LEIBNIZ FORMULAS FOR CYCLIC HOMOTOPY FIXED POINT SPECTRA Robert R. Bruner and John Rognes June 2nd 2004 Abstract. We analyze the homotopy fixed point spectrum of a T-equivariant* * com- mutative S-algebra R in homological terms. There is a homological homotop* *y fixed point spectral sequence with E2s,t= H-s (T; Ht(R; Fp)), which converges c* *ondition- ally to the continuous homology Hc*(RhT; Fp) of the homotopy fixed point * *spectrum. We show that there are Dyer-Lashof operations fifflQi acting on this alge* *bra spectral sequence, and that its differentials are completely determined by those o* *riginating on the vertical axis. More surprisingly, we show that for each class x in th* *e E2r-term of the spectral sequence there are 2r other classes in the E2r-term (obtaine* *d mostly by Dyer-Lashof operations on x) that are infinite cycles, i.e., survive to t* *he E1 -term. We apply this to completely determine the differentials in the homologica* *l homo- topy fixed point spectral sequences for the topological Hochschild homolo* *gy spectra R = T HH(B) of many S-algebras, including B = MU, BP , ku, ko and tmf. Si* *milar results apply for all finite subgroups C T, and for the Tate- and homot* *opy orbit spectra. This work is part of a homological approach to calculating topol* *ogical cyclic homology and algebraic K-theory of commutative S-algebras. 1. Introduction By an S-algebra we shall either mean one in the sense of [EKMM97], or a sym- metric ring spectrum in the sense of [HSS00]. For a connective S-algebra B, such as the sphere spectrum S, the complex bordism spectrum MU or the Eilenberg- Mac Lane spectrum of the integers Z, the algebraic K-theory spectrum K(B) can be very well approximated by the topological cyclic homology spectrum T C(B) of [BHM93], by the main theorem of [Du97]. The latter spectrum is obtained from the T-equivariant topological Hochschild homology spectrum X = T HH(B) as a homotopy limit of the fixed point spectra XC , indexed over finite cyclic subgr* *oups C of the circle group T. These fixed point spectra are in turn approximated by * *the homotopy fixed point spectra XhC = F (EC+ , X)C , whose homotopy groups can in principle be computed by the homotopical homotopy fixed point spectral sequence (1.1) E2s,t= H-s (C; ßt(X)) =) ßs+t(XhC ) . ______________ 1991 Mathematics Subject Classification. 19D55, 55P43, 55P91, 55S12, 55T05. Key words and phrases. Homotopy fixed points, Tate spectrum, homotopy orbits* *, commutative S-algebra, Dyer-Lashof operations, differentials, topological Hochschild homolo* *gy, topological cyclic homology, algebraic K-theory. Typeset by AM S-T* *EX 1 2 ROBERT R. BRUNER AND JOHN ROGNES This is derived from the tower of fibrations (with limit XhC ) that arises from* * the equivariant skeleton filtration on the free contractible C-space EC, by applying homotopy groups. Such computations presume a rather detailed knowledge of the homotopy groups ß*(X) of the T-equivariant spectrum in question. For example, [HM03] and [AuR02] deal with the cases when B is the valuation ring in a local number field and the Adams summand in p-complete connective topological K-theory, respectively. In most other cases the spectral sequence (1.1) cannot be calculated, because the homotopy groups ß*(X) are not sufficiently well known. It happens much more frequently that we are familiar with the homology groups H*(X; Fp). Applying mod p homology, rather than homotopy, to the tower of fibrations that approximates XhC leads to a homological homotopy fixed point spectral sequence (1.2) E2s,t= H-s (C; Ht(X; Fp)) =) Hcs+t(XhC ; Fp) . This spectral sequence converges conditionally [Bo99] to the (inverse) limit of* * the resulting tower in homology, which is not H*(XhC ; Fp), but a öc ntinuous" vers* *ion Hc*(XhC ; Fp) of it, for homology does not usually commute with limits. This continuous homology, considered as a comodule over the dual Steenrod al- gebra A* [Mi58], is nonetheless a powerful invariant of XhC . In particular, wh* *en X is bounded below and of finite type there is a strongly convergent spectral seq* *uence (1.3) Es,t2= Exts,tA*(Fp, Hc*(XhC ; Fp)) =) ßt-s(XhC )^p which can be obtained as an inverse limit of Adams spectral sequences [CMP87, 7.1]. Hence the continuous homology does in some sense determine the p-adic homotopy type of XhC . A form of the spectral sequence (1.3) was most notably applied in the proofs by W.H. Lin [LDMA80] and J. Gunawardena [AGM85] of the Segal conjecture for cyclic groups of prime order. The conjecture corresponds to the special case of the discussion above when B = S is the sphere spectrum, so X = T HH(S) = S is the T-equivariant sphere spectrum, which is split [LMS86, II.8] so that XhC * * ' F (BC+ , S) = D(BC+ ). The proven Segal conjecture [Ca84] then tells us that for each p-group C the comparison map XC ! XhC is a p-adic equivalence. The proof of the general (cyclic) case is by reduction to the initial case when C = Cp is* * of prime order, and therefore relies on the theorems of Lin and Gunawardena cited above. In this case, of course, we do not know ß*(X) = ß*(S) sufficiently well, but H*(X; Fp) = Fp is particularly simple. The proof of the theorems of Lin and Gunawardena now amounts to showing that although the natural homomorphism H*(XC ; Fp) ! Hc*(XhC ; Fp) of A*-comodules is not in itself an isomorphism, it does induce an isomorphism of E2-terms upon applying the functor Ext **A*(Fp, -* *). Returning to the general situation, we are therefore interested in studying * *(i) the differentials in the homological homotopy fixed point spectral sequence (1.2) a* *bove, and (ii) the A*-comodule extension questions relating its E1 -term to the abutm* *ent Hc*(XhC ; Fp). There will in general be non-trivial differentials in (1.2), but* * our main Theorem 1.5 below provides a very general and useful collapse result, as is ill* *ustrated by the examples in Section 6. The identification of the A*-comodule structure on the abutment plays a crucial role already in the case X = S, but requires furth* *er LEIBNIZ FORMULAS FOR CYCLIC HOMOTOPY FIXED POINT SPECTRA 3 study beyond that given here, and will be presented in the forthcoming Ph.D. th* *esis of Sverre Lunøe-Nielsen [L-N]. When B is a commutative S-algebra then so is X = T HH(B), and the tower of fibrations with limit XhC is one of commutative S-algebras [EKMM97, IX]. There- fore there are Dyer-Lashof operations acting on the spectral sequence (1.2) in * *this case, rather analogously to the action by Steenrod operations in the Adams spec* *tral sequence of a commutative S-algebra [BMMS86, IV]. In the latter case there are interesting relations between the Adams differentials and the Steenrod operatio* *ns, which propagate early differentials to higher degrees. The initial motivation f* *or the present article was to determine the analogous interaction between the differen* *tials and the Dyer-Lashof operations in the homological homotopy fixed point spectral sequence of a commutative S-algebra, hereafter denoted X = R. However, the analogy with the behavior of differentials in the Adams spectral sequence is