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Research on families of fixed point sets of G-manifolds in transformation group theory

Research Project

Project/Area Number 15540079
Research Category

Grant-in-Aid for Scientific Research (C)

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

Principal Investigator

KOMIYA Katsuhiro  Yamaguchi University, Faculty of Science, Professor, 理学部, 教授 (00034744)

Co-Investigator(Kenkyū-buntansha) ANDO Yoshifumi  Yamaguchi University, Faculty of Science, Professor, 理学部, 教授 (80001840)
MIYAZAWA Yasuyuki  Yamaguchi University, Faculty of Science, Associate Professor, 理学部, 助教授 (60263761)
SHIMA Hirohiko  Yamaguchi University, Faculty of Science, Professor, 理学部, 教授 (70028182)
NAITOH Hiroo  Yamaguchi University, Faculty of Science, Professor, 理学部, 教授 (10127772)
NAKAUCHI Nobumitu  Yamaguchi University, Faculty of Science, Associate Professor, 理学部, 助教授 (50180237)
Project Period (FY) 2003 – 2004
Project Status Completed (Fiscal Year 2004)
Budget Amount *help
¥3,600,000 (Direct Cost: ¥3,600,000)
Fiscal Year 2004: ¥1,700,000 (Direct Cost: ¥1,700,000)
Fiscal Year 2003: ¥1,900,000 (Direct Cost: ¥1,900,000)
Keywordscut-and-paste / SK-equivalence / SK-group / G-manifolds / fixed points / Euler characteristics / fibring over the circle / cobordism / Smith同値 / 接表現 / コボルディズム / 不動点多様体
Research Abstract

1.The cut-and-paste operation defines an equivalence relation on the set of smooth G-manifolds. This relation is called SK-equivalence. The set of SK-equivalence classes becomes a semigroup with the addition induced from the disjoint union of G-manifolds. Its Grothendieck group is called the SK-group of G-manifolds. We obtain a necessary and sufficient condition for the divisibility in the SK-group, if G is the cyclic group of order 2, or a finite abelian group of odd order. The condition is described in terms of the Euler characteristics of fixed point sets of G-manifolds.
2.A necessary and sufficient condition for a closed smooth manifold to be cobordant to the total space of fiber bundle over the circle is well-known In our research we extend this (non-equivariant) result to the equivariant case by making use of the result stated above..
3.Two linear representations of a group G are called Smith equivalent., if those two representations can occur as the tangential representations at fixed points of a homotopy G-sphere with exactly two fixed points. There are vast literatures on the question of which groups do and which groups do not have non-isomorphic Smith equivalent representations. Some of them give an affirmative answer and some of them give a negative answer. In our research we show that the question is affirmative if we restrict our attention to the restricted actions of a subgroup of index 2..

Report

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

    (13 results)

All 2005 2004 2003 Other

All Journal Article (11 results) Publications (2 results)

  • [Journal Article] The divisibility in the cut-and-paste group of G-manifolds and fibring over the circle within a cobordism class2005

    • Author(s)
      Katsuhiro Komiya
    • Journal Title

      Osaka Journal of Mathematics 42(1)(未定)

    • NAID

      120004844105

    • 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] The divisibility in the cut-and-paste group of G-manifolds and fibring over the circle within a cobordism class2005

    • Author(s)
      Katsuhiro Komiya
    • Journal Title

      Osaka J.Math. 42(1)(to appear)

    • NAID

      120004844105

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Smooth maps having only singularities with Boardman symbol (0,1)2004

    • Author(s)
      Yoshifumi Ando
    • Journal Title

      Topology and its applications 142

      Pages: 205-226

    • 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] Existence theorem of fold maps2004

    • Author(s)
      Yoshifumi Ando
    • Journal Title

      Japanese Journal of Mathematics 30

      Pages: 29-73

    • NAID

      10014334513

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Smooth maps having only singularities with Boardmann symbol (0,1)2004

    • Author(s)
      Yoshifumi Ando
    • Journal Title

      Topology and its applications 142

      Pages: 205-226

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Existence theorem of fold maps2004

    • Author(s)
      Yoshifumi Ando
    • Journal Title

      Japanese J.Math. 30

      Pages: 29-73

    • NAID

      10014334513

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Existence theorems of fold-maps2004

    • Author(s)
      Yoshifumi Ando
    • Journal Title

      Japanese Journal of Mathematics 30

      Pages: 29-73

    • NAID

      10014334513

    • Related Report
      2004 Annual Research Report
  • [Journal Article] Cutting and pasting of manifolds into G-manifolds2003

    • Author(s)
      Katsuhiro Komiya
    • Journal Title

      Kodai Mathematical Journal 26

      Pages: 230-243

    • NAID

      130003574434

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Cutting and pasting of families of submanifolds modeled on Z_2 manifolds2003

    • Author(s)
      Katsuhiro Komiya
    • Journal Title

      Tokyo Journal of Mathematics 26

      Pages: 403-411

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Cutting and pasting of manifolds into G-manifolds2003

    • Author(s)
      Katsuhiro Komiya
    • Journal Title

      Kodai Math.J. 26

      Pages: 230-243

    • NAID

      130003574434

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Cutting and pasting of families of submanifolds modeled on Z_2 manifolds2003

    • Author(s)
      Katsuhiro Komiya
    • Journal Title

      Tokyo J.Math. 26

      Pages: 403-411

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Publications] Katsuhiro Komiya: "Cutting and pasting of manifolds into G-manifolds"Kodai Mathematical Journal. 26. 230-243 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] Katsuhiro Komiya: "Cutting and pasting of families of submanifolds modeled on Z-2 manifolds"Tokyo Journal of Mathematics. 26. 403-411 (2003)

    • Related Report
      2003 Annual Research Report

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

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