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1999 Fiscal Year Final Research Report Summary

Study of contact structures and foliations on 3-manifolds

Research Project

Project/Area Number 09640130
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field Geometry
Research InstitutionChuo University

Principal Investigator

MITSUHATSU Yoshihiko  Chuo University, Faculty of Science & Eng., Professor, 理工学部, 教授 (70190725)

Co-Investigator(Kenkyū-buntansha) MIZUTANI Tadayoshi  Saitama University, Faculty of Science, Professor, 理学部, 教授 (20080492)
TAKAKURA Tatsuru  Chuo University, Faculty of Science & Eng., Lecturer, 理工学部, 講師 (30268974)
MATSUYAMA Yoshio  Chuo University, Faculty of Science & Eng., Professor, 理工学部, 教授 (70112753)
KANOA Yutaka  Hokkaido University, Faculty of Science, Instructor, 理学部, 助手 (30280861)
ONO Kaoru  Hokkaido University, Faculty of Science, Professor, 理学部, 教授 (20204232)
Project Period (FY) 1998 – 1999
KeywordsContact Structure / foliation / symplectic structure / bi-contact structure / projectively Anosoo flow / symplectic filling / geometric quantization / tight contact structure
Research Abstract

Mitsumatsu and Mizutani studied and constructed many examples of bi-contact structures with a research group of foliations. Especially, they constructed a bicontact structure on the 3-sphere which consists both of over-twisted ones. Still the realization problem of homotopy class of plane fields as such structures remains to be studied.
Ono has established in a colaboration with Fukaya foundamental theory in applying the J-curves to symplectic topology, overcoming the notorious problem of negative multiples. Major consequences from this are the definition of Gromov-Witten invariants for general symplectic manifolds and the positive solution for a version of the Arnold conjecture for the same class. He and Kanda also worked on applying Seiberg-witten theory to contact topology, colaborating with Ohta, and got toplogical constraints on the symplectic filling 4-manifolds around simple singularities. This streem of works continues and is expected to make a further progress, especially in relation with the last subject of study below.
Kanda studied contact structures in more toplogical way, and classified tight contact structures on 3-torus and showed nonexactness of Bennequin's inequality.
Takakura and Mitsumatsu have been searching for the formalism to study contact topology by using Lagrangian/Legendrian torus, instead of looking at J-curves in the symplectization. This is based on the theory of geometric quantization, on which Takakura has been working. They found that most of major concepts in the theory of algebraic functions in one variable can be suitably translated and planted in this framework. However, studying contact topology through this remains as the next step of research to go.

  • Research Products

    (12 results)

All Other

All Publications (12 results)

  • [Publications] K. Fukaya & K. Ono: "Arnold conjecture and Gromov-Witten invariant"Topology. 36. 933-1048 (1999)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] K. Fukaya & K. Ono: "Arnold conjecture and Gromov-Witten invariant for general symplectic manifolds"Fields Institute Communications A.M.S.,. 24. 173-190 (1999)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] H. Ohta & K. Ono: "Symple singularities and topology of symplectic-filling 4-manifolds"Comment. Math. Heb. 74. 575-590 (1999)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] Y. Kanda: "On the Thurston-Bennequin invariant of Legendrian knots and non exactness of Bennequin's inequality"Invest. Math. 133. 227-242 (1998)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] Y. Kanda: "The classification of tight contact structures on the 3-Toucs"Communication in Analysis and Geometry. 5-3. 413-438 (1997)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] 三松佳彦: "3 Dimensional Contact topology"日本数学会. 120 (2000)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] Kenji FUKAYA & Kaoru ONO: "Arndol Conjecture and Gromov-Wittern invariant"Topology. 36. 933-1048 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Kenji FUKAYA & Kaoru ONO: "Arnold Conjecture and Gromov-Wittern invariant for genernal symplectic nmanifolds"Fields Institute Communications, A.M.S.. 24. 173-190 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Hiroshi OHTA & Kaoru ONO: "Simple singularities and topology of symplectic filling 4-manifolds"Comment. Math. Helv.. 74. 575-590 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Yutake KANDA: "On the Thurston-Bemnequin invariant of Legendrian knots and non exactness of Bemnequin's isequality"Invest. Matn.. 133. 227-242 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Yutake KANDA: "The classification of tight contact structure on the 3-torus"Communications in Analysis and geometry. 5-3. 413-438 (1997)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Yoshihiko MITSUMATSU & Kaoru ONO: "3-dimensional contact topology"MEMOIR in Japanese, Math. Soc. Japan. Vol.1(to appear). (2000)

    • 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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