Finite symmetries of $S^4$

TL;DR

This paper explores the types of finite group actions on the 4-sphere, showing that orientation-preserving actions are linear, while exotic orientation-reversing actions can be topological but not linear, with implications for smoothability.

Contribution

It demonstrates that all orientation-preserving finite group actions on $S^4$ are linear, and constructs exotic topological actions that are not linear, analyzing their smoothability.

Findings

Orientation-preserving actions are subgroups of SO(5).

Exotic orientation-reversing actions on $S^4$ are constructed.

Local linearity and smoothability of these actions are analyzed.

Abstract

This paper discusses topological and locally linear actions of finite groups on $S^4$. Local linearity of the orientation preserving actions on $S^4$ forces the group to be a subgroup of $SO(5)$. On the other hand, orientation reversing topological actions of "exotic" groups $G$ (i.e. $G\not\subset O(5)$) on $S^4$ are constructed, and local linearity and stable smoothability of the actions are studied.