Deformation of cone-manifolds and topology of 3-mainfolds

本课题的目的是在以往的相关工作的基础上，从跨学科的观点发展锥流形的变形理论，并将其应用于三维流形的拓扑理论。课题的起点是1998年7月在东京工业大学举办的“锥流形与双曲几何”研讨会。通过对迄今为止相关工作的总结，我们再次肯定了锥结构变形方法的吸引力和威力。然后，反思在研讨会上对Dehn填充的几何理解的关注，我们通过两年的多次讨论和小型密集会议，发展了锥结构的变形理论和三维流形的拓扑，强调与规范理论，映射类，映射，Kleinian群和量子不变量的联系。我们还印制了一些非正式的小册子，收集了一些研究草案，并分发给机构和研究人员，以方便他们。2000年1月的总结性研讨会“3-流形的几何拓扑”总结了近两年来几何方法研究3-流形的成果，并讨论了研究的前景。最近3-流形的几个方面之间存在着许多相互作用，几何方法研究3-流形似乎进入了新的阶段。鉴于这些情况，我们的项目阐明了几何方法在寻找外观背后的数学结构变得重要的未来，其中的特点将更多地相互作用。

The aim of this project was to develop the deformation theory of cone-manifolds from interdisciplinary viewpoints and to apply it to the theory of topology of 3-manifolds, base on our previous related works.The starting point was the workshop "Cone-manifolds and hyperbolic geometry" which we held on July '98 at Tokyo Institute of Technology. Summarizing related works up to those days, we reconfirmed an attraction and power of the deformation method of cone structures. Then reflecting on much attention for geometric understanding of Dehn fillings in the workshop, we have developed the theory of deformation of cone structures and topology of 3-manifolds, emphasizing the connection with gauge theory, mapping classes, collapsings, Kleinian groups and quantum invariants, by having many discussions and small intensive meetings for two years. We also printed a few informal booklets which collected several research drafts, and distributed them to the institutions and researchers for their convenience. The concluding symposium "geometric topology of 3-manifolds" on January '00 summarized the results on 3-manifolds by geometric method in the last two years and discussed the prospect of the research.There are many interactions between several aspects of 3-manifolds recently and the study of 3-manifolds by geometric method seems to get into new stage. In view of these circumstance, our project clarified that geometric method in finding mathematical structures behind appearance became important for the future in which presentations of characteristics would interact more.