COMBINATORIAL SYMMETRIES OF THE m-DIMENSIONAL BALL

19

Proof of 1.14: From the hypothesis of 0.6 it follows that neither R',

3R', nor any 9b-(R') (in 1.14(a)) has any n-torsion in its integral

homology groups. So 1.14 follows from 1.15.

In step two below we shall need the following lemma. Let us fix

for the rest of this paper a block of £ having minimal dimension m-k-1.

We denote this block by b3(£). There are corresponding sets b,(£'),

b3(£ ),b3(Rf), etc. Note b,(£) is a PL sphere of dimension m-k-1.

Lemma 1.16. The integral homology groups H*(£',b3(£')), H*(R',b3(Rr))

are all torsion prime to n, and ty acts trivially on them.

Proof of 1.16: By the hypothesis of 0.6, the integral homology groups

of (£f,b3(£')) and (R',b3(R')) are torsion prime to n. Consider the

exact sequence of integral homology groups for the pair (Rl,b^(R')):

3 . . . 3

-H (b3(Rf)) -Hq(R') - H (R',b3(R')) -

A^ B^ c^

q q q

By 1.13j \\) acts trivially on A for all q. By 1.14(b) \p acts trivially

H

on B for all q. It follows, as in the proof of 1.15, that ty acts

trivially on C for all q. To see that ip acts trivially on H C£',b, (.£'))

q q 3

for all q, note ty acts trivially on H (£') and on Hn(b,(^')) by 1.8,

q q «-

1.14(b), 1.13j, and the construction of ty. So this same exact sequence

can be applied to the pair (£f,b3(£')) to show J\ acts trivially on the

integral homology groups H_(£',b,(£')) for all q.

q j

Completion of Step 2. First four lemmas are stated and proven.

These lemmas are used to construct a blocked surgery problem $ having F

for range space. Finally the surgery problem $ is related to a second

surgery problem t.

Lemma 1.17. Each pair (b. (£ ) ,3b . (.£Q))is an oriented Poincare duality

pair. Suppose the indices i,j are such that b-C£) = 3b. (£) and dimension