On the disk complexes of weakly reducible, unstabilized Heegaard splittings of genus three I - the Structure Theorem

TL;DR

This paper characterizes the structure of a specific subset of the disk complex in genus three Heegaard splittings and relates its components to isotopy classes of generalized splittings obtained via weak reductions.

Contribution

It introduces a detailed description of the shape of DVW(F) and establishes a function linking its components to isotopy classes of generalized Heegaard splittings.

Findings

Describes the shape of DVW(F) in genus three Heegaard splittings.

Establishes a correspondence between components of DVW(F) and isotopy classes of generalized splittings.

Provides insights into the embedding of thick levels in compression bodies.

Abstract

Let (V,W;F) be a weakly reducible, unstabilized, genus three Heegaard splitting in an orientable, irreducible 3-manifold M and DVW(F) the subset of the disk complex D(F) consisting of simplices having at least one vertex from V and at least one vertex from W. In this article, we describe the shape of DVW(F) and prove that there is a function from the components of DVW(F) to the isotopy classes of the generalized Heegaard splittings obtained by weak reductions from (V,W;F), where this function describes how the thick levels are embedded in the relevant compression bodies.