Stabilization for mapping class groups of 3-manifolds

TL;DR

This paper proves that the homology of the mapping class group of any compact, orientable 3-manifold stabilizes under connected sum operations, extending to various related groups and dimensions.

Contribution

It establishes homological stability for mapping class groups of 3-manifolds under connected sum and boundary connected sum, including quotient groups and related automorphism groups.

Findings

Homology stabilizes under connected sum operations

Stability extends to quotient groups by twists along spheres and disks

Methods apply to manifolds of other dimensions with puncture stabilization

Abstract

We prove that the homology of the mapping class group of any 3-manifold stabilizes under connected sum and boundary connected sum with an arbitrary 3-manifold when both manifolds are compact and orientable. The stabilization also holds for the quotient group by twists along spheres and disks, and includes as particular cases homological stability for symmetric automorphisms of free groups, automorphisms of certain free products, and handlebody mapping class groups. Our methods also apply to manifolds of other dimensions in the case of stabilization by punctures.