#### Volume 9, issue 2 (2009)

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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
Primary: 57M99