Fixed Points of Semi-free G-CW-complex with Prescribed Homotopy Type

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