GENERALIZED TATE HOMOLOGY 7
functor which assigns to each cosimplicial abelian group its normalized cochain
complex [BKl, §2]. Suppose that A is a cosimplicial simplicial abelian group
and that B = N.N• A is the double chain complex obtained by normalizing A
in both directions [BKl, 2.4]. Define a chain complex t(A) by letting t(A)n
be the product
ili-i=n
Bf
and using the standard formula [BKl, §3] for the
differential. The chain complex t(A) in general has entries in both positive and
negative degrees; we will let t+(A) stand for the sub chain complex which is
zero in negative dimensions, agrees with t(A) in dimensions greater than 0, and
contains in dimension
0
the cycles of t(A)
0

LEMMA 3.2. For any cosimplicial simplicial abelian group A there
is a
natural
chain map
17(A) : N. Tot( A) __. t+(A)
which induces an isomorphism on homology.
REMARK: Here Tot( A)
is
defined as in [BK2, X, §3]; it is dear that applying the
Tot functor to a cosimplicial simplicial abelian group yields a simplicial abelian
group.
PROOF OF 3.2: This result is in some sense implicit in [BKl] and [BK2, X,
§6] and is in any case very similar to the Eilenberg-Zilber theorem [Ml, p. 129].
We will only sketch the proof. The first step is to construct natural maps 77n :
Tot(A)n
--
t(A)n, (n
2:
0). Since Tot( -)n is represented by the cosimplical
simplicial abelian group Z 0
(~X ~[n])
[BK2, §3], such a map 77n is determined
universally by an element
en
E
t(Z 0
(~
X
~[n])n.
By definition, such an
en
is
specified by a collection
en(k) (k 2:
0), where
en(k)
lies in
N.Z0(~[k]
x
~[n])k+n
and
e0
(k) maps trivially to N.z 0
(~[k-
1] x
~[n])
under the maps induced
by the
k
standard collapses si :
~[k]
__.
~[k
- 1]. Choose
en(I)
to be the
fundamental cycle of
~[n]
x
~[k]
provided by the Eilenberg-MacLane shuffi.e
formula [Ml, p. 133]. A short calculation then shows that the maps 77n vanish
on degenerate elements of Tot(A) and combine to produce the desired chain
map 77(A) :
N.
Tot(A)
--
t+(A). The fact that 77(A) induces an isomorphism
on homology follows from the methods of [BKl]; both Tot(A) and t+(A) can
be expressed as inverse limits in a natural way, and 7J(A) induces equivalences
between the constituents of the corresponding towers of chain complexes.
PROOF OF 3.1: Let A denote the cosimplicial simplicial abelian group with
A~=
Homsets(EGp, Z
0
(EG x X)q) and B the cosimplicial simplicial abelian
group defined in a corresponding way with EG replaced by
~[0].
Note that
all of the cosimplicial operators of B are isomorphisms, so that N. Tot( B)
~
N.(z 0 (EG
X
X))~
t+(B)
~
t(B). The unique map EG __.
~[0]
induces a G-
map h : B --A of cosimplicial simplicial objects. By [BK2, X, 3.3(i)] the fixed
point set Tot(A)G is isomorphic to the simplicial function complex of G-maps
EG-- EG
X
X and thus ITot(A)GI is weakly equivalent to IZ 0 (EG x X)lhG.
Let F1 be the homotopy fiber (in the category of simplicial abelian groups) of the
composite Z 0 (EG XG X).!.:.. (Z 0 (EG
X
X))G
TO:.S.h)
Tot(A)G.
It
follows that
Tt(X) is naturally homotopy equivalent to IF1I· Let F2 be the homotopy fiber
(in the category of chain complexes) of the composite
N. (
Z 0 ( EG x
G
X))
N
~r)
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