E1 -ring structures for Tate spectra
J.E.McClure
1 Introduction.
Let G be a compact Lie group and kGa G spectrum (as defined in [3, Section I.2]*
*). Greenlees
and May ([2]) have definedan associated G-spectrum t(kG) called the Tate spectr*
*um. 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 thatif kG is homotopy-commutative then t(kG) will also be ho*
*motopy-
commutative (see [2,Proposition 3.5]). It is therefore natural to ask whether a*
*n 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 this*
* question.
On the positive side,we show that t(kG) inherits a structure which is somewh*
*at weaker
than an equivariant E1-ring structure, but which should be adequate for most pr*
*actical
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 defined*
* to be an
action!of!an equivariant operadequivalent to L(U) (see [3, Definitions VII.2.1 *
*and VII.1.2
and!Remark!VII.1.3]). Let us definean E10-ring structure to be an action of an *
*equivariant
operad!equivalent!to L(V );since G acts trivially on L(V ) we can rephrase this*
* 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 specia*
*lizes to
an!E01structure.!On the other hand,Remark VII.2.5 of [3] shows that if kG is an*
* E01-ring
spectrum!then the fixed point spectra (kG)H have(nonequivariant) E1 -ring struc*
*tures 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
The proof of Theorem 1 will show thatthe diagram in Proposition 3.5 of [2] i*
*s 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, DefinitionI.3.2]). Then t(kG) is defined to be the G-spectr*
*um
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. Suppos*
*e that
t(kG) has an equivariant E1-ring structure whose unit factors (up to equivarian*
*t 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 anequivariant E1 -ring structurewhose underly*
*ing ring
G-spectrum structure is compatible with thatof kG under the natural map kG !t(k*
*G ).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 e*
*quivariant
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 ofthe following two lemmmas, of which the*
* second
is well-known. Let usrecall from [3, Definition VII.2.7] that, given an equivar*
*iant operad C,
a C0space is an action of C in the category of based G-spaces; that is, it is a*
* basedG-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. Inparticular, this definition make*
*s sense if C is
a nonequivariant operad provided with the trivial G-action; it then says that C*
*acts on X
through G-maps.
Lemma 3 There is a nonequivariant E1 operad C for which "EG is an equivariant*
* C0space.
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 spectrumand X is a C0-space then hG^ X is a C-ring spectr*
*um.
Theorem 1 follows from Lemma 4(b)if we let hG be F(E G+;kG) and X be "EG.
Proof of Lemma 4. In each case, we specify the structural maps which constitute*
* 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 _F (Y+;kG)(j )! F (Y+;kG)
is the adjoint of the composite
Y+ ^ Cj_ F (Y+; kG)(j)^1 !(Y+)(j)^ Cj_ F(Y+;kG )(j )
= 0j
! Cj_ ((Y+)(j)^F (Y+;kG)(j))1_e!Cj_ k(j)G! 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 _(hG ^ X)(j)! hG ^ X
is the composite
0j^00j
Cj_ (hG^ X )(j=)Cj _(h(j)G^ X(j))!ffi(Cj_ h(j)G) ^ (Cj+^ X(j)) ! hG ^ X;
where ffi is the map given in Definition VI.3.5 of [3] and 0j,00jare the struct*
*ural 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 Hof G the H-fixed set (E"G)H is exactly S0;the same*
* is true
for ("EG)(js)ince the smash product of spaces commutes with H -fixed sets.
Let MapG denote based G-maps. Restriction to the G-fixed set gives a map
OE : MapG ("EG(j);E"G) ! Map (S0;S0)
which we claim is a weak equivalence. Assuming this for the moment, let C0jbe t*
*he space
OE1 (id). Then the spaces Cj0with 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 C00b eany non*
*equivariant
E1 operad and define C to be C0C00, acting on "EG via the projection C0C 00! C0.