mo* *re apparent than real, suggesting neither the survival to E1 of some classes nor * *the method of proof. In particular, there is no analog in the Adams spectral sequen* *ce of our main collapse result, Theorem 1.5. For each finite subgroup C T the homological spectral sequence for RhC is* * an algebra over the corresponding homological spectral sequence for RhT, as outlin* *ed in Section 7, so it will suffice for us to consider the circle homotopy fixed p* *oints RhT and the case C = T of the spectral sequence (1.2). Our first results in Section* *s 2-4 can then be summarized as follows. Theorem 1.4. (a) Let R be a T-equivariant commutative S-algebra. There is a natural A*-comodule algebra spectral sequence E2**= H-* (T; H*(R; Fp)) = P (y) H*(R; Fp) with y in bidegree (-2, 0), which is conditionally convergent to the continuous* * homo- logy Hc*(RhT; Fp) = limnH*(F (S2n+1+, R)T; Fp) of the homotopy fixed point spectrum RhT. (b) There are natural Dyer-Lashof operations fifflQi acting vertically on th* *is ho- mological homotopy fixed point spectral sequence. For each element x 2 E2r0,t Ht(R; Fp) we have the relation d2r(fifflQi(x)) = fifflQi(d2r(x)) for every integer i and ffl 2 {0, 1}. If d2r(x) = yr . ffix with ffix 2 Ht+2r-1* *(R; Fp), the right hand side fifflQi(d2r(x)) is yr . fifflQi(ffix). (c) The classes yn are infinite cycles, so the differentials from the vertic* *al axis E2r0,*propagate to each column by the relation d2r(yn . x) = yn . d2r(x) for all x 2 E2r0,*, 2r 2, n 0. Hence there are isomorphisms E2r** P (y) * *E2r0,* for all 2r 2, modulo y-torsion in filtrations -2r < s 0. For proofs, see Proposition 2.4, Proposition 4.1 and Lemma 4.3. The key idea is to identify the differentials in the homological homotopy fixed point spectr* *al sequence as obstructions to extending equivariant maps, as explained in Section* * 3. Note that the spectral sequence is concentrated in even columns, so all differe* *ntials of odd length (dr with r odd) must vanish. Our main theorem is the following collapse result. 4 ROBERT R. BRUNER AND JOHN ROGNES Theorem 1.5. Let R be a T-equivariant commutative S-algebra, suppose that x 2 Ht(R; Fp) survives to the E2r-term E2r0,t Ht(R; Fp) of the homological homotopy fixed point spectral sequence for R and write d2r(x) = yr . ffix. (a) For p = 2, the 2r classes x2 = Qt(x), Qt+1(x), . . ., Qt+2r-2(x) and Qt+2r-1(x) + xffix all survive to the E1 -term, i.e., are infinite cycles. (b) For p odd and t = 2m even, the 2r classes xp = Qm (x), fiQm+1 (x), . . ., Qm+r-1 (x) and xp-1 ffix all survive to the E1 -term, i.e., are infinite cycles. (c) For p odd and t = 2m - 1 odd, the 2r classes fiQm (x), Qm (x), . . ., fiQm+r-1 (x) and Qm+r-1 (x) - x(ffix)p-1 all survive to the E1 -term, i.e., are infinite cycles. This is proved in Section 5 as our Theorem 5.1. To be perfectly clear, in ca* *se (a) the classes are x2 = Qt(x), Qi(x) for t + 1 i t + 2r - 2, and Qt+2r-1(x) + * *xffix, in case (b) the classes are xp = Qm (x), fifflQi(x) for m + 1 i m + r - 1 a* *nd ffl 2 {0, 1}, and xp-1 ffix, and in case (c) the classes are fifflQi(x) for m * * i m+r -2 and ffl 2 {0, 1}, fiQm+r-1 (x), and Qm+r-1 (x) - x(ffix)p-1 . There are similar extensions of our results to the Tate constructions RtC = [EeC ^ F (EC+ , R)]C and homotopy orbit spectra RhC = EC+ ^C R, but to keep the exposition simple these are also only discussed in Section 7. As applications of our main results, we turn in Section 6 to the study of th* *e alge- braic K-theory spectrum K(MU) which interpolates between K(S) (which is Wald- hausen's A(*), related to high dimensional geometric topology) and K(Z) (which relates to the Vandiver and Leopoldt conjectures, and other number theory), by * *the methods of topological cyclic homology. Hence we must study the fixed- and homo- topy fixed point spectra of the commutative S-algebra R = T HH(MU), for various subgroups C of the circle group T. It is known that H*(MU; Fp) = P (bk | k 1), where P (-) denotes the polynomial algebra over Fp and |bk| = 2k, from which it follows ([MS93, 4.3] or [CS]) that H*(T HH(MU); Fp) = H*(MU; Fp) E(oebk | k 1), where E(-) denotes the exterior algebra over Fp and oe :H*(R; Fp) ! H*+1 (R; Fp) is the degree +1 operator induced by the circle action. Hence the homological homotopy fixed point spectral sequence for T HH(MU)hT begins E2**= P (y) P (bk | k 1) E(oebk | k 1) . There are differentials d2(bk) = y . oebk for all k 1, so by our Theorem 1.4 E4**= P (y) P (bpk| k 1) E(bp-1koebk | k 1) plus some classes (the image of oe) in filtration s = 0. By our Theorem 1.5, t* *he spectral sequence collapses completely at the E4-term, so that Hc*(T HH(MU)hT; Fp) = P (y) P (bpk| k 1) E(bp-1koebk | k 1) LEIBNIZ FORMULAS FOR CYCLIC HOMOTOPY FIXED POINT SPECTRA 5 plus some classes in filtration zero, as an algebra. The identification of the* * A*- comodule extensions remains, for which we refer to the cited Ph.D. thesis [L-N]* *. This provides the input for the inverse limit of Adams spectral sequences (1.3) conv* *erging to ß*(T HH(MU)hT)^p, which approximates the topological version T F (MU) of negative cyclic homology, and which determines the topological cyclic homology * *of MU by a fiber sequence T C(MU) -ß!T F (MU) -R-1--!T F (MU) . The fiber of the cyclotomic trace map K(MU) ! T C(MU) is equivalent to that of K(Z) ! T C(Z), by [Du97], which now is quite well known [Ro02], [Ro03]. Our theorem therefore provides a key input to the computation of K(MU). See Theorem 6.4(a). Similar applications are given for the connective Johnson-Wilson spectra B = BP , for p and n such that these are commutative S-algebras, and the (higher real) commutative S-algebras B = ko and tmf for p = 2. See Section 6. Lastly, we can also show the collapse at E4**of the homological homotopy fixed point spectral sequence for R = T HH(BP ), where BP is the p-local Brown-Peterson S-algebra [BJ02], without making the (presently uncertain) assumption that BP can be realized as a commutative S-algebra. See Theorem 6.4(b). This is possible by the homological approach, since the split surjection H*(MU; Fp) ! H*(BP ; Fp) prevails throughout the homological spectral sequences. 