Vol. 235, No. 1, 2008

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ISSN: 0030-8730
Generalized handlebody sets and non-Haken 3-manifolds

Jesse Edward Johnson and Terk Patel

Vol. 235 (2008), No. 1, 35–41

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.

curve complex, non-Haken 3-manifold
Mathematical Subject Classification 2000
Primary: 57M50
Received: 11 July 2007
Revised: 27 September 2007
Accepted: 28 September 2007
Published: 1 March 2008
Jesse Edward Johnson
Mathematics Department
Yale University
PO Box 208283
New Haven, CT 06520-8283
United States
Terk Patel
18 Dumas Street
Pondicherry - 605001