Heegaard

Heegaard is a surname associated with a Danish mathematician who introduced Heegaard splitting in the late 19th century. The term refers to a foundational idea in 3-manifold topology.

A Heegaard splitting of a closed orientable 3-manifold M is a decomposition into two handlebodies, H1 and H2, that intersect along their common boundary. This boundary is a closed orientable surface, called the Heegaard surface. The splitting is described by the genus of that surface, which is the genus of the splitting. The minimal genus among all possible splittings of M is the Heegaard genus of M. The two handlebodies are glued together by a homeomorphism of their boundary, and different gluing maps yield different manifolds.

A Heegaard diagram encodes the gluing on the Heegaard surface as a collection of simple closed curves. Typically two sets of curves, alpha and beta, are drawn on the same surface; these curves represent the attaching circles for the handlebodies and determine the gluing map up to isotopy.

In contemporary topology, Heegaard theory provides a practical framework for studying 3-manifolds. Heegaard Floer homology, developed by Ozsváth and Szabó, uses Heegaard diagrams to construct powerful invariants of 3-manifolds and knots. The concept has also inspired higher-dimensional analogues; for example, trisections of smooth 4-manifolds, introduced by Gay and Kirby, mirror the spirit of Heegaard splittings in a different dimension.