E1 -ring structures for Tate spectra J.E. McClure* 1 Introduction. Let G be a compact Lie group and kG a G spectrum (as defined in [3, Section I.2* *]). Greenlees and May ([2]) have defined an associated G-spectrum t(kG) called the Tate spect* *rum. They observe that if kG is a ring G-spectrum then there is an induced ring G-spectru* *m struc- ture on t(kG), and that if kG is homotopy-commutative then t(kG) will also be h* *omotopy- commutative (see [2, Proposition 3.5]). It is therefore natural to ask whether * *an equivariant E1 -ring structure on kG induces an equivariant E1 -ring structure on t(kG) (we* * will recall the definition in a moment). We offer both positive and negative answers to thi* *s question. On the positive side, we show that t(kG) inherits a structure which is somew* *hat weaker than an equivariant E1 -ring structure, but which should be adequate for most p* *ractical purposes. To explain this, let us recall from [3, Example VII.1.4] that to each* * G-universe U is associated an equivariant operad L(U). Let us fix a complete G universe U* * and let V denote the trivial G-universe UG . An equivariant E1 -ring structure is defin* *ed to be an action of an equivariant operad equivalent to L(U) (see [3, Definitions VII.2.1* * and VII.1.2 and Remark VII.1.3]). Let us define an E01-ring structure to be an action of an* * equivariant operad equivalent to L(V ); since G acts trivially on L(V ) we can rephrase thi* *s by saying that an E01-ring structure is an action of a nonequivariant E1 operad through * *G-maps. Since there is a map of operads L(V ) ! L(U), an equivariant E1 structure spec* *ializes to an E01structure. On the other hand, Remark VII.2.5 of [3] shows that if kG is a* *n E01-ring spectrum then the fixed point spectra (kG)H have (nonequivariant) E1 -ring stru* *ctures which are consistent as H varies; this is likely to be the point most relevant for ap* *plications. Our positive result is: Theorem 1 If kG is an E01-ring spectrum then so is t(kG); in particular all f* *ixed-point spectra (t(kG))H are nonequivariant E1 -ring spectra. ________________________________ *Partially supported by NSF grant 9207731-DMS 1 The proof of Theorem 1 will show that the diagram in Proposition 3.5 of [2] * *is a diagram of E01-ring spectra. To state our negative result we need to recall the definition of t(kG). Let* * EG be a contractible free G-CW complex and let E"G denote the G-space defined by the co* *fiber sequence EG+ ! S0 ! "EG (here + denotes a disjoint basepoint). Let F (EG+; kG) be the function spectrum* * of maps from EG+ to kG ([3, Definition I.3.2]). Then t(kG) is defined to be the G-spect* *rum F (EG+; kG) ^ "EG: Let us write for the natural map S0 ! "EG. Theorem 2 Let G be a finite cyclic group and let kG be any G-spectrum. Suppo* *se that t(kG) has an equivariant E1 -ring structure whose unit factors (up to equivaria* *nt homotopy) through 1G. Then t(kG) must be equivariantly contractible. This implies that if kG is a ring G-spectrum for which t(kG) is not equivari* *antly con- tractible, then t(kG) cannot have an equivariant E1 -ring structure whose under* *lying ring G-spectrum structure is compatible with that of kG under the natural map kG ! t* *(kG). In particular, the underlying ring G-spectrum structure of t(kG) cannot be that de* *fined in [3, Proposition 3.5]. Thus it seems that there is no natural way to give t(kG) an * *equivariant E1 -ring structure. I would like to thank Mike Hopkins for suggesting this problem to me. 2 Proof of Theorem 1. Theorem 1 is an immediate consequence of the following two lemmmas, of which th* *e second is well-known. Let us recall from [3, Definition VII.2.7] that, given an equiva* *riant operad C, a C0 space is an action of C in the category of based G-spaces; that is, it is * *a based G-space X