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Research on 3-manifolds by combinatorial and constructive methods

Research Project

Project/Area Number 15540091
Research Category

Grant-in-Aid for Scientific Research (C)

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

Principal Investigator

ISHLI Ippei  Keio University, Faculty of Science and Technology, Associate Professor, 理工学部, 助教授 (90051929)

Co-Investigator(Kenkyū-buntansha) MAEDA Yoshiaki  Keio University, Faculty of Science and Technology, Professor, 理工学部, 教授 (40101076)
OTA Katsuhiro  Keio University, Faculty of Science and Technology, Professor, 理工学部, 教授 (40213722)
MORIYOSHI Hitoshi  Keio University, Faculty of Science and Technology, Associate Professor, 理工学部, 助教授 (00239708)
SHIMOMURA Shun  Keio University, Faculty of Science and Technology, Professor, 理工学部, 教授 (00154328)
KAMETANI Yukio  Keio University, Faculty of Science and Technology, Associate Professor, 理工学部, 助教授 (70253581)
Project Period (FY) 2003 – 2004
Project Status Completed (Fiscal Year 2004)
Budget Amount *help
¥3,100,000 (Direct Cost: ¥3,100,000)
Fiscal Year 2004: ¥1,500,000 (Direct Cost: ¥1,500,000)
Fiscal Year 2003: ¥1,600,000 (Direct Cost: ¥1,600,000)
Keywords3-manifold / spine / Heegaard splitting / Seifert fibered space / hyperbolic manifold / topological invariant / symplectic manifold / エゴード分解 / グラフ
Research Abstract

In this research project, we have introduced a new topological invariant, called the "block number", for 3-manifolds, which estimates some kind of complexity of a 3-manifold just like as the Heegaard genus. The block number is defined by means of a flow-spine, and is enable us to classify 3-manifolds, and to parameterize 3-manifolds in each class by finitely many integers. Moreover the block number can be defined not only for a 3-manifold but also for a combed 3-manifold, a pair of a 3-manifold and a non-singular vector field on it. Using this invariant, we have gotten the following results :
1.The only combed 3-manifold having 0 as its block number is the canonical one on the product of the 2-sphere and the circle, and combed 3-manifolds with the block number 1 are canonical ones on lens spaces (including the 3-sphere).
2.The parameterization for 3-manifolds with the block number 2 was given. And, using the Reidemeister torsion, we have shown some results which imply that our parameterization uniformize the presentation of a combed 3-manifold.00
3.We have given a formula for calculating the value of the Thraev-Viro invariant for all Seifert fibered 3-manifolds.
On symplectic manifolds, we have gotten the following result :
4.If the universal covering space of a clsed symplectic manifold is contractible, the manifold does not admit any Riemannian metric with positive curvature.

Report

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

    (22 results)

All 2005 2004 Other

All Journal Article (15 results) Book (1 results) Publications (6 results)

  • [Journal Article] A new complexity for 3-manifolds2005

    • Author(s)
      Endoh M., Ishii I.
    • Journal Title

      Japanese J.Math. to appear

    • NAID

      10016620864

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Universal deformation formulae for three-dimensional solvable Lie groups2005

    • Author(s)
      P.Bieliavsky, P.Bonneau, Y.Maeda
    • Journal Title

      Lecture notes in Physics 662

      Pages: 127-141

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Annual Research Report 2004 Final Research Report Summary
  • [Journal Article] Universal deformation formula for three-dimensional solvable Lie group2005

    • Author(s)
      P.Bielivsky, P.Bonneau, Y.Maeda
    • Journal Title

      Lecture notes in Physics 662

      Pages: 127-141

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Quantum Field Theory and Noncommutative Geometry2005

    • Author(s)
      U.Carow-Watamura, Y.Maeda, S.Watamura
    • Journal Title

      Discrete Math. 662

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] A new complexity for 3-manifolds2005

    • Author(s)
      Endoh M., Ishii I.
    • Journal Title

      Japanese J.Math. (to appear)

