1999 Fiscal Year Final Research Report Summary
Deformation of cone-manifolds and topology of 3-mainfolds
| Project/Area Number |
10440017
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| Research Category |
Grant-in-Aid for Scientific Research (B)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Geometry
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| Research Institution | Tokyo Institute of Technology |
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Principal Investigator |
KOJIMA Sadayoshi Tokyo Institute of Technology, Material and Computing Sciences, Professor, 情報理工学研究科, 教授 (90117705)
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| Co-Investigator(Kenkyū-buntansha) |
SOMA Teruhiko Tokyo Denki University, Department of Mathematical Sciences, Professor, 理工学部, 教授 (50154688)
YOSHIDA Tomoyoshi Tokyo Institute of Technology, Department of Mathematics, Professor, 大学院・理工学研究科, 教授 (60055324)
MORITA Shigeyuki University of Tokyo, Department of Mathematical Sciences, Professor, 大学院・数理科学研究科, 教授 (70011674)
OHTSUKI Tomotada Tokyo Institute of Technology, Department of Mathematical and Computing Sciences, Associate Professor, 情報理工学研究科, 助教授 (50223871)
YAMAGUCHI Takao Kyushu University, Department of Mathematics, Professor, 大学院・数理学研究科, 教授 (00182444)
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| Project Period (FY) |
1998 – 1999
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| Keywords | hyperbolic geometry / cone-manifold / 3-manifold / deformation / geometric structure / rigidity / Dehn filling / knot |
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| Research Abstract |
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.
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