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:

(EG
X
a
X)-+
Z®(EG
x X) whose image
clearly lies in the G-fixed-points of
(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
Previous Page Next Page