On finite groups acting on spheres and finite subgroups of orthogonal groups

TL;DR

This survey reviews classical and recent results on finite groups acting on spheres, exploring their relationship with finite subgroups of orthogonal groups, including simple groups acting on spheres and homology spheres.

Contribution

It provides a comprehensive overview of finite group actions on spheres, connecting historical results with new insights and concepts in the field.

Findings

Finite groups acting on low-dimensional spheres are classified.

Comparison between finite groups acting on spheres and subgroups of orthogonal groups.

Finite simple groups can act on spheres and homology spheres in various dimensions.

Abstract

This is a survey on old and new results as well as an introduction to various related basic notions and concepts, based on two talks given at the International Workshop on Geometry and Analysis in Kemerovo (Sobolev Institute of Mathematics, Kemerovo State University) and at the University of Krasnojarsk in June 2011. We discuss finite groups acting on low-dimensional spheres, comparing with the finite subgroups of the corresponding orthogonal groups, and also finite simple groups acting on spheres and homology spheres of arbitrary dimension.