Heegaard splitting
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In mathematics, in the sub-field of geometric topology, a Heegaard splitting is a special structure on a 3-manifold that results from dividing it into two handlebodies. The importance of Heegaard splittings has grown in recent years as more connections and applications have been found.
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Definitions
To be precise, suppose that V and W are handlebodies of the same genus. Choose now a homeomorphism f from the boundary of V to the boundary of W. Now form the quotient space
- <math> M = V \cup_f W. <math>
This is clearly a three-manifold. The remarkable fact is that every closed, orientable three-manifold is obtained in this manner. This follows from deep results on the triangulability of three-manifolds due to Moise. This contrasts strongly with higher dimensional manifolds which need not admit smooth or piecewise linear structures.
The decomposition of M into two handlebodies is called a Heegaard splitting, and their common boundary is called the Heegaard surface of the splitting.
Note also that the gluing map f need only be specified up to taking a double coset in the mapping class group of the boundary of V. This connection with the mapping class group was first made by W. R. Lickorish.
Heegaard splittings can also be defined for compact 3-manifolds with boundary by replacing handlebodies with compression bodies. The gluing map is between the positive boundaries of the compression bodies.
A simple closed curve on a surface is essential if it is does not bound a disc on the surface.
A Heegaard splitting is reducible if there is an essential simple closed curve <math>\alpha<math> on H which bounds a disk in both V and in W. A splitting is irreducible if it is not reducible.
A Heegaard splitting is weakly reducible if there is are disjoint essential simple closed curves <math>\alpha<math> and <math>\beta<math> where <math>\alpha<math> on H bounding a disk in V and <math>\beta<math> bounding a disk in W. A splitting is strongly irreducible if it is not weakly reducible.
A generalized Heegaard splitting of M is a decomposition into compression bodies <math>V_i, W_i, i=1,\dots,n<math> and surfaces <math>H_i, i=1,\dots, n<math> such that <math>\partial_+ V_i = \partial_+ W_i = H_i<math> and <math>\partial_- W_i = \partial_- V_{i+1}<math>. The interiors of the compression bodies must be pairwise disjoint and their union must be all of <math>M<math>. The surface <math>H_i<math> forms a Heegaard surface for the submanifold <math>V_i \cup W_i<math> of <math>M<math>. (Note that here each Vi and Wi is allowed to have more than one component.)
A generalized Heegaard splitting is called strongly irreducible if each <math>V_i \cup W_i<math> is strongly irreducible.
History
The idea of a Heegaard splitting was introduced by Poul Heegaard in his 1898 thesis and was perhaps inspired by what is known today as Morse theory. While Heegaard splittings were studied extensively by mathematicians such as Wolfgang Haken and Friedhelm Waldhausen in the 1960s, it was not until a few decades later that the field was rejuvenated by Casson and Gordon, primarily through their concept of strong irreducibility.
The modern study of the topology of Heegaard splittings was pushed forward through the work of Hyam Rubinstein, Martin Scharlemann, Abigail Thompson, Jennifer Schultens, Yoav Moriah, Francis Bonahon, J.P. Otal, Klaus Johannson and others.
Applications and connections
Minimal surfaces
Heegaard splittings appeared in the theory of minimal surfaces first in the work of Blaine Lawson who proved that embedded minimal surfaces in compact manifolds of positive sectional curvature are Heegaard splittings. This result was extended by William Meeks to flat manifolds, except he proves that an embedded minimal surface in a flat three-manifold is either a Heegaard surface or totally geodesic.
Meeks and S. T. Yau went on to use results of Waldhausen to prove results about the topological uniqueness of minimal surface of finite topology in <math>R^3<math>. The final topological classification of embedded minimal surfaces in <math> R^3<math> was given by Meeks and Frohman. The result relied heavily on techniques developed for studying the topology of Heegaard splittings.
Heegaard Floer homology
Heegaard diagrams, which are simple combinatorial descriptions of Heegaard splittings, have been used extensively to construct invariants of three-manifolds. The most recent example of this is the Heegaard Floer homology of Peter Oszvath and Zoltan Szabo. The theory uses the <math>g^{th}<math> symmetric product of a Heegaard surface as the ambient space, and tori built from the boundaries of meridian disks for the two handlebodies as the Lagrangian submanifolds.