In the mathematical field of geometric topology, a Heegaard splitting // is a decomposition of a compact oriented 3-manifold that results from dividing it into two handlebodies.
Contents
Definitions
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.
The decomposition of M into two handlebodies is called a Heegaard splitting, and their common boundary H is called the Heegaard surface of the splitting. Splittings are considered up to isotopy.
The gluing map ƒ need only be specified up to taking a double coset in the mapping class group of H. This connection with the mapping class group was first made by W. B. 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 closed curve is called essential if it is not homotopic to a point, a puncture, or a boundary component.
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
A Heegaard splitting is weakly reducible if there are disjoint essential simple closed curves
A Heegaard splitting is minimal or minimal genus if there is no other splitting of the ambient three-manifold of lower genus. The minimal value g of the splitting surface is the Heegaard genus of M.
Generalized Heegaard splittings
A generalized Heegaard splitting of M is a decomposition into compression bodies
A generalized Heegaard splitting is called strongly irreducible if each
There is an analogous notion of thin position, defined for knots, for Heegaard splittings. The complexity of a connected surface S, c(S), is defined to be
Examples
Three-sphere: The three-sphere
Under the usual identification of
Stabilization: Given a Heegaard splitting H in M the stabilization of H is formed by taking the connected sum of the pair
Lens spaces: All have a standard splitting of genus one. This is the image of the Clifford torus in
Three-torus: Recall that the three-torus
Theorems
Alexander's Lemma: Up to isotopy, there is a unique (piecewise linear) embedding of the two-sphere into the three-sphere. (In higher dimensions this is known as the Schoenflies theorem. In dimension two this is the Jordan curve theorem.) This may be restated as follows: the genus zero splitting of
Waldhausen's Theorem: Every splitting of
Suppose now that M is a closed orientable three-manifold.
Reidemeister-Singer Theorem: For any pair of splittings
Haken's Lemma: Suppose that
Classifications
There are several classes of three-manifolds where the set of Heegaard splittings is completely known. For example, Waldhausen's Theorem shows that all splittings of
Splittings of Seifert fiber spaces are more subtle. 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.
As of 2008, the only hyperbolic three-manifolds whose Heegaard splittings are classified are two-bridge knot complements, in a paper of Tsuyoshi Kobayashi.
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
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 Ozsvath and Zoltán Szabó. The theory uses the
History
The idea of a Heegaard splitting was introduced by Heegaard (1898). 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 & Gordon (1987), primarily through their concept of strong irreducibility.