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 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 The idea of a Heegaard splitting was introduced by . 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 , primarily through their concept of strong irreducibility. See also Manifold decomposition Handle decompositions of 3-manifolds Compression body References 3-manifolds Minimal surfaces Geometric topology