Generalized handlebody sets and non-Haken 3-manifolds

TL;DR

This paper investigates the structure of boundary sets in the curve complex related to handlebody sets and non-Haken 3-manifolds, revealing their density and implications for the topology of the manifold.

Contribution

It introduces the concept of boundary sets in the curve complex and proves their 2-density, linking this property to the existence of non-separating incompressible surfaces.

Findings

Boundary sets are 2-dense in the curve complex.

Distance between boundary sets indicates presence of non-separating surfaces.

Characterization of non-Haken 3-manifolds via boundary set properties.

Abstract

In the curve complex for a surface, a handlebody set is the set of loops that bound properly embedded disks in a given handlebody bounded by the surface. A boundary set is the set of non-separating loops in the curve complex that bound two-sided, properly embedded surfaces. For a Heegaard splitting, the distance between the boundary sets of the handlebodies is zero if and only if the ambient manifold contains a non-separating, two sided incompressible surface. We show that the boundary set is 2-dense in the curve complex, i.e. every vertex is within two edges of a point in the boundary set.