Published online by Cambridge University Press: 03 August 2023
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.