Heegaard Splittings Of3-Manifolds

According to Moise and Bing’theorem, each compact orientable3-manifold admits a tri-angulation. Hence each compact orientable3-manifold has a Heegaard splitting. So we can understand3-manifolds by studying Heegaard splittings.The Heegaard distance is quite useful for studying a Heegaard splitting. A natural problem is how to construct a Heegaard splitting with a given distance. In recent years, some topologists have constructed some high distance Heegaard splittings by studying train tracks and automor-phisms of a surface. In this thesis, we study the subsurface projection of the curve complex and construct a geodesic connecting two separating essential simple closed curves in a surface. Through attaching2-handles along these two separating essential simple closed curves from d-ifferent sides of the surface and attaching handlebodies to the boundaries in a complicated way, we prove that for any given positive number n and genus g≥2, there is always a distance n, genus g Heegaard splitting.We can also study a Heegaard splitting through Dehn fillings. In recent years, some topolo-gists have studied the relationship between the minimal Heegaard splitting and Dehn fillings. At the same time, some Chinese topologists defined the distance non-decreasing Dehn fillings. In this thesis, after applying some results of the subsurface projection of a disk complex, we study the distance decreasing Dehn fillings and prove that for any distance at least3, genus2Heegaard splitting of a bounded3-manifold, there is always a distance non-decreasing Dehn filling.It is well known that the amalgamation and self amalgamation of3-manifolds produce new3-manifolds and the amalgamation and self-amalgamation of Heegaard splittings are also the Heegaard splittings of the new produced3-manifolds. In recent years, some topologists have studied the minimal Heegaard splitting of a3-manifold and its sub-manifolds. In this thesis, we study the stability of the self-amalgamation and the δ-stabilization of a Heegaard splitting and prove that:(1) when the Heegaard distance is at least3, the self-amalgamation of it is unstabilized;(2) when the Heegaard distance is at least6, the δ-stabilization of it is unstabilized.