2. A homological spectral sequence Let T C* be the circle group. As our model for a free contractible T-CW complex ET we take the unit sphere S1 C1 with the usual coordinatewise action by T. It has one T-equivariant cell in each even non-negative dimension. The equivariant 2n-skeleton is the odd sphere ET(2n) = S2n+1 Cn+1 , which is obtained from the equivariant (2n - 2)-skeleton ET(2n-2) = S2n-1 Cn by attaching a free T-equivariant 2n-cell T x D2n along the group action map ff :T* * x S2n-1 ! S2n-1 . Hence there is a T-equivariant filtration (2.1) ; S1 . . .S2n-1 S2n+1 . . . with colimit ET, and T-equivariant cofiber sequences S2n-1 ! S2n+1 ! T+ ^ S2n for each n 0. Let X be any spectrum with T-action, i.e., a naively T-equivariant spectrum. The homotopy fixed point spectrum of X is defined as the mapping spectrum XhT = F (ET+ , X)T of T-equivariant based maps from ET+ to X. The filtration (2.1) of ET = S1 induces a tower of fibrations (2.2) . .!.F (S2n+1+, X)T ! F (S2n-1+, X)T ! . .!.F (S1+, X)T = X ! * 6 ROBERT R. BRUNER AND JOHN ROGNES with the homotopy fixed point spectrum as its (homotopy) limit XhT ' holimnF (S2n+1+, X)T . The cofiber sequences above induce (co-)fiber sequences of spectra -2n X = F (T+ ^ S2n, X)T ! F (S2n+1+, X)T ! F (S2n-1+, X)T for each n 0. We now place F (S2n-1+, X)T in the two filtrations s = -2n and s = -2n + 1, * *for each n 0. Hence we obtain a chain of cofiber sequences of spectra: F (S2n+1+, X)T______//Fh(S2n-1+,hX)T____//_FQ(S2n-1+, X)T___//_F (S2n-3+, X)T QQ hhQQQ hhQQQ QQQQ | QQQQ | QQQQ | QQQQ | QQQQ | QQQQ | Q fflffl| QQQQ fflffl| Q fflffl| . . . -2n X * -2n+2 X Here the filtrations -2n - 1 s -2n + 2 are displayed. Next we apply mod p homology to this chain of cofiber sequences, to obtain a homologically indexed unrolled exact couple [Bo99, 0.1] with As,t= Hs+t(F (S2n-1+, X)T; Fp) for s = -2n and s = -2n + 1, and Es,t= Hs+t( -2n X; Fp) = Ht(X; Fp) for s = -2n and zero otherwise. Here Es,t= E1s,t= E2s,t. The E2-term of the associated spectral sequence can be expressed as the group cohomology E2s,t= H-s (T; Ht(X; Fp)) ~=H-s (T; Fp) Ht(X; Fp) of the circle group T, acting trivially on H*(X; Fp) as it must since T is path connected. We have H*(T; Fp) = P (y) with y in degree 2, where P (-) denotes the polynomial algebra, so E2**= P (y) H*(X; Fp) with y in bidegree (-2, 0) and Ht(X; Fp) in bidegree (0, t). See [GM95, 14.2] f* *or a discussion of this and related spectral sequences. Since As = 0 for s 0 we have A1 = colims As = 0. Therefore the spec- tral sequence is conditionally convergent, by [Bo99, 5.10], in this case to the* * limit A-1 = limsAs. Indexing the limit system by n in place of s, it can be written * *as (2.3) Hc*(XhT ; Fp) = limnH*(F (S2n+1+, X)T; Fp) , which we call the continuous homology of XhT . The spectral sequence will be strongly convergent to this target if the criterion RE1**= 0 of [Bo99, 7.4] is * *satisfied, for which it suffices that in each bidegree (s, t) we have Ers,t= E1s,tfor some* * finite r = r(s, t). LEIBNIZ FORMULAS FOR CYCLIC HOMOTOPY FIXED POINT SPECTRA 7 Proposition 2.4. There is a natural spectral sequence E2**= H-* (T; H*(X; Fp)) = P (y) H*(X; Fp) with y in bidegree (-2, 0), which is conditionally convergent to the continuous* * homo- logy Hc*(XhT ; Fp). We call this the homological homotopy fixed point spectral sequence. If H*(X; Fp) is finite in each degree, or the spectral sequence colla* *pses at a finite stage, then the spectral sequence is strongly convergent. Remark 2.5. Since homology does usually not commute with the formation of limits, the canonical map H*(XhT ; Fp) ! Hc*(XhT ; Fp) is usually not an isomorphism. The Segal conjecture provides striking examples * *of this phenomenon. As noted in the introduction, it is rather more traditional to apply the hom* *otopy group functor ß* to the tower of fibrations (2.2), to obtain an unraveled exact couple and a conditionally convergent (homotopical) homotopy fixed point spectr* *al sequence E2s,t= H-s (T; ßt(X)) =) ßs+t(XhT ) . However, this is not the spectral sequence that we will consider. Recent work * *by Ch. Ausoni and the second author [AuR02, x4], as well as current work by S. Lun* *øe- Nielsen (and the second author) [L-N] support the assertion that the homological spectral sequence is an interesting object. 3. Differentials We now make the differentials in the homological homotopy fixed point spectr* *al sequence more explicit, as obstructions to extending equivariant maps. Consider a class x 2 Ht(X; Fp), represented at the E2-term of the homological spectral sequence in bidegree (0, t). Let H = HFp be the mod p Eilenberg-Mac La* *ne spectrum. Then x can be represented as a non-equivariant map St ! H ^ X, or equivalently as a T-equivariant map x: S1+^ St ! H ^ X . Here T acts on S1+(freely off the base point) and X, but not on St or H. The condition that x 2 E20,t= Ht(X; Fp) survives to the E2r-term, i.e., that* * all differentials d2(x), . .,.d2r-2(x) vanish, is equivalent to x being in the imag* *e from Ht(F (S2r-1+, X)T; Fp) under the map induced by restriction along S1+ S2r-1+. This is in turn equivalent to the existence of a T-equivariant extension x0:S2r-1+^ St ! H ^ X of x along S1+ S2r-1+, in view of the natural equivariant equivalence H ^ F (S2r-1+, X) -'!F (S2r-1+, H ^ X) . (To establish this equivalence, note that the finite T-CW complex S2r-1+admits a T-equivariant Spanier-Whitehead dual. We are considering maps from free T-CW 8 ROBERT R. BRUNER AND JOHN ROGNES complexes into these spectra, so only the naive notion of a T-equivariant equiv* *alence is required.) Suppose that x 2 E2r0,thas survived to the E2r-term, so that such a T-equiva* *riant extension x0 exists. Then the differential d2r(x) 2 E2r-2r,t+2r-1 is the obstruction to extending x0 further along S2r-1+ S2r+1+to an equivariant map x00:S2r+1+^ St ! H ^ X . We put the obvious right adjoints of these maps together in a diagram, as below. S1+MM | MMMMxM | MMM ff+ fflffl|x0 MM&& (T x S2r-1 )+______//S2r-1+____//_F8(St,8H ^ X) r | | r r | | r x00 fflffl| fflffl|r (T x D2r)+ _______//S2r+1+ But since S2r+1+is obtained from S2r-1+by adjoining a free T-cell along the act* *ion map ff :T x S2r-1 ! S2r-1 , the obstruction to such an extension is precisely the obstruction to extending the equivariant map x0O ff+ from (T x S2r-1 )+ o* *ver (TxD2r)+ . Equivalently, the obstruction is that of finding a homotopy to a con* *stant map of the non-equivariant map ~x:S2r-1 ! F (St, H ^ X) given by regarding x0 as a non-equivariant map, and then restricting away from the disjoint base poin* *t. Its left adjoint again is then a map ~x:S2r-1 ^ St ! H ^ X . We summarize: Lemma 3.1. Let x 2 E2r0,t Ht(X; Fp) be represented by a T-equivariant map x :S1+^St ! H ^X, which extends to an equivariant map x0: S2r-1+^St ! H ^X. Then d2r(x) = yr .x~, where ~x2 Ht+2r-1(X; Fp) is represented by x0 considered * *as a non-equivariant map, restricted to the stable summand S2r-1 ^St of S2r-1+^St. The extended map x0 represents a class in the homology of F (S2r-1+, X)T, and considering x0 as a non-equivariant map amounts to following the map ': F (S2r-1+, X)T ! F (S2r-1+, X) that forgets the T-equivariance. There is a canonical map :DS2r-1+^ X ! F (S2r-1+, X) where DS2r-1+ = F (S2r-1+, S) is the functional dual of S2r-1+, and is a weak equivalence by Spanier-Whitehead duality, since S2r-1+ is a finite CW complex. See [LMS86, xIII.1]. Hence there is a natural isomorphism :H-* (S2r-1 ; Fp) H*(X; Fp) ! H*(F (S2r-1+, X); Fp) where we have identified H*(DS2r-1+; Fp) with H~-*(S2r-1+; Fp) = H-* (S2r-1 ; F* *p). We write H*(S2r-1 ; Fp) = E('2r-1), where '2r-1 is the canonical generator in degree (2r - 1) and E(-) denotes the exterior algebra. LEIBNIZ FORMULAS FOR CYCLIC HOMOTOPY FIXED POINT SPECTRA 9 Proposition 3.2. The composite map H*(F (S2r-1+, X)T; Fp)-'*!H*(F (S2r-1+, X); Fp) - * -* 2r-1 ~= H (S ; Fp) H*(X; Fp) takes any class x0 that is mapped to x 2 E2r0,t Ht(X; Fp) by the restriction m* *ap H*(F (S2r-1+, X)T; Fp) ! H*(F (S1+, X)T; Fp) = H*(X; Fp) to the sum ( -1*'*)(x0) = 1 x + '2r-1 ffix , where