A note on the Nielsen realization problem for connected sums of $S^2 \times S^1$

TL;DR

This paper investigates the Nielsen realization problem for certain 3-manifolds formed by connected sums of $S^2 imes S^1$, showing that finite group actions with trivial homology action are cyclic and discussing realizability of automorphisms by diffeomorphisms.

Contribution

It proves that for these 3-manifolds, finite groups acting trivially on homology are cyclic, and explores conditions under which automorphisms of the fundamental group are realizable by diffeomorphisms.

Findings

Finite group actions with trivial homology action are cyclic for $g > 1$.

No non-cyclic subgroup of the twist subgroup can be realized by diffeomorphisms.

Discussion on realizability of automorphisms of the fundamental group by diffeomorphisms.

Abstract

We consider finite group-actions on 3-manifolds $\cal H_g$ obtained as the connected sum of $g$ copies of $S^2 \times S^1$, with free fundamental group $F_g$ of rank $g$. We prove that, for $g > 1$, a finite group of diffeomorphisms of $\cal H_g$ inducing a trivial action on homology is cyclic. As a consequence, no non-cyclic subgroup of the twist subgroup of the mapping class group of $\cal H_g$ (generated by Dehn twists along embedded 2-spheres) can be realized by diffeomorphisms (in the sense of the Nielsen realization problem). We also discuss when a finite subgroup of the outer automorphism group ${\rm Out}(F_g)$ of the fundamental group of $\cal H_g$ can be realized by a group of diffeomorphisms of $\cal H_g$.