Stabilization, amalgamation and curves of intersection of Heegaard splittings

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.