEIT, ANDREW J. BLUMBERG, TEENA GERHARDT, MICHAEL A. HILL, TYLER LAWSON, AND MICHAEL A. MANDELL Abstract. W dscrib a construction of th cyclotomic structur on topological Hochschild homology (T HH) of a ring spctrum using th Hill Hopkins Ravnl multiplicativ norm. Our analysis taks plac ntirly in th catgory of quivariant orthogonal spctra, avoiding us of th Bökstdt cohrnc machinry. W ar also abl to dfin vrsions of topological cyclic homology (T C) and T R-thory rlativ to an arbitrary commutativ ring spctrum A. W dscrib spctral squncs computing th rlativ thory A T R in trms of T R ovr th sphr spctrum and vic vrsa. Furthrmor, our construction prmits a straightforward dfinition of th Adams oprations on T R and T C. Contnts 1. Introduction 1 2. Background on quivariant stabl homotopy thory 8 3. Cyclotomic spctra and topological cyclic homology Th construction and homotopy thory of th S 1 -norm Th cyclotomic trac A dscription of rlativ T HH as th rlativ S 1 -norm Th op-p-prcyclotomic structur on A N R T HH of ring C n -spctra Spctral squncs for A T R Adams oprations Madsn s rmarks 47 Rfrncs Introduction Ovr th last two dcads, th calculational study of algbraic K-thory has bn rvolutionizd by th dvlopmnt of trac mthods. In analogy with th Chrn charactr from topological K-thory to ordinary cohomology, thr xist Angltvit was supportd in part by an NSF All-Instituts Postdoctoral Fllowship administrd by th Mathmatical Scincs Rsarch Institut through its cor grant DMS , NSF grant DMS , and an Australian Rsarch Council Discovry Grant. Blumbrg was supportd in part by NSF grant DMS Grhardt was supportd in part by NSF grants DMS and DMS Hill was supportd in part by NSF grant DMS , DARPA grant FA , and th Sloan Foundation. Lawson was supportd in part by NSF grant DMS Mandll was supportd in part by NSF grants DMS , DMS

2 2 V.ANGELTVEIT, A.BLUMBERG, T.GERHARDT, M.HILL, T.LAWSON, AND M.MANDELL trac maps from algbraic K-thory to various mor homological approximations, which also can b mor computabl. For a ring R, Dnnis constructd a map to Hochschild homology K(R) HH(R) that gnralizs th trac of a matrix. Goodwilli liftd this trac map to ngativ cyclic homology K(R) HC (R) HH(R) and showd that, rationally, this map can oftn b usd to comput K(R). In his 1990 ICM addrss, Goodwilli conjcturd that thr should b a brav nw vrsion of this story involving topological analogus of Hochschild and cyclic homology dfind by changing th ground ring from Z to th sphr spctrum. Although th modrn symmtric monoidal catgoris of spctra had not yt bn invntd, Bökstdt dvlopd cohrnc machinry that nabld a dfinition of topological Hochschild homology (T HH) along ths lins. Furthr, h constructd a topological Dnnis trac map [6] K(R) T HH(R). Subsquntly, Bökstdt Hsiang Madsn [7] dfind topological cyclic homology (T C) and constructd th cyclotomic trac map to T C, lifting th topological Dnnis trac K(R) T C(R) T HH(R). Thy did this in th cours of rsolving th K-thory Novikov conjctur for groups satisfying a mild finitnss hypothsis. Subsquntly, sminal work of Mc- Carthy [32] and Dundas [11] showd that whn working at a prim p, T C oftn capturs a grat dal of information about K-thory. Hsslholt and Madsn (intr alia, [18]) thn usd T C to mak xtnsiv computations in K-thory, including a computational rsolution of th Quilln Lichtnbaum conjctur for crtain filds. Th calculational powr of trac mthods dpnds on th ability to comput T C(R), which ultimatly drivs from th mthods of quivariant stabl homotopy thory. Bökstdt s dfinition of T HH(R) closly rsmbls a cyclic bar construction, and as a consqunc T HH(R) is an S 1 -spctrum. Topological cyclic homology is constructd from this S 1 -action on T HH(R), via fixd point spctra T R n (R) = T HH(R) C p n. In fact, T HH(R) has a vry spcial quivariant structur: T HH(R) is a cyclotomic spctrum, which is an S 1 -spctrum quippd with additional data that modls th structur of a fr loop spac ΛX. Th cyclic bar construction can b formd in any symmtric monoidal catgory (A,, 1); w will lt N cyc dnot th rsulting simplicial (or cyclic) objct. Rcall that in th catgory of spacs, for a group-lik monoid M, thr is a natural map N cyc M Map(S 1, BM) = ΛBM (whr dnots gomtric ralization) that is a wak quivalnc on fixd points for any finit subgroup C n < S 1. Morovr, for ach such C n, th fr loop spac is quippd with quivalncs (in fact homomorphisms) (ΛBM) Cn = ΛBM of S 1 -spacs, whr (ΛBM) Cn is rgardd as an S 1 -spac (rathr than an S 1 /C n - spac) via pullback along th nth root isomorphism ρ n : S 1 = S 1 /C n.

3 TOPOLOGICAL CYCLIC HOMOLOGY VIA THE NORM 3 In analogy, a cyclotomic spctrum is an S 1 -spctrum quippd with compatibl quivalncs of S 1 -spctra t n : ρ nlφ Cn X X, whr LΦ Cn dnots th (lft drivd) gomtric fixd point functor. Th construction of th cyclotomic structur on T HH has classically bn on of th mor subtl and mystrious parts of th construction of T C. In a modrn symmtric monoidal catgory of spctra (.g., symmtric spctra or EKMM S- moduls), on can simply dfin T HH(R) as T HH(R) = N cyc R, but th rsulting quivariant spctrum did not hav th corrct homotopy typ. Only Bökstdt s original construction of T HH smd to produc th cyclotomic structur. Although this situation has not impdd th calculational applications, rlianc on th Bökstdt construction has limitd progrss in crtain dirctions. For on thing, it dos not sm to b possibl to us th Bökstdt construction to dfin T C rlativ to a ground ring that is not th sphr spctrum S. Morovr, th dtails of th Bökstdt construction mak it difficult to undrstand th quivarianc (and thrfor rlvanc to T C) of various additional algbraic structurs that aris on T HH, notably th Adams oprations and th coalgbra structurs. Th purpos of this papr is to introduc a nw approach to th construction of th cyclotomic structur on T HH using an intrprtation of T HH in trms of th Hill Hopkins Ravnl multiplicativ norm. Our point of dpartur is th obsrvation that th construction of th cyclotomic structur on T HH(R) ultimatly boils down to having good modls of th smash powrs R n = R R... R }{{} n of a spctrum R as a C n -spctrum such that thr is a suitably compatibl collction of diagonal quivalncs R Φ Cn R n. Th rcnt solution of th Krvair invariant on problm involvd th dtaild analysis of a multiplicativ norm construction in quivariant stabl homotopy thory that has prcisly this typ of bhavior. Although Hill Hopkins Ravnl studid th norm construction NH G for a finit group G and subgroup H, using th cyclic bar construction on can xtnd this construction to a norm N on associativ ring orthogonal spctra; such a construction first appard in th thsis of Martin Stolz [37]. For th following dfinition, w nd to introduc som notation. Lt S dnot th catgory of orthogonal spctra and lt SU dnot th catgory of orthogonal S 1 -spctra indxd on th complt univrs U. Finally, lt Ass dnot th catgory of associativ ring orthogonal spctra. Dfinition 1.1. Dfin th functor to b th composit functor N : Ass SU R I U R cyc N R,

