If n = odd integer there is a further restriction on the fixed point
set K of cp: TL^Bm - * Bm, which is not implied by 0.1-0.3, which shall be
recalled now. Let p denote an odd prime number, and (M,3M) a finite
simplical pair which is an orientable (with respect to Z -homology)
Z -homology manifold pair. The author introduces a characteristic class
EhP(M) E Hm+4i-1((M,3M) Z)
in [15], [10],and proves that h£(M) vanishes if M is the fixed point set
of PL TL -action on an oriented PL manifold. In particular
Z h?(K) I Hk+41_1((K,3K),Z)
is well defined for any odd prime divisor p of n, and for any P.L.
subset (K,3K) c (Bm,3Bm) which satisfies 0.1, 0.2; and the following is
0.4. If K c Bm is the fixed point set for a semi-free PL action
cp: TL xBm - Bm, then h£(K) = 0 for all odd prime divisors p of n.
In this paper the following characterization of fixed point sets of
odd order actions cp: TL *Bm •- » Bm is proven.
Theorem 0.5. Let K c Bm denote a PL subset of the m-dimensional ball,
and n an odd positive integer. Suppose K satisfies 0,1-0,4, and
dim(Bm) - dim00 _ 6. Then K c Bm is the fixed point set of a semi-free
PL action cp:Zn*Bm c Bm.
The above theorem is a special case of the following more general
theorem. Let (N,9N) denote a compact PL manifold pair, and K c N denote
a compact PL subset of N, with 3K = K n3N, satisfying:
0.6 (a) iri(N} = 0, TriC3N) = 0 for i-1,2,
(b) H*(N,Zn) = HQ(N,Zn) - Zn,
(c) (K,3K) satisfies 0.1, 0.2,
(d) dim(N)-dim(K) is even and greater than 5.
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