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
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