Exclusions of smooth actions on spheres of the non-split extension of $C_2$ by $SL(2,5)$

TL;DR

This paper investigates the limitations of smooth group actions on spheres by the specific nonsolvable group $SL(2,5).C_2$, proving that certain types of actions with fixed points are impossible in low dimensions.

Contribution

It classifies and excludes specific smooth actions of the non-split extension of $C_2$ by $SL(2,5)$ on spheres, especially those with fixed points.

Findings

No effective odd fixed point actions on low-dimensional spheres.

Excludes certain pseudofree one fixed point actions on spheres.

Identifies restrictions on group actions with fixed points in low dimensions.

Abstract

There are four groups $G$ fitting into a short exact sequence $ 1\rightarrow SL(2,5)\rightarrow G\rightarrow C_2\rightarrow 1, $ where $SL(2,5)$ is the special linear group of $(2\times 2)$-matrices with entries in the field of five elements. Except for the direct product of $SL(2,5)$ and $C_2$, there are two other semidirect products of these two groups and just one non-semidirect product $SL(2,5).C_2$, considered in this paper. It is known that each finite nonsolvable group can act on spheres with arbitrary positive number of fixed points. Clearly, $SL(2,5).C_2$ is a nonsolvable group. Moreover, it turns out that $SL(2,5).C_2$ possesses a free representation and as such, can potentially act pseudofreely with nonempty fixed point set on manifolds of arbitrarily large dimension. We prove that $SL(2,5).C_2$ cannot act effectively with odd number of fixed points on low-dimensional spheres. In the special case of effective one fixed point actions, we are able to exclude a wider class of spheres. Moreover, we prove that specific pseudofree one fixed point actions of $SL(2,5).C_2$ on spheres do not exist.