Heegaard splittings of compact 3-manifolds. (English) Zbl 0985.57005

Scharlemann, Martin

Daverman, R. J. (ed.) et al., Handbook of geometric topology. Amsterdam: Elsevier. 921-953 (2002).

From the introduction: This is a survey article on Heegaard splittings. After the basic construction is understood, some natural questions arise:
How universal is this construction? That is, how many closed 3-manifolds have such a structure? Is there a natural extension to 3-manifolds with boundary? This question is considered in Section 2. How unique is such a structure? That is, given two such structures on the same 3-manifold, how are they related? This question is addressed in Sections 6 and 7. How useful is the structure? That is, what information about the 3-manifold can be gleaned from the structure of a Heegaard splitting. Such questions are addressed in 5 and 8.
A useful earlier survey of the subject is [H. Zieschang, Astérisque 163-164, 247-280 (1988; Zbl 0674.57010), which focuses on Heegaard diagrams and on group presentations. I have relied heavily on its historical account. A central recent development has been an understanding of the importance of strongly irreducible Heegaard splittings, so their role has been chosen as a major theme of this survey.
For the entire collection see [Zbl 0977.00029].

MSC:

57-02	Research exposition (monographs, survey articles) pertaining to manifolds and cell complexes
57N10	Topology of general \(3\)-manifolds (MSC2010)

Keywords:

strongly irreducible Heegaard splittings