A. ADEM, R. L. COHEN, W. G. DWYER
A special case of this lemma, namely the case in which X
equivalent to the free loop space of a finite complex, was proved by Cohen and
1.1(4): The complex used to compute HC;
a subcomplex of the
one used to compute
One therefore has a long exact sequence (see [Go2]
As was discussed in [J], the connecting homomorphism
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
the transfer map defined in §2 for the Lie group 5
is just suspension because 5
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
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
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
[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 [
and of the
cyclic trace map defined in [B].