COMBINATORIAL SYMMETRIES OF THE m-DIMENSIONAL BALL 9

Outline of Section 5. The proof of Theorem 0.7 is completed by

showing that surgery can be completed on (£.,££•) even if aT(g.) f 0

0r aT(*i|«M.) * °-

It is convenient at this time to review the geometric interpretation

of the characteristic classes h^(K) of 0.4, as given in section 2 of

[10]. The natural map Z, . - Z , from the integers localized at p to

the integers mod p, induces a homomorphism 3: WT(Z. .) - W(Z ),where

WT(Zf •,) denotes the torsion subgroup of the Witt group of non-singular

forms over Z , having unit determinant in Z ^. Because each

(P)

IP)r

(g..gM )" (K) is a Z, -homology manifold, the invariant o"T(gi |*M ) lies

I i I i

in the sub-group w

T

(zrpO =• W

T

CQ)- Set

iii wp(g.) = B(CaT(gi|6Mi)).

Then by the definition in [10],the class h§(K) can be reconstructed from

the values w (g.) taken on by w ( ) on the characteristic variety

{g-;: M. -+ R/R i = l,2, ...,£}; and these values of w ( ) can be reconstructed

from h£(K). In particular, h£(K) = 0 » w (g.) = 0 for i=l,2,...,£ (see

[[10], 2.5]). Thus under the hypothesis of Theorem 0.7 we have w Cgj) = 0

for i=l,2,...,& and all p dividing n, but not necessarily o"T(g.,-w ) = 0.

The argument of section 5 starts by showing that the obstruction

to completing surgery on (t-,6t«) depends only on the values of

6T(g...M ), dim(N) and dim(K) (see 5.1 below). This part of the

argument relies on sections 3,4.

The next step in the argument of section 5 is the construction, for

each g.: M. -* • R/R, of a PL semi-free action cp*:1 xNl - N1 having Kf e NT

for fixed point set, satisfying the following (see 5.2 below).

(a) K1 = N! satisfies 0.6

(b) dim(N') - dim(K') = dim(N) - dim(K)

^ aTCSj|6Mj3 = aTCSi|6Mi)-

Here (in (c)) g'.: Ml + R'/Rf is some element in a characteristic variety

{g!: M| - R'/R' i=l, 2,. ..,£'} for R'/R', where R*,R', V , ( *!_, tV) are

defined for K' c N' as were R, R, t, (^,6^) for K c N.