【作者】 杜昆；

【导师】 邱瑞锋；

【作者基本信息】 大连理工大学 ， 基础数学， 2010， 博士

【摘要】 流形的分类是研究流形的重要课题之一,由于Kneser-Milnor定理,JSJ分解定理以及Thurston几何化猜想的提出,使得人们更加关注三维流形,并相信三维流形是可以完全拓扑分类的.从另一个角度研究三维流形,就是将两个三维流形沿着同胚的边界相粘,所得流形我们称之为三维流形的融合积.一般地,我们只考虑Haken流形的融合积.如果一个闭曲面S将三维流形M分成两个压缩体V和W,那么我们称V∪S W是M的一个Heegaard分解.由于任意一个紧致流形都存在Heegaard分解,Heegaard分解日渐成为组合三维流形的一个重要不变量,对三维流形的分类起着重要作用.但是,并不像我们期望的那样,任一三维流形的Heegaard分解都是唯一的.三维流形Heegaard分解的唯一性只是一种极为特殊的现象.本文主要研究了三维流形融合积的Heegaard分解,给出了三维流形自融合积的Heegaard分解如果满足一定条件,他的极小Heegaard分解在合痕意义下具有唯一性,并给出了三维流形融合积的Heegaard分解非退化的一个最好的条件.三维流形融合积的Heegaard分解是将两个流形沿闭曲面相粘,本文还讨论了将两个三维流形沿平环相粘,给出了他的具体的Heegaard结构,并证明了如果有一个平环是非分离的,那么我们有g(M)=g(M1)+g(M2).此结果蕴含了纽结的一个组合不变量tunnel number的超可加性.更多还原

【Abstract】 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.更多还原