Arc complexes, sphere complexes and Goeritz groups

Sangbum Cho; Yuya Koda; Arim Seo

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arXiv · 1403.7832

Arc complexes, sphere complexes and Goeritz groups

Sangbum Cho·Yuya Koda·Arim Seo

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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.

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Sangbum Cho, Yuya Koda, Arim Seo. 2014-03-30. Arc complexes, sphere complexes and Goeritz groups. https://arxiv.org/abs/1403.7832

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