Heegaard Splitting From The View Of Distance:A Summary

If a compact orientable 3-manifold is the union of two compression bodies along the boundary, then it is called a Heegaard splitting of this 3-manifold.the theory of Heegaard splittings mainly studies the stabilization problem of Heegaard splittings, genus prob-lem, Heegaard splittings and topological properties of three dimensional manifolds(such as fundamental group, incompressible surface), besides,geometry properties (hyperbolic structures on 3-manifolds, etc.)-In 2000, Hempel introduced some ideas in the theory of complex curve to the study of Heegaard splittings, and defined the concept of the dis-tance of Heegaard splittings, which makes us get some important progress in the study of the stabilization problem of Heegaard splittings, genus problem, Waldhausen’s Genus and Rank conjecture, the number of knot holes, etc. The author has learned the theory of Heegaard splittings comprehensively during the post-graduate, especially focused on reading the latest progress on the distance of Heegaard splittings. This article is mainly a summary on this study, meanwhile, the author proves that the a-stabilization of a Hee-gaard splitting with distance at least 5 is unstabilized, which is a simple generalization of the previous conclusions.