Classifications[edit]
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 are standard. The same holds for lens spaces (as proved by Francis Bonahon and Otal).
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.
Applications and connections[edit]
Minimal surfaces[edit]
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 Shing-Tung Yau went on to use results of Waldhausen to prove results about the topological uniqueness of minimal surfaces of finite genus in . The final topological classification of embedded minimal surfaces in was given by Meeks and Frohman. The result relied heavily on techniques developed for studying the topology of Heegaard splittings.
Heegaard Floer homology[edit]
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 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.
History[edit]
The idea of a Heegaard splitting was introduced by Poul 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 Andrew Casson and Cameron Gordon (1987), primarily through their concept of strong irreducibility.