Annuli In Handlebody And Several Properties Of The Heegaard Splitting

3-manifold theory is an important branch of low-dimensional topology. Basedon the combinatorial structures, complicated 3-manifolds can be decomposed intosome'simple'blocks by certain surfaces in 3-manifolds, such as Heegaard surfaces,incompressible surfaces, and normal surfaces, etc. We can then learn about the topo-logical properties and geometrical structures of 3-manifolds by analyzing these simplepieces. It is a powerful method in the study of 3-manifold. In this dissertation, we fo-cus on the embedding of maximal collections of essential annuli in a handlebody andseveral properties of the Heegaard splittings in 3-manifolds. Our research questionsinclude: the number of the maximal collection of annuli and the relations between theproperties of the Heegaard splittings and 3-manifolds. Some topological propertiesof 3-manifolds and surfaces in them, and geometrical structures of 3-mainfolds arederived by answers to the above questions.As we all know, handlebody is one kind of simple 3-manifold relatively. How-ever, as one element of the structure of Heegaard splitting, it's important out of ques-tion. One source of the subject comes from the embeddings of maximal incompress-ible surfaces in a handlebody. Since the number of maximal essential disks is alreadyclear, in this dissertation we emphasize on the embedding of maximal collections ofessential annuli in a handlebody. It is known that Rubinstein and Scharlemann solvethe number of maximal collections of essential annuli in H2, and all the possibilitiesare: 1, 2 or 3. Later on, F.C.Lei and J.Y.Tang show the upper bound of maximalcollections of essential annuli in Hn(n≥3) which is 4n ? 5, and the bound is thebest.Base on it, one part of this dissertation firstly show a best estimate of the lowerbound of the maximal collection of essential annuli in a handlebody, that is: Let Hnbe a handlebody of genus n≥3, then there exists a maximal collection of essentialannuli in Hn which contains exactly 2 annuli. Secondly, each possibility between thelower bound and the upper bound is constructed as an example, that is: Let Hn bea handlebody of genus n≥3. Then for each m, 2≤m≤4n ? 5, there exists amaximal collection of essential annuli in Hn which contains exactly m annuli. As an important tool in the research of geometrical structures and topologicalproperties, theories of the Heegaard splitting develop greatly in these years, and makegreat effect in solving the problems of 3-manifolds. The definition of distance ofHeegaard splitting which is first existed in Hempel's paper in recent years arousesthe interest of other scholars immediately. Not only it generalizes the definitions ofreducible, weakly reducible and disjoint curve property of Heegaard splitting, butalso it can be used as a tool to study incompressible surface, amalgamation and thetunnel number of knot. Therefore the research on it and the relative Heegaard splittingproperties is important for us to know more topological properties of 3-manifold.The other part of the dissertation focus on the study of the properties of Heegaardsplitting. Firstly, we generalize one conclusion of Thompson's and give a sufficientcondition of the Heegaard splitting with disjoint curve property. Secondly, we studythe bilongitude property and disjoint longitude property of the Heegaard splitting. Inthis part, the relations between bilongitude and the intersection number of two essen-tial disks is analyzed, and the manifold of genus 1 Heegaard splitting which satisfyingbilongitude property is classified. Simultaneously, for the manifolds of genus 1 and2 Heegaard splitting satisfying disjoint longitudinal property, the classification is alsosummarized. At last, we give an if and only if condition of the Heegaard splitingsatisfying disjoint (A2,D2).