4 4 V.ANGELTVEIT, A.BLUMBERG, T.GERHARDT, M.HILL, T.LAWSON, AND M.MANDELL with N cyc R rgardd as an orthogonal S 1 -spctrum indxd on th standard trivial univrs R. Hr IR U dnots th chang of univrs functor (s Dfinition 2.6). Sinc both th cyclic bar construction and th chang of univrs functor prsrv commutativ ring orthogonal spctra, th norm abov also prsrvs commutativ ring orthogonal spctra. In th following proposition, provd in Sction 4, Com and Com U dnot th catgoris of commutativ ring orthogonal spctra and commutativ ring orthogonal S 1 -spctra, rspctivly. Proposition 1.2. N rstricts to a functor N : Com Com U that is th lft adjoint to th forgtful functor from commutativ ring orthogonal S 1 -spctra to commutativ ring orthogonal spctra. Th forgtful functor from commutativ ring orthogonal S 1 -spctra to commutativ ring orthogonal spctra is th composit of th chang of univrs functor IU R and th functor that forgts quivarianc. Th proof of th abov proposition idntifis N : Com Com U as th composit functor R I U R (R ), which is lft adjoint to th forgtful functor. Hr dnots th tnsor of a commutativ ring orthogonal spctrum with an unbasd spac, and w rgard ( ) S 1 as a functor from commutativ ring orthogonal spctra to commutativ ring orthogonal spctra with an action of S 1. Th dp aspct of th Hill Hopkins Ravnl tratmnt of th norm functor is thir analysis of th lft drivd functors of th norm. As part of this analysis thy show that th norm NH G prsrvs crtain wak quivalncs. For our norm N into SU, w work with th homotopy thory dfind by th F-quivalncs of orthogonal S 1 -spctra, whr an F-quivalnc is a map that inducs an isomorphism on all th homotopy groups at th fixd point spctra for th finit subgroups of S 1. W prov th following thorm in Sction 4. Proposition 1.3. Assum that R is a cofibrant associativ ring orthogonal spctrum and R is ithr a cofibrant associativ ring orthogonal spctrum or a cofibrant commutativ ring orthogonal spctrum. If R R is a wak quivalnc, thn N R N R is an F-quivalnc in SU. Of cours th conclusion holds if R is a cofibrant commutativ ring orthogonal spctrum as wll; th point of Proposition 1.3 is to compar cofibrant rplacmnts in associativ and commutativ ring orthogonal spctra. As a consqunc w obtain th following additional obsrvation about th adjunction in th commutativ cas. S Proposition 4.10 for a mor prcis statmnt. Proposition 1.4. Th functor N : Com Com U is Quilln lft adjoint to th forgtful functor (for an appropriat modl structur with wak quivalncs th F-quivalncs on th codomain); in particular, its lft drivd functor xists and is lft adjoint to th right drivd forgtful functor.

5 TOPOLOGICAL CYCLIC HOMOLOGY VIA THE NORM 5 Our first main thorm is that whn R is a cofibrant associativ ring orthogonal spctrum, N R is a cyclotomic spctrum. To b prcis, w us th point-st modl of cyclotomic spctra from [5], which provids a dfinition ntirly in trms of th catgory of orthogonal S 1 -spctra. Thorm 1.5. Lt R b a cofibrant associativ or cofibrant commutativ ring orthogonal spctrum. Thn N R has a natural structur of a cyclotomic spctrum. Exprts will rcogniz that on can giv a dirct construction of th cyclotomic trac inducd by th inclusion of objcts in a spctral catgory nrichd in orthogonal spctra (.g., s [4]). W rviw this construction in Sction 5. Proposition 1.4, which dscribs N as th homotopical lft adjoint to th forgtful functor, suggsts a gnralization of our construction of T HH that taks ring orthogonal C n -spctra as input. For commutativ ring orthogonal C n -spctra, w can dfin NC n as th lft adjoint to th forgtful functor. Howvr, to xtnd to th non-commutativ cas, w nd an xplicit construction. W giv such a construction in Sction 8 in trms of a cyclic bar construction, which w dnot as N cyc,cn R. Its gomtric ralization N cyc,cn R has an S 1 -action, and by promoting it to th complt univrs w obtain a gnuin orthogonal S 1 -spctrum that w dnot as NC n R. Th following proposition is a consistncy chck. Proposition 1.6. Lt R b a commutativ ring orthogonal C n -spctrum. Thn NC n R is isomorphic to th lft adjoint of th forgtful functor from commutativ ring orthogonal S 1 -spctra to commutativ ring orthogonal C n -spctra. Again, w can dscrib th lft adjoint in trms of a tnsor N C n R = I U R (R C n S 1 ), whr th rlativ tnsor R Cn S 1 may b xplicitly constructd as th coqualizr (i R) C n S 1 (i R) S 1 of th canonical action of C n on S 1 and th action map (i R) C n i R, whr i dnots th chang-of-group functor to th trivial group. Choosing an appropriatly subdividd modl of th circl producs th isomorphism btwn th two dscriptions. As abov, by cofibrantly rplacing R w can comput th lft-drivd functor of NC n, and in this cas NC n R is a p-cyclotomic spctrum (s Dfinition 3.1) providd ithr n is prim to p or R is C n -cyclotomic (q.v. Dfinition 8.7 blow). This lads to th obvious dfinition of T C Cn R. This C n -rlativ T HH (and th associatd constructions of T R and T C) is xpctd to b both intrsting and comparativly asy to comput for som of th quivariant spctra that aris in Hill Hopkins Ravnl, in particular th ral cobordism spctrum MU R. W can also considr anothr kind of rlativ construction, namly in th situation whr R is an algbra ovr an arbitrary commutativ ring orthogonal spctrum A. Dfinition 1.1 can b xtndd to th rlativ stting; th quivariant indxd product can b carrid out in any symmtric monoidal catgory, and th homotopical analysis in th cas of A-moduls is givn in Sction 6. Dfinition 1.7. Lt A b a cofibrant commutativ ring orthogonal spctrum, and dnot by A-Alg th catgory of A-algbras. W dfin th A-rlativ norm functor AN : A-Alg A S 1-Mod U

