On finite simple groups acting on homology spheres with small fixed point sets

TL;DR

This paper classifies finite simple groups that can act smoothly on homology spheres with small fixed point sets, establishing finiteness results and identifying specific groups and dimensions where such actions occur.

Contribution

It proves only finitely many finite simple groups can act on homology spheres with bounded fixed point set dimension and explicitly identifies these groups for fixed point sets of dimension at most one.

Findings

Only finitely many finite simple groups act with small fixed point sets.

The groups A_5, PSL_2(7), and possibly PSU_3(3) act on homology spheres with 1-dimensional fixed points.

Specific dimensions for these actions are identified, mainly in dimensions 2, 3, and 5.

Abstract

A finite nonabelian simple group does not admit a free action on a homology sphere, and the only finite simple group which acts on a homology sphere with at most 0-dimensional fixed point sets ("pseudofree action") is the alternating group A_5 acting on the 2-sphere. Our first main theorem is the finiteness result that there are only finitely many finite simple groups which admit a smooth action on a homology sphere with at most d-dimensional fixed points sets, for a fixed d. We then go on proving that the finite simple groups acting on a homology sphere with at most 1-dimensional fixed point sets are the alternating group A_5 in dimensions 2, 3 and 5, the linear fractional group PSL_2(7) in dimension 5, and possibly the unitary group PSU_3(3) in dimension 5 (we conjecture that it does not admit any action on a homology 5-sphere but cannot exclude it at present). Finally, we discuss the situation for arbitrary finite groups which admit an action on a homology 3-sphere.