Heegaand spliffings and hyperbolic structures of 3-manifolds

Heegaand spliffings 和 3 流形的双曲结构

基本信息

批准号：
09440033
负责人：
SAKUMA Makoto
金额：
$ 7.04万
依托单位：
Osaka University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (B)
财政年份：
1997
资助国家：
日本
起止时间：
1997 至 1999
项目状态：
已结题

项目摘要

The study of Heegaard splittings of 3-manifolds has been one of the most important themes in 3-manifold theory, and we already have deep understanding of the Heegaard splittings of "non-hyperbolike" 3-manifolds. However, our understanding of those of hyperbolic manifolds is far from satisfaction. In particular, as far as we know, no relationship between the hyperbolic structures and the Heegaard splittigs had been known.In this project we have proved that the hyperbolic structure of a 2-bridge knot complement is intimately related with its bridge structure, which is a kind of Heegaard splitting. In fact, we have given a concrete construction of the hyperbolic structure of a 2-bridge knot complement by using the 2-bridge structure. To be more precise, we have constructed a continuous family of hyperbolic cone-manifold structures on a 2-bridge knot complement which have singularities along the upper and lower tunnels, where the cone angle varies from 0 to 2π. The cone-manifold structure with cone angle 0 corresponds to a rational boundary group of the quasi-Fuchsian once-punctured torus space and that with cone angle 2π gives the hyperbolic structure of the 2-bridge knot complement. To establish this result, we have given an explicit formulation and a full proof to (a part of) the theory announced by Jorgensen on the quasi-Fuchsian once-punctured torus groups, and generalized the theory to that for the groups outside the quasi-Fuchsian once-punctured space. The computer program "OPTI" developed by Masaaki Wada for this project visualizes Jorgensen's theory and its generalization, and it has been an indispensable tool not only for this project but also for the study of Teichmuller spaces. We hope the result we have obtained in this project is the beginning of the study of the relationship between the hyperbolic structures and the Heegaard splittings of 3-manifolds.

对3个manifolds的Heegaard分裂的研究一直是3个模板理论中最重要的主题之一，我们已经对“非杂酚型” 3个manifolds的Heegaard分裂有深刻的了解。但是，我们对双曲线歧管的理解远非满意。据我们所知，双曲线结构与Heegaard splittigs之间没有任何关系。在这个项目中，我们规定，2桥结的完成的双曲线结构与其桥梁结构密切相关，这是一种Heegaard剥离。实际上，我们通过使用2桥结构给出了2桥结完成的双曲线结构的混凝土结构。更确切地说，我们已经在2桥结的完成上构建了一个连续的双曲线锥体结构，该结构沿着上下隧道具有奇异性，其中锥角度从0到2π不等。具有锥角度0的锥体式结构对应于准芬斯曾经启动的圆环空间的一个理性边界组，并且具有锥形角2π的圆锥形结构构成了2桥结的双曲线结构。为了建立这一结果，我们给了Jorgensen在准富奇西曾经曾经启动过的圆环群体上宣布的理论（一部分）的明确公式和完整的证据，并将该理论推广到了准富奇西亚曾经启用的空间之外的基团。 Masaaki Wada为此项目开发的计算机程序“ Opti”可视化Jorgensen的理论及其概括，它不仅是该项目的必不可少的工具，而且对于Teichmuller空间的研究也是必不可少的工具。我们希望我们在该项目中获得的结果是研究双曲线结构与3个manifolds的Heegaard分裂之间关系的开始。