§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.

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