Generalized handlebody sets and non-Haken 3-manifolds

Details Generalized handlebody sets and non-Haken 3-manifolds Generalized handlebody sets and non-Haken 3-manifolds

2007-07-04

arxiv.org/abs/0707.0518v1

Mathematics

Geometric Topology

6 pages

Scientific paper

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.

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