COMBINATORIAL SYMMETRIES OF THE m-DIMENSIONAL BALL

3

Theorem 0.7. Let K c N be as in 0.6, and n an odd positive integer.

Then there is a semi-free PL action tp: 7L xN - » N having K c N for fixed

point set if and only if h§(K) = 0 for all odd prime divisors p of n.

A complete classification of the group actions of 0.7 is given in §6.

Organization of Paper

There are six sections to the paper, which shall be outlined in a

moment.

The following reading procedure is recommended. First read the

outlines of the sections provided immediately below. Next read section 1;

read step 2 and lemmas 2.5, 2.6 (but not proofs) of section 2; read

lemma 3.2 (but not proof) of section 3; read lemma 4.0 and the first two

steps in its proof in section 4; read lemmas 5.1, 5.2 (but not proofs)

and the completion of the proof of 0.7 given in section 5. Finally read

all the steps in the proofs for lemmas 2.5, 2.6, 3.2, 4.0, 5.1. 5.2.

Outline of Section 1. The author has reduced the proof of 0.5

to completing surgery on a "blocked" normal map t (see [13]). This

reduction is reviewed in detail in this section, and also adapted to

the proof of 0.7.

Both the image blocks and the domain blocks of t are Poincare

duality pairs with fundamental group 7L . The domain blocks are not

manifolds, and the framing information is given in the category of

spherical fibrations. Thus the surgery procedures on the various blocks

of t need to be carried out in the Poincare duality category as discussed

in [12].

In the special case that K is a PL manifold, both the domain and

range of t are block spaces over a cell structure for K, but having

Poincare duality pairs for blocks instead of PL manifold pairs for blocks

as in [26]. Thus t can be identified with a mapping f: K - » l.i,.i ?n)

into F. Quinn's surgery classifying spaces (see [5] and [14] for a

description of these spaces).

If K is not a PL manifold, then the block structure of t is somewhat

more exotic. Choose a triangulation T for N which also triangulates K.