2

A. ADEM, R. L. COHEN, W. G. DWYER

THEOREM 1.1. Let G be a compact Lie group and E a spectrum with an action

of G. Then there exists a spectrum fiG(E) with the following properties:

(1) E ~--+

iJG

(E) is a functor from the category of spectra with an action of

G to the category of spectra. This functor carries weak equivalences to

weak equivalences.

(2) If G is a finite group, X is a G-GW complex, and E is the spectrum H A

(X+) with the induced action of G, then there is a natural isomorphism

(3) If G is a finite group, X is a G-CW complex, and E is the suspension

spectrum of X+ with the induced action ofG, then fiG(E) is equivalent

to the fiber of the Kahn-Priddy transfer map [KP}

( 4) If G is the circle group

S1

,

X is a G-CW complex, and E is the spectrum

H A X+ with the induced action of G, then 7r.(fiG(E)) is the periodic

cyclic homology as defined by Jones

{J

J

and Good willie {Go2} of the the

chain complex S.(X).

REMARK: In (3) above,

Q(Y)

=

limO"E"(Y). For any G-space

Z,

zhG

denotes

----+

the homotopy fixed point set MapG(EG, Z).

Part (3) of this theorem reflects the fact that the spectra fiG (E) are in fact

constructed in terms of homotopy fibers of certain transfer maps. To be more

precise, assume that G is a compact Lie group with Lie algebra

g.

Given a

G-space X, one can use the adjoint representation of G on g to construct an

induced vector bundle

ad: EG

Xa

(g

x X)-+ EG

Xa

X.

Let (EG

XG

X)ad

denote the Thorn space of this vector bundle. The Becker-

Schultz transfer (or umkehr) map [BS] can be viewed as a map of spectra

which when G is finite agrees with the Kahn-Priddy transfer mentioned in 1.1(3).

(The definition of the above homotopy fixed point spectrum will be given in §2.)

Now let E be any spectrum with an action of

G.

Using the Becker-Schultz

construction we will produce an E-transfer map

which agrees with the Becker-Schultz transfer when E is E

00

X. The spectrum

fiG (E) is defined to be the stable fiber of

T£.

The generalized Tate homology

groups of

G

with coefficients in

E

are defined by

A. ADEM, R. L. COHEN, W. G. DWYER

THEOREM 1.1. Let G be a compact Lie group and E a spectrum with an action

of G. Then there exists a spectrum fiG(E) with the following properties:

(1) E ~--+

iJG

(E) is a functor from the category of spectra with an action of

G to the category of spectra. This functor carries weak equivalences to

weak equivalences.

(2) If G is a finite group, X is a G-GW complex, and E is the spectrum H A

(X+) with the induced action of G, then there is a natural isomorphism

(3) If G is a finite group, X is a G-CW complex, and E is the suspension

spectrum of X+ with the induced action ofG, then fiG(E) is equivalent

to the fiber of the Kahn-Priddy transfer map [KP}

( 4) If G is the circle group

S1

,

X is a G-CW complex, and E is the spectrum

H A X+ with the induced action of G, then 7r.(fiG(E)) is the periodic

cyclic homology as defined by Jones

{J

J

and Good willie {Go2} of the the

chain complex S.(X).

REMARK: In (3) above,

Q(Y)

=

limO"E"(Y). For any G-space

Z,

zhG

denotes

----+

the homotopy fixed point set MapG(EG, Z).

Part (3) of this theorem reflects the fact that the spectra fiG (E) are in fact

constructed in terms of homotopy fibers of certain transfer maps. To be more

precise, assume that G is a compact Lie group with Lie algebra

g.

Given a

G-space X, one can use the adjoint representation of G on g to construct an

induced vector bundle

ad: EG

Xa

(g

x X)-+ EG

Xa

X.

Let (EG

XG

X)ad

denote the Thorn space of this vector bundle. The Becker-

Schultz transfer (or umkehr) map [BS] can be viewed as a map of spectra

which when G is finite agrees with the Kahn-Priddy transfer mentioned in 1.1(3).

(The definition of the above homotopy fixed point spectrum will be given in §2.)

Now let E be any spectrum with an action of

G.

Using the Becker-Schultz

construction we will produce an E-transfer map

which agrees with the Becker-Schultz transfer when E is E

00

X. The spectrum

fiG (E) is defined to be the stable fiber of

T£.

The generalized Tate homology

groups of

G

with coefficients in

E

are defined by