Arc complexes, sphere complexes, and Goeritz groups (Q737123)

Let \((V, W; \Sigma)\) be a genus \(g\) Heegaard splitting of a \(3\)-manifold \(M = M_1\sharp (S^2\times S^1)\) which is obtained by gluing a genus \((g-1)\) Heegaard splitting \((V_1, W_1;\Sigma_1)\) of \(M_1\) and a genus \(1\) Heegaard splitting \((V_2, W_2;\Sigma_2)\) of \(S^2\times S^1\), where \(V\) (resp. \(W\)) is a boundary connected sum \(V_1\natural V_2\) (resp. \(W_1\natural W_2)\) and \(\Sigma\) is a connected sum \(\Sigma_1\sharp\Sigma_2\). In this setting, we have a sphere complex \(\mathcal{H}\) for \((V, W;\Sigma)\). Using a reducing disk \(E_0\), the authors introduce a simplicial complex \(\mathcal{H}_{E_0}\) whose vertices are equivalence classes of Haken spheres with suitable condition, and they observe that \(\mathcal{H}_{E_0}\) is isomorphic to a full subcomplex \(\mathcal{A}^*_{g-1, 2}\) of the arc complex \(\mathcal{A}_{g-1, 2}\) of a genus \((g-1)\) surface with two open disks removed. Then the authors provide a sufficient condition for a full subcomplex of the arc complex of a compact orientable surface to be contractible. Applying this they prove that if \((V_1, W_1;\Sigma_1)\) is a genus \((g-1)\) Heegaard splitting (\(g\geq 2\)) with Hempel distance at least two, then the sphere complex \(\mathcal{H}\) for \((V, W,\Sigma)\) is isomorphic to \(\mathcal{A}^*_{g-1, 2}\), which is a \((4g-5)\)-dimensional contractible complex. Then the authors direct their attention to the Goeritz group of the Heegaard splitting \((V, W;\Sigma)\), and prove that if the Heegaard splitting \((V_1, W_1;\Sigma_1)\) of \(M_1\) has Hempel distance at least four, then the Goeritz group of the Heegaard splitting \((V, W;\Sigma)\) is finitely generated.