§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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