On topological actions of finite, non-standard groups on spheres

TL;DR

This paper demonstrates that for dimensions greater than 5, there exist finite groups that can act topologically on spheres in a faithful way, but are not realizable as linear actions, revealing new topological group actions beyond classical linear representations.

Contribution

The paper proves the existence of finite groups with faithful topological sphere actions that are not isomorphic to subgroups of the orthogonal group for all dimensions greater than 5.

Findings

Existence of non-linear finite group actions on spheres in dimensions > 5

Such groups are not isomorphic to subgroups of O(d+1)

Open problem remains for smooth actions

Abstract

The standard actions of finite groups on spheres S^d are linear actions, i.e. by finite subgroups of the orthogonal group O(d+1). We prove that, in each dimension d>5, there is a finite group G which admits a faithful, topological action on a sphere S^d but is not isomorphic to a subgroup of O(d+1). The situation remains open for smooth actions.