多个三维流形融合亏格的变化

发布时间：2018-02-21 02:15

本文关键词： 三维流形 Heegaard分解融合亏格　出处：《哈尔滨工业大学》2017年硕士论文　论文类型：学位论文

【摘要】：在低维拓扑学中,对于三维流形的探究是最主流的,其分类问题更是其中的一个基础性的方向。在众多的三维流形分析手法中,本文主要采纳的是其中的组合方案,即通过站在三维流形的融合的视角来分析三维流形。组合方案是研究三维流形的主要手法之一,它的特点是可以很容易地对流形的属性进行辩识,这是一种很便利的研究方案。Heegaard属性是三维流形的一种很重要的组成结构,所以探讨它是十分有必要的。Heegaard分解理论的主要理念是:首先在目标流形内部找到某些个合适的曲面,然后沿着它们切开,如此目标流形就可以被剖析成不定数量的相对容易辩识的“简明个体块”,最后对诸如此类的“简明个体块”施以粘合工作,以还原成原始的目标流形。人们只需研究各个压缩体的属性以及子流形间的相连线索就可以分析到目标流形的属性及构造。十几年前,著名数学家Hempel定义了一个重要概念—Heegaard距离,而后有很多与此相关的研究结果被提出,这使得Heegaard分解理论的成长空间更上了一个高度。Heegaard亏格作为Heegaard分解理论中的一个重要拓扑指标,人们对其投入的精力也一直存在。近些年来,三维流形融合过程中的亏格变化成为了一个热点问题。在本文中,我们要对带边三维流形中的本质曲面进行探讨工作,通过分析流形及其内部的曲面的Heegaard结构,使用Heegaard分解的细化和融合来提供多个三维流形实施融合前后的Heegaard亏格稳定的充分条件。值得一提的是,流形中的本质曲面于其中扮演着关键角色,因为它的欧拉示性数是与我们提供的条件唯一相关的。
[Abstract]:In low dimensional topology, is the most mainstream for the research of three-dimensional manifolds, the classification problem is one of the fundamental direction. In many 3-manifolds analysis methods, this paper mainly adopt the combination of these solutions, namely by standing in the fusion of 3-manifolds from the perspective of 3-manifolds. The combination scheme is one of the main techniques of three-dimensional manifolds, its characteristic is can be carried out easily on the manifold identification attribute, which is a research program very convenient.Heegaard property is a very important component of the three-dimensional manifold structure, so it is the main idea of necessary.Heegaard decomposition theory is first find some suitable surface in the target manifold, then cut along them, "concise blocks so the target manifold can be parsed into an indefinite number of relatively easy identification", At the end of the "simple blocks" and so on to the adhesive work, to restore the original target manifold. People only need to study various compact attributes and submanifolds between connected leads can be analysis to property and structure of the target manifold. Ten years ago, the famous mathematician Hempel defines an important concept - Heegaard distance then, there are a lot of research and the related results are presented, which makes the Heegaard decomposition theory of growth space on a more highly.Heegaard genus as an important topological index Heegaard decomposition theory, people also have the inputs of energy. In recent years, the three-dimensional manifold fusion genus changes in the process it has become a hot topic. In this paper, we want to discuss the essence of the work surface in a 3-manifold with boundary, through the analysis of the surface and interior of the Heegaard manifold The use of Heegaard decomposition refinement and fusion to provide a number of three-dimensional manifold implementation before and after the integration of Heegaard deficiency of sufficient conditions for lattice stability. It is worth mentioning that the essence of the surface manifold in which plays a key role, because it is the Euler number and we offer only relevant.

【学位授予单位】：哈尔滨工业大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O189

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