Volume 8 (2004)

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ISSN (electronic): 1364-0380
ISSN (print): 1465-3060

The disjoint curve property

Saul Schleimer

Geometry & Topology 8 (2004) 77–113

DOI: 10.2140/gt.2004.8.77

arXiv: math.GT/0401399

Abstract

A Heegaard splitting of a closed, orientable three-manifold satisfies the disjoint curve property if the splitting surface contains an essential simple closed curve and each handlebody contains an essential disk disjoint from this curve. A splitting is full if it does not have the disjoint curve property. This paper shows that in a closed, orientable three-manifold all splittings of sufficiently large genus have the disjoint curve property. From this and a solution to the generalized Waldhausen conjecture it would follow that any closed, orientable three manifold contains only finitely many full splittings.

Keywords

Heegaard splittings, disjoint curve property, Waldhausen Conjecture

Mathematical Subject Classification

Primary: 57M99

Secondary: 57M27, 57N10

References
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Publication

Received: 29 May 2002
Revised: 21 January 2004
Accepted: 13 December 2003
Published: 22 January 2004
Proposed: Cameron Gordon
Seconded: David Gabai, Joan Birman

Authors
Saul Schleimer
Department of Mathematics
University of Illinois at Chicago
851 South Morgan Street
Chicago
Illinois 60607
USA