d2r(x) = yr.ffix in E2r-2r,t+2r-1. Suppressing the power of y we may some* *what imprecisely write this formula as '*(x) = 1 x + '2r-1 d2r(x) . The case r = 1 says d2(x) = y . oex, and follows e.g. from [Ro98, 3.3]. Proof. This is really a corollary to Lemma 3.1, but for the observation that the restriction of the non-equivariant x0 to the subspace St S2r-1+^ St equals the restriction of the non-equivariant x to the same subspace St S1+^ St, which in turn corresponds to x 2 E2r0,tunder the identification H*(F (S1+, X)T; Fp) = H*(X; Fp). Remark 3.3. Lemma 3.1 says that the differential in the homotopy fixed point spectral sequence is essentially the T-equivariant root invariant for H ^ X. A corresponding description of the (Mahowald) C2-equivariant root invariant for S can be found in [BG95, 2.5]: Let Sn+kff denote the C2-equivariant sphere that is the one point compactification of Rn Rk(-1), where C2 acts trivially on Rn and by negation on Rk(-1). Given a non-equivariant (stable) map x : Sn ! S0, let x0 : Sn+kff ! S0 be a C2-equivariant extension of x with k maximal. Then the C2-equivariant root invariant of x contains the non-equivariant map x0: Sn+k !* * S0 underlying x0. 4. Commutative S-algebras Now suppose that X = R is a (naively) T-equivariant commutative S-algebra, i.e., a commutative S-algebra with a continuous point-set level action by the c* *ircle group T. We shall be concerned with the homotopy fixed points of R, rather than its genuine fixed points, so only this weak notion of an equivariant spectrum w* *ill be needed [GM95, x1]. Our principal example is R = T HH(B), the topological Hochschild homology spectrum of another commutative S-algebra B. The cyclic structure on topological Hochschild homology then provides the relevant T-action [EKMM97, IX]. In this situation the homotopy fixed point spectrum RhT = F (ET+ , R)T is al* *so a commutative S-algebra. Writing ~: R^R ! R for the T-equivariant multiplication map of R, the corresponding multiplication map for RhT is given by the composite F (ET+ , R)T ^ F (ET+ , R)T-^! F (ET+ ^ ET+ , R ^ R)T -~#--#-!F (ET T + , R) . 10 ROBERT R. BRUNER AND JOHN ROGNES Here ^ smashes together two T-equivariant maps 1 ET+ ! R, and considers the resulting (T x T)-equivariant map as a T-equivariant map by the diagonal action. The map ~# composes on the left by ~: R ^ R ! R, while the map # composes on the right by the space level diagonal map : ET+ ! ET+ ^ ET+ . Since ~ is commutative and is cocommutative, the resulting multiplication on RhT is also strictly commutative. Writing j :S ! R for the T-equivariant unit map of R, the corresponding unit map for RhT is the composite S ! F (ET+ , S)T -j#-!F (ET+ , R)T . Here the definition of the first map relies on the fact that T acts trivially o* *n S. Commutative S-algebras are E1 ring spectra, and are in particular also H1 ring spectra. Hence there are Dyer-Lashof operations Qi acting on their mod p homology algebras [BMMS86, xIII.1]. Recall that Qi is a natural transformation Qi: Ht(R; Fp) ! Ht+iq(R; Fp) for all integers t, where q = 2p - 2. We also include their composites fiQi wit* *h the homology Bockstein operation fi :Ht(R; Fp) ! Ht-1(R; Fp) as generators of the Dyer-Lashof algebra. For p = 2 the standard notation is to write Q2i and Q2i-1 for the operations that would otherwise be called Qi and fiQi, respectively. The homological homotopy fixed point spectral sequence of Proposition 2.4 is derived by applying homology to the tower (2.2). Now that X = R, each spectrum F (S2n+1+, R)T is a commutative S-algebra, for the same reasons as we just indi- cated for RhT, and each fibration in the tower is a map of commutative S-algebr* *as. Therefore the spectral sequence is one of commutative (A*-comodule) algebras ov* *er the Dyer-Lashof algebra. We can make this action quite explicit, as follows. Proposition 4.1. Let R be a T-equivariant commutative S-algebra, and let Er** be its homological homotopy fixed point spectral sequence. Then for each eleme* *nt x 2 E2r0,t Ht(R; Fp) we have the relation d2r(fifflQi(x)) = fifflQi(d2r(x)) , for every integer i and ffl 2 {0, 1}. Here the right hand side should be inter* *preted as follows. If d2r(x) = yr . ffix with ffix 2 Ht+2r-1(R; Fp) then fifflQi(d2r(* *x)) = yr . fifflQi(ffix). The case r = 1 also appears as [AnR, 5.9]. Proof. Let x 2 Ht(R; Fp) and suppose that x survives to the E2r-term. Then there exists an extension x0 2 Ht(F (S2r-1+, R)T; Fp) of x over the restriction map, * *and z0 = fifflQi(x0) is an extension of z = fifflQi(x) over the same map, by natura* *lity. The maps ' and from Proposition 3.2 are both maps of commutative S-algebras, and therefore induce (A*-comodule) algebra homomorphisms '* and * that commute with the Dyer-Lashof operations. Thus (4.2) ( -1*'*)(fifflQi(x0)) = 1 fifflQi(x) + '2r-1 ffiz LEIBNIZ FORMULAS FOR CYCLIC HOMOTOPY FIXED POINT SPECTRA 11 where d2r(fifflQi(x)) = yr . ffiz, is equal to fifflQi(( -1*'*)(x0)) = fifflQi(1 x + '2r-1 ffix) where d2r(x) = yr . ffix. Now the Dyer-Lashof operations on the homology of the smash product DS2r-1+^ R are given by a Cartan formula, and on the tensor factor H*(DS2r-1+; Fp) ~=H-* (S2r-1 ; Fp) the operation fifflQi corresponds to the Ste* *enrod operation fifflP -i, by [BMMS86, III.1.2]. But the latter operations all act tr* *ivially on H*(S2r-1 ; Fp), except for P 0= 1, so the Cartan formula gives fifflQi(1 x + '2r-1 ffix) = 1 fifflQi(x) + '2r-1 fifflQi(ffi* *x) . Identifying this with (4.2) and comparing the coefficients of '2r-1 we obtain t* *he identity ffiz = fifflQi(ffix) , as claimed. Any spectrum X can be considered as a module over the sphere spectrum S, and any action by T on X may be taken to be in the category of S-modules. It follows that the homological homotopy fixed point spectral sequence for X is a module o* *ver the corresponding spectral sequence for S, which is an algebra spectral sequence by our previous remarks (since S is a commutative S-algebra). In fact the homological homotopy fixed point spectral sequence for S is part* *ic- ularly simple, since H*(S; Fp) = Fp is concentrated in degree 0, so the spectral sequence collapses to E2**= P (y) , which is concentrated on the horizontal axis. Hence each power of y is an infin* *ite cycle, i.e., dr(yn ) = 0 for all r and n. Since the spectral sequence for X is a module over the one for S, the Leibniz formula for the module pairing immediately yields the following result. Lemma 4.3. Let X be any T-equivariant spectrum. The differentials in the ho- mological homotopy fixed point spectral sequence converging to Hc*(XhT ; Fp) sa* *tisfy the relation d2r(yn . x) = yn . d2r(x) for all x 2 E2r0,* H*(X; Fp), 2r 2 and n 0. Hence the spectral sequence is completely determined by the differentials that originate on the vertical axis. Remark 4.4. A proof by induction on r shows that each class in E2r-2n,thas the form yn . x for a class x 2 E2r0,t Ht(X; Fp). The E2r-term may therefore only contain y-torsion of height strictly less than r, and concentrated in filt* *rations -2r < s 0. In Section 7 we shall remark on an analogous homological Tate spectral sequence, where