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.

近年来，“低维拓扑理论”已经远远超出了“几何”的框架，在群论、复分析、动力系统、甚至理论物理、计算机科学等数学之外的领域中发现了密切的关系。关系，有许多（通常非常巨大）组合结构，例如，在 Teichmuller 空间上给出坐标的火车轨道，理想单元对双曲 3-流形的规范分解（Epstein-Penner）、Young 图（Jones）对 Hecke 代数表示的构造、自动群理论（Thurston）以及 Haken 的法向曲面理论与这些现象相关，似乎低维拓扑和低维拓扑的最新发展。计算机使我们能够直接、具体地对待这些对象。鉴于这些情况，在本研究中，我们打算从几何可以组合的角度来具体研究2维和3维流形。说起来，我们研究了以下主题。・通过 Heegaard 分裂分析 3 流形和结（特别是使用 Rubinstein-Scharlemann 引入的“图形”），并获得有关结解开隧道的有用信息，・研究 3 上的双曲结构-通过三角测量的流形，特别是从非常简单的双曲结构开始的2桥结补上的双曲结构，・研究关系3-流形的某种黎曼度量的模空间和几何结构之间，・研究算法（计算机可以处理的），以将给定 Heegaard 分裂的附加同胚分解为规范的 Dehn 扭曲。