(?)-reducible, Annular And Toroidal Handle Additions

In this paper, we will consider some problems about the degenerating handle additions in the 3-manifold theory by combinatorial method.The 3-manifold is the main subject in the research of the low-dimention topology. Since the hyperbolic 3-manifold is simple and basal, we can obtain some properties of complicated 3-manifold by studying it. In this thesis, our main aim is to estimate the minimal geometrical intersection number of two degenerating slope on hyperbolic 3-manifold, this is also one of the hotspots in the 3-manifold theory.M.Scharlemann and Y-Q wu have proved the theorem: suppose M is a hyperbolic 3-manifold with boundary, and α, β are two separating essential curves on a genus g > 1 boundary component of M, if M[α], M[β] are both non-hyperbolic, then A(α, β) ≤ 14. On base of this theorem, we will give more fined results. That is: Let M be a hyperbolic 3-manifold with boundary, suppose that α, β are two separating slopes on (?) M. If M[α] is (?)-reducible or annular while M[P) is (?)-reducible, then A(α,β) < 8; if M[α] is toroidal while M [β] is (?)-reducible, then A(α,β)<10.The structure of this paper as follows:In the first chapter, we are devoted to introduce the development and methods of the reseach of 3-manifold theory, also present the problems that will be studied.In the second chapter, we expatiate the basic definitions and theorems of 3-manifolds.In the third chapter, we expatiate the basic definitions and correlative conclusions of the graph theory, especialy the graghs on 2-sphere and torus.In the last chapter, we expatiate the basic definitions and correlative conclusions of hyperbolic 3-manifold and handle additions, then prove the lemmas and central theorem of the whole paper.