with based G-maps (Cj)+ ^j X(j)! X (here (j)denotes j-fold smash product) satisfying the same compatability condit* *ions that are used to define an equivariant C-space. In particular, this definition makes* * sense if C is a nonequivariant operad provided with the trivial G-action; it then says that C* * acts on X through G-maps. 2 Lemma 3 There is a nonequivariant E1 operad C for which "EG is an equivarian* *t C0 space. Lemma 4 Let C be any equivariant operad. (a) If kG is a C-ring spectrum (that is, if it has an equivariant action of C) * *then so is F (Y+; kG) for any G-space Y . (b) If hG is a C-ring spectrum and X is a C0-space then hG ^ X is a C-ring spec* *trum. Theorem 1 follows from Lemma 4(b) if we let hG be F (EG+; kG) and X be "EG. Proof of Lemma 4. In each case, we specify the structural maps which constitut* *e the C-action; the fact that they satisfy the necessary compatibility relations is a* * straightforward application of the methods of [3, Sections VI.1-VI.3]. For part (a) the structural map j : Cj |xF (Y+; kG)(j)! F (Y+; kG) is the adjoint of the composite Y+ ^ Cj |xF (Y+; kG)(j)^1--!(Y+)(j)^ Cj |xF (Y+; kG)(j) ~= (j) (j) 1|xe (j) 0j -! Cj |x((Y+) ^ F (Y+; kG) )--! Cj |xkG -! kG; here is the diagonal map of Y , the isomorphism is that of [3, Proposition VI.* *1.5], e is the evaluation map, and 0jis the structural map of kG. For part (b) the structural map j : Cj |x(hG ^ X)(j)! hG ^ X is the composite 0j^0* *0j Cj |x(hG ^ X)(j)= Cj |x(h(j)G^ X(j))!-ffi(Cj |xh(j)G) ^ (Cj+ ^ X(j)) ---! * * hG ^ X; where ffi is the map given in Definition VI.3.5 of [3] and 0j, 00jare the struc* *tural maps for hG and X. QED Proof of Lemma 3. First let us observe that "EG is nonequivariantly contractibl* *e and that for any nontrivial subgroup H of G the H-fixed set (E"G)H is exactly S0; the sa* *me is true for (E"G)(j)since the smash product of spaces commutes with H-fixed sets. 3 Let Map G*denote based G-maps. Restriction to the G-fixed set gives a map OE : Map G*(E"G(j); "EG) ! Map *(S0; S0) which we claim is a weak equivalence. Assuming this for the moment, let C0jbe t* *he space OE-1(id). Then the spaces C0jwith the evident composition operations fl form an* * operad C0 and "EG is a C00-space. The only thing preventing C0from being a nonequivariant* * E1 operad is that the action of j on C0jmay not be free. To remedy this let C00be any non* *equivariant E1 operad and define C to be C0x C00, acting on "EG via the projection C0x C00!* * C0. It only remains to prove the claim that OE is a weak equivalence. First we o* *bserve that the reduced diagonal map : "EG ! "EG(j) is a weak equivalence on each fixed-point set, and is therefore a G-homotopy eq* *uivalence by the equivariant Whitehead theorem. It follows that * : Map G*(E"G; "EG) ! Map G*(E"G(j); "EG) is a homotopy equivalence, so it suffices to verify the claim when j = 1. To handle this case, we map the cofiber sequence EG+ ! S0 ! "EG into "EG to get a fiber sequence Map G*(E"G; "EG) ! Map G*(S0; "EG) ! Map G*(EG+; "EG): The middle term is equal to S0, so it suffices to show that the third term is w* *eakly con- tractible. For this we recall that the functor Map G*(EG+; -) takes G-maps whi* *ch are nonequivariant weak equivalences to weak equivalences (for example, this follow* *s from [1, XI.5.6] since Map G*(EG+; -) is a special case of the holim construction). Sin* *ce E"G is nonequivariantly contractible we see that Map G*(EG+; "EG) is weakly contractib* *le and we are done. QED 3 Proof of Theorem 2. As motivation for the proof of Theorem 2, we first explain why the operad C0 co* *nstructed in the proof of Lemma 3 is not equivalent to the linear isometries operad LU. L* *et G = Z=2 4 for simplicity and consider the G x 2-spaces LU2 and C02. Let H be the diagonal* * copy of Z=2 in G x 2 = Z=2 x Z=2. We claim