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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”正在走出“几何”的框架，与群论、复分析、动力系统，甚至与理论物理、计算机科学等数学之外的领域有着密切的联系。在关系中，有许多（非常巨大的，在一般情况下）组合结构，例如，火车轨道，使坐标Teichmuller空间，规范分解的双曲3-流形的理想细胞（爱泼斯坦-彭纳），建设的代表Hecke代数的杨图（琼斯），自动群论（瑟斯顿），和正常的表面理论的哈肯。与这些现象相联系的是，近年来低维拓扑学和计算机的发展使我们能够直接具体地处理这些问题，鉴于这些情况，本研究拟从几何能组合的观点来研究2维和3维流形。具体而言，我们研究了以下主题。·用Heegaard分裂分析三维流形和纽结·通过三角剖分研究3-流形上的双曲结构，特别是从一个非常简单的双曲结构开始研究2-桥纽结补上的双曲结构，·研究三维流形的某种黎曼度量的模空间与几何结构之间的关系;·研究将给定Heegaard分裂的附加同胚分解为标准Dehn扭的算法（计算机可以处理）。