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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.

三维流形的Heegaard分裂一直是三维流形理论中的重要课题之一，我们对“非双曲”三维流形的Heegaard分裂已经有了深入的了解。然而，我们对双曲流形的理解还远远不能令人满意。特别是，据我们所知，双曲结构与Heegaard分裂之间的关系还不清楚，本文证明了2-桥纽结补的双曲结构与它的桥结构密切相关，这是一种Heegaard分裂。实际上，我们已经利用2-桥结构给出了2-桥纽结补的双曲结构的具体构造。更精确地说，我们构造了一个连续的双曲锥流形结构族，它在2-桥纽结补上沿沿着上下隧道具有奇点，其中锥角从0到2π变化.锥角为0的锥流形结构对应于拟Fuchsian一次穿孔环面空间的一个有理边界群，锥角为2π的锥流形结构给出了2-桥纽结补的双曲结构.为了建立这一结果，我们对Jorgensen关于拟Fuchsian一次穿孔环面群的理论给出了一个明确的表述和充分的证明，并将这一理论推广到了拟Fuchsian一次穿孔环面群以外的群.由Masaaki Wada为该项目开发的计算机程序“OPTI”可视化了Jorgensen的理论及其推广，它不仅是该项目的不可或缺的工具，也是Teichmuller空间研究的不可或缺的工具。我们希望我们在这个项目中所得到的结果是研究三维流形的双曲结构与Heegaard分裂之间关系的开始。