    • NAID

      10016620864

    • Related Report
      2004 Annual Research Report
  • [Journal Article] Vertex-disjoint cycles containing specified vertices in a bipartite graph2004

    • Author(s)
      G.Chen, H.Enomoto 他
    • Journal Title

      J.Graph Theory 46

      Pages: 145-166

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Annual Research Report 2004 Final Research Report Summary
  • [Journal Article] On minimally 3-connected graphs on a surface2004

    • Author(s)
      K.Ota
    • Journal Title

      AKCE Int.J.Graphs Comb. 1

      Pages: 29-33

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Annual Research Report 2004 Final Research Report Summary
  • [Journal Article] Chromatic numbers and cycle parities of quadrangulations on nonorientable closed surfaces2004

    • Author(s)
      A.Nakamoto, S.Negami, K.Ota
    • Journal Title

      Discrete Math. 285

      Pages: 211-218

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Annual Research Report 2004 Final Research Report Summary
  • [Journal Article] Lower estimates for the growth of the fourth and the second Painleve transcendents transcendents2004

    • Author(s)
      S.Shimomura
    • Journal Title

      Proc.Edinburgh Math.Soc. 47

      Pages: 231-249

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Chromatic numbers and cycle parities of quadrangulations on nonorientable closed surfaces2004

    • Author(s)
      A.Nakamoto, S.Negami, K.Ota
    • Journal Title

      Lecture Notes in Physics 285

      Pages: 211-218

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Planar triangulations which quadrangulate other surfaces2004

    • Author(s)
      A.Nakamoto, S.Negami, K.Ota, J.Siran
    • Journal Title

      European J.Combin. 25

      Pages: 817-833

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Lower estimates for the growth of the fourth and the second Painleve transcendents2004

    • Author(s)
      S.Shimomura
    • Journal Title

      Proc.Edinburgh Math.Soc. 47

      Pages: 231-249

    • Related Report
      2004 Annual Research Report
  • [Journal Article] A new complexity for 3-manifolds

    • Author(s)
      M.Endoh, I.Ishii
    • Journal Title

      Japanese J.Math. (to appear)

    • NAID

      10016620864

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Systematic singular triangulations for all Seifert 3-manifolds

    • Author(s)
      T.Taniguchi, K.Tsuboi, M.Yamashita
    • Journal Title

      Tokyo J.Math. (to appear)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Meromorphic solutions of Riccati differential equations with doubly periodic coefficients

    • Author(s)
      S.Shimomura
    • Journal Title

      J.Math.Anal.Appl. (to appear)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Book] 量子的な微分幾何2004

    • Author(s)
      大森英樹, 前田吉昭
    • Total Pages
      370
    • Publisher
      シュプリンガーフェアラーク社
    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Annual Research Report 2004 Final Research Report Summary
  • [Publications] M.Endoh, I.Ishii: "A new complexity for 3-manifolds"Tokyo J.Math.. (to appear).

    • Related Report
      2003 Annual Research Report
  • [Publications] K.Kawarabayashi, A.Nakamoto, K.Ohta: "2-connected 7-covering of 3-connected graphs on surfaces"J.Graph Theory. 43. 26-36 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] R.Mori, A.Nakamoto, K.Ohta: "Diagonal flips in Hamiltonian triangulations on the sphere"Graphs and Combinatorics. 19. 413-418 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] K.Kawarabayashi, A.Nakamoto, K.Ohta: "Sabgraphs of graphs on surfaces with high representability"J.Combinatorial Theory. 19. 207-229 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] K.Ishizaki, I.Laine, S.Shimomura, K.Tohge: "Riccati differential equations with elliptic coefficients, II"Tohoku J.Math.. 55. 99-108 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] S.Shimomura: "Lower estimates for the growth of Painlave transcendents"Funkcial.Ekvac.. 46. 287-295 (2003)

    • Related Report
      2003 Annual Research Report

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Published: 2003-04-01   Modified: 2016-04-21  

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