P (y) is replaced by P (y, y-1 ) and the issue of y-to* *rsion classes becomes void. 5. Infinite cycles The Dyer-Lashof operations satisfy instability conditions [BMMS86, III.1.1] * *that are in a sense dual to those of the Steenrod operations. For a class x 2 Ht(R; * *Fp) the 12 ROBERT R. BRUNER AND JOHN ROGNES lowest nontrivial operation is Qt(x) = x2 when p = 2, Qm (x) = xp when p is odd and t = 2m is even, and fiQm (x) when p is odd and t = 2m - 1 is odd. Similarly, the lowest nontrivial operation on ffix 2 Ht+2r-1(R; Fp) with d2r(x) = yr . ffi* *x is Qt+2r-1(ffix) = (ffix)2 when p = 2, fiQm+r (ffix) when p is odd and t = 2m is e* *ven, and Qm+r-1 (ffix) = (ffix)p when p is odd and t = 2m - 1 is odd. Thus there is in each case a sequence of (2r - 1) Dyer-Lashof operations fifflQi whose action* * on x can be nontrivial, but whose action on ffix must be trivial. By Proposition * *4.1, this sequence of operations on x will survive past the E2r-term, at least to the E2r+2-term. It is the main point of the present article to show that these clas* *ses, and one more öc mpanion class", then in fact go on indefinitely to survive to t* *he E1 -term, i.e., are infinite cycles! Theorem 5.1. Let R be a T-equivariant commutative S-algebra, suppose that x 2 Ht(R; Fp) survives to the E2r-term E2r0,t Ht(R; Fp) of the homological homotopy fixed point spectral sequence for R, and write d2r(x) = yr . ffix. (a) For p = 2, the 2r classes x2 = Qt(x), Qt+1(x), . . ., Qt+2r-2(x) and Qt+2r-1(x) + xffix all survive to the E1 -term, i.e., are infinite cycles. (b) For p odd and t = 2m even, the 2r classes xp = Qm (x), fiQm+1 (x), . . ., Qm+r-1 (x) and xp-1 ffix all survive to the E1 -term, i.e., are infinite cycles. (c) For p odd and t = 2m - 1 odd, the 2r classes fiQm (x), Qm (x), . . ., fiQm+r-1 (x) and Qm+r-1 (x) - x(ffix)p-1 all survive to the E1 -term, i.e., are infinite cycles. Proof. The argument proceeds by considering a universal example. Recall that a class x 2 E2r0,tis represented by a T-equivariant map x: S1+^ St ! H ^ R that admits an equivariant extension x0:S2r-1+^ St ! H ^ R. Let X = Dp(S2r-1+^ St) = E p n p (S2r-1+^ St)^p be the p-th extended power of S2r-1+^ St. Somewhat abusively, we write H~*(S2r-1+^ St; Fp) = Fp{x, ffix} with |x| = t * *and |ffix| = t + 2r - 1. Then the homology of the p-th extended power is H*(X; F2) = F2{xffix, Qi(x) | i t, Qi(ffix) | i t + 2r - 1} for p = 2, H*(X; Fp) = Fp{xp-1 ffix, fifflQi(x) | i m + ffl, fifflQi(ffix) | i * *m + r} for p odd and t = 2m even, and H*(X; Fp) = Fp{x(ffix)p-1 , fifflQi(x) | i m, fifflQi(ffix) | i m + r -* * 1 + ffl} LEIBNIZ FORMULAS FOR CYCLIC HOMOTOPY FIXED POINT SPECTRA 13 for p odd and t = 2m - 1 odd. Throughout i is an integer and ffl 2 {0, 1}. The equivariant extension x0 induces an equivariant map Dp(x0): X = Dp(S2r-1+^ St) ! Dp(H ^ R) . The commutative S-algebra structures on H and R combine to form one on H ^ R, and the associated H1 structure includes, in particular, a T-equivariant struc* *ture map ,p :Dp(H ^ R) ! H ^ R that extends the p-fold multiplication map on H ^R. Taken together, these produ* *ce an equivariant map 0) 1^,p ~^1 H ^ Dp(S2r-1+^ St) -1^Dp(x-----!H ^ Dp(H ^ R) ---! H ^ H ^ R --! H ^ R , where ~ is the multiplication on H. Applying homotopy we have a homomorphism (5.2) H*(X; Fp) = H*(Dp(S2r-1+^ St); Fp) ! H*(R; Fp) which, by definition, takes the classes generating H*(X; Fp) to the classes wit* *h the same names in H*(R; Fp). Now X = Dp(S2r-1+^ St) is a T-equivariant retract of the free commutative S-algebra ` P ' Dj(S2r-1+^ St) j 0 on the space S2r-1+^ St, so the homological homotopy fixed point spectral seque* *nce for X is a direct summand of the one for P . Thus the formula from Proposition * *4.1 for the d2r-differentials in the spectral sequence for P is also applicable in* * the spectral sequence for X. Now consider the homological homotopy fixed point spectral sequence for X = Dp(S2r-1+^ St), first for p = 2 and then for p odd. We shall show in each case that the 2r classes in E2r0,* H*(X; Fp), with names as listed in the statement* * of the theorem, are infinite cycles. By naturality of the homotopy fixed point spe* *ctral sequence with respect to the map H ^ X ! H ^ R from (5.2), it follows that the * *2r target classes listed in E2r0,* H*(R; Fp) are also infinite cycles. This will * *complete the proof of the theorem. (a) Let p = 2. The homological homotopy fixed point spectral sequence for X has E2**= P (y) F2{xffix, Qi(x) | i t, Qi(ffix) | i t + 2r - 1} and nontrivial differentials d2r(xffix) = yr . (ffix)2 and d2r(Qi(x)) = yr . Qi(ffix) for all i t + 2r - 1, together with their y-multiples. This leaves E2r+2**= P (y) F2{Qi(x) | t i < t + 2r - 1, Qt+2r-1(x) + xffix} 14 ROBERT R. BRUNER AND JOHN ROGNES yr . Qt+2r(ffix)kkVV ... VVV VVVV d2rVVVVV VVVVVV VV yr . (ffix)2kkWWWmm_______________~Qt+2r(x)______________________* *_________________________________________________________________________@ WWWWWW_________d2r__________________________________* *_________________________________________________________________________@ WWWW _____________________________________* *_________________________________________________________________________@ 2rWWWW WWWW ____________________________* *_________________________________________________________________________@ d W ____________________* *_________________________________________________________________________@ Qt+2r-1(x) ~xffix Qt+2r-2(x) .. . Qt(x) = x2 Figure 1. The case p = 2. plus some y-torsion classes from E2**in filtrations -2r < s 0. Hence there a* *re no classes remaining in the entire quadrant with filtration s -2r and vertical degree * > |xffix| = 2t + 2r - 1. All further differentials on the classes in E* *2r+20,*on the vertical axis land in this zero region, since already E20,*starts in degree* * 2t with the lowest class Qt(x) = x2. Thus all further differentials from the vertical a* *xis are zero, and the spectral sequence collapses at E2r+2**= E1**. (b) Let p be odd and t = 2m even. The homological homotopy fixed point spectral sequence for X has E2**= P (y) Fp{xp-1 ffix, fifflQi(x) | i m + ffl, fifflQi(ffix) | i * * m + r} and nontrivial differentials d2r(fifflQi(x)) = yr . fifflQi(ffix) for i m + r. This leaves E2r+2**= P (y) Fp{xp-1 ffix, Qi(x) | m + ffl i < m + r} plus some y-torsion classes in filtrations -2r < s 0. Hence there are no clas* *ses left in the region where s -2r and the vertical degree is * > |Qm+r-1 (x)|. Now, x was also a class in the E2r-2 -term, with d2r-2(x) = 0, so by inducti* *on over r we may assume (by naturality from the case of (r - 1)) that the classes fifflQi(x) with m + ffl i < m + (r - 1) are infinite cycles. This leaves the* * three classes xp-1 ffix, fiQm+r-1 (x) and Qm+r-1 (x) in E2r+20,*that are not y-torsio* *n, and could therefore imaginably support a differential after d2r. But the first two * *classes LEIBNIZ FORMULAS FOR CYCLIC HOMOTOPY FIXED POINT SPECTRA 15 yr . Qm+r (ffix)kkWW ... WWWW WWWWd2rWWWW WWWWWW WWW yr . fiQm+r (ffix)kkWW Qm+r (x) WWWWWWWd2rWWWWW WWWWWW WW fiQm+r (x) Qm+r-1 (x) fiQm+r-1 (x) .. p-1 . x ffix Qm (x) = xp Figure 