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 × 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.

The Powell conjecture and reducing sphere complexes

“…These structures have also been examined for other 3‐manifolds. In the case that is a genus two Heegaard surface for an arbitrary 3‐manifold , , which is also called the Haken sphere complex in the literature, has been studied and characterized by Cho, Koda, and Seo in and by Cho and Koda in , in which they prove the surprising fact that for the genus two Heegaard splitting of many lens spaces, is not connected. Most recently, Cho and Koda completed the classification of the Goeritz groups of all 3‐manifolds admitting a genus two Heegaard splitting .…”

Twisted book decompositions and the Goeritz groups

“…For the higher genus Heegaard splittings of the 3-sphere, the problem of existence of finite generating sets of the Goeritz groups still remains open. For other works on finite generating sets of Goeritz groups, see [17,18,8].…”

The mapping class groups of reducible Heegaard splittings of genus two

“…In [6] a finite presentation of the genus-2 Goeritz group of S 2 × S 1 was obtained, and in [7] finite presentations of the genus-2 Goeritz groups were obtained for the connected sums whose summands are S 2 × S 1 or lens spaces. We refer the reader to [15], [16], [24], [18] and [9] for finite presentations or finite generating sets of the Goeritz groups of several Heegaard splittings and related topics.…”