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Complex analytic approach towards topology studies on the mapping class ganups for surfaces

Research Project

Project/Area Number 14540065
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field Geometry
Research InstitutionThe University of Tokyo

Principal Investigator

KAWAZUMI Nariya  The University of Tokyo, Graduate School of Mathematical Sciences, Associate Professor, 大学院・数理科学研究科, 助教授 (30214646)

Co-Investigator(Kenkyū-buntansha) MATSUMOTO Yukio  The University of Tokyo, Graduate School of Mathematical Sciences, Professor, 大学院・数理科学研究科, 教授 (20011637)
MORITA Shigeyuki  The University of Tokyo, Graduate School of Mathematical Sciences, Professor, 大学院・数理科学研究科, 教授 (70011674)
HASHIMOTO Yoshitake  Osaka City University, Graduate School of Sciences, Associate Professor, 大学院・理学研究科, 助教授 (20271182)
SHIBUKAWA Youichi  Hokkaido University, Graduate School of Sciences, Assistant Professor, 大学院・理学研究科, 助手 (90241299)
AKITA Toshiyuki  Hokkaido University, Graduate School of Sciences, Associate Professor, 大学院・理学研究科, 助教授 (30279252)
大場 清  お茶の水女子大学, 理学部, 助手 (80242337)
Project Period (FY) 2002 – 2004
Project Status Completed (Fiscal Year 2004)
Budget Amount *help
¥4,200,000 (Direct Cost: ¥4,200,000)
Fiscal Year 2004: ¥1,600,000 (Direct Cost: ¥1,600,000)
Fiscal Year 2003: ¥1,200,000 (Direct Cost: ¥1,200,000)
Fiscal Year 2002: ¥1,400,000 (Direct Cost: ¥1,400,000)
KeywordsRiemann surface / moduli space / mapping class group / Magnus expansion / Morita-Mumford class / Stasheff associahedron / automorphism group of a free group / Johnson homomorphism / マグナス表現 / スタシェフのアソシアヘドロン / 超楕円的写像類群 / 調和積分論 / 擬等角変形 / 積分周期 / 森田マシフォード類
Research Abstract

We discovered a close relation between Stasheff associahedrons and (generalized) Magnus expansions of a free group. A certain part of the twisted Morita-Mumford classes can be extended to the automorphism group of a free group. It is parametrized by Stasheff associahedrons "infinitesimally" and "combinatorially" how the extended Johnson maps are far from true group homomorphisms.
We extended our theory on harmonic Magnus expansions to the universal family of Riemann surfaces. This yields another series of canonical 1 forms on the universal family than what we have already obtained on the moduli space. As a corollary, we obtained a proof that the first Jonson map and the (0,3)-twisted Morita-Mumford class coincides with each other as differential forms on the moduli space.
The Magus representation of the automorphism group of a free group was constructed in an intrinsic manner. Here 'intrinsic' means 'with no use of Fox' free differentials.'
We co-organized a workshop entitled "Toward the future of the topological study of manifolds" in November 2004.

Report

(4 results)
  • 2004 Annual Research Report   Final Research Report Summary
  • 2003 Annual Research Report
  • 2002 Annual Research Report
  • Research Products

    (11 results)

All 2005 2004 2003 2002 Other

All Journal Article (10 results) Publications (1 results)

  • [Journal Article] Word representation of cords on a punctured plane2005

    • Author(s)
      Yukio Matsumoto, Seiichi Kamada
    • Journal Title

      Topology and its applications 146-147

      Pages: 21-50

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Signatures of foliated surface bundles and the symplectomorphism groups of surfaces2005

    • Author(s)
      D.Kotschick, S.Morita
    • Journal Title

      Topology 44

      Pages: 131-149

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Word representaion of cords on a punctured plane2005

    • Author(s)
      Yukio Matsumoro, Seiichi Kamada
    • Journal Title

      Topology and its appl. 146-147

      Pages: 21-50

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Vertex-face correspondence of Boltzmann weights related to sl(m|n)2004

    • Author(s)
      Youichi Shibukawa
    • Journal Title

      Journal of Physics A : Mathematical and General 37

      Pages: 2115-2120

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Vertex-face correspondence of Boltzmann weights related to sl(m|n)2004

    • Author(s)
      Youichi Shibukawa
    • Journal Title

      Journal of Physics A 37

      Pages: 2115-2120

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Generators for the tautological algebra of the moduli space of curves2003

    • Author(s)
      Shigeyuki Morita
    • Journal Title

      Topology 42

      Pages: 787-819

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Weierstrass points and Morita-Mumford classes on hyperelliptic mapping class group2002

    • Author(s)
      Nariya Kawazumi
    • Journal Title

      Topology and its applications 125

      Pages: 363-383

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Nilpotency and triviality of mod p Morita-Mumford classes of mapping class groups of surfaces2002

    • Author(s)
      Toshiyuki Akita
    • Journal Title

      Nagoya Math.J. 165

      Pages: 1-22

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Weierstrass points and Marita-Mumford classes, on hyperelliptic mapping class groups2002

    • Author(s)
      Nariya Kawazumi
    • Journal Title

      Topology and its appl. 125

      Pages: 363-383

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Nilpotency and triviality of mod p Marita-Mumford classes of mapping class groups of surfaces2002

    • Author(s)
      Toshiyuki Akita
    • Journal Title

      Nagoya Math.J. 165

      Pages: 1-22

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Publications] N.kawazumi: "Weierstrass points and Morita-Mumford classes on hyperelliptri mapping class groups"Topology and its applications. 125. 363-383 (2002)

    • Related Report
      2002 Annual Research Report

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Published: 2002-04-01   Modified: 2016-04-21  

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