Abstract
Haken’s theorem tells us that a Heegaard splitting of a reducible 3-manifold is necessarily reducible. Scharlemann’s Strong Haken Theorem says more: An essential 2-sphere in a 3-manifold can always be isotoped to intersect a given Heegaard surface in a single simple closed curve. Scharlemann’s proof of the Strong Haken Theorem can be reinterpreted through the use of sphere complexes. This is joint work with Sebastian Hensel.
