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