Topological symmetries of simply-connected four-manifolds and actions of automorphism groups of free groups

TL;DR

This paper investigates the symmetries of simply-connected four-manifolds, proving restrictions on group actions and showing that certain automorphism groups act trivially or through small quotients.

Contribution

It establishes that compact Lie groups acting effectively and homologically trivially are abelian of rank at most two, and that automorphism groups of free groups and special linear groups have limited or trivial actions.

Findings

Any such Lie group is abelian of rank ≤ 2.

Automorphism groups of free groups (n≥4) act through Z/2 on the manifold.

SL(n,Q) (n≥4) acts trivially on the manifold.

Abstract

Let $M$ be a simply connected closed $4$-manifold. It is proved that any (possibly finite) compact Lie group acting effectively and homologically trivially on $M$ by homeomorphisms is an abelian group of rank at most two. As applications, let $\mathrm{Aut}(F_{n})$ be the automorphism group of the free group of rank $n.$ We prove that any group action of $\mathrm{Aut}% (F_{n})$ $(n\geq 4)$ on $M\neq S^{4}$ by homologically trivial homeomorphisms factors through $\mathbb{Z}/2.$ Moreover, any action of $% \mathrm{SL}_{n}(\mathbb{Q})$ $(n\geq 4)$ on $M\neq S^{4}$ by homeomorphisms is trivial.