§0. Introduction to the Problem

Notation: Z ; additive cyclic group of order n.

TL ; multiplicative cyclic group of order n.

B ; unit ball in m-dimensional Euclidean space,

cp:ZnxBm - • Bm; P.L. group action on Bm by TL .

A semi-free group action cp:ZSxB - + Bm is one such that for any

x G B the orbit set {cp(t,x) |tGZZ } contains exactly one point or n

points. The set of fixed points {xGBm |cp(t,x)=x vtG TL } will be denoted K.

Set aK = Kn8Bm.

P. A. Smith has studied semi-free group actions cp: TLn*Bm - Bm in

terms of their fixed point set (see [2 7]). He has proven that K satisfies

these two properties.

0.1. (K,9K) is a Z -homology manifold pair, of dimension k ^ m.

0 if i0, and

0

±. Hi(K,Zn) =

zn if i-o

Hi((K,9K),Zn)

0 if i^k

Zn if i-k

It is also well known that K must satisfy the following

0.3. If n has an odd divisor, then m-k = 0 mod 2.

The author has shown in [13] that if K = B satisfies 0.1, 0.2, 0.3,

and in addition satisfies H*(K,Z2) = HQ(K,Z2) = Z2, then K = Bm is the

fixed point set of some P.L. semi-free group action cp: TL xB - Bm

(provided m-k _ 6). For example, if K c Bm satisfies 0.1, 0.2, 0.3, and

n = even, then H^(K,Z2) = HQ(K,Z2) = Z2 can be deduced from 0.2.

In the rest of this paper it is assumed that n is an odd integer.

Received by the editors May 3, 1982 and, in revised form May 2, 1986.

The author was supported in part by the NSF.