Strong Haken via Sphere Complexes

Abstract

We give a short proof of Scharlemann's Strong Haken Theorem for closed $3$-manifolds (and manifolds with spherical boundary). As an application, we also show that given a decomposing sphere $R$ for a $3$-manifold $M$ that splits off an $S^2\times S^1$ summand, any Heegaard splitting of $M$ restricts to the standard Heegaard splitting on the summand.

13 pages, 4 figures

References in corpus (2)

• The mapping class group of connect sums of $S^2\times S^1$
• A Strong Haken's Theorem

Cited by in corpus (2)

• Strong Haken via Thin Position
• Genus three Goeritz groups of connected sums of two lens spaces