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Stabilization, amalgamation and curves of intersection of Heegaard splittings

Ryan Derby-Talbot

Algebraic & Geometric Topology 9 (2009) 811–832
Abstract

We address a special case of the Stabilization Problem for Heegaard splittings, establishing an upper bound on the number of stabilizations required to make a Heegaard splitting of a Haken 3–manifold isotopic to an amalgamation along an essential surface. As a consequence we show that for any positive integer n there are 3–manifolds containing an essential torus and a Heegaard splitting such that the torus and splitting surface must intersect in at least n simple closed curves. These give the first examples of lower bounds on the minimum number of curves of intersection between an essential surface and a Heegaard surface that are greater than one.

Keywords
Heegaard splitting, incompressible surface
Mathematical Subject Classification 2000
Primary: 57M99
References
Publication
Received: 22 July 2008
Revised: 21 December 2008
Accepted: 18 March 2009
Published: 26 April 2009
Authors
Ryan Derby-Talbot
Department of Mathematics
The American University in Cairo
113 Sharia Kasr El Aini
PO Box 2511
Cairo, 11511
Egypt