从距离的观点看Heegaard分解:综述

发布时间：2018-07-03 03:55

本文选题：Heegaard分解 + 稳定化　；参考：《华东师范大学》2015年硕士论文

【摘要】：任意紧致可定向三维流形都是两个压缩体沿着正边界的并,这被称之为三维流形上的Heegaard分解。]Heegaard分解理论主要研究Heegaard分解的稳定化问题、亏格问题以及Heegaard分解与三维流形的拓扑性质(如基本群、不可压缩曲面等)、几何性质(三维流形上的双曲结构等)。Hempel于2000年将曲线复形理论中的一些思想引入Heegaard分解的研究中,并定义了Heegaard分解的距离的概念,这使得人们在Heegaard分解的稳定化问题、亏格问题、Waldhausen的Genus and Rank猜想、纽结洞数等方面的研究中取得了一些重要的进展。作者硕士研究生期间对三维流形的Heegaard分解理论进行了较为系统的学习,尤其重点阅读了关于Heegaard分解距离方面的最新研究进展。本文主要是是作者关于这方面学习的一个总结,同时,作者证明了距离至少为5的Heegaard分解的边界稳定化是不可稳定化的,这是以往结论的简单推广。
[Abstract]:Any compact orientable three-dimensional manifold is a union of two compressible bodies along a positive boundary, which is called Heegaard decomposition on a three-dimensional manifold.] Heegaard decomposition theory mainly studies the stabilization of Heegaard decomposition. Genus problem and topological properties of Heegaard decomposition and three-dimensional manifolds (such as fundamental groups, incompressible surfaces, etc.), geometric properties (hyperbolic structures on three-dimensional manifolds, etc.) .Hempel introduced some ideas of curve complex theory into the study of Heegaard decomposition in 2000. The concept of the distance of Heegaard decomposition is defined, which makes people make some important progress in the study of Heegaard decomposition stabilization problem, Genus and Rank conjecture of Waldhausen, knot hole number and so on. The author studied the theory of Heegaard decomposition of three-dimensional manifold systematically during his master's degree and especially read the latest research progress on the distance of Heegaard decomposition. This paper is mainly a summary of the author's study in this field. At the same time, the author proves that the boundary stabilization of Heegaard decomposition with a distance of at least 5 is not stable, which is a simple generalization of the previous conclusions.
【学位授予单位】：华东师范大学
【学位级别】：硕士
【学位授予年份】：2015
【分类号】：O189.3

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