In the mathematical field of geometric topology, a Heegaard splitting (Danish: [ˈhe̝ˀˌkɒˀ] ⓘ) is a decomposition of a compact oriented 3-manifold that results from dividing it into two handlebodies.
Let V and W be handlebodies of genus g, and let ƒ be an orientation reversing homeomorphism from the boundary of V to the boundary of W. By gluing V to W along ƒ we obtain the compact oriented 3-manifold Every closed, orientable three-manifold may be so obtained; 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.
Assuming smoothness the existence of a Heegaard splitting also follows from the work of Smale about handle decompositions from Morse theory.
Heegaard splittings can also be defined for compact 3-manifolds with boundary by replacing handlebodies with compression bodies.
A closed curve is called essential if it is not homotopic to a point, a puncture, or a boundary component.
[1] A Heegaard splitting is reducible if there is an essential simple closed curve
A Heegaard splitting is stabilized if there are essential simple closed curves
It follows from Waldhausen's Theorem that every reducible splitting of an irreducible manifold is stabilized.
A Heegaard splitting is weakly reducible if there are disjoint essential simple closed curves
bounds a disk in W. A splitting is strongly irreducible if it is not weakly reducible.
A generalized Heegaard splitting is called strongly irreducible if each
There is an analogous notion of thin position, defined for knots, for Heegaard splittings.
, where the index runs over the Heegaard surfaces in the generalized splitting.
A generalized Heegaard splitting is thin if its complexity is minimal.
There are several classes of three-manifolds where the set of Heegaard splittings is completely known.
The same holds for lens spaces (as proved by Francis Bonahon and Otal).
Here, all splittings may be isotoped to be vertical or horizontal (as proved by Yoav Moriah and Jennifer Schultens).
Cooper & Scharlemann (1999) classified splittings of torus bundles (which includes all three-manifolds with Sol geometry).
It follows from their work that all torus bundles have a unique splitting of minimal genus.
All other splittings of the torus bundle are stabilizations of the minimal genus one.
A paper of Kobayashi (2001) classifies the Heegaard splittings of hyperbolic three-manifolds which are two-bridge knot complements.
Computational methods can be used to determine or approximate the Heegaard genus of a 3-manifold.
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.
The final topological classification of embedded minimal surfaces in
The result relied heavily on techniques developed for studying the topology of Heegaard splittings.
The most recent example of this is the Heegaard Floer homology of Peter Ozsvath and Zoltán Szabó.
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.
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 Andrew Casson and Cameron Gordon (1987), primarily through their concept of strong irreducibility.