6

A. ADEM, R. L. COHEN, W. G. DWYER

map a is an equivariant. Hence if Z is connected, a is a weak G-equivalence and

so induces an equivalence on homotopy fixed points.

PROOF OF 1.1(3): Assume that

X

is finite and that

G

acts freely on

X.

(If

the action on

X

is not free, replace

X

as above by

EG

x

X.

If

X

is not finite,

restrict to finite subcomplexes and make a direct limit argument. We leave it to

the reader to fill in these details

(CI].)

Choose as above an equivariant embedding

e :

X

-t

V

of

X

into a representation space

V.

Let

B

be the quotient space

XjG.

Suppose the order of the finite group G is n. Consider the Kahn-Priddy

transfer map

[KP]

TKP:

B-+ F(V,n)

Xl;,.

xn

-t

C(V,X+)

defined by

rK p(b)

= ( e(x

1

), · · · ,

e(xn))

x (x1

, · · · ,

xn),

where the

x/s

run through

the orbit in X represented by b E B.

It

is clear that the image of

TK p

lies

in the fixed points of

C(V,

X+). Moreover, a check of the definition of the

map

a:

C(V,X+)-+ nvEv(X+)

as given in

(M2]

shows that the composition

1"KP

a

B--+ C(V,X+)G-+ (OvEv

(X+))G-+

Qa(X+)

is the map

t'Jc

defined above.

(Note that since G is a finite group Bad= B+.) The desired result is immediate.

§3. RELATIONSHIP TO CLASSICAL TATE HOMOLOGY

In this section we will prove 1.1(2).

It

is convenient to work simplicially (Ml],

and so we will assume that

X

is a simplicial set with an action of the finite

group

G

and that

EG

is a contractible simplicial set on which

G

acts freely

(Ml, p. 83]. The first step is to give a homotopy-theoretic construction of the

positive-dimensional part of the classical Tate homology of (the realization of)

X.

Consider the simplicial free abelian group Z 0

X

which in each dimension n

is the free abelian group on the set

Xn

of n-simplices of

X.

As in [Ml, p. 98],

the homotopy groups of the realization IZ®XI are the integral homology groups

of

lXI.

One can associate to an n--simplex u of

EG

x a

X

the sum of the n--simplices

in

EG

x

X

which are in the free G-orbit u represents. This association extends

additively to a simplicial map

tr:

Z®

(EG

X

a

X)-+

Z®(EG

x X) whose image

clearly lies in the G-fixed-points of Z®

(EG

x X). Let

"f

denote the composition

oftr

with the obvious map

(Z®(EG

x X))G-+ (Z®X)G and let

TzG(X)

denote

the homotopy fiber of the map IZ®(EGxaX)I-+ IZ®XIhG given by composing

1"11 with the inclusion of the fixed point set into the homotopy fixed point set.

PROPOSITION 3.1. For any simplicial set X with an action of the finite group

G there

are

natural isomorphisms

for all i

~

0.

This requires a lemma. Let

N.

be the normalization functor which assigns

to each simplicial abelian group its normalized chain complex and

N•

the dual

A. ADEM, R. L. COHEN, W. G. DWYER

map a is an equivariant. Hence if Z is connected, a is a weak G-equivalence and

so induces an equivalence on homotopy fixed points.

PROOF OF 1.1(3): Assume that

X

is finite and that

G

acts freely on

X.

(If

the action on

X

is not free, replace

X

as above by

EG

x

X.

If

X

is not finite,

restrict to finite subcomplexes and make a direct limit argument. We leave it to

the reader to fill in these details

(CI].)

Choose as above an equivariant embedding

e :

X

-t

V

of

X

into a representation space

V.

Let

B

be the quotient space

XjG.

Suppose the order of the finite group G is n. Consider the Kahn-Priddy

transfer map

[KP]

TKP:

B-+ F(V,n)

Xl;,.

xn

-t

C(V,X+)

defined by

rK p(b)

= ( e(x

1

), · · · ,

e(xn))

x (x1

, · · · ,

xn),

where the

x/s

run through

the orbit in X represented by b E B.

It

is clear that the image of

TK p

lies

in the fixed points of

C(V,

X+). Moreover, a check of the definition of the

map

a:

C(V,X+)-+ nvEv(X+)

as given in

(M2]

shows that the composition

1"KP

a

B--+ C(V,X+)G-+ (OvEv

(X+))G-+

Qa(X+)

is the map

t'Jc

defined above.

(Note that since G is a finite group Bad= B+.) The desired result is immediate.

§3. RELATIONSHIP TO CLASSICAL TATE HOMOLOGY

In this section we will prove 1.1(2).

It

is convenient to work simplicially (Ml],

and so we will assume that

X

is a simplicial set with an action of the finite

group

G

and that

EG

is a contractible simplicial set on which

G

acts freely

(Ml, p. 83]. The first step is to give a homotopy-theoretic construction of the

positive-dimensional part of the classical Tate homology of (the realization of)

X.

Consider the simplicial free abelian group Z 0

X

which in each dimension n

is the free abelian group on the set

Xn

of n-simplices of

X.

As in [Ml, p. 98],

the homotopy groups of the realization IZ®XI are the integral homology groups

of

lXI.

One can associate to an n--simplex u of

EG

x a

X

the sum of the n--simplices

in

EG

x

X

which are in the free G-orbit u represents. This association extends

additively to a simplicial map

tr:

Z®

(EG

X

a

X)-+

Z®(EG

x X) whose image

clearly lies in the G-fixed-points of Z®

(EG

x X). Let

"f

denote the composition

oftr

with the obvious map

(Z®(EG

x X))G-+ (Z®X)G and let

TzG(X)

denote

the homotopy fiber of the map IZ®(EGxaX)I-+ IZ®XIhG given by composing

1"11 with the inclusion of the fixed point set into the homotopy fixed point set.

PROPOSITION 3.1. For any simplicial set X with an action of the finite group

G there

are

natural isomorphisms

for all i

~

0.

This requires a lemma. Let

N.

be the normalization functor which assigns

to each simplicial abelian group its normalized chain complex and

N•

the dual