It only remains to prove theclaim that OE is a weak equivalence. First we ob*
*serve 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;E"G) ! MapG ("EG(j);E"G)
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;E"G) ! MapG (S0; "EG) ! Map G(EG+; "EG):
The middle term is equal to S0,so it suffices to show that the third term is we*
*akly con-
tractible. For this we recall that the functor Map G(EG+ ;) takes G-maps which*
* 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). Since*
* "EG is
nonequivariantly contractible we see that Map G(EG+; "EG) is weakly contractibl*
*e and we
are done. QED
3 Proof of Theorem 2.
As motivation for the proof of Theorem 2, we first explain why the operad C0con*
*structed
in the proof of Lemma 3 is not equivalent to the linear isometries operad LU. L*
*et G = Z=2
for simplicity and consider the G 2-spaces LU2 and C02. Let H be the diagonal *
*copy of
Z=2 in G 2= Z=2 Z=2. We claim that LU2 has Hfixed points but C02has none; this
certainly implies that LU2 and C02are not G 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 2-equivariant map
"EG(2)!E"G
(with 2acting trivially on the target) which extends the identity map of S0, an*
*d passing to
H-fixed points would give a (nonequivariant) map (E"G(2))H !S0 which extends th*
*e identity
map of S0. But this is imp ossible since (E"G(2))H is contractible:there is a (*
*nonequivariant)
homeomorphism
E"G ! ("EG(2))H
which takes x to x ^ gx, where gis the generator of G.
The proof of Theorem 2 is a variant of the same idea. For simplicity,we begi*
*n with the
case G = Z=2. Suppose that t(kG) has an equivariant E1 -ring structure whose un*
*itj factors
through 1G. Then there is a G-homotopy commutative diagram ofG-spectra
1_1G(2)
LU2_ 2?(S0G)(2) ! LU2 _ 2(1G"EG)(2)! LU2_ 2 t(kG)(2)
?y2 ??y0
2
j
SG0 ! t(kG);
where 2 and 02are the structural maps for SG0and 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_ 2 (S0)(2)=1G(LU2+^2 (S0G)(2))
which carries 2to the composite
1Gss
1G(LU2+^(S 0)(2)) = 1G(LU2=2)+ ! 1GS0;
here ss is the evidentpro jection (LU2=2)+ ! S0. Thus the adjoint of the diagra*
*mab ove
has the form
1^2 (2)
LU2+ ^2 (S0)(2) ! LU2+^ "EG(2)! 1G(LU2_ 2 t(kG)(2))
#=
(LU2=2)+ # 1G02
# ss
"j
S0 ! 1Gt(kG)
For our purposes,the important thing about this diagram is that "jffi ss fac*
*tors, upto
G-homotopy, through LU2+^E"G(2). Precomposing with the projection
LU2+^ (S0)(2)! LU2+ ^2 (S0)(2)
we see that the composite
"j
(1) LU2+ ^(S0)(2)=(LU2)+ ss!S0! 1Gt(kG)
(where we have again written ss for the evident projection) factors up to G 2-*
*homotopy
through LU2+^ "EG(2). Now let H be the diagonal copy of Z=2 in G 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 2acts trivially) we see that the composite
H "jG
(2) (LUH2)+ ss! S0! (1Gt(kG))G
factors up to (nonequivariant) homotopy through
LUH2+^ ("EG(2))H:
But we have shown in the first paragraph of this section that ("EG(2))H is cont*
*ractible, so the
composite (2) is (nonequivariantly) homotopy trivial. We also showed in the fir*
*st paragraph
that LUH2is contractible, so ssH is an equivalence, and we conclude that
"jG: S0! (1Gt(kG))G
is homotopy trivial. Thismeans that "jis G-homotopy trivial, and passing to the*
* adjointwe
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,2replaced by n,*
* and
H replaced by the subgroup of G n generated by (g;oe), where oe is an n-cycle.*
* QED
References
[1]A.K. Bousfield and D.M. Kan, Homotopy Limits, Completions and Localizations.
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 stablehomotopy theory.*
* Springer
Lecture Notes in Mathematics v. 1213 (1986).