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Geometry and topology of 4-manifolds

Research Project

Project/Area Number 16204003
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

MATSUMOTO Yukio  University of Tokyo, Graduate School of Mathematical Sciences, Professor, 大学院数理科学研究科, 教授 (20011637)

Co-Investigator(Kenkyū-buntansha) ASHIKAGA Tadashi  Tohoku Gakuin Univ., Dept. of Tech., Prof., 工学部, 教授 (90125203)
IMAYOSHI Yoichi  Osaka City Univ., Grad. School of Sci., Prof., 大学院理学研究科, 教授 (30091656)
SAEKI Osamu  Kyushu Univ., Grad. School of Math. Sci., Prof., 大学院数理学研究院, 教授 (30201510)
KAMADA Seiichi  Hiroshima Univ., Grad. School of Sci., Prof., 大学院理学研究科, 教授 (60254380)
ENDO Hisaaki  Osaka Univ., Grad. School of Sci., Associate Prof., 大学院理学研究科, 教授 (20323777)
小野 薫  北海道大学, 大学院理学研究科, 教授 (20204232)
志摩 亜希子  東海大学, 理学部, 助教授 (50317765)
Project Period (FY) 2004 – 2006
Project Status Completed (Fiscal Year 2006)
Budget Amount *help
¥20,930,000 (Direct Cost: ¥16,100,000、Indirect Cost: ¥4,830,000)
Fiscal Year 2006: ¥7,800,000 (Direct Cost: ¥6,000,000、Indirect Cost: ¥1,800,000)
Fiscal Year 2005: ¥6,890,000 (Direct Cost: ¥5,300,000、Indirect Cost: ¥1,590,000)
Fiscal Year 2004: ¥6,240,000 (Direct Cost: ¥4,800,000、Indirect Cost: ¥1,440,000)
Keywords4-manifolds / signature / Lefschetz fibration / Riemann surface / 正則ファイバー空間 / モノドロミー / チャート理論 / 初等手術理論 / 退化 / 局所符号数 / シンプレクティック幾何 / フラックス予想
Research Abstract

Since a decade ago, there have been many research activities on 4-manifolds having the structure of Lefschetz fibrations. The concept of Lefschetz pencils was introduced in algebraic geometry in 1920's, but recently the concept has become the central theme of investigations mainly due to the results of Donaldson and Gompf according to which Lefschetz fibrations are the underlying structure of the symplectic structure. The structure of Lefschetz fibrations is topologically described by their monodromy representations taking the value in the mapping class groups, while the structure is, as mentioned above, closely related to algebraic geometry and symplectic geometry. Thus the study of Lefschetz fibrations is considered to be a meeting place of many research areas, such as 4-manifolds, mapping class groups of surfaces, algebraic geometry, symplectic geometry ad so on. In the present research, we took the above mentioned situation in consideration, and put stress on communication between different research areas. For example, we held a research seminar every summer during the research period entitled "Topology and Algebraic Geometry". Also by the same idea, we held two seminars entitled "Topology and Symplectic Geometry".
We have many new results on Lefschetz fibrations, fibering structure of low dimensional manifolds, monodromy, combinatorics of charts, mapping class groups (Teichmuller modular groups), degenerations of Riemann surfaces and the construction of splitting families, topological and analytic local signature, Dedekind sums, and symplectic geometry. By these results, we have obtained some new prospects of research.

Report

(4 results)
  • 2006 Annual Research Report   Final Research Report Summary
  • 2005 Annual Research Report
  • 2004 Annual Research Report
  • Research Products

    (27 results)

All 2006 2005 2004

All Journal Article (23 results) Book (4 results)

  • [Journal Article] Morse functions with sphere fibers2006

    • Author(s)
      O.Saeko
    • Journal Title

      Hiroshima Math. J. 36

      Pages: 141-170

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] Elimination of definite fold2006

    • Author(s)
      O.Saeko
    • Journal Title

      Kyushu J. Math. 60

      Pages: 363-382

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] Stable mapping class groups of 4-manifolds with Goundary2006

    • Author(s)
      O.Saeko
    • Journal Title

      Trans. Amer. Math. Soc. 358

      Pages: 2091-2104

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] On splittability of steller singular filer with three branches2006

    • Author(s)
      K.Ahara, S.Takamara
    • Journal Title

      Tokyo J. of Math. 29

      Pages: 1-17

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] Various aspects of degenerate families of Rirmann surfaces2006

    • Author(s)
      T.Ashikaga, H.Endo
    • Journal Title

      Sugaka Expositions 19

      Pages: 171-196

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] Hyplane-ploygonal model of hypenbolic plane2006

    • Author(s)
      K.Ahara
    • Journal Title

      FORMA 21

      Pages: 5-18

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] Morse functions with sphere fibers2006

    • Author(s)
      O.Saeki
    • Journal Title

      Hiroshima Math. J. 36

      Pages: 141-170

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2006 Annual Research Report 2006 Final Research Report Summary
  • [Journal Article] Elimination of definite fold2006

