Volume 7, issue 2 (2007)

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Saddle tangencies and the distance of Heegaard splittings

Tao Li

Algebraic & Geometric Topology 7 (2007) 1119–1134

arXiv: math.GT/0701396

Abstract

We give another proof of a theorem of Scharlemann and Tomova and of a theorem of Hartshorn. The two theorems together say the following. Let M be a compact orientable irreducible 3–manifold and P a Heegaard surface of M. Suppose Q is either an incompressible surface or a strongly irreducible Heegaard surface in M. Then either the Hempel distance d(P) 2genus(Q) or P is isotopic to Q. This theorem can be naturally extended to bicompressible but weakly incompressible surfaces.

Keywords
Heegaard splitting, incompressible surface, curve complex, sample layout
Mathematical Subject Classification 2000
Primary: 57N10
Secondary: 57M50
References
Publication
Received: 7 January 2007
Revised: 1 June 2007
Accepted: 25 July 2007
Published: 2 August 2007
Authors
Tao Li
Department of Mathematics
Boston College
Chestnut Hill, MA 02467
USA
http://www2.bc.edu/~taoli/