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)

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)