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

G

t(h)

G . . .

N.(Z 0 (EG x X)) -+ t(A) , It follows essentially by mspect10n that the

homology groups of F2 are the classical Tate hyperhomology groups of

G

with

coefficients in N.(Z 0 X) ""S.(IXI). The proposition is now a consequence of

the existence of a commutative diagram

N.(h)

N.((Z 0 (EG

x

X))G)

---+

N. Tot(AG)

q(BG)

1 1

t(h)

---+

in which the right-hand vertical arrow (which is the composite of 77(AG) and the

inclusion t+(AG) -+ t(AG)) is a homology equivalence in non-negagtive dimen-

siOns.

Let

SP00

denote the infinite symmetric product construction, either in the

category of topological spaces or in the category of simplicial sets. Let X+ be

the space obtained by adjoining a disjoint G-fixed basepoint to

X.

By

[Sp]

there

is a weak G-equivalence between ISP

00

(X+)I and SP

00

1X+I

Define a simplicial transfer tr' :

SP

00

(EG xa X+) -+

SP00

(EG x X+) as

follows: for each n-simplex in EG xa X+ take the G-orbit in

SP

00

(EG

x

X+)

which it represents. The map tr' is a simplicial analogue of the transfer defined

by

L.

Smith

[Sm].

There is an evident natural G-equivariant group completion

map

gp: SP

00

(X+)-+

Z

0 X which sends the added basepoint"+" to 0 (this

map is an isomorphism on homotopy in strictly positive dimensions).

The following diagram then commutes :

tr

1

SP00

(EG xa X+)

---+

SP

00

(EG

X

X+)

gp

1

lgp

Z0(EGxaX)

--+

tr

Z0(EG

X

X).

Both transfers fall into the fixed-point sets. Let Tip(X) denote the homotopy

fiber of the map

SP00

1EG Xa X+l-+

SP00

IEG x XlhG -+ SP

00

IXIhG corre-

sponding to tr'. By using the homotopy invariance property of homotopy fixed

point sets [BK2, XI, 5.6] we obtain by 3.1 the following proposition.

PROPOSITION

3.3. For any simplicial set

X

with an action of the finite group

G there are natural isomorphisms

for all i

0.

PROOF OF

1.1(2): Assume as in the proof of 1.1(3) (§2) that X is a finite G-

CW complex on which

G

acts freely and that

X

---+

V

is an embedding of

X

into an orthogonal representation space of G. Let B be the quotient XjG. Take

as a model of the Eilenberg-MacLane space K(Z, m) the space

SP

00

(Sm).

The

G

t(h)

G . . .

N.(Z 0 (EG x X)) -+ t(A) , It follows essentially by mspect10n that the

homology groups of F2 are the classical Tate hyperhomology groups of

G

with

coefficients in N.(Z 0 X) ""S.(IXI). The proposition is now a consequence of

the existence of a commutative diagram

N.(h)

N.((Z 0 (EG

x

X))G)

---+

N. Tot(AG)

q(BG)

1 1

t(h)

---+

in which the right-hand vertical arrow (which is the composite of 77(AG) and the

inclusion t+(AG) -+ t(AG)) is a homology equivalence in non-negagtive dimen-

siOns.

Let

SP00

denote the infinite symmetric product construction, either in the

category of topological spaces or in the category of simplicial sets. Let X+ be

the space obtained by adjoining a disjoint G-fixed basepoint to

X.

By

[Sp]

there

is a weak G-equivalence between ISP

00

(X+)I and SP

00

1X+I

Define a simplicial transfer tr' :

SP

00

(EG xa X+) -+

SP00

(EG x X+) as

follows: for each n-simplex in EG xa X+ take the G-orbit in

SP

00

(EG

x

X+)

which it represents. The map tr' is a simplicial analogue of the transfer defined

by

L.

Smith

[Sm].

There is an evident natural G-equivariant group completion

map

gp: SP

00

(X+)-+

Z

0 X which sends the added basepoint"+" to 0 (this

map is an isomorphism on homotopy in strictly positive dimensions).

The following diagram then commutes :

tr

1

SP00

(EG xa X+)

---+

SP

00

(EG

X

X+)

gp

1

lgp

Z0(EGxaX)

--+

tr

Z0(EG

X

X).

Both transfers fall into the fixed-point sets. Let Tip(X) denote the homotopy

fiber of the map

SP00

1EG Xa X+l-+

SP00

IEG x XlhG -+ SP

00

IXIhG corre-

sponding to tr'. By using the homotopy invariance property of homotopy fixed

point sets [BK2, XI, 5.6] we obtain by 3.1 the following proposition.

PROPOSITION

3.3. For any simplicial set

X

with an action of the finite group

G there are natural isomorphisms

for all i

0.

PROOF OF

1.1(2): Assume as in the proof of 1.1(3) (§2) that X is a finite G-

CW complex on which

G

acts freely and that

X

---+

V

is an embedding of

X

into an orthogonal representation space of G. Let B be the quotient XjG. Take

as a model of the Eilenberg-MacLane space K(Z, m) the space

SP

00

(Sm).

The