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
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