Generating the Goeritz group of $S^3$

TL;DR

This paper proves that an expanded set of generators, including eyeglass twists and conjugates of Powell's generators, suffices to generate the Goeritz group of the 3-sphere for any genus Heegaard splitting.

Contribution

It extends Powell's original generating set by including all eyeglass twists and conjugates, providing a comprehensive generating set for the Goeritz group.

Findings

Expanded generators suffice for the Goeritz group

Includes all eyeglass twists and conjugates

Completes the generating set for any genus splitting

Abstract

In 1980 J. Powell \cite{Po} proposed that five specific elements sufficed to generate the Goeritz group for any genus Heegaard splitting of the 3-sphere. Here we prove that a natural expansion of Powell's proposed generators, to include all eyeglass twists and all topological conjugates of Powell's generators does suffice.