that LU2 has H fixed points but C02has none;* * this certainly implies that LU2 and C02are not G x 2-equivalent. To see that LU2 has* * H-fixed points we need only show that there is an H-equivariant linear isometry from U * * U to U; but this is obvious since as an H-representation U U is a complete H-universe,* * and is therefore H-isomorphic to U. (We note for later use that (LU2)H is in fact cont* *ractible by [3, Lemma II.1.5]). On the other hand, if C02had an H-fixed point then there wo* *uld be a G x 2-equivariant map "EG(2)! "EG (with 2 acting trivially on the target) which extends the identity map of S0, a* *nd passing to H-fixed points would give a (nonequivariant) map (E"G(2))H ! S0 which extends t* *he identity map of S0. But this is impossible since (E"G(2))H is contractible: there is a (* *nonequivariant) homeomorphism E"G ! (E"G(2))H which takes x to x ^ gx, where g is the generator of G. The proof of Theorem 2 is a variant of the same idea. For simplicity, we beg* *in with the case G = Z=2. Suppose that t(kG) has an equivariant E1 -ring structure whose un* *it j factors through 1G. Then there is a G-homotopy commutative diagram of G-spectra 1|x1G(2) 1 (2) (2) LU2 |x2(S0G)(2)?-----! LU2 |x2(G "EG) ! LU2 |x2t(kG)? ?y2 ?y0 2 j S0G - ! t(kG); where 2 and 02are the structural maps for S0Gand t(kG). Next we recall that the* * upper-left corner of this diagram is an equivariant suspension spectrum, so that we may pa* *ss to the adjoint to get a G-homotopy commutative diagram of spaces. More precisely, [3, * *Proposition VI.5.3] gives an isomorphism LU2 |x2(S0)(2)~=1G(LU2+ ^2 (S0G)(2)) which carries 2 to the composite 1Gss1 0 1G(LU2+ ^ (S0)(2)) = 1G(LU2=2)+ ---! G S ; here ss is the evident projection (LU2=2)+ ! S0. Thus the adjoint of the diagra* *m above 5 has the form 1^2 (2) (2) 1 (2) LU2+ ^2 (S0)(2)-----! LU2+ ^ "EG ! G (LU2 |x2t(kG) ) # = (LU2=2)+ # 1G02 # ss "j 1 S0 -! G t(kG) For our purposes, the important thing about this diagram is that "jO ss fact* *ors, up to G-homotopy, through LU2+ ^ "EG(2). Precomposing with the projection LU2+ ^ (S0)(2)! LU2+ ^2 (S0)(2) we see that the composite "j1 (1) LU2+ ^ (S0)(2)= (LU2)+ -ss!S0 -! G t(kG) (where we have again written ss for the evident projection) factors up to G x 2* *-homotopy through LU2+ ^ "EG(2). Now let H be the diagonal copy of Z=2 in G x 2. Passing * *to the H-fixed points of (1) (and noting that the H-fixed points of 1Gt(kG) are the sa* *me as the G-fixed points since 2 acts trivially) we see that the composite H 0 "jG 1 G (2) (LUH2)+ -ss-!S --! (G t(kG)) factors up to (nonequivariant) homotopy through LUH2+^ (E"G(2))H : But we have shown in the first paragraph of this section that (E"G(2))H is cont* *ractible, so the composite (2) is (nonequivariantly) homotopy trivial. We also showed in the fir* *st paragraph that LUH2 is contractible, so ssH is an equivalence, and we conclude that j"G: S0 ! (1Gt(kG))G is homotopy trivial. This means that "jis G-homotopy trivial, and passing to th* *e adjoint we see that j itself is G-homotopy trivial. But j is the unit of the equivariant E* *1 ring t(kG), so t(kG) must be equivariantly contractible, as was to be shown. So far we have assumed that G is Z=2. When G is cyclic of order n, with gene* *rator g, one need only repeat the same argument with LU2 replaced by LUn, 2 replaced by * *n, and H replaced by the subgroup of G x n generated by (g; oe), where oe is an n-cycl* *e. QED 6 References [1]A.K. Bousfield and D.M. Kan, Homotopy Limits, Completions and Localizatio* *ns. Springer Lecture Notes in Mathematics v. 304 (1972). [2]J.P.C. Greenlees and J.P. May, Generalized Tate, Borel and CoBorel Cohomolo* *gy. Preprint. [3]L.G. Lewis, J.P. May, and M. Steinberger, Equivariant stable homotopy theor* *y. Springer Lecture Notes in Mathematics v. 1213 (1986). 7