A proof of Powell's conjecture on the Goeritz group of $S^3$

TL;DR

This paper proves Powell's conjecture that the Goeritz group of any genus g Heegaard splitting of the 3-sphere (g ≥ 3) is generated by four specific elements, using properties of topologically minimal surfaces.

Contribution

It confirms Powell's conjecture for all g ≥ 3 and provides a new proof of the topological index of genus g Heegaard surfaces in the 3-sphere.

Findings

Goeritz group of genus g splitting is generated by four elements for g ≥ 3

Heegaard surface of the 3-sphere is topologically minimal

Genus g Heegaard surface has topological index 2g-1

Abstract

For a genus $g$ Heegaard splitting of the $3$-sphere, the Goeritz group is defined to be the group of isotopy classes of diffeomorphisms of the $3$-sphere that preserve the splitting setwise. In this paper, we prove the following conjecture proposed by Powell: For every $g \ge 3$, the Goeritz group of a genus $g$ Heegaard splitting is generated by four specific elements. Our proof relies crucially on the fact that a Heegaard surface of the $3$-sphere is topologically minimal, that is, its disk complex has nontrivial homotopy group in some dimension. Along the way, we also give a new proof of the fact that a genus $g$ Heegaard surface of the $3$-sphere has topological index $2g-1$.