Strong Haken via Thin Position

Abstract

We use thin position of Heegaard splittings to give a new proof of Haken's Lemma that a Heegaard surface of a reducible manifold is reducible and of Scharlemann's ``Strong Haken Theorem'': a Heegaard surface for a 3-manifold may be isotoped to intersect a given collection of essential spheres and discs in a single loop each. We also give a reformulation of Casson and Gordon's theorem on weakly reducible Heegaard splittings, showing that they exhibit additional structure with respect to certain incompressible surfaces. This article could also serve as an introduction to the theory of generalized Heegaard surfaces and it includes a careful study of their behaviour under amalgamation.