Pseudofree group actions on spheres

TL;DR

This paper investigates the conditions under which certain finite groups can act pseudofreely and locally linearly on spheres, extending previous classifications and identifying new restrictions based on sphere dimension and orientation.

Contribution

It proves that dihedral groups cannot act pseudofreely and locally linearly on spheres when the dimension is divisible by 4, and explores open cases for other dimensions and orientation-reversing actions.

Findings

Dihedral groups do not act pseudofreely on spheres when dimension is divisible by 4.

Classification of finite group actions on spheres is extended with new restrictions.

Open problems remain for dimensions congruent to 2 mod 4 and for orientation-reversing actions.

Abstract

R. S. Kulkarni showed that a finite group acting pseudofreely, but not freely, preserving orientation, on an even-dimensional sphere (or suitable sphere-like space) is either a periodic group acting semifreely with two fixed points, a dihedral group acting with three singular orbits, or one of the polyhedral groups, occurring only in dimension 2. It is shown here that the dihedral group does not act pseudofreely and locally linearly on an actual n-sphere when n is congruent to 0 mod 4. The possibility of such an action when n is congruent to 2 mod 4 and n>2 remains open. Orientation-reversing actions are also considered.