Strong Haken via Thin Position
arXiv:2402.14077 · doi:10.1007/s40590-024-00688-3
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.
Final version, includes structure theorem for weakly reducible Heegaard splittings
References in corpus (7)
- Stabilization for mapping class groups of 3-manifolds
- Heegaard splittings of exteriors of, two bridge knots
- An algorithm to determine the Heegaard genus of simple 3-manifolds with non-empty boundary
- Thin position for knots, links, and graphs in 3-manifolds
- A Strong Haken's Theorem
- Equivariant Heegaard genus of reducible 3-manifolds
- Strong Haken via Sphere Complexes