Automorphisms of the 3-sphere that preserve a genus two Heegaard splitting

TL;DR

This paper provides an updated proof of Goeritz's 1933 theorem, identifying generators for the automorphism group of the 3-sphere that preserve a genus two Heegaard splitting, using its action on a connected 2-complex.

Contribution

It offers a modern, detailed proof of a classical result and analyzes the automorphism group via its action on a specific 2-complex, a problem still open for higher genus.

Findings

Finite set of generators for the automorphism group

Group action on a connected 2-complex elucidates structure

Updated proof of classical theorem

Abstract

An updated proof of a 1933 theorem of Goeritz, exhibiting a finite set of generators for the group of automorphisms of the 3-sphere that preserve a genus two Heegaard splitting. The group is analyzed via its action on a certain connected 2-complex. (The analogous problem for higher genus Heegaard splittings appears to remain unresolved.)