Some Results On Amalgamated Heegaard Splittings Of 3-manifolds

In the field of combinatorial 3-manifold theory, one of the most important methodsis the Heegaard splitting theory. Complicated 3-manifolds can be decomposed into twocompression bodies by a certain surface which is called the Heegaard surface. In thepast several decades, the Heegaard splitting theory has developed greatly, and has playedan important role in solving the problems of 3-manifolds. The definition of distanceof Heegaard splitting was first introduced by Hempel in 2001. This definition not onlygeneralizes the definitions of reducibleness, weakly reducibleness of Heegaard splitting,but also plays a major role in the study of incompressible surface, amalgamation of 3-manifold with boundary etcetera. Based on it, the research on the change of the Heegaardgenus of the surface sum of 3-manifold with boundary is gradually becoming the focus ofthe 3-manifold theory in recent years.From the study of the essential surface in 3-manifold and the distance of Heegaardsplitting, this paper mainly performs the change of genus in the process of amalgamation,annulus sum and self-amalgamation of 3-manifold with boundary by using the thin posi-tion and amalgamation of Heegaard splitting theory, the special properties of annulus inmanifold and the theorem proved by Scharlemann-Tomova. In this dissertation, the mainresults are some sufficient conditions for the genus of the surface sum not to go down.Concretely, the main contents are as follows:1. We study the change of the genus in the process of amalgamation of 3-manifold.By using the thin position theory of Heegaard splitting developed by Scharlemann-Thompson and the amalgamation theory developed by Schultens, we give a sufficientcondition for the genus of amalgamation to be non-degenerate. In particular, this suffi-cient condition only relates on the Euler characteristic of the essential surface with bound-ary in the manifold, it has nothing to do with the gluing maps between surfaces and thedistance of the factor manifold.2. We study the problem of additivity of Heegaard genera of one side non-separatingannulus sum of 3-manifolds with boundary. By using the special properties of annulus inmanifold and the theorem proved by Scharlemann-Tomova which says that the Heegaarddistance is bounded by twice of the genus of the manifold, we give a sufficient condition for the Heegaard genera of the annulus sum to be additive, the condition is that the Hee-gaard distance is sufficiently large. Under this condition, we prove that when the genus ofthe boundary components where the annulus lies is not less than 2, the manifold obtainedfrom the annulus sum has a unique minimal Heegaard splitting up to isotopy.3. We research the application of the annulus sum in the knot theory, the connectedsum of knots is a particular kind of annulus sum, it relates to an important knot invariantcalled the tunnel number of knot. We give a sufficient condition for the tunnel number tobe super-additive and any minimal Heegaard splitting of the knot exterior to be weaklyreducible.4. We discuss the self-amalgamation of the 3-manifold with boundary. By using theproperties of the boundary stabilization of Heegaard splitting, we give a lower bound forthe genus of the self-amalgamation of 3-manifold. And under some conditions, we provethat the self-amalgamation of the Heegaard splitting is minimal.