18 LOWELL JONES

Lemma 1.15. Let R!, \p be as in 1.14. Let X denote a subcomplex of R!

which is the union of some of the b.(R'). Then i p acts trivially on the

Z(-)-homology groups

integers by adding

Z(-)-homolog y group of X. Here Z(—) is the ring obtained from the

1

Proof of 1.15: The proof uses double induction. By 1.13j, and induction,

ty acts trivially on the Z(-)-homology of any of the following spaces.

(a) b.(R'), for any b.(R') contained in R!.

(b) bi(Rl) f l Rf. _1, for any b^R') in R! but not in R'._r

(c) X n R!^.

Let {b.(Rf)|i€l} denote the subcomplexes of X which are as in (b)

above. So X = (XflR! -,) U ( U b.(R')). Proceeding inductively over the

3~L i€l X !

index set I, we assume \p acts trivially on the Z (--)-homology groups of

X ^ (XnRl -. ) U C U b.(R'))

3 i€l,

where I denotes the smallest £ integers of the index set I. Note that

(c) above shows this is true for I - 0. Consider the Mayer-Vietoris

sequence for the triple (X , ,X ,b. (R')) , where I.'JU {j} = L+i. This

is a long exact sequence of Z(-)-homology groups

3 . 3

- HqObj(R')) - Hq(X£)

+

Hq(bj(R')) - Hq(X£+1) +

K B. Cn

q q q

By induction on £ (this is the second induction hypothesis), \p acts

trivially on H (X ) for all q. Then by (a), (b) above, i| ; must act

trivially on A ,B . Let t: C -» • C be a generator of the action by i p on

C . Then 3(t(a)-a) = 0 for any a € C . THen, by exactness of the

sequence above, t(a) = a + y, where y is in the image (B +C ). So t(y) =y.

H H

Then computing:

a = tn(a) + ny * n-y = 0 * y = 0.

This shows ^ acts trivially on C above.