%0 Journal Article
%T Arc complexes, sphere complexes and Goeritz groups
%A Sangbum Cho
%A Yuya Koda
%A Arim Seo
%J Mathematics
%D 2014
%I arXiv
%X 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.
%U http://arxiv.org/abs/1403.7832v2