Generalized Heegaard splittings and the disk complex

TL;DR

This paper explores the structure of generalized Heegaard splittings in 3-manifolds, introducing an equivalence relation and special subsets of the disk complex to classify splittings and their weak reductions.

Contribution

It defines an equivalence relation on generalized Heegaard splittings, introduces the concept of equivalent clusters in the disk complex, and establishes canonical functions relating these structures.

Findings

Established a relation between weak reductions and generalized Heegaard splittings.

Proved the bijection of the canonical function $\

,

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 $\mathcal{D}_{\mathcal{VW}}(F)$ to the set of the isotopy classes of the generalized Heegaard splittings obtained by weak reductions and describe what $\Omega$ is.