Geometry and Topology of 3-Manifolds

三流形的几何和拓扑

基本信息

批准号：
12440015
负责人：
KOJIMA Sadayoshi
金额：
$ 7.17万
依托单位：
Tokyo Institute of Technology
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (B)
财政年份：
2000
资助国家：
日本
起止时间：
2000 至 2002
项目状态：
已结题

项目摘要

This project was aimed to develop the interdisciplinary study of 3-manifolds which interacts geometry and topology based on the connection between several structures related mainly with hyperbolic geometry. We have in fact promoted our project in conjunction with the activity of the Topology Research Congress.We made certain significant progresses during these three research years which turned out to be rather surprising than what we had expected. The study of myself together with Mizushima and Tan on circle packings on surfaces with projective structures has clarified an expected global structure of their moduli space in the light of the deformation theory of hyperbolic 3-manifolds. The study of Yoshida on SU(2) conformal field theory has established successfully an explicit description of a basis of conformal blocks, and has approached to the fundamental connection between the geometry and topology of 3-manifolds suggested for instance by the volume con-jecture. Also, the global diagram in the 3-manifold topological invariant world suggested by Ohtsuki was completed quite recently in the most universal way by Habiro and Le. In addition, there have been down-to-earth progresses by other collaborators such as Morita's study on the mapping class group of surfaces, Matsumoto's on foliations, Sakuma's on knots and geometric structures, Soma's on bounded cohomology,In conclusion, our research has clarified the object on which we should now focus for finding the mathematical principle behind the interaction between many structures observed in the 3-manifold theory. Note in addition, the theme was fortunately funded for further study as the part II.

该项目旨在发展三流形的跨学科研究，该研究基于主要与双曲几何相关的几种结构之间的联系，使几何学和拓扑学相互作用。事实上，我们已经结合拓扑研究大会的活动来推动我们的项目。在这三年的研究中，我们取得了一些重大进展，结果超出了我们的预期。我与 Mizushima 和 Tan 一起对射影结构表面上的圆堆积进行的研究，根据双曲 3 流形的变形理论阐明了其模空间的预期全局结构。吉田对SU(2)共形场理论的研究成功地建立了共形块基础的明确描述，并接近了例如体积猜想所暗示的3-流形的几何和拓扑之间的基本联系。此外，Ohtsuki 提出的 3 流形拓扑不变世界中的全局图最近由 Habiro 和 Le 以最普遍的方式完成。此外，其他合作者也取得了脚踏实地的进展，例如 Morita 对曲面映射类群的研究、Matsumoto 对叶状结构的研究、Sakuma 对结和几何结构的研究、Soma 对有界上同调的研究，总而言之，我们的研究已经阐明我们现在应该关注的对象是寻找三流形理论中观察到的许多结构之间相互作用背后的数学原理。另外请注意，幸运的是，该主题作为第二部分获得了进一步研究的资助。