$G$-surgery on 3-dimensional Manifolds for Homology Equivalences

Abstract

For a finite group $G$ and a $G$-map $f:X\to Y$ of degree one, where $X$ and $Y$ are compact, connected, oriented, 3-dimensional, smooth $G$-manifolds, we give an obstruction element $\sigma (f)$ in a $K$-theoretic group called the Bak group, with the property: $\sigma (f)=0$ guarantees that one can perform $G$-surgery on $X$ so as to convert $f$ to a homology equivalence $f^{\prime }:X^{\prime }\to Y$. Using this obstruction theory, we determine the $G$-homeomorphism type of the singular set of a smooth action of $A_5$ on a 3-dimensional homology sphere having exactly one fixed point, where $A_5$ is the alternating group on five letters.