On topological actions of finite groups on S^3

TL;DR

This paper investigates topological actions of finite groups on the 3-sphere and Euclidean space, showing that such actions are closely related to subgroups of orthogonal groups, with specific classifications for simple and general groups.

Contribution

It proves that the only nonabelian simple groups acting topologically on S^3 are A_5 and the dodecahedral group, and that all finite groups acting on R^3 are subgroups of SO(3).

Findings

Only A_5 and the dodecahedral group act topologically on S^3.

All finite groups acting on R^3 are subgroups of SO(3).

Topological actions are closely related to orthogonal group subgroups.

Abstract

We consider orientation-preserving actions of a finite group G on the 3-sphere S^3 (and also on Euclidean space R^3). By the geometrization of finite group actions on 3-manifolds, if such an action is smooth then it is conjugate to an orthogonal action, and in particular G is isomorphic to a subgroup of the orthogonal group SO(4) (or of SO(3) in the case of R^3). On the other hand, there are topological actions with wildly embedded fixed point sets; such actions are not conjugate to smooth actions but one would still expect that the corresponding groups G are isomorphic to subgroups of the orthgonal groups SO(4) (or of SO(3), resp.). In the present paper, we obtain some results in this direction; we prove that the only finite, nonabelian simple group with a topological action on S^3, or on any homology 3-sphere, is the alternating or dodecahedral group A_5 (the only finite, nonabelian simple subgroup of SO(4)), and that every finite group with a topological, orientation-preserving action on Euclidean space R^3 is in fact isomorphic to a subgroup of SO(3).