Geometry & Topology, Vol. 8 (2004)
Paper no. 3, pages 77--113.
The disjoint curve property
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.
Heegaard splittings, disjoint curve property, Waldhausen Conjecture
AMS subject classification.
Secondary: 57M27, 57N10.
Submitted to GT on 29 May 2002.
(Revised 21 January 2004.)
Paper accepted 13 December 2003.
Paper published 22 January 2004.
Notes on file formats
Department of Mathematics, UIC
851 South Morgan Street
Chicago, Illinois 60607, USA
Email: saul at math dot uic dot edu
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