Combinatorial structures of low dimensional manifolds

低维流形的组合结构

基本信息

批准号：
10640076
负责人：
KOBAYASHI Tsuyoshi
金额：
$ 2.43万
依托单位：
Nara Women's University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
1998
资助国家：
日本
起止时间：
1998 至 1999
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-10640076/
关键词：
3-manifold knot Heegaard splitting triangulation Riemannian metric Dehn twist Knot 絡み目層状化分岐曲面 train track 双曲構造擬等角変形

项目摘要

Recently "low dimensional topology theory" is far going out of the framework of "geometry" and finding out intimate relations between group theory, complex analysis, dynamical system, and even some fields out of mathematics like theoretical physics, and computer science. Within the relations, there are many (very huge, in general) combinatorial structures, for example, train tracks which give coordinates on Teichmuller spaces, canonical decomposition of hyperbolic 3-manifolds by ideal cells (Epstein-Penner), constructions of representations of Hecke algebra by Young diagram (Jones), automatic group theory (Thurston), and normal surface theory by Haken. In connection with these phenomena, it seems that recent development of low dimensional topology and of computer enables us to treat these objects directly and concretely.In view of these situations, in this research, we intended to study 2 and 3 dimensional manifolds from geometrical can combinatorial viewpoint. Concretely speaking, we studied about the following topics.・Analyzing 3-manifolds and knots via Heegaard splitting (particularly, with using "graphic" introduced by Rubinstein- Scharlemann), and obtaining useful informations on unknotting tunnels of knots,・Studying hyperbolic structures on 3-manifolds via triangulations, particularly on hyperbolic structures on 2-bridge knot complements starting from a very simple hyperbolic structure,・Studying about the relations between moduli spaces of certain kind of Riemannian metrics of 3-manifolds and geometric structures,・Studying about algorithms (that the computer can handle) to decompose the attaching homeomorphisms of the given Heegaard splittings into canonical Dehn twists.

最近，“低维拓扑理论”远远超出了“几何”的框架，并在群体理论，复杂分析，动态系统甚至某些领域之间找到了诸如理论物理学和计算机科学等数学的领域之间的亲密关系。在关系中，例如，有许多（一般而言，一般而言）组合结构，例如，在Teichmuller空间上提供坐标的火车轨道，由理想细胞（Epstein-Penner）（Epstein-Penner）对双曲线3个manifords的规范分解，由年轻的[Daivainga）组成的（jones）组合（jones）理论（thurston and thrurston and thurston and thurston and thrurston and y hecke代表）的结构。与这些现象有关，看来低维拓扑和计算机的最新发展使我们能够直接和具体地处理这些对象。在这些情况下，在这项研究中，我们打算研究来自几何歧管的2和3尺寸歧管。 Concretely speaking, we studied about the following topics.・Analyzing 3-manifolds and knots via Heegaard splitting (particularly, with using "graphic" introduced by Rubinstein- Scharlemann), and obtaining useful information on unknotting tunnels of knots,・Studying hyperbolic structures on 3-manifolds via triangulations, particularly on hyperbolic structures on 2-bridge knot completions starting from a非常简单的双曲线结构，・研究了某些类型的3个序列和几何结构的riemannian指标的模量空间之间的关系，・研究计算机（计算机可以处理的算法）将其分解为给定的Heegaard分布的算法（可以处理的），以分解为平方远处的同型同形。