Nielsen Realization for sphere twists on 3-manifolds

TL;DR

This paper investigates when subgroups of sphere twists in 3-manifolds can be realized by diffeomorphisms, establishing conditions for realization and implications for the Burnside problem in 3-manifold topology.

Contribution

It characterizes the subgroups of sphere twists that can be realized by diffeomorphisms and applies this to the Burnside problem for 3-manifolds.

Findings

Nontrivial subgroup G is realizable iff G is cyclic and M is a connected sum of lens spaces.

Diff(M) contains no infinite torsion group if M is reducible and not a connected sum of lens spaces.

Provides a complete characterization of Nielsen realization for sphere twists in 3-manifolds.

Abstract

For a 3-manifold M, the twist group Twist(M) is the subgroup of the mapping class group Mod(M) generated by twists about embedded 2-spheres. We study the Nielsen realization problem for subgroups of Twist(M). We prove that a nontrivial subgroup G<Twist(M) is realized by diffeomorphisms if and only if G is cyclic and M is a connected sum of lens spaces. We also apply our methods to the Burnside problem for 3-manifolds and show that Diff(M) does not contain an infinite torsion group when M is reducible and not a connected sum of lens spaces.