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

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 $\mathrm{Aut}$ $(n\ge 4)$ on $M\ne S^4$ by homologically trivial homeomorphisms factors through $\mathbb{Z}/2.$ Moreover, any action of $(n\ge 4)$ on $M\ne S^4$ by homeomorphisms is trivial.

Final version, to appear in the Quarterly Journal of Mathematics

References in corpus (2)

• Groups acting on manifolds: around the Zimmer program
• Euler characteristics and actions of automorphism groups of free groups