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.
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