Chapter 18 – Heegaard Splittings of Compact 3-Manifolds

By M. Scharlemann. 2000

124 citations recorded by the source corpus.

1. Background 2 2. Heegaard splittings and their guises 3 2.1. Splittings from triangulations 3 2.2. Splitting 3-manifolds with boundary 3 2.3. Splittings as handle decompositions Heegaard diagrams 4 2.4. Splittings as Morse functions and as sweep-outs 6 3. Structures on Heegaard splittings 8 3.1. Stabilization 8 3.2. Reducible splittings 9 3.3. Weakly reducible splittings 12 4. Heegaard splittings in nature: Seifert manifolds 15 4.1. Vertical splittings 17 4.2. Horizontal splittings 17 5. Connections with group presentations 19 6. Uniqueness 21 6.1. The 3-sphere 21 6.2. Seifert manifolds 22 6.3. Genus and the Casson-Gordon examples 23 6.4. Other uniqueness results 24 7. The stabilization problem 24 8. Normal surfaces and decision problems 31 8.1. Normal surfaces and Heegaard splittings 31 8.2. Special case: Normal surfaces in a triangulation 33 References 36

Subfields: Computational Theory and Mathematics, Geometry and Topology, Mathematical Physics

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Citations in the corpus

The Heegaard Genus of Amalgamated 3-Manifolds (2003, methodology): “From F , we construct (as in [7], [8] or [4]) a generalised Heegaard splitting {F1, .”
Proximity in the curve complex: boundary reduction and bicompressible surfaces (2004, background): “Proof. The following are classical facts about Heegaard splitting s (cf [ Sc ]): If Q is a sphere then P is reducible, hence d(P) = 0. If Q is a disk then P is ∂-reducible so d(P) ≤ 1. If neither occurs, then M is irreducible and ∂-irreducible, which is what we henceforth assume.”
Holomorphic disks and topological invariants for closed three-manifolds (2001, background): “For modern surveys on the theory of Heegaard splittings, see [34] and [41].”
On the geometric and the algebraic rank of graph manifolds (2007, unknown): “The following theorem summarizes the discussion in [Scharlemann and Thompson 1994], [Scharlemann 2002] and [Scharlemann and Schultens 1999, Lemma 2].”