Heegaard Splittings Of Amalgamated 3-manifold

The classification of manifolds is an important topic of studying manifolds. Because of Kneser-Milnor theorem, JSJ decomposition theorem and the Thurston geometrization conjecture, now people are interested in 3-manifold, and believed that 3-manifolds can be completely topological classification.From another point of view, we call 3-manifold obtained by gluing two 3-manifolds along their homeomorphic boundary the amalgamated 3-manifold. In general, we consider only the amalgamated Haken 3-manifold.If a closed surface S cuts a 3-manifold into two compression bodies V and W, then we call V Us W the Heegaard splitting of M.Since any 3-manifold has a Heegaard splitting, it becomes an important invariant of combinatorial 3-manifolds and play an important role for classifying 3-manifolds. However, it is not as we expected, any Heegaard splitting of 3-manifold is unique. The uniqueness of Heegaard splitting of 3-manifold is only a very special phenomenon.We mainly discuss the Heegaard splittings of amalgamated 3-manifold, prove that if the self-amalgamated 3-manifold satisfies some conditions, then the minimal Heegaard splitting is unique up to isotopy. We still show the best condition of amalgamated 3-manifold which the Heegaard genus is not degenerate.The amalgamated 3-manifold is obtained by gluing two closed surfaces. In this paper, we also discuss the 3-manifolds obtained by gluing annuli, prove that if one of annuli is not separating, then we have g(M)= g(M1)+g(M2). This result implies a combinatorial knot invariant:the super additivity of tunnel number.