Richness of Smith equivalent modules for finite gap Oliver groups

Abstract

Let $G$ be a finite group not of prime power order. Two real $G$-modules $U$ and $V$ are $\mathcal{P}(G)$-connectively Smith equivalent if there exists a homotopy sphere with smooth $G$-action such that the fixed point set by $P$ is connected for all Sylow subgroups $P$ of $G$, it has just two fixed points, and $U$ and $V$ are isomorphic to the tangential representations as real $G$-modules respectively. We study the $\mathcal{P}(G)$-connective Smith set for a finite Oliver group $G$ of the real representation ring consisting of all differences of $\mathcal{P}(G)$-connectively Smith equivalent $G$-modules, and determine this set for certain nonsolvable groups $G$.