    • Author(s)
      O.Saeki
    • Journal Title

      Kyushu J. Math. 60

      Pages: 363-382

    • NAID

      130000063146

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2006 Annual Research Report 2006 Final Research Report Summary
  • [Journal Article] Stable mapping class groups of 4-manifolds with boundary2006

    • Author(s)
      O.Saeki
    • Journal Title

      Trans. Amer. Math. Soc. 358

      Pages: 2091-2104

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] On splittability of steller singular fiber with three branches2006

    • Author(s)
      K.Ahara, S.Takamura
    • Journal Title

      Tokyo J. Math. 29

      Pages: 1-17

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] Various aspects of degenerate families of Riemann surfaces2006

    • Author(s)
      T.Ashikaga, H.Endo
    • Journal Title

      Sugaku Expositions 19

      Pages: 171-196

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2006 Annual Research Report 2006 Final Research Report Summary
  • [Journal Article] Hyplane-polygonal model of hyperbolic plane2006

    • Author(s)
      K.Ahara
    • Journal Title

      FORMA 21

      Pages: 5-18

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] Stable mapping class goups of 4-manifolds with boundary2006

    • Author(s)
      O.Saeki
    • Journal Title

      Trans Amer Math. Soc. 358

      Pages: 2091-2104

    • Related Report
      2006 Annual Research Report
  • [Journal Article] On Splittability of Stellar Singular Fiber with three branches2006

    • Author(s)
      K.Ahara, S.Takamura
    • Journal Title

      Tokyo J. of Math. 29

      Pages: 1-17

    • Related Report
      2006 Annual Research Report
  • [Journal Article] Hyplane-polygonalmodel of hyperbolic plane2006

    • Author(s)
      K.Ahara
    • Journal Title

      FORMA 21

      Pages: 5-18

    • Related Report
      2006 Annual Research Report
  • [Journal Article] Chart description and a new proof of the classification theorem of genus one Lefschetz fibrations2005

    • Author(s)
      S.Kamada
    • Journal Title

      J.Math.Soc.Japan 57

      Pages: 537-555

    • NAID

      10016420091

    • Related Report
      2005 Annual Research Report
  • [Journal Article] On surface braids of index four with at most two crossings2005

    • Author(s)
      T.Nagase
    • Journal Title

      Fundamenta Mathematicae 188

      Pages: 167-193

    • Related Report
      2005 Annual Research Report
  • [Journal Article] Word representation of cords on a punctured plane2005

    • Author(s)
      S.Kamada
    • Journal Title

      Topology Appl. 146-147

      Pages: 21-50

    • Related Report
      2004 Annual Research Report
  • [Journal Article] Enveloping monoidal quandles2005

    • Author(s)
      S.Kamada
    • Journal Title

      Topology Appl. 146-147

      Pages: 133-148

    • Related Report
      2004 Annual Research Report
  • [Journal Article] Local splitting families of hyperelliptic pencils II.2004

    • Author(s)
      T.Arakawa
    • Journal Title

      Nagoya Math.J. 175

      Pages: 103-124

    • Related Report
      2004 Annual Research Report
  • [Journal Article] A reducibility problem for monodromy of some surface bundles2004

    • Author(s)
      Y.Imayoshi
    • Journal Title

      J. of Knot Theory and its Ramifications 13-5

      Pages: 597-616

    • Related Report
      2004 Annual Research Report
  • [Journal Article] リーマン面の退化族の諸相2004

    • Author(s)
      遠藤 久顕
    • Journal Title

      数学 56巻1号

      Pages: 49-72

    • NAID

      10013123370

    • Related Report
      2004 Annual Research Report
  • [Journal Article] Pull back relation for non-spherical knots2004

    • Author(s)
      V.Blanloeil
    • Journal Title

      J.Knot Theory Ramifications 13-5

      Pages: 689-701

    • Related Report
      2004 Annual Research Report
  • [Book] Splitting deformations of degenerations of complex curves2006

    • Author(s)
      S.Takamara
    • Total Pages
      590
    • Publisher
      Springer-Vezlag
    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Book] Splitting Deformation of Degenerations of Complex Curves2006

    • Author(s)
      S.Takamura
    • Total Pages
      590
    • Publisher
      Springer-Verlag
    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Book] Splitting Deformations of Degenerations of Complex Curves2006

    • Author(s)
      S.Takamura
    • Total Pages
      590
    • Publisher
      Springer-Verlag
    • Related Report
      2006 Annual Research Report
  • [Book] Topology of singular fibers of differentiable maps2004

    • Author(s)
      O.Saeki
    • Total Pages
      146
    • Publisher
      Springer-Verlag, Lecture Notes in Math.
    • Related Report
      2004 Annual Research Report

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Published: 2004-04-01   Modified: 2016-04-21  

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