GENERALIZED TATE HOMOLOGY
11
Here
~P
denotes isomorphism after p-completion, and the sum is taken over
all
conjugacy classes of nontrivial subgroups H G.
REMARK:
Notice that this theorem says that the Tate homology of :E00
X+
de-
pends at least p-adically only on the singular subspace of X. In particular, if X
is a free G-space, the p-adic Tate homology of its suspension spectrum is zero.
We now turn our attention to the case in which G
=
8
1
,
the circle group,
and to the proof of 1.1(
4).
Our machinery actually leads to the definition of
generalized cyclic homology theories, and we will make some observations about
this.
Assume
X
is a finite G-cW complex. As observed in (DHK] and (J], the
chain complex
S,.(X)
of
X
has a natural cyclic structure. We will write
HC .. (X),
HC;(X),
and
HC,.(X)
respectively to denote the cyclic, negative cyclic, and
periodic cyclic homology groups of the cyclic module
S .. (X).
We refer the reader
to (Go2] and (J] for a description of these theories. It has been proved by
Goodwillie (Gol] that the groups
HC,.(X)
are the homotopy groups of the
spectrum H.t\(ES1
x
5
1
X+)· We will now describe the other two types of cyclic
homology in terms of the homotopy groups of spectra.
LEMMA
4.4. There is
a
natural isomorphism
HC:;(X) ~
1r,.(H .t\
X+)h
51

PROOF:
This lemma is proved in two steps. First, by (DHK] (Gol] we may
assume without loss of generality that
X
is the realization of a cyclic set [(.
By applying the free abelian group functor Z
® -
from the category of cyclic
sets to the category of cyclic modules ( cf. §3) and then taking realization, we
end up with an
5
1-space IZ ®
Kl
which is homotopy equivalent to the zero
space 0
00
(H .t\ X+). Moreover we can make this equivalence a weak equivariant
equivalence, and in particular produce an equivalence of infinite loop spaces
q,:
noo(H
.t\
X+)hSl
-+
IZ
®
Ilhsl.
The construction of
q,
is carried out using the techniques in the proof of 1.1(2).
The second step in the proof is then to show that
1r ..
1
Z ®
J(
lhS
1
is isomorphic in
positive dimensions to
HC;(X)
or equivalently to the negative cyclic homology
of the cyclic module Z®K. This is done by recalling the second quadrant double
chain complex for computing HC; constructed in
[J].
One compares this double
complex with the one obtained by first giving ES1 a cyclic decomposition and
then using this to construct a cocyclic cyclic abelian group Hoiilcyc( ES1
,
Z
®
K)
whose total space is IZ ®
J(
lh
51

Here "HOIIlcyc" denotes morphisms preserving
the cyclic structure. An argument analogous to the one which was used to
prove 1.1(2) proves that in positive total dimensions the above double complex
computes
1r.(Tot Homcyc(ES1
,
Z
®I)~
1r,.(IZ ® Klh 51
)
~
1r,.(H .t\ X+).
As in the proof of 1.1(2), the restriction to positive dimensions can be removed
by working with arbitrarily high suspensions
of
X.
This completes the proof of
the lemma.
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