12
A. ADEM, R. L. COHEN, W. G. DWYER
REMARK:
A special case of this lemma, namely the case in which X
is
weakly
equivalent to the free loop space of a finite complex, was proved by Cohen and
Jones in
[CJ].
PROOF OF
1.1(4): The complex used to compute HC;
is
a subcomplex of the
one used to compute
HC •.
One therefore has a long exact sequence (see [Go2]
or [J])
As was discussed in [J], the connecting homomorphism
6
is induced on the
chain level by the 5 1-transfer map. Translating that discussion to the language
we are using here gives that the connecting map 6 is given by the composition
6: HCq-1(X)
~
7rq(H
/1.
L:(E51
Xsr
X+))=
7rq(E5~
1\sr
(H
/1.
X+)td
-+
7rq((H
/1.
X+)h 5
')
~ HC';(X).
T'
Here
T
is
the transfer map defined in §2 for the Lie group 5
1.
Note that
(-)ad
is just suspension because 5
1
is abelian. By the definition of the Tate homology
spectrum, this immediately gives the desired result.
Given the above it makes sense to define generalized cyclic homology groups
as follows. Let E be a spectrum representing a generalized homology theory E •.
Let X be a space with an 5 1-action. Define the E. generalized cyclic homology
groups of X by the formulas
and
By [Gol), lemma 3.4 and theorem 1.1(3) these definitions agree with the
definitions of the usual cyclic homology theories when E = H. Moreover the
homotopy exact sequence for the transfer map gives an exact sequence
analogous to the standard one above. However, the generalized "periodic cyclic
homology"
EC.(X)
need no longer be periodic- periodicity fails in general, for
instance, if E is the sphere spectrum.
We end by remarking that the case in which E is the sphere spectrum S has
proved very important in the study of Waldhausen's I-theory
[CCGH], [CJ],
[B]. In particular the negative cyclic stable homotopy of the free loop space
SC; (AX+) is the target of the Chern character map defined in [
CJ]
and of the
cyclic trace map defined in [B].
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