6 6 V.ANGELTVEIT, A.BLUMBERG, T.GERHARDT, M.HILL, T.LAWSON, AND M.MANDELL by R IR U cyc N A R. Hr A S 1 dnots IR U A, constructd by applying th point-st chang of univrs functor IR U to A rgardd as a commutativ ring orthogonal -spctrum (on th univrs R ) with trivial S 1 -action. Thn A S 1 is a commutativ ring orthogonal S 1 -spctrum (on th univrs U) and A S 1-Mod U dnots th catgory of A S 1- moduls in SU. W writ A T HH(R) for th undrlying non-quivariant spctrum of A N R; this spctrum was dnotd thh A (R) in [12, IX.2.1]. Whn R is a commutativ A-algbra, A N R is naturally a commutativ A S 1-algbra. Th functor AN : A-Com A S 1-Com U is again lft adjoint to th forgtful functor. Howvr, du to th subtltis of th bhavior of IR U whn applid to cofibrant commutativ ring orthogonal spctra rgardd as S 1 -spctra with trivial action, A S 1 is in gnral not cyclotomic and so nithr is A N R. In particular, w always hav A S 1 = A T HH(A) vn though A may not xtnd to a cyclotomic spctrum. (S also Exampls 6.3 and 7.6 blow.) Instad, w must sttl for th following wakr analogu of Thorm 1.5. Hr an op-p-prcyclotomic structur asks mrly for an analogu of th cyclotomic structur map (in th opposit dirction) that is not ncssarily an quivalnc. W prov th following thorm in Sction 7. Thorm 1.8. Lt R b an A-algbra. Thn A N R is an op-p-prcyclotomic spctrum with structur map in th catgory of A S 1-moduls. Nonthlss, w can dfin th A-rlativ topological cyclic homology op A T C(R) as a homotopy limit ovr th Frobnius and wrong-way rstriction maps. Th rlativ topological cyclic homology is th targt of an A-rlativ cyclotomic trac K(R) op A T C(R), factoring though th usual cyclotomic trac K(R) T C(R), ssntially by construction. Thorm 1.9. Lt R b a cofibrant associativ A-algbra or a cofibrant commutativ A-algbra. Thr is an A-rlativ cyclotomic trac map K(R) op A T C(R) making th following diagram commut in th stabl catgory K(R) T C(R) T HH(R) op A T C(R) AT HH(R). Using th idntification N A = IR U (A ) in th commutativ contxt, th map S 1 inducs a map of quivariant commutativ ring orthogonal spctra N A A S 1. Just as in th non-quivariant cas, w can idntify A N R as xtnsion of scalars along this map. Proposition Lt R b an associativ A-algbra. Thr is a natural isomorphism AN R = N R N S 1 A A S 1.

7 TOPOLOGICAL CYCLIC HOMOLOGY VIA THE NORM 7 Whn R is a cofibrant associativ A-algbra or cofibrant commutativ A-algbra, this inducs a natural isomorphism in th stabl catgory AN Th quivariant homotopy groups π Cn so π Cn ( A N R = N R L A N A S 1. (N R) ar th T R-groups T R n (R) and R) ar by dfinition th rlativ T R-groups A T R n (R). Th Künnth spctral squnc of [23] can b combind with th prvious thorm to comput th rlativ T R-groups from th absolut T R-groups and Macky functor Tor. Mor oftn w xpct to us th rlativ thory to comput th absolut thory. Nonquivariantly, th isomorphism (1.11) T HH(R) A = A T HH(R A) givs ris to a Künnth spctral squnc Tor A (R SR op ), (A (R), A (R)) = A (T HH(R)). An Adams spctral squnc can thn in thory b usd to comput th homotopy groups of T HH(R). For formal rasons, th isomorphism (1.11) still holds quivariantly, but now w hav thr diffrnt vrsions of th non-quivariant Künnth spctral squnc (non of which hav quit as lgant an E 2 -trm) which w us in conjunction with quation (1.11). W discuss ths in Sction 9. A furthr application of our modl of T HH and T C is a construction, whn R is commutativ, of Adams oprations on N R and A N R that ar compatibl (in th absolut cas) with th cyclotomic structur. McCarthy xplaind how Adams oprations can b constructd on any cyclic objct that, whn viwd as a functor from th cyclic catgory, factors through th catgory of finit sts (and all maps). As a consqunc, it is possibl to construct Adams oprations on T HH of a commutativ monoid objct in any symmtric monoidal catgory of spctra. An advantag of our formulation is that w can asily vrify th quivarianc of ths oprations and in particular show thy dscnd to T C. W prov th following thorm in Sction 10. Thorm Lt A b a commutativ ring orthogonal spctrum and R a commutativ A-algbra. Thr ar Adams oprations ψ r : A N R A N R. Whn r is prim to p, th opration ψ r is compatibl with th rstriction and Frobnius maps on th p-cyclotomic spctrum T HH(R) and so inducs a corrsponding opration on T R(R) and T C(R). W hav organizd th papr to contain a brif rviw with rfrncs to much of th background ndd hr. Sction 2 is mostly rviw of [29] and [20, App. B], and Sction 3 is in part a rviw of [5, 4]. In addition, th main rsults in Sction 4 ovrlap significantly with [37], although our tratmnt is vry diffrnt: w rly on [20] to study th absolut S 1 -norm whras [37] dirctly analyzs th construction by using a somwhat diffrnt modl structur and focuss on th cas of commutativ ring orthogonal spctra. Acknowldgmnts. Th authors would lik to thank Lars Hsslholt, Mik Hopkins, and Ptr May for many hlpful convrsations. W thank Aaron Royr and Erni Fonts for hlping to idntify a srious rror in a prvious draft. W thank Cary Malkiwich for many hlpful suggstions rgarding th prvious draft. This

8 8 V.ANGELTVEIT, A.BLUMBERG, T.GERHARDT, M.HILL, T.LAWSON, AND M.MANDELL projct was mad possibl by th hospitality of AIM, th IMA, MSRI, and th Hausdorff Rsarch Institut for Mathmatics at th Univrsity of Bonn. 2. Background on quivariant stabl homotopy thory In this sction, w brifly rviw ncssary dtails about th catgory of orthogonal G-spctra and th gomtric fixd point and norm functors. Our primary sourcs for this matrial ar th monograph of Mandll-May [29] and th appndics to Hill Hopkins Ravnl [20]. S also [5, 2] for a rviw of som of ths dtails. W bgin with two subsctions discussing th point-st thory followd by two subsctions on homotopy thory and drivd functors Th point-st thory of quivariant orthogonal spctra. Lt G b a compact Li group. W dnot by T G th catgory of basd G-spacs and basd G- maps. Th smash product of G-spacs maks this a closd symmtric monoidal catgory, with function objct F (X, Y ) th basd spac of (non-quivariant) maps from X to Y with th conjugation G-action. In particular, T G is nrichd ovr G-spacs. W will dnot by U a fixd univrs of G-rprsntations [29, II.1.1], by which w man a countabl dimnsional vctor spac with linar G-action and G-fixd innr product that contains R, is th sum of finit dimnsional G-rprsntations, and that has th proprty that any G-rprsntation that occurs in U occurs infinitly oftn. Lt V G (U) dnot th st of finit dimnsional G-innr product spacs which ar isomorphic to a G-vctor subspac of U. Excpt in this sction, w always assum that U is a complt G-univrs, maning that all finit dimnsional irrducibl G-rprsntations ar in U. For V, W in V G (U), dnot by I G (V, W ) th spac of (non-quivariant) isomtric isomorphisms V W, rgardd as a G- spac via conjugation. Lt IG U b th catgory nrichd in G-spacs with VG (U) as its objcts and I G (V, W ) as its morphism G-spacs; w writ just I G whn U is undrstood. Dfinition 2.1 ([29, II.2.6]). An orthogonal G-spctrum is a G-quivariant continuous functor X : I G T G quippd with a structur map σ V,W : X(V ) S W X(V W ) that is a natural transformation of nrichd functors I G I G T G and that is associativ and unital in th obvious sns. A map of orthogonal G-spctra X X is a natural transformation that commuts with th structur map. W dnot th catgory of orthogonal G-spctra by S G. Whn ncssary to spcify th univrs U, w includ it in th notation as SU G. Th catgory of orthogonal G-spctra is nrichd ovr basd G-spacs, whr th G-spac of maps consists of all natural transformations (not just th quivariant ons). Tnsors and cotnsors ar computd lvlwis. Th catgory of orthogonal G-spctra is a closd symmtric monoidal catgory with unit th quivariant sphr spctrum S G (with S G (V ) = S V ). For tchnical rasons, it is oftn convnint to giv an quivalnt formulation of orthogonal G-spctra as diagram spacs. Following [29, II.4], w considr th catgory J G which has th sam objcts as I G but morphisms from V to W givn by th Thom spac of th complmnt bundl of linar isomtris from V to W.

