GENERALIZED TATE HOMOLOGY 9

Kahn-Priddy transfer

TKP

combines with the diagonal map of

SP

00

(Sm)

to give

a transfer map

whose image lies in the fixed point set.

It follows from the proof of 1.1(3) and

a little manipulation that passing to the limit in m and V with r(m, V) and

extracting homotopy fixed points in the range will give the transfer map of §2

whose fiber is :HG(H A X+)· (A similar result holds with H replaced by any

other spectrum on which G acts trivially.) For any based space Z there is a

natural map

h : C(V, Z)

-+

SP

00

(Z)

obtained by adding coordinates. One can

check that the following diagram commutes on the point set level:

SP

00

(Sm)

A

B+

r(m,V)

C(V,SP

00

(sm) A

X+)

61

lh

SP

00

(Sm

A

B+)

tr 1(m)

SP

00

(Sm

A X+)

---+

where

tr' (

m) is constructed from the map

tr'

above in the obvious way and the

map labeled 8 is the standard pairing. Now loop the diagram down m times and

pass to the limit in m and V. The vertical arrows become weak equivalences

and, as noted above, the upper horizontal map (after passing to homotopy fixed

points in the range) determines the zero space in the n-spectrum corresponding

to HG(H A X+)· It follows that this zero space can be computed as the homotopy

fiber of the map

lim tr'(m)

~nmSP

00

(Sm AB+)

_-+ __ _.

~(nmspoo(sm AX+))hG.

m m

However, the standard pairing

6

produces by adjointness a commutative diagram

~nmSP 00 (Sm

A

B+)

m

tr'

---+

I~tr'(m)

lim(nmspoo(sm A

X+))

--+

m

in which the vertical arrows induce isomorphisms on homotopy in positive dimen-

sions. Proposition 3.4 then implies that

Hy(H

A

X+)

is isomorphic to

ilf(X)

for

i

0. The general theorem is proven by applying this result to an arbitrarily

high suspension of X; it is easy to see that suspending X has essentially the effect

of shifting both the classical Tate homology and the generalized Tate homology

up by one in dimension.

Kahn-Priddy transfer

TKP

combines with the diagonal map of

SP

00

(Sm)

to give

a transfer map

whose image lies in the fixed point set.

It follows from the proof of 1.1(3) and

a little manipulation that passing to the limit in m and V with r(m, V) and

extracting homotopy fixed points in the range will give the transfer map of §2

whose fiber is :HG(H A X+)· (A similar result holds with H replaced by any

other spectrum on which G acts trivially.) For any based space Z there is a

natural map

h : C(V, Z)

-+

SP

00

(Z)

obtained by adding coordinates. One can

check that the following diagram commutes on the point set level:

SP

00

(Sm)

A

B+

r(m,V)

C(V,SP

00

(sm) A

X+)

61

lh

SP

00

(Sm

A

B+)

tr 1(m)

SP

00

(Sm

A X+)

---+

where

tr' (

m) is constructed from the map

tr'

above in the obvious way and the

map labeled 8 is the standard pairing. Now loop the diagram down m times and

pass to the limit in m and V. The vertical arrows become weak equivalences

and, as noted above, the upper horizontal map (after passing to homotopy fixed

points in the range) determines the zero space in the n-spectrum corresponding

to HG(H A X+)· It follows that this zero space can be computed as the homotopy

fiber of the map

lim tr'(m)

~nmSP

00

(Sm AB+)

_-+ __ _.

~(nmspoo(sm AX+))hG.

m m

However, the standard pairing

6

produces by adjointness a commutative diagram

~nmSP 00 (Sm

A

B+)

m

tr'

---+

I~tr'(m)

lim(nmspoo(sm A

X+))

--+

m

in which the vertical arrows induce isomorphisms on homotopy in positive dimen-

sions. Proposition 3.4 then implies that

Hy(H

A

X+)

is isomorphic to

ilf(X)

for

i

0. The general theorem is proven by applying this result to an arbitrarily

high suspension of X; it is easy to see that suspending X has essentially the effect

of shifting both the classical Tate homology and the generalized Tate homology

up by one in dimension.