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.

3-流形的Heegaard分裂研究一直是3-流形理论中最重要的主题之一，我们对“非双曲”3-流形的Heegaard分裂已经有了深入的了解，但我们对3-流形的Heegaard分裂的理解还很有限。双曲流形远不能令人满意。特别是，据我们所知，双曲结构和Heegaard splittigs之间的关系尚不清楚。在这个项目中我们已经证明了。 2 桥结补的双曲结构与其桥结构密切相关，这是一种 Heegaard 分裂。事实上，我们已经利用以下公式给出了 2 桥结补的双曲结构的具体构造。更准确地说，我们在 2 桥结补上构建了一系列连续的双曲锥流管结构，该结构沿上隧道和下隧道具有奇点，锥角从 0 到 0 变化。 2π。锥角为 0 的锥流形结构对应于准 Fuchsian 一次穿孔环面空间的有理边界群，锥角为 2π 的锥流形结构给出了 2 桥结补的双曲结构。我们对 Jorgensen 提出的拟 Fuchsian 一次穿孔环面群的理论（部分）给出了明确的表述和充分的证明，并推广了该理论和田正明为这个项目开发的计算机程序“OPTI”将乔根森的理论及其推广可视化，它不仅是这个项目不可或缺的工具，也是整个项目不可或缺的工具。我们希望在这个项目中获得的结果是双曲结构和 Heegaard 分裂之间关系研究的开始。 3-歧管。