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.

ยง2. 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.

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.

ยง2. 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.