A converse to the P. A. Smith theorem for nonunitary homology spheres

If p is an odd prime and F is the fixed point set of a smooth Z_p action on Sⁿ or Dⁿ, then F is a smooth manifold with a unitary structure. Conversely; most Z_p homology disks or spheres with unitary structures are fixed point sets of smooth Z_p actions on Dⁿ or Sⁿ for suitable n. The results of this paper show that an arbitrary oriented mod p homology disk or sphere is the fixed point set of a smooth Z_p action on some Zl/2]-homology disk or sphere. This result is in general the best possible.Partially supported by NSF Grants MCS 81-04852 and MCS 83-00669