• Search Research Projects
  • Search Researchers
  • How to Use
  1. Back to previous page

The autornorphisrn group of smcoth G-manifokls andals applications

Research Project

Project/Area Number 18540077
Research Category

Grant-in-Aid for Scientific Research (C)

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

Principal Investigator

ABE Kojun  Shinshu University, Faculty of Science, Associate Professor (30021231)

Project Period (FY) 2006 – 2007
Project Status Completed (Fiscal Year 2007)
Budget Amount *help
¥2,230,000 (Direct Cost: ¥1,900,000、Indirect Cost: ¥330,000)
Fiscal Year 2007: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000)
Fiscal Year 2006: ¥800,000 (Direct Cost: ¥800,000)
Keywordsdiffeornorphisrn group / smooth G-manifold / smooth orbifold / first homology group / Lipshcitz homeomorphism group
Research Abstract

Let M be a smooth manifold which has a smooth action of a compact Lie group G. Let Lip_G(M) denote the group of equivariant Lipschitz homeomorphisms of M. We can introduce two kind of topologies on Lip_G(M). Let Lip_G(M), (resp. L_G(M)) denote the identity component of the identity of Lip_G(M) when Lip_G(M) has the compact open Lipschitz topology (resp. compact open topology). We investigate the first homology group of H_I(Lip_G(M)_O).
When M is a principal G-manifold or G is a finite group, it is known that the group LiP_G(M)_O is perfect. When M is the complex n-dimensional space C^n with the canonical U(n)-action,H_1(L_G(M)) is isomorphic to the vector space of some real valued functions which has continuous moduli.
During the study duration, we investigate the case where M has a G-manifold with codimension one orbit. First we proved that Lip_G(V)_o is perfect when V is a representation space with codimension one orbit. By using this result and analyzing the behavior of equivariant Lipschitz homeomorphism around the singular orbit, we calculate H_1(Lip_G(M)-O) for any smooth Gmanifold with codimension one orbit. The result shows that the first homology group of the automorphism group of smooth Gmanifold is quite depends on the category such as smooth, compact open or compact open Lipschitz. We also calculate the first homology group of the group of equivariant diffeomorphisms of 3-dimensional smooth S^Imanifold. The result has recently published in the foreign journal.

Report

(3 results)
  • 2007 Annual Research Report   Final Research Report Summary
  • 2006 Annual Research Report
  • Research Products

    (22 results)

All 2008 2007 2006 Other

All Journal Article (11 results) (of which Peer Reviewed: 3 results) Presentation (9 results) Remarks (2 results)

  • [Journal Article] The first homology of the group of equivariant diffeomorphisms and its applications2008

    • Author(s)
      Kojun Abe and Kazuhiko Fukui
    • Journal Title

      Jour. Topology 1

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
    • Peer Reviewed
  • [Journal Article] The first homology of the group of equivariant diffeornorphisrns and its applications2008

    • Author(s)
      Kojun, Abe, Kazuhiko, Fukui
    • Journal Title

      Jour. Topology 1

      Pages: 461-476

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Journal Article] The first homology of the group of equivariant diffe. omorphisms and its applications.2008

    • Author(s)
      Kojun Abe and Kazuhiko Fukui
    • Journal Title

      Journal of Topology 1

    • Related Report
      2007 Annual Research Report
    • Peer Reviewed
  • [Journal Article] On the structure of equivariant Lipschitz homeomorphism groups of G-manifolds with codirnension one orbit2007

    • Author(s)
      Kojun, Abe
    • Journal Title

      RIMS Kokyuroku 1569

      Pages: 132-137

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Journal Article] 余次元1軌道を持つG-多様体の同変リプシッツ同相群の構造2007

    • Author(s)
      阿部 孝順
    • Journal Title

      数理解析研究所攻究録 1569

    • Related Report
      2007 Annual Research Report
  • [Journal Article] On the first homology of the group of equivariant Lipschitz homeomorphisms2006

    • Author(s)
      Kojun Abe, Kazuhiko Fukui and Takeshi Miura
    • Journal Title

      Jour. Math. Soc. Japan 58

    • NAID

      10017178257

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
    • Peer Reviewed
  • [Journal Article] On Lie algebras of vector fields of manifolds with singularities2006

    • Author(s)
      Kojun Abe and Suguru Fujiwara
    • Journal Title

      数理解析研究所講究録 1517

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Journal Article] On the first homology of the group of equivariant Lipschitz Hornernorphisms2006

