Nonexistence of Smooth Effective One Fixed Point Actions of Finite Oliver Groups on Low-dimensional Spheres

TL;DR

This paper investigates the conditions under which finite Oliver groups can act smoothly and effectively with exactly one fixed point on low-dimensional spheres, providing new strategies to exclude such actions for specific groups.

Contribution

It introduces new methods to exclude smooth effective one fixed point actions of certain finite Oliver groups on low-dimensional spheres, expanding understanding of group actions on spheres.

Findings

Excluded smooth effective one fixed point actions for groups of order up to 216

Excluded actions for groups of the form A_5×C_n with n=3,5,7

Developed strategies for analyzing group actions on spheres

Abstract

According to the work of Laitinen, Morimoto, Oliver and Pawa\l{}owski, a finite group $G$ has a smooth effective one fixed point action on some sphere if and only if $G$ is an Oliver group. For some finite Oliver groups $G$ of order up to $216$, and for $G=A_5\times C_n$ for $n=3,5,7$, we present a strategy of excluding of smooth effective one fixed point $G$-actions on low-dimensional spheres.