9 TOPOLOGICAL CYCLIC HOMOLOGY VIA THE NORM 9 Proposition 2.2 ([29, II.4.3]). Th catgory S G of orthogonal G-spctra is quivalnt to th catgory of J G -spacs, i.., th continuous quivariant functors from J G to T G. Th symmtric monoidal structur is givn by th Day convolution. This dscription provids simpl formulas for suspnsion spctra and dsuspnsion spctra in orthogonal G-spctra. Dfinition 2.3 ([29, II.4.6]). For any finit-dimnsional G-innr product spac V w hav th shift dsuspnsion spctrum functor dfind by F V : T G S G (F V A)(W ) = J G (V, W ) A. This is th lft adjoint to th valuation functor which valuats an orthogonal G-spctrum at V. Rmark 2.4. In [20], th dsuspnsion spctrum F V S 0 is dnotd as S V and F 0 A is dnotd as Σ A in a nod to th classical notation. (Thy writ S V A for F V A = F V S 0 A.) Sinc th catgory SU G is symmtric monoidal undr th smash product, w hav catgoris of associativ and commutativ monoids, i.., algbras ovr th monads T and P that crat associativ and commutativ monoids in symmtric monoidal catgoris (.g., s [12, II.4] for a discussion). Notation 2.5. Lt Ass G and Com G dnot th catgoris of associativ and commutativ ring orthogonal G-spctra. For a fixd objct A in Com G, thr is an associatd symmtric monoidal catgory A-Mod G of A-moduls in orthogonal G-spctra, with product th A-rlativ smash product A. As in Notation 2.5, thr ar catgoris A-Alg G of A-algbras, and A-Com G of commutativ A-algbras [29, III.7.6]. W now turn to th dscription of various usful functors on orthogonal G- spctra. W bgin by rviwing th chang of univrs functors. In contrast to th classical framwork of coordinat-fr quivariant spctra [26], orthogonal G- spctra disntangl th point-st and homotopical rols of th univrs U. A first manifstation of this occurs in th bhavior of th point-st chang of univrs functors. Dfinition 2.6 ([29, V.1.2]). For any pair of univrss U and U, th point-st chang of univrs functor I U U : S G U S G U is dfind by I U U X(V ) = J (Rn, V ) O(n) X(R n ) for V in V G (U ), whr n = dim V. Ths functors ar strong symmtric monoidal quivalncs of catgoris: Proposition 2.7 ([29, V.1.1,V.1.5]). Givn univrss U, U, U, (1) IU U (2) I U U (3) I U U is naturally isomorphic to th idntity. IU U is naturally isomorphic to IU U. is strong symmtric monoidal.

10 10 V.ANGELTVEIT, A.BLUMBERG, T.GERHARDT, M.HILL, T.LAWSON, AND M.MANDELL W ar particularly intrstd in th chang of univrs functors associatd to th univrss U and U G. Th lattr of ths univrss is isomorphic to th standard trivial univrs R. Not that th catgory of orthogonal G-spctra on R is just th catgory of orthogonal spctra with G-actions. Givn a subgroup H < G, w can rgard a G-spac X(V ) as an H-spac ι HX(V ). Th spac-lvl construction givs ris to a spctrum-lvl chang-of-group functor. Dfinition 2.8 ([29, V.2.1]). For a subgroup H < G, dfin th functor by for V in V H (ι HU), whr n = dim(v ). ι H : S G U S H ι H U (ι HX)(V ) = J H (R n, V ) O(n) ι H(X(R n )) As obsrvd in [29, V.2.1, V.1.10], for V in V G (U), (ι HX)(ι HV ) = ι H(X(V )). In contrast to th catgory of G-spacs, thr ar two rasonabl constructions of fixd-point functors: th catgorical fixd points, which ar basd on th dscription of fixd points as G-quivariant maps out of G/H, and th gomtric fixd points, which commut with suspnsion and th smash product (on th lvl of th homotopy catgory). Again, th dscription of orthogonal G-spctra as J G - spacs in Proposition 2.2 provids th asist way to construct th catgorical and gomtric fixd point functors [29, V]. For any normal H G, lt JG H (U, V ) dnot th G/H-spac of H-fixd points of J G (U, V ). Givn any orthogonal spctrum X, th collction of fixd points {X(V ) H } forms a JG H-spac. W can turn this collction into a J G/H-spac in two ways. Thr is a functor q : J G/H JG H inducd by rgarding G/Hrprsntations as H-trivial G-rprsntations via th quotint map G G/H. Dfinition 2.9 ([29, V.3]). For H a normal subgroup of G, th catgorical fixd point functor ( ) H : S G U S G/H U H is computd by rgarding th J H G -spac {X(V )H } as a J G/H -spac via q. On th othr hand, thr is an quivariant continuous functor φ: JG H J G/H inducd by taking a G-rprsntation V to th G/H-rprsntation V H. Dfinition 2.10 ([29, V.4]). For H a normal subgroup of G, lt Fix H dnot th functor from orthogonal G-spctra (=J G -spacs) to JG H -spacs dfind by (Fix H X)(V ) = (X(V )) H. Th gomtric fixd point functor Φ H ( ): S G U S G/H U H is constructd by taking Φ H (X) to b th lft Kan xtnsion of th J H G -spac Fix H X along φ. Rmark Hill Hopkins Ravnl [20, B.190] call th point-st gomtric fixd point functor th monoidal gomtric fixd point functor and dfin it using th coqualizr JG H(V, W ) F W H S0 (X(V )) H F V H S 0 (X(V )) H, V,W <U V <U

11 TOPOLOGICAL CYCLIC HOMOLOGY VIA THE NORM 11 drivd from applying th gomtric fixd point functor abov to th tautological prsntation of X: J G (V, W ) F W S 0 X(V ) F V S 0 X(V ), V,W <U noting that Φ H F V A = F V H A H for a G-spac A. Although Φ H dos not prsrv coqualizrs in gnral, it dos prsrv th coqualizrs prsrvd by Fix H, and Fix H prsrvs th canonical coqualizr diagram sinc it is lvlwis split. Thus, th dfinition abov agrs with th dfinition in [20, B.190]. V <U Both fixd-point functors ar lax symmtric monoidal [29, V.3.8, V.4.7] and so dscnd to catgoris of associativ and commutativ ring orthogonal G-spctra. Proposition Lt H G b a normal subgroup. Lt X and Y b orthogonal G-spctra. Thr ar natural maps Φ H X Φ H Y Φ H (X Y ) and X H Y H (X Y ) H that xhibit Φ H and ( ) H as lax symmtric monoidal functors. Thrfor, thr ar inducd functors and Φ H, ( ) H : Ass G Ass G/H Φ H, ( ) H : Com G Com G/H. For a commutativ ring orthogonal G-spctrum A, a corollary of Proposition 2.12 is that th fixd-point functors intract wll with th catgory of A-moduls. Corollary Lt A b a commutativ ring orthogonal G-spctrum. Th fixdpoint functors rstrict to functors and Φ H : A-Mod G (Φ H A)-Mod G/H ( ) H : A-Mod G A H -Mod G/H. Rmark W can xtnd ths constructions to subgroups H < G that ar not normal by considring th normalizr NH and quotint W H = NH/H. Howvr, sinc w do not nd this gnrality hrin, w do not discuss it furthr. Lt z G b an lmnt in th cntr of G. Thn multiplication by z is a natural automorphism on objcts of SR G or on objcts of A-ModG R. In particular, it will induc a natural automorphism IR U z of N H GX or of AN G HX, as dscribd in Sctions 4 and 7. Proposition Lt z b an lmnt in th cntr of G, and K a normal subgroup. Thn for any X SR G, w hav an idntification Φ K (IR U z) = IUK R z whr z = zk G/K. In particular, for z K th map Φ K (IR U z) is th idntity. Proof. Using th tautological prsntation of I U R X and naturality, it suffics to vrify this idntity on orthogonal spctra of th form F V Y for a G-rprsntation V V G (U); on such spctra, th map I U R z : F V Y F V Y is givn by f y (f z 1 ) (z y). Th rsult follows from th fact th fixd point functor ( ) K