2. The case p odd and t = 2m even. fiQm+r-1 (x) and Qm+r-1 (x) are so close to the horizontal edge of the vanishing region that all differentials after d2r must vanish on these classes. The third class xp-1 ffix has odd degree, so an even length differential on * *it must land in an even degree. The only even degree classes in filtrations s -2r are* * the y-multiples of Qi(x) for m i < m+r, of which Qm (x) = xp is in lower degree t* *han that of xp-1 ffix. The remaining possible target classes Qi(x) for m < i < m +* * r all have nontrivial Bockstein images fiQi(x), but fi(xp-1 ffix) = 0 in H*(X; Fp* *). Therefore, by naturality of the differential with respect to the Bockstein oper* *ation, all of these targets for a differential on xp-1 ffix are excluded. Thus also xp* *-1 ffix is an infinite cycle. (c) Let p be odd and t = 2m - 1 odd. The homological homotopy fixed point spectral sequence for X has E2**= P (y) Fp{x(ffix)p-1 , fifflQi(x) | i m, fifflQi(ffix) | i m + r* * - 1 + ffl} and nontrivial differentials d2r(x(ffix)p-1 ) = yr . (ffix)p and d2r(fifflQi(x)) = yr . fifflQi(ffix) for i m + r - 1 + ffl. This leaves E2r+2**= P (y) Fp{fifflQi(x) | m i < m + r - 1 + ffl, Qm+r-1 (x) - x(ffix* *)p-1 } 16 ROBERT R. BRUNER AND JOHN ROGNES yr . fiQm+r (ffix)kkWW ... WWW W WWWd2rWWWWW WWWWWW WW yr . (ffix)pkkWWWnn________________fiQm+r_(x)_______________________* *_________________________________________________________________________@ WWWWWW___________d2r___________________________________* *_________________________________________________________________________@ WWWWW _______________________________________* *_________________________________________________________________________@ 2r WW WWWWWW _____________________________* *_________________________________________________________________________@ d W ___________________* *____________________________________________ Qm+r-1 (x) x(ffix)p-1 fiQm+r-1 (x) .. . fiQm (x) Figure 3. The case p odd and t = 2m - 1 odd. plus y-torsion classes in filtrations -2r < s 0. Hence there are no classes l* *eft in the region where s -2r and the vertical degree is * > |Qm+r-1 (x)|. Again considering x as a class in E2r-20,*and using induction on r we may as* *sume that the classes fifflQi(x) for m i < m + r - 2 + ffl and Qm+r-2 (x) - x(ffi0* *x)p-1 are infinite cycles. Here ffi0x is defined by d2r-2(x) = yr-1 . ffi0x. The fa* *ct that d2r-2(x) = 0 gives ffi0x = 0, so in fact all the classes fifflQi(x) for m i <* * m + r - 1 in E2r+2**are infinite cycles. This leaves only the two classes fiQm+r-1 (x) and Qm+r-1 (x) - x(ffix)p-1 , * *but these are so close to the horizontal border of the vanishing region that all di* *fferen- tials after d2r must be zero on them. 6. Examples Our Theorem 5.1 has applications to the homological homotopy fixed point spe* *c- tral sequence for the commutative S-algebra R = T HH(B) given by the topological Hochschild homology of a commutative S-algebra B. The T-homotopy fixed point spectrum T HH(B)hT is closely related to the topological model T F (B) for the negative cyclic homology of B, which in turn is very close to the topological c* *yclic homology T C(B) [BHM93] and algebraic K-theory K(B) of B [Du97]. These spec- tral sequences therefore have significant interest. First consider the connective Johnson-Wilson spectra B = BP , for some prime p and integer 0 m < 1. So ß*BP = BP*=(vn | n m) , LEIBNIZ FORMULAS FOR CYCLIC HOMOTOPY FIXED POINT SPECTRA 17 where BP* = Z(p)[vn | n 1] and v0 = p, and æ P (~,2, . .,.~,2, ~, , . .). for p = 2, H*(BP ; Fp) = 1 m m+1 P (~,k| k 1) E(~øk| k m) for p odd. The latter is a sub-algebra of the dual Steenrod algebra A* = H*(HFp; Fp) [Mi58* *]. Suppose that p and m are such that BP admits the structure of a commutative S-algebra. This is so at least for m 2 {0, 1, 2}, when BP <-1> = HF* *p, BP <0> = HZ(p) and BP <1> = `, respectively, where ` is the Adams summand of p-local connective topological K-theory ku(p). When p = 2, ` = ku(2). Then the Bökstedt spectral sequence E2**= HH*(H*(B; Fp)) =) H*(T HH(B); Fp) has E2-term æ H (BP ; F ) E(oe,~2, . .,.oe,~2, oe,~ , . .).for p = 2, E2**= * 2 1 m m+1 H*(BP ; Fp) E(oe,~k| k 1) (oe~øk| k m) for p odd. For x 2 H*(B; Fp), oex 2 HH1(H*(B; Fp)) is represented by the Hochschild 1- cycle 1 x. The operator oe is a differential (oe2 = 0) and a graded derivati* *on (oe(xy) = xoe(y) + (-1)|y|oe(x)y). Here (-) denotes the divided power algebra. For p odd, Bökstedt found differentials dp-1 (flj(oe~øk)) = oe,~k+1. flj-p(oe~øk) for j p, and in all cases the spectral sequence collapses at the Ep-term. So æ H (BP ; F ) E(oe,~2, . .,.oe,~2, oe,~ , . .). for p = 2, E1**= * 2 1 m m+1 H*(BP ; Fp) E(oe,~1, . .,.oe,~m) Pp(oe~øk| k m)for p odd. Here Pp(-) denotes the truncated polynomial algebra of height p. If BP , and thus T HH(BP ), is a commutative S-algebra, then (oe,~k)2 = oe,~k+1for p = 2 and (oe~øk)p = oe~øk+1for p odd, so H*(T HH(BP ); Fp) æ H (BP ; F ) E(oe,~2, . .,.oe,~2) P (oe,~ fo)r p = 2, = * 2 1 m m+1 H*(BP ; Fp) E(oe,~1, . .,.oe,~m) P (oe~øm)for p odd. For more references and details on the calculation up to this point, see [AnR, * *x5]. We now consider the homological homotopy fixed point spectral sequence for R = T HH(B). It starts with E2**= P (y) H*(T HH(B); Fp) and by Lemma 3.1 it has first differentials d2(x) = y . oex for all x 2 H*(T HH(B); Fp). Here oex 2 Ht+1(T HH(B); Fp) is the image of x s1 2 Ht(T HH(B); Fp) H1(T; Fp) under the circle action map ff :T HH(B) ^ T+ ! T HH(B) , where s1 2 H1(T; Fp) is the canonical generator. By Lemma 4.3 we have similar differentials d2(yn . x) = yn+1 . oex for all n 0. Hence we can find the columns of E4**in the homological homotopy fixed point spectral sequence by passing to the homology of E20,*= H*(T HH(B); Fp) with respect to the operator oe, at least to the left of the vertical axis. 18 ROBERT R. BRUNER AND JOHN ROGNES Proposition 6.1. The homological homotopy fixed point spectral sequence for R = T HH(B) with B = BP , for p and m such that B is a commutative S- algebra, collapses after the d2-differentials, with the following E1 -term: (a) For p = 2, E1**= P (y) P (~,41, . .,.~,4m, ~,2m+1, ,0m+2, . .). E(~,21oe,~21, . * *.,.~,2moe,~2m) plus some classes in filtration s = 0, where ,0k+1= ~,k+1+ ~,koe,~kfor k m + * *1. (b) For p odd, E1**= P (y) P (~,pk| 1 k m) P (~,k+1| k m) E(øk0+1| k m) E(~,p-1koe,~k| 1 k m) plus some classes in filtration s = 0, where øk0+1= ~øk+1- ~øk(oe~øk)p-1 for k * * m. Proof. (a) For B = BP and p = 2 we have E20,*= H*(T HH(BP ); F2) = P (~,21, . .,.~,2m, ~,m+1, . .). E(oe,~21, . .,.oe,~2m)* * P (oe,~m+1) . Here oe :~,2k7! oe,~2kfor 1 k m and oe :~,k+17! oe,~k+1 for k m. We have oe,~k+1= (oe,~k)2 for k m + 1. So the squares (~,2k)2 = ~,4kand ~,2m+1, as well as the companion classes de* *fined by ,0k+1= ~,k+1+ ~,koe,~k for k m + 1, are d2-cycles, while E(~,2k, oe,~2k) has homology E(~,2koe,~2k) * *for each k, and E(~,m+1) P (oe,~m+1) has homology F2. Hence the homological spectral sequence has E4**= P (y) P (~,41, . .,.~,4m, ~,2m+1, ,0m+2, . .). E(~,21oe,~21, . * *.,.