Generalized handlebody sets and non-Haken 3-manifolds

Jesse Edward Johnson and Terk Patel

Vol. 235 (2008), No. 1, 35–41

DOI: 10.2140/pjm.2008.235.35

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 nonseparating 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 nonseparating, two sided incompressible surface. We show that every vertex in the curve complex is within two edges of a point in the boundary set.

Keywords

curve complex, non-Haken 3-manifold

Mathematical Subject Classification 2000

Primary: 57M50

Milestones

Received: 11 July 2007

Revised: 27 September 2007

Accepted: 28 September 2007

Published: 1 March 2008