Fixed Point Sets and the Fundamental Group I: Semi-free Actions on G-CW-Complexes

TL;DR

This paper extends Smith theory to semi-free group actions on finite CW-complexes, linking fixed point properties to K-theoretical obstructions and providing examples of their effects.

Contribution

It generalizes the converse of Smith theory to broader settings and identifies K-theoretical obstructions as key factors.

Findings

The converse of Smith theory holds when certain K-theoretical obstructions vanish.

Examples demonstrate how different K-theoretical obstructions influence fixed point properties.

Extension of Smith theory to various homotopy types of CW-complexes.

Abstract

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