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
Previous Page Next Page