GENERALIZED TATE HOMOLOGY

5

moreover, if e0 is the basepoint of E (necessarily fixed by the action of G), the

inclusion

5°

-+

E+ determined by e0 induces maps Q(S0

)

-+

Q(E+)

-+

E and

(Q(S0 ))ha-+ (Q(E+))ha-+ Eha which are null homotopic. It follows that the

map

tE

induces a map

where ( EG + /\a

E)

ad

denotes the quotient of ( EG

x

a E)

ad

by ( EG

X {

e0

} )ad

=

BGad. If E ={En} is an omega spectrum with an action of G then the source

spaces of the maps

TEn

fit together to form a spectrum which we will denote

(EG+ /\a E)ad. Similarly, the target spaces fit together to form a spectrum

whose structure maps are induced in the obvious way by the structure maps

of the spectrum E. We call this second spectrum the homotopy fixed point

spectrum of the action of G on

E

and denote it by Eha. Notice furthermore

that the maps

TEn

respect the structure maps of these two spectra and hence

induce a map of spectra

If

E

is simply a spectrum with an action of G (not necessarily an omega spec-

trum) we define Eha to be (E')ha, where E' is the associated omega spectrum,

and obtain as before a transfer map TE·

DEFINITION

2.2: If G is a compact Lie group and E a spectrum with an action

of G, we define

(1)

the generalized Tate homology spectrum

Ha(E)

to be the (stable) fiber

of the map of spectra given by the transfer TE : (EG+ /\a E)ad

-+

Eha,

and

(2)

the generalized Tate homology groups

H~(E)

to be the homotopy groups

of

Ha(E).

We will now prove 1.1(3). To do this it is convenient to use configuration space

approximations to loop spaces, as described in [M2]. To be more precise, if

V

is a finite dimensional vector space let

This configuration spaces has a free symmetric group

('En)

action given by per-

muting coordinates. Given a based space Z, let C(V, Z) be the complex

C(V, Z) =

u

F(V, n)

X!;n

zn

I"'

nO

where the components in this union are patched together by a basepoint relation

as described in [M2]. For

Z

path connected, there is a well known weak ho-

motopy equivalence [M2] [H] a :

C(V, Z)

-+

nv'Ev

(Z). If

V

is a G-orthogonal

representation space and

Z

is a G-space with a basepoint that is fixed under the

action, then C(V, Z) has an induced G-action and it is easily observed that the

5

moreover, if e0 is the basepoint of E (necessarily fixed by the action of G), the

inclusion

5°

-+

E+ determined by e0 induces maps Q(S0

)

-+

Q(E+)

-+

E and

(Q(S0 ))ha-+ (Q(E+))ha-+ Eha which are null homotopic. It follows that the

map

tE

induces a map

where ( EG + /\a

E)

ad

denotes the quotient of ( EG

x

a E)

ad

by ( EG

X {

e0

} )ad

=

BGad. If E ={En} is an omega spectrum with an action of G then the source

spaces of the maps

TEn

fit together to form a spectrum which we will denote

(EG+ /\a E)ad. Similarly, the target spaces fit together to form a spectrum

whose structure maps are induced in the obvious way by the structure maps

of the spectrum E. We call this second spectrum the homotopy fixed point

spectrum of the action of G on

E

and denote it by Eha. Notice furthermore

that the maps

TEn

respect the structure maps of these two spectra and hence

induce a map of spectra

If

E

is simply a spectrum with an action of G (not necessarily an omega spec-

trum) we define Eha to be (E')ha, where E' is the associated omega spectrum,

and obtain as before a transfer map TE·

DEFINITION

2.2: If G is a compact Lie group and E a spectrum with an action

of G, we define

(1)

the generalized Tate homology spectrum

Ha(E)

to be the (stable) fiber

of the map of spectra given by the transfer TE : (EG+ /\a E)ad

-+

Eha,

and

(2)

the generalized Tate homology groups

H~(E)

to be the homotopy groups

of

Ha(E).

We will now prove 1.1(3). To do this it is convenient to use configuration space

approximations to loop spaces, as described in [M2]. To be more precise, if

V

is a finite dimensional vector space let

This configuration spaces has a free symmetric group

('En)

action given by per-

muting coordinates. Given a based space Z, let C(V, Z) be the complex

C(V, Z) =

u

F(V, n)

X!;n

zn

I"'

nO

where the components in this union are patched together by a basepoint relation

as described in [M2]. For

Z

path connected, there is a well known weak ho-

motopy equivalence [M2] [H] a :

C(V, Z)

-+

nv'Ev

(Z). If

V

is a G-orthogonal

representation space and

Z

is a G-space with a basepoint that is fixed under the

action, then C(V, Z) has an induced G-action and it is easily observed that the