12 12 V.ANGELTVEIT, A.BLUMBERG, T.GERHARDT, M.HILL, T.LAWSON, AND M.MANDELL taks multiplication by z to multiplication by z, and th functor JG K inducs maps JG K(V, V ) J G/K(V K, V K ) taking z to z. J G/K 2.2. Th point-st thory of th norm. Cntral to our work is th ralization by Hill, Hopkins, and Ravnl [20] that a tractabl modl for th corrct quivariant homotopy typ of a smash powr can b formd as a point-st construction using th point-st chang of univrs functors. It is corrct insofar as thr is a diagonal map which inducs an quivalnc onto th gomtric fixd points (s Sction 2.3 blow). Thy rfr to this construction as th norm aftr th norm map of Grnls-May [16], which in turn is namd for th norm map of Evns in group cohomology [13, Chaptr 6]. Th point of dpartur for th construction of th norm is th us of th changof-univrs quivalncs to rgard orthogonal G-spctra on any univrs U as G- objcts in orthogonal spctra. (Good xplicit discussions of th intrrlationship can b found in [29, V.1] and [36, 2.7].) W now giv a point-st dscription of th norm following [36] and [10]; ths dscriptions ar quivalnt to th dscription of [20, A.3] by th work of [10]. For th construction of th norm, it is convnint to us BG to dnot th catgory with on objct, whos monoid of ndomorphisms is th finit group G. Th catgory S BG of functors from BG to th catgory S of (non-quivariant) orthogonal spctra indxd on th univrs R is isomorphic to th catgory SR G of orthogonal G-spctra indxd on th univrs R. W can thn us th chang of univrs functor IR U to giv an quivalnc of SBG with th catgory SU G of orthogonal G-spctra indxd on U. Dfinition Lt G b a finit group and H < G b a finit indx subgroup with indx n. Fix an ordrd st of cost rprsntativs (g 1,..., g n ), and lt α: G Σ n H b th homomorphism α(g) = (σ, h 1,..., h n ) dfind by th rlation gg i = g σ(i) h i. Th indxd smash-powr functor is dfind as th composit G H : S BH S BG Th norm functor S BH n S B(Σn H) α S BG. is dfind to b th composit N G H : S H U S G U X I U R ( G H(IU R X)). This dfinition dpnds on th choic of cost rprsntativs; howvr, any othr choic givs a canonically naturally isomorphic functor (th isomorphism inducd by prmuting factors and multiplying ach factor by th appropriat lmnt of H). As obsrvd in [20, A.4], in fact it is possibl to giv a dscription of th norm which is indpndnt of any choics and is dtrmind instad by th univrsal proprty of th lft Kan xtnsion. Altrnativly, Schwd [36, 9.3] givs anothr way of avoiding th choic abov, using th st G : H of all choics of ordrd sts

13 TOPOLOGICAL CYCLIC HOMOLOGY VIA THE NORM 13 of cost rprsntativs; G : H is a fr transitiv Σ n H-st and th inclusion of (g 1,..., g n ) in G : H inducs an isomorphism G HX = G : H + Σn H X n. In our work, G will b th cyclic group C nr < S 1 and H = C r (usually for r = 1), and w hav th obvious choic of cost rprsntativs g k = 2π(k 1)i/nr, ltting us tak advantag of th xplicit formulas. In th cas r = 1, w hav th following. Proposition Lt G b a finit group and U a complt G-univrs. Th norm functor is givn by th composit N G : S S G U X I U R X G, whr X G dnots th smash powr indxd on th st G. Whn daling with commutativ ring orthogonal G-spctra, th norm has a particularly attractiv formal dscription [20, A.56], which is a consqunc of th fact that th norm is a symmtric monoidal functor. Thorm Lt G b a finit group and lt H b a subgroup of G. Th norm rstricts to th lft adjoint in th adjunction N G H : Com H Com G : ι H, whr ι H dnots th chang of group functor along H < G. Th rlationship of th norm with th gomtric fixd point functor is ncodd in th diagonal map [20, B.209]. Proposition Lt G b a finit group, H < G a subgroup, and K G a normal subgroup. Lt X b an orthogonal H-spctrum. Thn thr is a natural diagonal map of orthogonal G/K-spctra : N G/K HK/K ΦH K X Φ K N G H X. (Hr w supprss th isomorphism H/H K = HK/K from th notation.) Proof. Th construction of is th sam as [20, Proposition B.209] aftr gnralizing th corrsponding spac-lvl diagonal. To do this, first not that for any basd H-spac A, thr is a natural isomorphism : N G/K = HK/KAH K (NH G A) K. For this, it is convnint to modl th spac-lvl norm as follows. Th spac NH GA is isomorphic to th subspac of tupls a = (a g ) g G g G A such that a hg = ha g. Th lft G-action is givn by (k a) g = a gk. Undr this idntification, N G/K HK/K AH K consists of tupls b = (b [g] ) [g] G/K of lmnts in A H K such that b [hg] = hb [g] for h H. Similarly, (NH GA)K consists of tupls a = (a g ) g G such that a hg = ha g for h H and a gk = a g for k K. This allows us to dfin th bijction by ( b) g = b [g].

