GENERALIZED TATE HOMOLOGY
5
moreover, if e0 is the basepoint of E (necessarily fixed by the action of G), the
inclusion

-+
E+ determined by e0 induces maps Q(S0
)
-+
Q(E+)
-+
E and
(Q(S0 ))ha-+ (Q(E+))ha-+ Eha which are null homotopic. It follows that the
map
tE
induces a map
where ( EG + /\a
E)
ad
denotes the quotient of ( EG
x
a E)
ad
by ( EG
X {
e0
} )ad
=
BGad. If E ={En} is an omega spectrum with an action of G then the source
spaces of the maps
TEn
fit together to form a spectrum which we will denote
(EG+ /\a E)ad. Similarly, the target spaces fit together to form a spectrum
whose structure maps are induced in the obvious way by the structure maps
of the spectrum E. We call this second spectrum the homotopy fixed point
spectrum of the action of G on
E
and denote it by Eha. Notice furthermore
that the maps
TEn
respect the structure maps of these two spectra and hence
induce a map of spectra
If
E
is simply a spectrum with an action of G (not necessarily an omega spec-
trum) we define Eha to be (E')ha, where E' is the associated omega spectrum,
and obtain as before a transfer map TE·
DEFINITION
2.2: If G is a compact Lie group and E a spectrum with an action
of G, we define
(1)
the generalized Tate homology spectrum
Ha(E)
to be the (stable) fiber
of the map of spectra given by the transfer TE : (EG+ /\a E)ad
-+
Eha,
and
(2)
the generalized Tate homology groups
H~(E)
to be the homotopy groups
of
Ha(E).
We will now prove 1.1(3). To do this it is convenient to use configuration space
approximations to loop spaces, as described in [M2]. To be more precise, if
V
is a finite dimensional vector space let
This configuration spaces has a free symmetric group
('En)
action given by per-
muting coordinates. Given a based space Z, let C(V, Z) be the complex
C(V, Z) =
u
F(V, n)
X!;n
zn
I"'
nO
where the components in this union are patched together by a basepoint relation
as described in [M2]. For
Z
path connected, there is a well known weak ho-
motopy equivalence [M2] [H] a :
C(V, Z)
-+
nv'Ev
(Z). If
V
is a G-orthogonal
representation space and
Z
is a G-space with a basepoint that is fixed under the
action, then C(V, Z) has an induced G-action and it is easily observed that the
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