Reducing spheres of genus-2 Heegaard splitting of $S^3$

TL;DR

This paper presents an algorithm to construct all reducing spheres for a genus-2 Heegaard splitting of the 3-sphere, providing an alternative proof of the finite generation of the Goeritz group G_2.

Contribution

It introduces a constructive algorithm for reducing spheres and offers a new proof of G_2's finite generation based on stabilizer properties.

Findings

Algorithm to generate any reducing sphere from a standard one

Alternative proof of G_2's finite generation

Assumption of stabilizer's finite generation

Abstract

The Goeritz group of the standard genus-g Heegaard splitting of the three sphere, $G_g$, acts on the space of isotopy classes of reducing spheres for this Heegaard splitting. Scharlemann MR2199366 (2007c:57020) uses this action to prove that $G_2$ is finitely generated. In this article, we give an algorithm to construct any reducing sphere from a standard reducing sphere for a genus-2 Heegaard splitting of the $S^3$. Using this we give an alternate proof of the finite generation of $G_2$ assuming the finite generation of the stabilizer of the standard reducing sphere.