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### A new proof of Scharlemannâ€™s Strong Haken theorem

**Geometry/Topology**

Speaker: | Jennifer Schultens, UC Davis |

Location: | Zoom |

Start time: | Tue, Mar 30 2021, 1:10PM |

Given a Heegaard splitting of a reducible 3-manifold, Haken's theorem tells us that the Heegaard splitting is reducible. Scharlemann's Strong Haken theorem tells us that more is true: Given an essential 2-sphere S in a 3-manifold M, any Heegaard splitting of M can be isotoped to meet S in a single simple closed curve. As it turns out, Scharlemann's proof can be simplified by referring to the sphere complex, which encodes the essential 2-spheres in a 3-manifold and some aspects of their positioning with respect to each other. This is joint work with Sebastian Hensel.