10 A. ADEM, R. L. COHEN, W. G. DWYER
§4.
Two EXAMPLES
In this section we analyze two examples of generalized Tate homology. In
the first example the group
G
is a finite p-group and the spectrum involved is
the suspension spectrum of a finite G-cW complex. In this case we will use
Carlsson's theorem (the Segal conjecture [Ca]) to give a complete p-prirnary
calculation of the generalized Tate homology. In the second example the group
is the circle group 5
1
and the spectrum involved isH t\X+ for X a finite G-cW
complex. In this case we relate the corresponding generalized Tate homology
theory to the periodic cyclic homology theory of Goodwillie [Go2] and Jones
[J].
We begin with the assumption that G is a finite p-group, where
p
is
a prime
number. Let X be a finite G-cW complex. We recall torn Dieck's theorem
[tD]
describing the fixed points Qa(X+
)G.
THEOREM
4.1
[tD].
Suppose that G is
a
finite p-group and that X is
a
finite
G-GW complex. For H
a
subgroup of G, let N(H) G be the normalizer
subgroup of H, and let W(H)
=
N(H)/H be tl1e corresponding Weyl group.
Then there is
a
homotopy equivalence of infinite loop spaces
¢:IT
Q((EW(H) Xw(H) xH)+)-+ Qa(X+)G
H
where the product is taken over all conjugacy classes of subgroups H G.
Moreover the restriction of¢ to the factor corresponding to the trivial subgroup
is given by the Kahn-Priddy transfer map
We now recall Carlsson's theorem [Ca].
THEOREM
4.2 [Ca]. For G a finite p- group and X
a
finite G-CW complex,
the natural map
is
a
weak equivalence when completed at the prime p.
Consider the Tate homology spectrum fiG (E
00
X+) as defined in §2. By def-
inition, its corresponding zero space
noofiG
(E
00
X+) is the fiber of the map of
infinite loop spaces r : Q(EG xa X+)
-+
Qa(X+)hG
~
Q(X+)hG. Thus by
combining theorems 4.1 and 4.2 we obtain the following calculation of the p-adic
completion of the Tate homology spectrum:
THEOREM
4.3. For G
a
finite p-group and X
a
finite G-CW complex there is
an isomorphism
H~(E
00
X+)
=p
EB
7T!(EW(H) Xw(H) xH).
H;t{e}
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