Arc complexes, sphere complexes and Goeritz groups

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.

16 pages, 4 figures; minor changes, typos corrected

References in corpus (2)

• Automorphisms of the 3-sphere that preserve a genus two Heegaard splitting
• Primitive disk complexes for lens spaces

Cited by in corpus (1)

• Connected primitive disk complexes and genus two Goeritz groups of lens spaces