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1999 Fiscal Year Final Research Report Summary

Deformation of cone-manifolds and topology of 3-mainfolds

Research Project

Project/Area Number 10440017
Research Category

Grant-in-Aid for Scientific Research (B)

Allocation TypeSingle-year Grants
Section一般
Research Field Geometry
Research InstitutionTokyo Institute of Technology

Principal Investigator

KOJIMA Sadayoshi  Tokyo Institute of Technology, Material and Computing Sciences, Professor, 情報理工学研究科, 教授 (90117705)

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)
Project Period (FY) 1998 – 1999
Keywordshyperbolic geometry / cone-manifold / 3-manifold / deformation / geometric structure / rigidity / Dehn filling / knot
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.

  • Research Products

    (16 results)

All Other

All Publications (16 results)

  • [Publications] S Kojima: "Hyperbolic 3-manifolds singular along knots"Chaos, Solitons and Fractals. 9. 765-777 (1998)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] S. Kojima: "Deformations of hyperbolic 3-cone-manifolds"Journal of Defferenctial Geometry. 49. 469-516 (1998)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] Kojima, Nishi & Yamashita: "Configuration spaces of points on the circle and hyperbolic Dehn fillings"Topology. 38. 479-516 (1999)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] S. Morita: "Structure of the mapping class groups of surfaces : a survey and a prospect"Geometry and Topology Monographs. 2. 349-406 (1998)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] T. Soma: "Non-zero degree maps to hyperbolic 3-manifolds"Journal of Differential Geometry. 49. 517-546 (1998)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] T. Ohtsuki: "The perturbative SO(3) invariant of rational homology 3-spheres recovers from the universal perturbative invariant"to appear in Topology.

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] 小島定吉: "トロポロジー入門"共立出版. 220 (1998)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] 吉田朋好: "ディラック作用素の指数定理"共立出版. 252 (1998)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] S. Kojima: "Hyperbolic 3-manifolds singular along knots"Chaos, Solitons and Fractal. 9. 765-777 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] S. Kojima: "Deformations of hyperbolic 3-cone-manifolds"Journal of Differential Geometry. 49. 469-516 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] S. Kojima, H. Nishi and Y. Yamashita: "Configuration spaces of points on the circle and hyperbolic Dehn fillings"Topology. 38. 497-516 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] S. Morita: "Structure of the mapping class groups of surfaces : a survey and a prospect"Geometry and Topology. 2. 349-406 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] T. Soma: "Non-zero degree maps to hyperbolic 3-manifolds"Journal of Differential Geometry. 49. 517-546 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] T. Ohtuski: "The perturbative SO(3) invariant of rational homology 3-spheres recovers from the universal perturbative invariant"(to appear in Topology).

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] S. Kojima: "Introduction to topology (book in Japanese)"Kyoritsu Shuppan. 220 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] T. Yoshida: "Index throrem for Dirac operators (book in Japanese)"Kyoritsu Shuppan. 255 (1998)

    • Description
      「研究成果報告書概要(欧文)」より

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Published: 2001-10-23  

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