GENERALIZED TATE HOMOLOGY 3
REMARK: Greenlees was the first to prove that Tate cohomology is a repre-
sentable functor, and he has studied its representing spectrum in some detail.
We urge the reader to consult his paper [Gr]. We would like to thank him for
making his work available to us and for pointing out common results. Our work
complements his in some ways; one of our intentions is to show that defining
Tate homology in terms of a classical stable homotopy transfer map naturally
leads to a more general notion that allows for compact Lie groups and arbitrary
spectra.
Organization of the paper. Section 2 contains the definition of generalized
Tate homology and the proof of 1.1(3). Section 3 is concerned with the proof
of 1.1(2). Finally, Section 4 treats in detail one example of generalized Tate
homology, describes the proof of 1.1( 4), and concludes with some remarks about
generalized cyclic homology.
§2. GENERALIZED TATE HOMOLOGY
In this section we will give the construction of the Tate homology spectrum
:fiG(E) and prove 1.1(3). Our basic idea is to use the transfer construction of
Becker and Schultz
(BS]
and keep track of equivariance.
First we will establish some notation. Let
G
be a compact Lie group. If
W
is a finite dimensional representation of G we will denote by
sw
the one point
compactification of the underlying vector space of
W.
The space
sw
is a sphere,
and the action of
G
on
W
induces an action of
G
on
sw
which fixes the (base-)
point 'at infinity. Given a based G-space
Y
let
EwY
be the smash product
sw /\ Y
and
awy
the space Map.(Sw, Y) of basepoint preserving maps from
sw
to
Y.
It is clear that
EWY
has a diagonal G-action and that
nwy
has a
G-action given by conjugation of maps. (The fixed point set (nwy)G of this
latter action is the space of G-equivariant maps
sw
--+
Y.) We define Qa(Y)
by the formula
Qa(Y) = lirnOwEw(Y)
--+
w
where the limit is taken over all finite dimensional representations W of G. As
usual the space QY
is
obtained by restricting the indexing set in the above direct
limit to be the set of trivial finite-dimensional representations of
G.
If ( : E--+ B is
a vector bundle over
B,
let
B'
denote the Thorn space of(.
Now suppose that
G
is a compact Lie group and that
X
is a finite, free, G-CW
complex. Let
B
=
X/G
be the orbit space, and p:
X--+ B
the natural projection
map. Choose an equivariant embedding
e : X
---
V
of
X
into a finite dimensional
representation space
V
of
G.
This induces an embedding
p
x e :
X
--+
B
x
V
with normal bundle, say, v. (Note that the notion of normal bundle makes sense
here. Over any point b of B the map
p
X
e induces an equivariant embedding
eb :
p- 1
(b)
---
V. Since
p- 1
(b)
is a free, transitive G-space, it is clear that
eb
has
a well-defined normal bundle
Vb.
As b varies the bundles
Vb
can be assembled
into a global normal bundle
v
for
X
in
B
x V.)
Denote the projection map B
x
V
--+
B
by
1r.
There is evidently a Thom-
Pontryagin collapse map
Ev B+
=
B'lr
--+
X
11
but ( cf.
[BS])
unless
G
is finite
this
is
not quite the transfer map we want.
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