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

Abstract

Smith theory says that the fixed point set of a semi-free action of a group $G$ on a contractible space is ${\mathbb {Z}}_p$ -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.