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2006 Fiscal Year Final Research Report Summary

Geometry of Groups and Moduli Spaces

Research Project

Project/Area Number 16204005
Research Category

Grant-in-Aid for Scientific Research (A)

Allocation TypeSingle-year Grants
Section一般
Research Field Geometry
Research InstitutionUNIVERSITY OF TOKYO

Principal Investigator

MORITA Shigeyuki  University of Tokyo, Graduate School of Mathematical Sciences, Professor, 大学院数理学研究科, 教授 (70011674)

Co-Investigator(Kenkyū-buntansha) FURUTA Mikio  University of Tokyo, Graduate School of Mathematical Sciences, Professor, 大学院数理学研究科, 教授 (50181459)
TSUBOI Takashi  University of Tokyo, Graduate School of Mathematical Sciences, Professor, 大学院数理学研究科, 教授 (40114566)
KAWAZUMI Nariya  University of Tokyo, Graduate School of Mathematical Sciences, Associate Professor, 大学院数理学研究科, 助教授 (30214646)
FUJIWARA Koji  Tohoku University, Graduate School of Science, Associate Professor, 大学院理学研究科, 助教授 (60229078)
MITSUMATSU Yoshihiko  Chuo University, Department of Science & Technology, Professor, 理工学部, 教授 (70190725)
Project Period (FY) 2004 – 2006
Keywordsmapping class group / Riemann surface / moduli space / Floer homotopy type / diffeomorphism group / Teichmuller space / hyperbolic group / Morita-Mumford class
Research Abstract

We investigated mapping class group of surfaces and moduli space of Riemann surfaces as well as related groups and moduli spaces mainly from the viewpoint of topology. In a joint work with Penner, the representative constructed a canonical 1-cocycle on the Teichmuller space. Kawazumi, together with Penner and Bene, realized higher Johnson homomorphisms combinatorially on the Teichmuller space. Furuta developed theory of Floer homotopy type associated to the Seiberg-Witten theory and Tsuboi obtained a remarkable result concerning the structure of real analytic diffeomorphism group of manifolds. Fujiwara, Kohno, and Matsumoto obtained deep results in combinatorial structure of hyperbolic as well as the mapping class group, study of configuration spaces and arithmetic mapping class group, respectively. Mitsumatsu, Kitano, Akita with-Kawazumi, Hirose, and Murakami obtained interesting results, respectively, in the studies of 3-dimensional contact geometry, twisted Alexander polynomials, integral Morita-Mumford classes, mapping class group of surfaces in 4-manifolds, and volumes in hyperbolic geometry. Also the representative studied the structure of the Lie algebra consisting of symplectic derivations of the tensor algebra, without unit, generated by the homology of surfaces and, as an application, constructed unstable homology classes of genus 1 moduli spaces. Based on these results, further study of our theme has come into view.

  • Research Products

    (13 results)

All 2007 2006 Other

All Journal Article (12 results) Book (1 results)

  • [Journal Article] Characteristic classes of foliated surface bunales with area-preserving holonomy2007

    • Author(s)
      MORITA, Shigegaki, D. Kotschick
    • Journal Title

      Journal of Differential Geometry 75

      Pages: 273-302

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Nilpotency of the Bauer-Furate stable homotopy Seiberg-Nitten invariants2007

    • Author(s)
      FUROIA, Hilao, Kamstani Minami
    • Journal Title

      Geomtrg and Topology Monographs 10

      Pages: 147-154

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Characteristic classes of foliated surface bundles with area-preserving holonomy2007

    • Author(s)
      Shigeyuki MORITA, Dieter Kotschick
    • Journal Title

      Journal of Differential Geometry 75

      Pages: 273-302

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] Nilpotency of the Bauer-Furuta stable homotopy Seiberg-Witten invariants2007

    • Author(s)
      Mikio FURUTA, Kametani, Minami
    • Journal Title

      Geometry and Topology Monographs 10

      Pages: 147-154

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] On the group of foliation preserving diffcomorphisms2006

    • Author(s)
      TSUBOI, Takashi
    • Journal Title

      Foliations 2005, Lodz

      Pages: 411-430

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] JSJ-decampositions of finitely psesented groups and complexes of groups2006

    • Author(s)
      FUJIWARA, Koji, Papasogla
    • Journal Title

      Geometric and Functional Analysis 16

      Pages: 120-125

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] On the group of foliation preserving diffeomorphisms2006

    • Author(s)
      Takashi TSUBOI
    • Journal Title

      Foliations 2005 Lodz

      Pages: 411-430

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] JSJ-decomposition of finitely presented groups and complexes of curves2006

    • Author(s)
      Koji FUJIWARA, Papasoglu
    • Journal Title

      Geometric and Functional Analysis 16

      Pages: 70-125

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] Torelligroups, extended Johnson homomorphisms, and new cycles on the modali space of curves

    • Author(s)
      MORITA, Shigegaki, R. Penner
    • Journal Title

      Math. Proc. Cambridge Philosophicel society (to appear)

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Twisted Morita-Mumford classes on braid groups

    • Author(s)
      KAWAZUMI, Nariya
    • Journal Title

      Geometry and Topology Honographs (to appear)

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Torelli Groups, extended Johnson homomorphisms, and new-cycles on the moduli space of curves

    • Author(s)
      Shigeyuki MORITA, Robert Penner
    • Journal Title

      Math. Proceedings Cambridge Philosophical Society (to appear)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] Twisted Morita-Munford classes on braid groups

    • Author(s)
      Nariya KAWAZUMI
    • Journal Title

      Geometry and Topology Monographs (to appear)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Book] わじれAlexander不変量(数学メモクール第5巻)2006

    • Author(s)
      北野晃朗, 合田洋, 森藤孝之
    • Total Pages
      183
    • Publisher
      日本数学会
    • Description
      「研究成果報告書概要(和文)」より

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Published: 2008-05-27  

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