# The mapping class groups of reducible Heegaard splittings of genus two

## TL;DR

This paper proves that the mapping class groups of genus-2 Heegaard splittings in all lens spaces are finitely presented, providing explicit presentations and implications for the fundamental groups of associated splitting spaces.

## Contribution

It establishes the finite presentability of the mapping class groups for all lens spaces' genus-2 splittings, including explicit group presentations.

## Key findings

- Mapping class groups of genus-2 splittings in lens spaces are finitely presented.
- Explicit presentations of these groups are provided.
- Fundamental groups of the spaces of these splittings are finitely presented.

## Abstract

The manifold which admits a genus-$2$ reducible Heegaard splitting is one of the $3$-sphere, $\mathbb{S}^2 \times \mathbb{S}^1$, lens spaces and their connected sums. For each of those manifolds except most lens spaces, the mapping class group of the genus-$2$ splitting was shown to be finitely presented. In this work, we study the remaining generic lens spaces, and show that the mapping class group of the genus-$2$ Heegaard splitting is finitely presented for any lens space by giving its explicit presentation. As an application, we show that the fundamental groups of the spaces of the genus-$2$ Heegaard splittings of lens spaces are all finitely presented.