On finite simple groups acting on homology spheres

TL;DR

This paper proves that in each dimension, only finitely many finite simple groups can act faithfully and smoothly on homology spheres, extending classical linear bounds to topological actions.

Contribution

It establishes a finiteness result for finite simple groups acting on homology spheres, generalizing Jordan's linear bound to smooth topological actions.

Findings

Finitely many simple groups act on homology spheres in each dimension

Extension of Jordan's bound from linear to smooth topological actions

Specific analysis of groups acting on 3-, 4-, and 5-dimensional homology spheres

Abstract

It is a consequence of the classical Jordan bound for finite subgroups of linear groups that in each dimension n there are only finitely many finite simple groups which admit a faithful, linear action on the n-sphere. In the present paper we prove an analogue for smooth actions on arbitrary homology n-spheres: in each dimension n there are only finitely many finite simple groups which admit a faithful, smooth action on some homology sphere of dimension n, and in particular on the n-sphere. We discuss also the finite simple groups which admit an action on a homology sphere of dimension 3, 4 or 5.