Stabilization, amalgamation and curves of intersection of Heegaard splittings

Derby-Talbot, Ryan

doi:10.2140/agt.2009.9.811

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Stabilization, amalgamation and curves of intersection of Heegaard splittings

Ryan Derby-Talbot

Algebraic & Geometric Topology 9 (2009) 811–832

DOI: 10.2140/agt.2009.9.811

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

Volume 9, issue 2 (2009)