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].

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].