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.

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.