The disjoint curve property

Schleimer, Saul

doi:10.2140/gt.2004.8.77

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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 2000

Primary: 57M99

Secondary: 57M27, 57N10

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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

Volume 8, issue 1 (2004)