Generalized Heegaard splittings and the disk complex

Abstract

Let $M$ be an orientable, irreducible $3$-manifold and $(\mathcal{V},\mathcal{W};F)$ a weakly reducible, unstabilized Heegaard splitting of $M$ of genus at least three. In this article, we define an equivalent relation $\sim$ on the set of the generalized Heegaard splittings obtained by weak reductions and find special subsets of the disk complex $\mathcal{D}(F)$ named by the "equivalent clusters", where we can find a canonical function $\Phi$ from the set of equivalent clusters to the set of the equivalent classes for the relation $\sim$. As an application, we prove that if $F$ is topologically minimal and the topological index of $F$ is at least three, then there is a $2$-simplex in $\mathcal{D}(F)$ formed by two weak reducing pairs such that the equivalent classes of the generalized Heegaard splittings obtained by weak reductions along the weak reducing pairs for the relation $\sim$ are different. In the last section, we prove $\Phi$ is a bijection if the genus of $F$ is three. Using it, we prove there is a canonical function $\Omega$ from the set of components of ${\mathcal{D}}_{\mathcal{V}\mathcal{W}}(F)$ to the set of the isotopy classes of the generalized Heegaard splittings obtained by weak reductions and describe what $\Omega$ is.

40 pages, 5 figures, This article is the generalization of the authour's previous article arXiv:1412.2228 to arbitrarily high genus cases

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• On critical Heegaard splittings of tunnel number two composite knot exteriors
• Normalizing Topologically Minimal Surfaces II: Disks
• Normalizing Topologically Minimal Surfaces I: Global to Local Index
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