Heegaard splittings of Haken manifolds have bounded distance

Abstract

Given a Heegaard splitting and an incompressible surface S and a Heegaard splitting of an irreducible manifold, I shall use a generalization of Haken's lemma proved by Kobayashi in order to define a pair of simple closed curves on the splitting surface such that each bounds a disc in one of the handlebodies of the splitting. By modifying the proof of Kobayashi's lemma, I shall show that the sequence of boundary compressions used to isotope S places a bound on the distance between these two simple closed curves in the complex of curves. This will then place a bound on the distance of the Heegaard splitting.