14 14 V.ANGELTVEIT, A.BLUMBERG, T.GERHARDT, M.HILL, T.LAWSON, AND M.MANDELL For any particular commutativ ring orthogonal spctrum A, th indxd smashpowr construction of Dfinition 2.16 can b carrid out in th symmtric monoidal catgory A-Mod. Dnot th A-rlativ indxd smash-powr by ( A ) G H. For X an A-modul with H-action, w undrstand ( A ) G HX to b ( A ) G HX := α X n, whr th nth smash powr is ovr A and α is as in Dfinition This is an A-modul (in SR G ). W thn hav th following dfinition of th A-rlativ norm functor: Dfinition Lt A b a commutativ ring orthogonal spctrum. Writ A H for th commutativ ring orthogonal H-spctrum IR U A obtaind by rgarding A (with trivial H-action) as an objct of S BH and applying th chang of univrs functor, and similarly for A G. Th A-rlativ norm functor is dfind to b th composit AN G H : A H -Mod H U A G -Mod G U X I U R (( A) G H(I R X)). Th thory of th diagonal map in th A-rlativ contxt is somwhat mor complicatd than in th absolut stting; w xplain th dtails in Sction Homotopy thory of orthogonal spctra. W now rviw th homotopy thory of orthogonal G-spctra with a focus on discussing th drivd functors associatd to th point-st constructions of th prcding sction. W bgin by rviwing th various modl structurs on orthogonal G-spctra. All of ths modl structurs ar ultimatly drivd from th standard modl structur on T G (th catgory of basd G-spacs), which w bgin by rviwing. Following th notational convntions of [29], w start with th sts of maps and I = {(G/H S n 1 ) + (G/H D n ) + } J = {(G/H D n ) + (G/H (D n I)) + }, whr n 0 and H varis ovr th closd subgroups of G. Rcall that thr is a compactly gnratd modl structur on th catgory T G in which I and J ar th gnrating cofibrations and gnrating acyclic cofibrations (.g., [29, III.1.8]). Th wak quivalncs and fibrations ar th maps X Y such that X H Y H is a wak quivalnc or fibration for ach closd H < G. Transporting this structur lvlwis in V G (U), w gt th lvl modl structur in orthogonal G-spctra. Proposition 2.21 ([29, III.2.4]). Fix a G-univrs U. Thr is a compactly gnratd modl structur on SU G in which th wak quivalncs and fibrations ar th maps X Y such that ach map X(V ) Y (V ) is a wak quivalnc or fibration of G-spacs. Th sts of gnrating cofibrations and acyclic cofibrations ar givn by IG U = {F V i i I} and JG U = {F V j j J}, whr V varis ovr V G (U). U Th lvl modl structur is primarily scaffolding to construct th stabl modl structurs. In ordr to spcify th wak quivalncs in th stabl modl structurs, w nd to dfin quivariant homotopy groups.

15 TOPOLOGICAL CYCLIC HOMOLOGY VIA THE NORM 15 Dfinition Fix a G-univrs U. Th homotopy groups of an orthogonal G-spctrum X ar dfind for a subgroup H < G and an intgr q as colim π q((ω V X(V )) H ) q 0 πq H V <U (X) = colim π 0 ((Ω V R q X(V )) H ) q < 0, R q <V <U (s [29, III.3.2]). Ths ar th homotopy groups of th undrlying G-prspctrum associatd to X (via th forgtful functor from orthogonal G-spctra to prspctra). W dfin th stabl quivalncs to b th maps X Y that induc isomorphisms on all homotopy groups. Proposition 2.23 ([29, III.4.2]). Fix a G-univrs U. Th standard stabl modl structur on SU G is th compactly gnratd symmtric monoidal modl structur with th cofibrations givn by th lvl cofibrations, th wak quivalncs th stabl quivalncs, and th fibrations dtrmind by th right lifting proprty. Th gnrating cofibrations ar givn by IG U as abov, and th gnrating acyclic cofibrations K ar th union of JG U and crtain additional maps dscribd in [29, III.4.3]. This modl structur lifts to a modl structur on th catgory Ass G U of associativ monoids in orthogonal G-spctra. Thorm 2.24 ([29, III.7.6.(iv)]). Fix a G-univrs U. Thr ar compactly gnratd modl structurs on Ass G U in which th wak quivalncs ar th stabl quivalncs of undrlying orthogonal G-spctra indxd on U, th fibrations ar th maps which ar stabl fibrations of undrlying orthogonal G-spctra indxd on U, and th cofibrations ar dtrmind by th lft lifting proprty. To obtain a modl structur on commutativ ring orthogonal spctra, w also nd th positiv variant of th stabl modl structur. W dfin th positiv lvl modl structurs in trms of th gnrating cofibrations (I U G )+ I U G and (J U G )+ J U G, consisting of thos maps F V i and F V j such that th rprsntation V contains a nonzro trivial rprsntation; ths also xtnd to a positiv stabl modl structur. Thorm 2.25 ([29, III.5.3]). Fix a G-univrs U. Thr ar compactly gnratd modl structurs on Com G U in which th wak quivalncs ar th stabl quivalncs of th undrlying orthogonal G-spctra, th fibrations ar th maps which ar positiv stabl fibrations of undrlying orthogonal G-spctra indxd on U, and th cofibrations ar dtrmind by th lft lifting proprty. W will also us a variant of th standard stabl modl structur that can b mor convnint whn working with th drivd functors of th norm. W rfr to this as th positiv complt stabl modl structur. S [20, B.4] for a comprhnsiv discussion of this modl structur, and [39, A] for a brif rviw. In ordr to dscrib this, dnot by (I ι H U H )+ and (J ι H U H )+ gnrating cofibrations for th positiv stabl modl structur on orthogonal H-spctra indxd on th univrs ι H U. Thorm 2.26 ([20, B.63]). Fix a G-univrs U. Thr is a compactly gnratd symmtric monoidal modl structur on S G with gnrating cofibrations and acyclic

16 16 V.ANGELTVEIT, A.BLUMBERG, T.GERHARDT, M.HILL, T.LAWSON, AND M.MANDELL cofibrations th sts {G + H i i (I ι H U H )+, H < G} and {G + H j j (J ι H U H )+, H < G} rspctivly. Th wak quivalncs ar th stabl quivalncs, and th fibrations ar dtrmind by th right lifting proprty. W thn hav corrsponding positiv complt modl structurs for Com G and Ass G. Thorm 2.27 ([20, B.130], [20, B.136 ( v3)]). Fix a G-univrs U. Thr ar compactly gnratd modl structurs on Ass G U and ComG U in which th wak quivalncs ar th stabl quivalncs of th undrlying orthogonal G-spctra, th fibrations ar th maps which ar positiv complt stabl fibrations of undrlying orthogonal G-spctra indxd on U, and th cofibrations ar dtrmind by th lft-lifting proprty. For a fixd objct A in Com G U, thr ar also liftd modl structurs on th catgoris A-Mod G U of A-moduls, A-Alg G U of A-algbras, and A-Com G U of commutativ A-algbras in both th stabl and positiv complt stabl modl structurs ([29, III.7.6] and [20, B.137]). Thr ar also liftd modl structurs on th catgory A-Mod G U of A-moduls whn A is an objct of Ass G U, but w will not nd ths. Part of th following is [20, B.137]; th rst follows by standard argumnts. Thorm Fix a G-univrs U. Lt A b a commutativ ring orthogonal G-spctrum indxd on U. Thr ar compactly gnratd modl structurs on th catgoris A-Mod G U and A-Alg G U in which th fibrations and wak quivalncs ar cratd by th forgtful functors to th stabl, complt stabl, and positiv complt stabl modl structurs on SU G. Thr ar compactly gnratd modl structurs on A-Com G U in which th fibrations and wak quivalncs ar cratd by th forgtful functors to th positiv stabl and positiv complt stabl modl structurs on A-Mod G U. Finally, whn daling with cyclotomic spctra, w nd to us variants of ths modl structurs whr th stabl quivalncs ar dtrmind by a family of subgroups of G. Rcall from [29, IV.6.1] th dfinition of a family: a family F is a collction of closd subgroups of G that is closd undr taking closd subgroups (and conjugation). W say a map X Y is an F-quivalnc if it inducs an isomorphism on homotopy groups π H for all H in F. All of th modl structurs dscribd abov hav analogus with rspct to th F-quivalncs (.g., s [29, IV.6.5]), which ar built from sts I and J whr th clls (G/H S n 1 ) + (G/H D n ) + and (G/H D n ) + (G/H (D n I)) + ar rstrictd to H F. W rcord th situation in th following omnibus thorm. Thorm Thr ar stabl, positiv stabl, and positiv complt stabl compactly gnratd modl structurs on th catgoris SU G and AssG U whr th wak quivalncs ar th F-quivalncs. Thr ar positiv stabl and positiv complt stabl compactly gnratd modl structurs on th catgory Com G U whr th wak quivalncs ar th F-quivalncs. Lt A b a commutativ ring orthogonal G-spctrum. Thr ar stabl, positiv stabl, and positiv complt stabl compactly gnratd modl structurs on th catgoris A-Mod G U, A-Alg G U whr th wak quivalncs ar th F-quivalncs. Thr ar positiv stabl and positiv complt stabl compactly gnratd modl structurs on A-Com G U whr th wak quivalncs ar th F-quivalncs.

