Fixed point sets and the fundamental group I: semi-free actions on G-CW-complexes

Smith theory says that the fixed point set of a semi-free action of a group G on a contractible space is Zp-acyclic for any prime factor p of the order of G. Jones proved the converse of Smith theory for the case G is a cyclic group acting semi-freely on contractible, finite CW-complexes. We extend the theory to semi-free group actions on finite CW-complexes of given homotopy types, in various settings. In particular, the converse of Smith theory holds if and only if a certain K-theoretical obstruction vanishes. We also give some examples that show the geometrical effects of different types of K-theoretical obstructions.