Topological symmetries of simply connected 4-manifolds and actions of automorphism groups of free groups

Abstract

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, when $b_{2}(M)\gt2$. 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 $\mathrm{Aut}(F_{n})$ and $\mathrm{GL}_{n}\mathbb{Z}$, n > = 4, on $M\neq S^{4}$ factors through $\mathbb{Z}/2$, if the group action is by homologically trivial homeomorphisms.