A remark about dihedral group actions on spheres

TL;DR

This paper proves that finite dihedral groups cannot act pseudofreely and locally linearly on spheres of dimension greater than 2, resolving a question posed in 1982.

Contribution

It establishes a non-existence result for certain group actions on spheres, specifically dihedral groups acting on spheres of dimension 2k with k > 1.

Findings

Finite dihedral groups do not act pseudofreely and locally linearly on spheres when dimension is greater than 2.

Answers a long-standing question from 1982 about group actions on spheres.

Provides a theoretical proof for the non-existence of such actions.

Abstract

We show that a finite dihedral group does not act pseudofreely and locally linearly on a 2k-dimensional sphere, if k > 1. This answers a question of R. S. Kulkarni from 1982.