Fixed Point Sets and the Fundamental Group II: Euler Characteristics

TL;DR

This paper extends Oliver's results by demonstrating that the Euler characteristic determines whether a given homotopy type can be realized as a fixed point set of a finite G-CW-complex, highlighting the roles of fundamental groups and component structure.

Contribution

It generalizes the obstruction theory for fixed point sets to include fixed point sets of specified homotopy types, incorporating fundamental group and component considerations.

Findings

Euler characteristic determines fixed point set realizability.

Trace maps in K_0 reveal fundamental group roles.

Fixed point set structure depends on component configuration.

Abstract

For a group $G$ of not prime power order, Oliver showed that the obstruction for a finite CW-complex $F$ to be the fixed point set of a contractible finite $G$-CW-complex is the Euler characteristic $\chi(F)$. He also has the similar results for compact Lie group actions. We show that the analogous problem for $F$ to be the fixed point set of a finite $G$-CW-complex of some given homotopy type is still determined by the Euler characteristic. Using trace maps in $K_0$, we also see that there are interesting roles for the fundamental group and the component structure of the fixed point set.