~,2moe,~2m) plus the image of oe in filtration s = 0. By our Theorem 5.1(a) applied to the classes x = ~,2kfor 1 k m, in even degree t = |x|, the classes x2 = ~,4kand Qt+1(x) + xoex = ~,2koe,~2kare infinit* *e cycles, for Qt+1(~,2k) = 0 by the Cartan formula. Similarly, by Theorem 5.1(a) applied to the classes x = ~,kfor k m + 1, in* * odd degree t = |x|, the classes x2 = ~,2kand Qt+1(x) + xoex = ~,k+1+ ~,koe,~k= ,0k+* *1are infinite cycles. For Qt+1(~,k) = ~,k+1by [BMMS86, III.2.2 and I.3.6]. The extra classes in filtration s = 0 are y-torsion, hence infinite cycles. * *Therefore the E4-term above is generated as an algebra by infinite cycles, so the homolog* *ical spectral sequence collapses at this stage. (b) For B = BP and p odd we have E20,*= H*(T HH(BP ); Fp) = P (~,k| k 1) E(~øk| k m) E(oe,~k| 1 k m) P (oe~øm* *) . Here oe :~,k7! oe,~kfor 1 k m, oe :~,k+17! 0 for k m and oe :~øk7! oe~økf* *or k m. We have oe~øk+1= (oe~øk)p for k m. LEIBNIZ FORMULAS FOR CYCLIC HOMOTOPY FIXED POINT SPECTRA 19 So the p-th powers ~,pkfor 1 k m, the classes ~,k+1 for k m, and the companion classes defined by øk0+1= ~øk+1- ~øk(oe~øk)p-1 for k m, are d2-cycles, while Pp(~,k) E(oe,~k) has homology E(~,p-1koe,~k) * *for each 1 k m, and E(~øm) P (oe~øm) has homology Fp. Hence the homological spectral sequence has E4**= P (y) P (~,pk| 1 k m) P (~,k+1| k m) E(øk0+1| k m) E(~,p-1koe,~k| 1 k m) plus some classes in filtration s = 0. Applying our Theorem 5.1(b) to the classes x = ~,kfor 1 k m, in even deg* *ree t = |x|, the classes xp = ~,pkand xp-1 oex = ~,p-1koe,~kare infinite cycles. Similarly, applying Theorem 5.1(c) to the classes x = ~økfor k m, in odd degree t = |x| = 2pk - 1, the classes fiQpk(x) = ~,k+1and Qpk(x) - x(oex)p-1 = ~øk+1- ~øk(oe~øk)p-1 = øk0+1are infinite cycles, for Qpk(~øk) = ~øk+1and fi~øk+* *1= ~,k+1 by [BMMS86, III.2.3 and I.3.6]. Hence the E4-term above is generated as an algebra by infinite cycles, and t* *he homological spectral sequence collapses after the d2-differentials. For convenience in the comparison with ko, we make the case B = ku at p = 2 explicit: Corollary 6.2. The homological homotopy fixed point spectral sequence for R = T HH(ku) at p = 2 collapses after the d2-differentials, with E1**= P (y) P (~,41, ~,42, ~,23, ,04, . .). E(~,21oe,~21, ~,2* *2oe,~22) plus some classes in filtration s = 0, where ,0k+1= ~,k+1+ ~,koe,~kfor k 3. Proposition 6.3. The homological homotopy fixed point spectral sequence for R = T HH(B) collapses after the d2-differentials, in both of the cases: (a) B = ko and p = 2, when E1**= P (y) P (~,81, ~,42, ~,23, ,04, . .). E(~,41oe,~41, ~,2* *2oe,~22) plus classes on the vertical axis, and (b) B = tmf and p = 2, when E1**= P (y) P (~,161, ~,82, ~,43, ~,24, ,05, . .). E(~,81oe,~81, ~* *,42oe,~42, ~,23oe,~23) plus classes on the vertical axis. Proof. (a) For B = ko with H*(B; F2) = (A==A1)* = P (~,41, ~,22, ~,3, . .).we h* *ave H*(T HH(ko); F2) = P (~,41, ~,22, ~,3, . .). E(oe,~41, oe,~22) P (* *oe,~3) . See [AnR, 6.2(a)]. 20 ROBERT R. BRUNER AND JOHN ROGNES As in the proof of Proposition 6.1, the squares ~,81, ~,42and ~,23, as well * *as the classes ,0k+1= ~,k+1+ ~,koe,~kfor k 3 are d2-cycles, while E(~,41, oe,~41) and E(~,22* *, oe,~22) have homology E(~,41oe,~41) and E(~,22oe,~22), respectively. The homology of E(~,3) * * P (oe,~3) is F2. So E4**= P (y) P (~,81, ~,42, ~,23, ,04, . .). E(~,41oe,~41, ~,2* *2oe,~22) plus some classes in filtration s = 0. By Theorem 5.1(a), all of these algebra generators are in fact infinite cycl* *es, so the homological spectral sequence collapses, as claimed. (b) For B = tmf with H*(B; F2) = (A==A2)* = P (~,81, ~,42, ~,23, ~,4, . .).w* *e have H*(T HH(tmf); F2) = P (~,81, ~,42, ~,23, ~,4, . .). E(oe,~81, oe,~42, oe,* *~23) P (oe,~4) . See [AnR, 6.2(b)]. This gives the E2-term of the homological spectral sequence, and as before its homology with respect to the oe-operator is E4**= P (y) P (~,161, ~,82, ~,43, ~,24, ,05, . .). E(~,81oe,~81, ~* *,42oe,~42, ~,23oe,~23) plus some classes in filtration s = 0. By Theorem 5.1(a), all of these algebra generators are in fact infinite cycl* *es, so the homological spectral sequence collapses, as claimed. Theorem 6.4. The homological homotopy fixed point spectral sequence for R = T HH(B) collapses after the d2-differentials, in both of the cases: (a) B = MU, with E1**= P (y) P (bpk| k 1) E(bp-1koebk | k 1) plus classes in filtration zero, and (b) B = BP , with E1**= P (y) P (,pk| k 1) E(,p-1koe,k | k 1) plus classes in filtration zero. (When p = 2, substitute ,2kfor ,k.) Note that we do not need to assume that BP is a commutative S-algebra for the result in part (b). Proof. The integral homology algebra of MU is H*(MU; Z) = Z[bk | k 1], where bk in degree 2k is the stabilized image of the generator fik+1 2 H2k+2(BU(1); Z* *), under the zero-section identification BU(1) ' MU(1). So H*(MU; Fp) = P (bk | k 1) is concentrated in even degrees, and the E2-term of the Bökstedt spectral seque* *nce is E2**= HH*(H*(MU; Fp)) = H*(MU; Fp) E(oebk | k 1) . All the algebra generators are in filtrations s 1, so the spectral sequence c* *ol- lapses at this stage. There are no algebra extensions, since for p = 2, (oebk)* *2 = LEIBNIZ FORMULAS FOR CYCLIC HOMOTOPY FIXED POINT SPECTRA 21 Q2k+1(oebk) = oeQ2k+1(bk) = 0, where Q2k+1(bk) = 0 because it has odd degree. For p odd, (oebk)2 = 0 by graded commutativity, because oebk has odd degree. Th* *us H*(T HH(MU); Fp) = H*(MU; Fp) E(oebk | k 1) . This much can also be read off from [MS93, 4.3], or from Cohen and Schlichtkrul* *l's formula T HH(MU) ' MU ^ SU+ [CS]. The homological homotopy fixed point spectral sequence has E2-term E2**= P (y) P (bk | k 1) E(oebk | k 1) . Its homology with respect to the d2-differential, satisfying d2(bk) = y . oebk,* * is E4**= P (y) P (bpk| k 1) E(bp-1koebk | k 1) plus the usual y-torsion on the vertical axis. By Theorem 5.1(a) and (b), the a* *lgebra generators of this E4-term are all infinite cycles. Hence the spectral sequence collapses at this stage. (b) The Brown-Peterson spectrum BP was originally constructed to have mod p homology æ P (,2k| k 1) for p = 2, H*(BP ; Fp) = P (,k | k 1) for p odd. This equals the sub-algebra (A==E)* of A* that is dual to the quotient algebra A==E = A=AfiA of A. Hereafter we focus on the odd-primary case; the reader should substitute ,2kfor ,k when p = 2. The spectrum BP is known to be an (associative) S-algebra, and to receive an S-algebra map from MU [BJ02, 3.5]. This map induces a split surjective algebra homomorphism H*(MU; Fp) ! H*(BP ; Fp) in homology, which maps bpk-1 to ,k for k 1 and takes the remaining algebra generators bi to 0 for i 6= pk - 1. For the homology of BP injects into H*(HZ(p); Fp) and at the level of second spaces the composite map of spectra MU ! BP ! HZ(p)is a p-local equivalence MU(1) ! K(Z(p), 2). The generator fii+1 2 H~2i+2(MU(1); Fp) maps to bi 2 H2i(MU; Fp), while the corresponding generator fii+1 2 ~H2i+2(K(Z(p), 2); Fp) m* *aps to ,k 2 H2i(HZ(p); Fp) when i = pk - 1 and to 0 otherwise [Mi58, x5]. This prov* *es the claim. The Bökstedt spectral sequence for BP has E2-term E2**= HH*(H*(BP ; Fp)) = H*(BP ; Fp) E(oe,k | k 1) . Note that the map MU ! BP induces a surjection of Bökstedt spectral sequence E2-terms. Thus the fact that the Bökstedt spectral sequence for MU collapses at E2 with no algebra extensions implies the corresponding statement for BP , also without the assumption that BP is a commutative S-algebra. We can conclude that H*(T HH(BP ); Fp) = H*(BP ; Fp) E(oe,k | k 1) . The homological homotopy fixed point spectral sequence has E2-term E2**= P (y) P (,k | k 1) E(oe,k | k 1) . 