Topologically Minimal Property And The Stability Distance Of Heegaard Surfaces

One way to study3-manifolds is to detect the properties of embedded surfaces. The incompressible surfaces and strongly irreducible surfaces are such two important surfaces. The American mathematician David Bachman discussed the high dimensional homotopy groups of the disk complexes, and generalized the concepts of incompressible surfaces and strongly irreducible surfaces by defining topologically minimal surfaces and their topological indices. Topologically minimal surfaces are embedded surfaces whose disk complex are empty or non-contractible. They are minimal surfaces in the topology category, which can be viewed as the analogue of those in differential geometry. Especially, incompressible surfaces and strongly irreducible surfaces can be viewed as index0and1topologically minimal surfaces respectively, and many properties of these two types of surfaces are also applicable to topologically minimal surfaces with other indices.One of basic questions is to ask whether a3-manifold contains topologically minimal surfaces and if it does contain we may determine or estimate the index. However, there are not too many known topologically minimal surfaces with high index. Since high index topological minimal surfaces are all weakly reducible, we are interested in which weakly reducible surfaces are topologically minimal.In this thesis, we focus on a type of important weakly reducible surfaces, that is, the standard Heegaard surfaces of self-amalgamated manifolds. We give a sufficient con-dition and a necessary condition for self-amalgamated Heegaard surfaces to be index2topologically minimal. As a corollary, we show a sufficient condition of the standard Hee-gaard surfaces of surface bundles to be index2topologically minimal. Based on these results, we show a necessary condition for certain self-amalgamated Heegaard surfaces to be topologically minimal. We show that if the original Heegaard splittings is complicated enough. then the self-amalgamated Heegaard surface could not be topologically minimal.John Hempel introduced the concept of the distance of a Heegaard splitting by his research on the curve complexes of the Heegaard surfaces. It is a generalization of concepts of reducibility, weakly reducibility and strongly irreducibility. Heegaard distance, as an invariant of Heegaard splittings, is often used to describe the complexity of the Heegaard splitting and the construction of the3-manifold. Based on Hempel’s idea, we introduced a new tool to describe the complexity by defining the stability distance of a Heegaard splitting. We show a method to construct stabilized Heegaard splitting by operating finite easy Dehn surgeries and we give an alternative proof of Lickorish-Wallace Theorem, which is the fundamental theorem of the Dehn surgery theory of3-manifolds.