Contemporary Mathematics

Volume 96, 1989

Generalized Tate Homology, Homotopy Fixed Points

and the Transfer

A. ADEM,

R. L.

COHEN AND

w.

G.

DWYER

Department of Mathematics, Stanford University

Department of Mathematics, University of Notre Dame

Abstract. We define and study the generalized Tate homology of a compact Lie

group G with coefficients in a spectrum upon which G acts.

§1.

INTRODUCTION

Let G be a finite group and X a G-CW complex. The cellular chain complex

S.(X)

is

a graded Z[G]-module, and the hyperhomology of G with coefficients

in S.(X) (denoted H;;(X))

is

known as the equivariant homology of X. The

groups

Hf

(X) can be identified with the ordinary homology groups of the Borel

construction EG x

G

X or equivalently with the homotopy groups of the spectrum

H A (EG

xa

X)+, where His the Eilenberg-MacLane spectrum {1(Z, n)}.

Define the equivariant Tate homology of X, denoted

fi?

(X), to be the Tate

hyperhomology of G with coefficients in S.(X). (Up to regrading, the Tate

hyperhomology of G with coefficients in a Z[G] chain complex S. is the Tate

hypercohomology

[Sw]

of the cochain complex

S*

obtained by reversing the

indices of

S.

in sign. The regrading is such that the Tate hyperhomology of

G

with coefficients in a single module

M

concentrated in degree 0 agrees in strictly

positive dimensions with the ordinary group homology of G with coefficients in

M.)

One goal of this paper is to prove that the equivariant Tate homolog)' of

X

can also be expressed in a natural way as the homotopy of a certain spectrum.

In fact, we will make a much more general construction. Let G be a compact Lie

group. A spectrum E with an action of G is by definition a spectrum

{En}

in

the usual sense { cf. [A]) together with an action of G on each space with respect

to which the suspension maps are equivariant. A weak equivalence between two

such objects is an equivariant map which is a weak equivalence on the underlying

spectra.

During the preparation of this work all three authors were supported by N.S.F. grants and the

second author by an N.S.F-P.Y.I award.

1980 Mathematics subject classifications. Primary

SSN~S;

Secondary 55P42

1

©

1989 American Mathematical Society

0271-4132/89 $1.00

+

$.25 per page

http://dx.doi.org/10.1090/conm/096/1022669

Volume 96, 1989

Generalized Tate Homology, Homotopy Fixed Points

and the Transfer

A. ADEM,

R. L.

COHEN AND

w.

G.

DWYER

Department of Mathematics, Stanford University

Department of Mathematics, University of Notre Dame

Abstract. We define and study the generalized Tate homology of a compact Lie

group G with coefficients in a spectrum upon which G acts.

§1.

INTRODUCTION

Let G be a finite group and X a G-CW complex. The cellular chain complex

S.(X)

is

a graded Z[G]-module, and the hyperhomology of G with coefficients

in S.(X) (denoted H;;(X))

is

known as the equivariant homology of X. The

groups

Hf

(X) can be identified with the ordinary homology groups of the Borel

construction EG x

G

X or equivalently with the homotopy groups of the spectrum

H A (EG

xa

X)+, where His the Eilenberg-MacLane spectrum {1(Z, n)}.

Define the equivariant Tate homology of X, denoted

fi?

(X), to be the Tate

hyperhomology of G with coefficients in S.(X). (Up to regrading, the Tate

hyperhomology of G with coefficients in a Z[G] chain complex S. is the Tate

hypercohomology

[Sw]

of the cochain complex

S*

obtained by reversing the

indices of

S.

in sign. The regrading is such that the Tate hyperhomology of

G

with coefficients in a single module

M

concentrated in degree 0 agrees in strictly

positive dimensions with the ordinary group homology of G with coefficients in

M.)

One goal of this paper is to prove that the equivariant Tate homolog)' of

X

can also be expressed in a natural way as the homotopy of a certain spectrum.

In fact, we will make a much more general construction. Let G be a compact Lie

group. A spectrum E with an action of G is by definition a spectrum

{En}

in

the usual sense { cf. [A]) together with an action of G on each space with respect

to which the suspension maps are equivariant. A weak equivalence between two

such objects is an equivariant map which is a weak equivalence on the underlying

spectra.

During the preparation of this work all three authors were supported by N.S.F. grants and the

second author by an N.S.F-P.Y.I award.

1980 Mathematics subject classifications. Primary

SSN~S;

Secondary 55P42

1

©

1989 American Mathematical Society

0271-4132/89 $1.00

+

$.25 per page

http://dx.doi.org/10.1090/conm/096/1022669