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

§4.

Two EXAMPLES

In this section we analyze two examples of generalized Tate homology. In

the first example the group

G

is a finite p-group and the spectrum involved is

the suspension spectrum of a finite G-cW complex. In this case we will use

Carlsson's theorem (the Segal conjecture [Ca]) to give a complete p-prirnary

calculation of the generalized Tate homology. In the second example the group

is the circle group 5

1

and the spectrum involved isH t\X+ for X a finite G-cW

complex. In this case we relate the corresponding generalized Tate homology

theory to the periodic cyclic homology theory of Goodwillie [Go2] and Jones

[J].

We begin with the assumption that G is a finite p-group, where

p

is

a prime

number. Let X be a finite G-cW complex. We recall torn Dieck's theorem

[tD]

describing the fixed points Qa(X+

)G.

THEOREM

4.1

[tD].

Suppose that G is

a

finite p-group and that X is

a

finite

G-GW complex. For H

a

subgroup of G, let N(H) G be the normalizer

subgroup of H, and let W(H)

=

N(H)/H be tl1e corresponding Weyl group.

Then there is

a

homotopy equivalence of infinite loop spaces

¢:IT

Q((EW(H) Xw(H) xH)+)-+ Qa(X+)G

H

where the product is taken over all conjugacy classes of subgroups H G.

Moreover the restriction of¢ to the factor corresponding to the trivial subgroup

is given by the Kahn-Priddy transfer map

We now recall Carlsson's theorem [Ca].

THEOREM

4.2 [Ca]. For G a finite p- group and X

a

finite G-CW complex,

the natural map

is

a

weak equivalence when completed at the prime p.

Consider the Tate homology spectrum fiG (E

00

X+) as defined in §2. By def-

inition, its corresponding zero space

noofiG

(E

00

X+) is the fiber of the map of

infinite loop spaces r : Q(EG xa X+)

-+

Qa(X+)hG

~

Q(X+)hG. Thus by

combining theorems 4.1 and 4.2 we obtain the following calculation of the p-adic

completion of the Tate homology spectrum:

THEOREM

4.3. For G

a

finite p-group and X

a

finite G-CW complex there is

an isomorphism

H~(E

00

X+)

=p

EB

7T!(EW(H) Xw(H) xH).

H;t{e}

§4.

Two EXAMPLES

In this section we analyze two examples of generalized Tate homology. In

the first example the group

G

is a finite p-group and the spectrum involved is

the suspension spectrum of a finite G-cW complex. In this case we will use

Carlsson's theorem (the Segal conjecture [Ca]) to give a complete p-prirnary

calculation of the generalized Tate homology. In the second example the group

is the circle group 5

1

and the spectrum involved isH t\X+ for X a finite G-cW

complex. In this case we relate the corresponding generalized Tate homology

theory to the periodic cyclic homology theory of Goodwillie [Go2] and Jones

[J].

We begin with the assumption that G is a finite p-group, where

p

is

a prime

number. Let X be a finite G-cW complex. We recall torn Dieck's theorem

[tD]

describing the fixed points Qa(X+

)G.

THEOREM

4.1

[tD].

Suppose that G is

a

finite p-group and that X is

a

finite

G-GW complex. For H

a

subgroup of G, let N(H) G be the normalizer

subgroup of H, and let W(H)

=

N(H)/H be tl1e corresponding Weyl group.

Then there is

a

homotopy equivalence of infinite loop spaces

¢:IT

Q((EW(H) Xw(H) xH)+)-+ Qa(X+)G

H

where the product is taken over all conjugacy classes of subgroups H G.

Moreover the restriction of¢ to the factor corresponding to the trivial subgroup

is given by the Kahn-Priddy transfer map

We now recall Carlsson's theorem [Ca].

THEOREM

4.2 [Ca]. For G a finite p- group and X

a

finite G-CW complex,

the natural map

is

a

weak equivalence when completed at the prime p.

Consider the Tate homology spectrum fiG (E

00

X+) as defined in §2. By def-

inition, its corresponding zero space

noofiG

(E

00

X+) is the fiber of the map of

infinite loop spaces r : Q(EG xa X+)

-+

Qa(X+)hG

~

Q(X+)hG. Thus by

combining theorems 4.1 and 4.2 we obtain the following calculation of the p-adic

completion of the Tate homology spectrum:

THEOREM

4.3. For G

a

finite p-group and X

a

finite G-CW complex there is

an isomorphism

H~(E

00

X+)

=p

EB

7T!(EW(H) Xw(H) xH).

H;t{e}