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2007 Fiscal Year Final Research Report Summary

The study of Low-dimensicnal manifolds with various geometric structures

Research Project

Project/Area Number 18540081
Research Category

Grant-in-Aid for Scientific Research (C)

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

Principal Investigator

UE Masaaki  Kyoto University, Graduate School of Science, Professor (80134443)

Co-Investigator(Kenkyū-buntansha) FUJII Michihiko  Kyoto University, Graduate School of Science, Associate Professor (60254231)
KATO Tsuyoshi  Kyoto University, Graduate School of Science, Associate Professor (20273427)
KATO Shin'ichi  Kyoto University, Graduate School of Science, Professor (90114438)
USHIKI Shigehiro  Kyoto University, Graduate School of Hunan and Environmental Studies, Professor (10093197)
NISHIWADA Kimimasa  Kyoto University, Graduate School of Science, Professor (60093291)
Project Period (FY) 2006 – 2007
Keywords3-manifold / 4-manifold / Seiberg-Witten theory / Heegaard Floer homology / homology 3-sphere
Research Abstract

The head investigator Ue continued the research of 3 and 4-manifolds. In particular he considered Fukumoto-Furuta invariant for rational homology 3-spheres coming from Seiberg-Witten theory (which coincides with the Neumann-Siebenmann invariant in case of plumbed 3-manifolds) and Ozsvath-Szabo's d-invariant defined by Heegaard Floer homology. He showed that these two invariants coincide for spherical 3-manifolds, and also for certain plumbed 3-manifolds. This implies that under such conditions both invariants are the integral 〓 of the classical
Rochlin invariant and also homology cobordism invariants. But they are not the sane in general, and it is sill open whether there exists an invariant satisfying the above two conditions. He also gave certain constraits for the signature of 4-manifolds bounding Seifert rational homology 3-spheres in terms of the above invariants.
The investigator Tsuyoshi Kato estimated the growth of the Casson handles embedded in the K3 surface by Yang-Milis Gauge theory, and also analized the entropy of iterations by families of maps. Fujii found the confluence phenomena of singular points of ordinary differential equations induced by deformations of hyperbolic 2-cone-manifolds, and descried harmonic vector fields on hyperbola 3-cone-manifolds in terms of hypergeometric functions. Shin'ichi Kato established the relative = symmetric space version of Jaquet's theorem, which claims that every irreducible admissible representation of p-adic reductive groups is embedded to an induced representation for irreducible cusp representation of a parabolic subgroup. Ushiki considered transgression operators over the space of distributions induced by complex dynamical systems over the Riemann sphere and showed that the Fredholm determinant is represented by Artin-Mazurzeta function.

  • Research Products

    (8 results)

All 2007 Other

All Journal Article (6 results) (of which Peer Reviewed: 3 results) Presentation (2 results)

  • [Journal Article] An expresson of harmonic vertor fields on hyperbolic 3-cone-manifolds In terms of hypergeometric functions2007

    • Author(s)
      藤井 道彦
    • Journal Title

      Publ. RIMS 43

      Pages: 727-761

    • Description
      「研究成果報告書概要(和文)」より
    • Peer Reviewed
  • [Journal Article] An expression of harmonic vertor fields on hyperbolic 3-cone-manifolds in terms of hypergeometric functions2007

    • Author(s)
      Michihiko, Fujii
    • Journal Title

      Publ. RIMS 43(refereed)

      Pages: 727-761

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] The Fukumoto-Furuta and the Ozsvath-Szabo invariants for spherical 3-manifolds

    • Author(s)
      上 正明
    • Journal Title

      Proceedings of the Postnikov memorial Conference (掲載確定)

    • Description
      「研究成果報告書概要(和文)」より
    • Peer Reviewed
  • [Journal Article] Growth of Casson handles and trasversality for ASD moduli spaces

    • Author(s)
      加藤 毅
    • Journal Title

      Geometry and Topology (掲載確定)

    • Description
      「研究成果報告書概要(和文)」より
    • Peer Reviewed
  • [Journal Article] The Fukumoto-Furuta and the Ozsvath-Szabo invariants for spherical 3-manifolds

    • Author(s)
      Masaaki, Ue
    • Journal Title

      Proceedings of the Postnikov memorial conference (refereed)(to appear)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] Growth of Casson handles and trasversality for ASD moduli spaces

    • Author(s)
      Tsuyoshi, Kato
    • Journal Title

      Geometry and Topology (refereed to appear)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Presentation] The Fukumoto-Furuta and the Ozsvath-Szabo invariants for spherical 3-manifolds2007

    • Author(s)
      上 正明
    • Organizer
      Postnikov Memorial Conference
    • Place of Presentation
      Stefan Banach International Mathematical Center, Poland
    • Year and Date
      2007-06-22
    • Description
      「研究成果報告書概要(和文)」より
  • [Presentation] The Fukumoto-Furuta and the Ozsvath-Szabo invariants for spherical 3-manifolds2007

    • Author(s)
      Masaaki, Ue
    • Organizer
      Postnikov Memorial Conference
    • Place of Presentation
      Stefan Banach international Mathernatical Center, Poland
    • Year and Date
      2007-06-22
    • Description
      「研究成果報告書概要(欧文)」より

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Published: 2010-02-04  

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