17 TOPOLOGICAL CYCLIC HOMOLOGY VIA THE NORM 17 W ar most intrstd in cas of G = S 1 and th familis F Fin of finit subgroups of S 1 and F p of p-subgroups {C p n} of S 1 for a fixd prim p Drivd functors of fixd points and th norm. W now discuss th us of th modl structurs dscribd in th prvious sction to construct th drivd functors of th catgorical fixd point, gomtric fixd point, and norm functors. W bgin with th catgorical fixd point functor. Sinc this is a right adjoint, w hav right-drivd functors computd using fibrant rplacmnt (in any of our availabl stabl modl structurs): Thorm Lt H G b a normal subgroup. Thn th catgorical fixd point functor ( ) H : SU G SG/H is a Quilln right adjoint; in particular, it prsrvs U H fibrations and wak quivalncs btwn fibrant objcts in th stabl and positiv complt stabl modl structurs on SU G. As th fibrant objcts in th modl structurs on associativ and commutativ ring orthogonal spctra ar fibrant in th undrlying modl structurs on orthogonal G-spctra, w can driv th catgorical fixd points by fibrant rplacmnt in any of th sttings in which w work. In contrast, th gomtric fixd point functor admits a Quilln lft drivd functor (s [29, V.4.5] and [20, B.197]). Thorm Lt H b a normal subgroup of G. Th functor Φ H ( ) prsrvs cofibrations and wak quivalncs btwn cofibrant objcts in th stabl, positiv stabl, and positiv complt stabl modl structurs on S G U. Sinc th cofibrant objcts in th liftd modl structurs on Ass G U ar cofibrant whn rgardd as objcts in SU G [29, III.7.6], an immdiat corollary of Thorm 2.31 is that w can driv Φ H by cofibrant rplacmnt whn working with associativ ring orthogonal G-spctra. In contrast, th undrlying orthogonal G-spctra associatd to cofibrant objcts in Com G, in ithr of th modl structurs w study, ar ssntially nvr cofibrant and th point-st functor Φ G dos not always agr on ths with th gomtric fixd point functor on th quivariant stabl catgory. Th first part of th following thorm is [20, B.104]; th statmnt in th cas of A-moduls is similar and discussd in Sction 6. Thorm Th norm NH G ( ) prsrvs wak quivalncs btwn cofibrant objcts in any of th various stabl modl structurs on S H, Ass H, and Com H. Lt A b a commutativ ring orthogonal spctrum. Thn th A-rlativ norm AN G ( ) prsrvs wak quivalncs btwn cofibrant objcts in any of th various stabl modl structurs on A-Mod, A-Alg, and A-Com. Th utility of th positiv complt modl structur is th following homotopical vrsion of Thorm 2.18 [20, B.135]. Thorm Lt H b a subgroup of G. Th adjunction N G H : Com H Com G : ι H is a Quilln adjunction for th positiv complt stabl structurs. Finally, w hav th following rsult about th drivd vrsion of th diagonal map [20, B.209]. W not th strngth of th conclusion: th diagonal map is an isomorphism on cofibrant objcts, not just a wak quivalnc.

18 18 V.ANGELTVEIT, A.BLUMBERG, T.GERHARDT, M.HILL, T.LAWSON, AND M.MANDELL Thorm 2.34 ([20, B.209]). Lt H b a normal subgroup of G. Th diagonal map : Φ H X Φ G N G H X is an isomorphism of orthogonal spctra (and in particular a wak quivalnc) whn X is cofibrant in any of th stabl modl structurs on S H, or whn X is a cofibrant objct in Ass H. Along th lins of Proposition 2.19, w also nd th following mor gnral statmnt, which ssntially follows from th argumnt of [20, B.209] using th isomorphism givn in th proof of Proposition 2.19 to start th induction. Thorm Lt G b a finit group, H < G a subgroup, and K G a normal subgroup. Lt X b an orthogonal H-spctrum. Th diagonal map of orthogonal G/K-spctra : N G/K HK/K ΦH K X Φ K N G H X. is an isomorphism of orthogonal spctra (and in particular a wak quivalnc) whn X is cofibrant in any of th stabl modl structurs on S H or whn X is a cofibrant objct in Ass H. W also nd th commutativ ring orthogonal spctrum vrsion of Thorm Thorm Th diagonal map : X Φ G N G X is an isomorphism of orthogonal spctra whn X is a cofibrant commutativ ring orthogonal spctrum. Proof. Th induction in [20, B.209] and monoidality of both sids rducs th statmnt to th cas whn X = (F V B + ) (m) /Σ m whr V is a finit-dimnsional (nonquivariant) innr product spac and B is th disk D n or sphr S n 1 in particular, whn B is a compact Hausdorff spac. In gnral, for a (non-quivariant) orthogonal spctrum T th diagonal map is constructd as follows: for vry (nonquivariant) innr product spac Z, th univrsal proprty of th indxd smash product givs a map of basd G-spacs N G (T (Z)) (N G T )(Ind G Z), which rstricts on th diagonal to a map (2.37) T (Z) (N G T (Ind G Z)) G = (Fix G (N G T ))(Ind G Z), and thn (passing to th lft Kan xtnsion P φ along th fixd point functor φ: JG G J on th right) inducs a map T (Z) (Φ G (N G T ))((Ind G Z) G ) = (Φ G (N G T ))(Z). Whn T is a cll of th form F V B +, th map in (2.37) factors as T (Z) = J (V, Z) B + J G G (Ind G V, Ind G Z) B + (J G (Ind G V, Ind G Z) N G (B) + ) G = (Fix G (N G T ))(Ind G Z). Th first map T (Z) = J (V, Z) B + J G G (IndG V, Ind G Z) B + inducs an isomorphism T P φ (J G G (Ind G V, ) B + ) = J ((Ind G V ) G, ) B +. By passing to quotints, w s that likwis in th cas of intrst, T = X = (F V B + ) (m) /Σ m = FV mb m + /Σ m,