    • Author(s)
      Kojun, Abe, Kazuhiko,Fukui, Takeshi, Miura
    • Journal Title

      Jour. Math. Soc. Japan 58

      Pages: 1-16

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Journal Article] On Lie algebras of vector fields of manifolds with singularities2006

    • Author(s)
      Kojun, Abe
    • Journal Title

      RIMS KOkyfiroku 1517

      Pages: 1-9

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Journal Article] On the structure of the group of equivariant Lipschitz homeomorphisms of G-manifolds with codimension one orbit2006

    • Author(s)
      Kojun Abe, Kazuhiko Fukui, Takeshi Miura
    • Journal Title

      Jour. Math. Soc. Japan 58・1

      Pages: 1-15

    • Related Report
      2006 Annual Research Report
  • [Journal Article] On Lie algebras of vector fields of manifolds with singularities.2006

    • Author(s)
      Kojun Abe, Suguru Fujiwara
    • Journal Title

      数理解析研究所講究禄 1517

      Pages: 1-9

    • Related Report
      2006 Annual Research Report
  • [Presentation] Diffeornorphism groups of smooth orbifolds and its applications2008

    • Author(s)
      Kojun, Abe
    • Organizer
      Geometric Group Theory Seminar
    • Place of Presentation
      Ohio State Univ
    • Year and Date
      2008-03-05
    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] On the structure of equivariant Lipschitz horneornorphism group of G-manifold with codimension one orbit2008

    • Author(s)
      Kojun, Abe, Kaznhiko, Fukui
    • Organizer
      On the homeomorphism group and the surrounding research
    • Place of Presentation
      Kyoto Sangyo Univ
    • Year and Date
      2008-02-19
    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] On equivariant Lipschitz homeomorphisms.2007

    • Author(s)
      Kojun Abe and Kazuhiko Fukui
    • Organizer
      8th Conference on Geometry and Topology of Manifolds
    • Place of Presentation
      Przemysl(Poland)
    • Year and Date
      2007-04-30
    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] On equivariant Lipschitz horneornorphisrns2007

    • Author(s)
      Kojun, Abe, Kazuhiko, Fukui
    • Organizer
      8th Conference on Geometry and Topology of Manifolds
    • Place of Presentation
      Przemysl (Poland)
    • Year and Date
      2007-04-30
    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] On the structure of equivariant Lipschitz horneornorphism group of G-manifold with codirnension one orbit2007

    • Author(s)
      Kojun, Abe
    • Organizer
      Method of Transformation group theory
    • Place of Presentation
      RIMS Kyoto Univ
    • Year and Date
      2007-04-30
    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] On equivariant Lipschitz homeomorphisms of G-manifolds with codimension one orbit2007

    • Author(s)
      阿部 孝順
    • Organizer
      8th Conference on Geometry and Topology of Manifolds
    • Place of Presentation
      State High School of East Europe(Poland)
    • Year and Date
      2007-04-30
    • Related Report
      2007 Annual Research Report
  • [Presentation] On the first homology group of Lipschitz homeornorphism group2006

    • Author(s)
      Kojun, Abe
    • Organizer
      On the horneornorphism group and the surrounding research
    • Place of Presentation
      Kyoto Institute Technology
    • Year and Date
      2006-12-22
    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] The equivqriant diffeornorphism groups and their applications2006

    • Author(s)
      Kojun, Abe
    • Organizer
      Lie groups and the related fields
    • Place of Presentation
      Shinshu Univ
    • Year and Date
      2006-10-08
    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] On the differnorphism groups of smooth orbifolds and application to Fuchsian group2006

    • Author(s)
      Kojun, Abe
    • Organizer
      Groups in Geometry and Topology Univ
    • Place of Presentation
      Malaga (Spain)
    • Year and Date
      2006-06-23
    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Remarks] 「研究成果報告書概要(和文)」より

    • URL

      http://math.shinshu-u.ac.jp/~kabe/index-j.html

    • Related Report
      2007 Final Research Report Summary
  • [Remarks]

    • URL

      http://math.shinshu-u.ac.jp/~kabe/index-j.html

    • Related Report
      2007 Annual Research Report

URL: 

Published: 2006-04-01   Modified: 2016-04-21  

Information User Guide FAQ News Terms of Use Attribution of KAKENHI

Powered by NII kakenhi