Finite symmetries of $S^4$

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. $G\subset ̸O(5)$) on $S^4$ are constructed, and local linearity and stable smoothability of the actions are studied.