22 ROBERT R. BRUNER AND JOHN ROGNES Again the map MU ! BP induces a surjection of E2-terms, so the d2-differentials satisfy d2(,k) = y . oe,k and d2(y) = 0, and are derivations. This leaves E4**= P (y) P (,pk| k 1) E(,p-1koe,k | k 1) plus some y-torsion on the vertical axis, and the map from the E4-term of the spectral sequence for MU is still surjective. Thus the spectral sequence for BP also collapses at this stage. 7. Generalizations and comments In this section we note some generalizations of our results, and also comment on the relation to related patterns of differentials in other spectral sequence* *s. The generalizations are of two sorts. First, we can replace the homotopy fixed poi* *nts construction by the Tate construction or the homotopy orbits. Second, we can change the group of equivariance. We consider these in order. First, there are spectral sequences similar to the one considered here for t* *he Tate construction XtT = [EeT ^ F (ET+ , X)]T (denoted tT(X)T in [GM95] and bH(T, X) in [AuR02]) and the homotopy orbit spectrum XhT = ET+ ^T X. Proposition 7.1. There is a natural spectral sequence Eb2**= bH-*(T; H*(X; Fp)) = P (y, y-1 ) H*(X; Fp) with y in bidegree (-2, 0), which is conditionally convergent to the continuous* * homo- logy Hc*(XtT; Fp). We call this the homological Tate spectral sequence. If H*(X; Fp) is finite in each degree, or the spectral sequence collapses at a fin* *ite stage, then the spectral sequence is strongly convergent. Proposition 7.2. There is a natural spectral sequence E2**= H*(T; H*(X; Fp)) = P (y-1 ) H*(X; Fp) with y-1 in bidegree (2, 0), which is strongly convergent to H*(XhT; Fp). We * *call this the homological homotopy orbit spectral sequence. (Note that for XhT the continuous homology is the same as the ordinary homology.) Further, the middle and right hand maps of the (homotopy) norm cofiber se- quence XhT -N! XhT ! XtT ! 2XhT induce the homomorphisms of E2-terms given by tensoring H*(X; Fp) with the short exact sequence of P (y)-modules 0 ! P (y) ! P (y, y-1 ) ! 2P (y-1 ) ! 0. Thus the homological Tate spectral sequence is an upper half plane spectral se- quence whose E2-term is obtained by continuing the y-periodicity in the homo- logical homotopy fixed point spectral sequence into the first quadrant, and the homological homotopy orbit spectral sequence (shifted 2 degrees to the right fr* *om Proposition 7.2) has the quotient of these as its E2-term. LEIBNIZ FORMULAS FOR CYCLIC HOMOTOPY FIXED POINT SPECTRA 23 Proposition 4.1 and Theorem 5.1 apply equally well to all three spectral se- quences. For details, see the thesis of Lunøe-Nielsen [L-N]. Second, we could also consider these three spectral sequences for the action* * of a finite cyclic subgroup C of T. For example, there is the homological Tate spect* *ral sequence bE2**= bH-*(C; H*(X; Fp)) converging conditionally to Hc*(XtC ; Fp). The analogue of Lemma 4.3 still hol* *ds, so that there are isomorphisms bEr**~=bH-*(C; Fp) bEr0,* for all r 2 (and now y is invertible, so there is no y-torsion), and all diff* *erentials are determined by those originating on the vertical axis Ebr0,*. In turn, the * *latter differentials are determined by those in the T-equivariant case, by naturality * *with respect to the restriction map XtT ! XtC . Therefore the collapsing results in Theorem 5.1 also hold in these cases. See [L-N] for more details. These latter spectral sequences, for finite subgroups C T, are essential i* *n the analysis of the topological model T F (B) for the negative cyclic homology of B* *, and the topological cyclic homology T C(B). Though the differentials here allow us to determine E1**in the cases of inte* *rest (see Section 6), there are still A*-comodule extensions hidden by the filtratio* *n. These are of course of critical importance for the analysis of the Adams spectr* *al sequence (1.3). A more elaborate study of the geometry of the universal examples used in Section 5 allows these to be recovered. This too can be found in [L-N]. Finally, it is interesting to compare the formulas for differentials here to* * analo- gous results in other spectral sequences. The first to be considered was the Ad* *ams spectral sequence, where the results are due to Kahn [Ka70], Milgram [Mi72], Mäkinen [Mä73], and the first author [BMMS86, Ch. VI]. For simplicity, let us assume p = 2 in this discussion, as there are several cases to be considered at odd primes ([BMMS86, VI.1.1]). Suppose that x is in the Er-term of the Adams spectral sequence E**2= ExtA (H*(R; F2), F2) =) ß*(R)^2, where R is a commutative S-algebra. The commutative S-algebra structure of R induces Steenrod operations in the E2-term of the Adams spectral sequence, which are the analog in this situation of the Dyer-Lashof operations in H*(R; F2). (* *In fact, under the Hurewicz homomorphism, they map to the Dyer-Lashof operations.) Then, in most cases we have (7.3) d*(Sqjx) = Sqjdr(x) +` aSqj-v x , where A +`B denotes whichever of A or B is in the lower filtration, or their su* *m, if they are in the same filtration. The subscript in d* is then the difference* * in filtrations between the right and left hand sides. In this formula, a is an in* *finite cycle in the Adams spectral sequence for the homotopy groups of spheres, and a and v are determined by j and the degree of x. When the first half of the right hand side dominates we have d2r-1(Sqjx) = Sqjdr(x) , 24 ROBERT R. BRUNER AND JOHN ROGNES and this formula resembles the formula d2r(fifflQi(x)) = fifflQi(d2r(x)) of Proposition 4.1, in that both essentially say that the relevant differential* * com- mutes with the Dyer-Lashof operations. The fact that the length of the differen* *tial increases from r to (2r - 1) when we apply the squaring operation in the Adams spectral sequence reflects the difference between the homotopy fixed point filt* *ration and the Adams filtration, and the way in which they interact with the extended * *pow- ers. A more extreme difference occurs when the second term aSqj-v x is involved. In the homological homotopy fixed point spectral sequence this term disappears, essentially because the element a 2 ß*S is mapped to 0 by the Hurewicz homo- morphism. Homotopical homotopy fixed point spectral sequences, as in [AuR02], will have differential formulas with two parts, as in the Adams spectral sequen* *ce. Such two part formulas for differentials reflect universally hidden extensions * *in the following sense. The differential (7.3) arises from decomposing the boundary of the cell on w* *hich Sqjx is defined into two pieces. One of the pieces carries Sqjdr(x) and the ot* *her carries aSqj-v x. 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