Geometry & Topology, Vol. 8 (2004) Paper no. 3, pages 77--113.

The disjoint curve property

Saul Schleimer

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 [Thompson, 1999]. 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

AMS subject classification. Primary: 57M99. Secondary: 57M27, 57N10.

DOI: 10.2140/gt.2004.8.77

E-print: arXiv:math.GT/0401399

Submitted to GT on 29 May 2002. (Revised 21 January 2004.) Paper accepted 13 December 2003. Paper published 22 January 2004.

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Saul Schleimer
Department of Mathematics, UIC
851 South Morgan Street
Chicago, Illinois 60607, USA
Email: saul at math dot uic dot edu
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