On finite simple and nonsolvable groups acting on closed 4-manifolds

TL;DR

This paper classifies finite simple groups capable of acting on closed 4-manifolds under specific conditions, revealing that only certain alternating groups and PSL(2,7) can occur, with new results for the case when the second Betti number is two.

Contribution

It provides a complete classification of finite simple groups acting on 4-manifolds with trivial homology, especially clarifying the case when the second Betti number is two.

Findings

Only A5, A6, and PSL(2,7) admit such actions when homologically trivial.

For b2(M)=2, only A5 can occur among simple groups.

A short list of nonsolvable groups includes all candidates for these actions.

Abstract

We show that the only finite nonabelian simple groups which admit a locally linear, homologically trivial action on a closed simply connected 4-manifold $M$ (or on a 4-manifold with trivial first homology) are the alternating groups $A_5$, $A_6$ and the linear fractional group PSL(2,7) (we note that for homologically nontrivial actions all finite groups occur). The situation depends strongly on the second Betti number $b_2(M)$ of $M$ and has been known before if $b_2(M)$ is different from two, so the main new result of the paper concerns the case $b_2(M)=2$. We prove that the only simple group that occurs in this case is $A_5$, and then give a short list of finite nonsolvable groups which contains all candidates for actions of such groups.