19 TOPOLOGICAL CYCLIC HOMOLOGY VIA THE NORM 19 th diagonal map factors as an isomorphism X P φ (J G G (Ind G V m, ) Σm B m + ) = J ((Ind G V m ) G, ) Σm B m + followd by a map P φ (J G G (Ind G V m, ) Σm B m + ) Φ G (N G X) that is th inducd map on lft Kan xtnsion from a map of J G G -spacs J G G (Ind G V m, ) Σm B m + (J G (Ind G V m, ) Σ G m N G (B m ) + ) G. Thus, it suffics to show that th lattr map is an isomorphism. This amounts to showing that for ach G-innr product spac W, th map J G G (Ind G V m, W ) Σm B m + (J G (Ind G V m, W ) Σ G m N G (B m ) + ) G is a homomorphism, but sinc both sids ar compact Hausdorff spacs, it amounts to showing that th map is a bijction. Th map is clarly an injction. To s that it is a surjction, w not that any non-baspoint x of J G (Ind G V m, W ) N G Σ m N G (B m ) + is rprsntd by a collction of points b h B m (indxd on h G) and isomtris φ h : V m W (indxd on h G) such that h φ h : Ind G V m W is injctiv. Th point x is G-fixd if for vry g G, thr xist an lmnt σ(g) in N G Σ m such that (2.38) g ((φ h ), ( b h )) = ((φ h ) σ(g) 1, σ(g) ( b h )). If w writ σ(g) also in coordinats σ(g) = (σ h (g)), whr (φ h ) σ(g) 1 = (φ h σ h (g) 1 ) and σ(g) ( b h ) = (σ h (g) b h ), thn (2.38) bcoms g φ g 1 h = φ h σ h (g) 1 bg 1 h = σ h (g) b h. for all g, h G, whr w hav writtn h ( ) to dnot th action of h on W (and likwis w us ( ) h blow to dnot th action of h on Ind G V m ). Lt φ h = h φ 1 = φ h σ h (h) 1 b h = σ h (h) b h = b 1, Thn ((φ h ), ( b h )) also rprsnts th lmnt x, with ( b h ) clarly a diagonal lmnt. Sinc (g φ ) h = (g φ g 1 ) h = g φ g 1 h = g g 1 h φ 1 = h φ 1 = φ h, w also hav (φ h ) in th imag of J G G (IndG V m, W ).

20 20 V.ANGELTVEIT, A.BLUMBERG, T.GERHARDT, M.HILL, T.LAWSON, AND M.MANDELL 3. Cyclotomic spctra and topological cyclic homology In this sction, w rviw th dtails of th catgory of p-cyclotomic spctra and th construction of topological cyclic homology (T C). Th diagonal maps that naturally aris in th contxt of th norm go in th opposit dirction to th usual cyclotomic structur maps, and so w also xplain how to construct T C from ths op -cyclotomic spctra. In th following, fix a prim p and a complt S 1 -univrs U Background on p-cyclotomic spctra. In this sction, w brifly rviw th point-st dscription of p-cyclotomic spctra from [5, 4]; w rfr th radr to that papr for mor dtaild discussion. Dfinition 3.1 ([5, 4.5]). A p-prcyclotomic spctrum X consists of an orthogonal S 1 -spctrum X togthr with a map of orthogonal S 1 -spctra t p : ρ pφ Cp X X. Hr ρ p dnots th p-th root isomorphism S 1 S 1 /C p. A p-prcyclotomic spctrum is a p-cyclotomic spctrum whn th inducd map on th drivd functor ρ plφ Cp X X is an F p -quivalnc. A morphism of p-cyclotomic spctra consists of a map of orthogonal S 1 -spctra X Y such that th diagram ρ pφ Cp X ρ pφ Cp Y X Y commuts. Rmark 3.2. A cyclotomic spctrum is an orthogonal spctrum with p-cyclotomic structurs for all prims p satisfying crtain compatibility rlations; s [5, 4.7 8] for dtails. Following [5, 5.4 5], w hav th following wak quivalncs for p-prcyclotomic spctra. Dfinition 3.3. A map of p-prcyclotomic spctra is a wak quivalnc whn it is an F p -quivalnc of th undrlying orthogonal S 1 -spctra. Proposition 3.4 ([5, 5.5]). A map of p-cyclotomic spctra is a wak quivalnc if and only if is a wak quivalnc of th undrlying (non-quivariant) orthogonal spctra Constructing T R and T C from a cyclotomic spctrum. In this sction, w giv a vry rapid rviw of th dfinition of T R and T C in trms of th point-st catgory of cyclotomic spctra dscribd abov. Th intrstd radr is rfrrd to th xcllnt tratmnt in Madsn s CDM nots [28] for mor dtails on th construction in trms of th classical (homotopical) dfinition of a cyclotomic spctrum. For a p-prcyclotomic spctrum X, th collction {X C p n } of (point-st) catgorical fixd points is quippd with functors F, R: X C p n X C p n 1

21 TOPOLOGICAL CYCLIC HOMOLOGY VIA THE NORM 21 for all n, dfind as follows. Th Frobnius maps F ar simply th obvious inclusions of fixd points, and th rstriction maps R ar constructd as th composits X C p n = (ρ p X Cp ) C p n 1 (ρ pω)c pn 1 (ρ pφ Cp X) C p n 1 (tp)c p n 1 X C p n 1, whr th map ω is th usual map from catgorical to gomtric fixd points [29, V.4.3]. Th Frobnius and rstriction maps satisfy th idntity F R = R F. Whn X is fibrant in th F p -modl structur (of Thorm 2.29), w thn dfin T R(X) = holim R X C p n and T C(X) = holim R,F X C p n. In gnral, w dfin T R and T C using a fibrant rplacmnt that prsrvs th p-prcyclotomic structur; such a functor is providd by th main thorms of [5, 5], which construct modl structurs on p-prcyclotomic and p-cyclotomic spctra whr th fibrations ar th fibrations of th undrlying orthogonal S 1 -spctra in th F p -modl structur. Altrnativly, an xplicit construction of a fibrant rplacmnt functor on orthogonal spctra that prsrvs prcyclotomic structurs is givn in [3, 4.6 7]. Proposition 3.5 (cf. [5, 1.4]). A wak quivalnc X Y of p-prcyclotomic spctra inducs wak quivalncs T R(X f ) T R(Y f ) and T C(X f ) T C(Y f ) of orthogonal spctra, whr ( ) f dnots any fibrant rplacmnt functor in p- cyclotomic spctra. Rmark 3.6. W do not yt hav an abstract homotopy thory for multiplicativ objcts in cyclotomic spctra, and th xplicit fibrant rplacmnt functor Q I of [3, 4.6] is lax monoidal but not lax symmtric monoidal. As a consqunc, at prsnt w do not know how to convrt a p-cyclotomic spctrum which is also a commutativ ring orthogonal S 1 -spctrum into a cyclotomic spctrum that is a fibrant commutativ ring orthogonal S 1 -spctrum Op-prcyclotomic spctra. For our construction of T HH basd on th norm (in th nxt sction), th diagonal map X Φ G N G X is in th opposit dirction of th cyclotomic structur map ndd in th dfinition of a cyclotomic spctrum. In th cas whn X is cofibrant (or a cofibrant ring or cofibrant commutativ ring orthogonal spctrum), th diagonal map is an isomorphism and so prsnts no difficulty; in th cas whn X is just of th homotopy typ of a cofibrant orthogonal spctrum, th fact that th structur map gos th wrong way ncssitats som tchnical manuvring in ordr to construct T R and T C. Dfinition 3.7. An op-p-prcyclotomic spctrum X consists of an orthogonal S 1 - spctrum X togthr with a map of orthogonal S 1 -spctra γ : X ρ pφ Cp X. An op-p-cyclotomic spctrum is an op-p-prcyclotomic spctrum whr th structur map is an F p -quivalnc. A map of op-p-prcyclotomic spctra is a map of orthogonal S 1 -spctra that commuts with th structur map. A map of op-pprcyclotomic spctra is a wak quivalnc whn it is an F p -quivalnc of th undrlying orthogonal S 1 -spctra. Not that th dfinition abov uss a condition on th point-st gomtric fixd point functor rathr than th drivd gomtric fixd point functor. Such a dfinition works wll whn w rstrict to thos op-p-cyclotomic spctra X whr th