Chapter 18 – Heegaard Splittings of Compact 3-Manifolds

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

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