Finite presentations of the mapping class groups of once-stabilized Heegaard splittings

TL;DR

This paper proves that the mapping class group of the once-stabilized Heegaard splitting of certain 3-manifolds with high distance is finitely presented, contributing to understanding the algebraic structure of these groups.

Contribution

It establishes finite presentability of the mapping class group for once-stabilized Heegaard splittings with high distance, a new result in 3-manifold topology.

Findings

Mapping class group of once-stabilized splitting is finitely presented

High-distance condition is crucial for finite presentability

Advances understanding of algebraic properties of 3-manifold splittings

Abstract

Let $g \ge 2$ and assume that we are given a genus $g$ Heegaard splitting of a closed orientable $3$-manifold with the distance greater than $2g+2$. We prove that the mapping class group of the once-stabilization of such a Heegaard splitting is finitely presented.