Reducible Handle Additions

Nowadays, the 3-manifold theory is the one of the hotspots in the research of the low-dimension topology. Today, the main means of the study of the 3-manifold are the algebraic method, geometrical method,and combinatorial method. In this paper, we mainly make use of the combinatorial method.In the research of the 3-manifold theory,the combinatorial method mainly study the Heegaard splittings of the 3-manifolds , Dehn surgery, and the relationship of the incompressible surfaces of the 3-manifolds. And the same time, the problems of the Heegaard splitting, Dehn surgery can sum up the problems of the handle additions.After A.Cordon and R.Litherland's setting up the method of the graph theory, almost all of the cases can be well estimated in the research of the Dehn surgery. At present, people more focus their attention on the matter of the general handle additions of the 3-manifolds, especially, people care about the matter of the handle ~ additions of the separating curves.M.Scharlermann and Y-Q Wu proved the theorem: Suppose M is a hyperbolic 3-manifold, let α,β be the separatly curves on a genus g > 1 boundary component of M, if M[α],M[β] are nonhyperbolic, then△(α,β) ≤ 14.In this paper, about some cases of given theorem, we can obtain the finer conclusion on this base, namely: Suppose M is a hyperbolic 3-manifold, let α,β are the separatly essential cueves of the same boundary component of M. If M[α],M[β] are reducible, then △(α,β)≤8.The structure of this paper as follows:In the first chapter, we exptiate the basic definations and theorems of 3-manifolds.In the second chapter, we exptiate the basic definations and correlative conclusions of the graph theory.In the last chapter, we prove the central theorem of the whole paper.