Arc complexes, sphere complexes and Goeritz groups

TL;DR

This paper investigates the properties of certain Heegaard splittings in 3-manifolds, establishing conditions under which their Goeritz groups are finitely generated and analyzing the contractibility of related complexes.

Contribution

It introduces a new sufficient condition for contractibility of subcomplexes of the arc complex and applies this to show finitely generated Goeritz groups for specific Heegaard splittings.

Findings

Goeritz group of certain splittings is finitely generated

Provided a generalization of Hatcher's contractibility result

Constructed infinitely many Heegaard splittings with contractible complexes

Abstract

We show that if a Heegaard splitting is obtained by gluing a splitting of Hempel distance at least 4 and the genus-1 splitting of $S^2 \times S^1$, then the Goeritz group of the splitting is finitely generated. To show this, we first provide a sufficient condition for a full subcomplex of the arc complex for a compact orientable surface to be contractible, which generalizes the result by Hatcher that the arc complexes are contractible. We then construct infinitely many Heegaard splittings, including the above-mentioned Heegaard splitting, for which suitably defined complexes of Haken spheres are contractible.