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Study of equivariant homology surgery theory and its applications

Research Project

Project/Area Number 15540076
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field Geometry
Research InstitutionOKAYAMA UNIVERSITY

Principal Investigator

MORIMOTO Masaharu  OKAYAMA UNIVERSITY, Graduate School of Natural Science and Technology, Professor, 大学院・自然科学研究科, 教授 (30166441)

Co-Investigator(Kenkyū-buntansha) SHIMAKAWA Kazuhisa  OKAYAMA UNIVERSITY, Graduate School of Natural Science and Technology, Professor, 大学院・自然科学研究科, 教授 (70109081)
NAKAJIMA Atsushi  OKAYAMA UNIVERSITY, Graduate School of Natural Science and Technology, Professor, 大学院・自然科学研究科, 教授 (30032824)
IKEHATA Shuichi  OKAYAMA UNIVERSITY, Graduate School of Natural Science and Technology, Professor, 大学院・自然科学研究科, 教授 (20116429)
SASAKI Toru  OKAYAMA UNIVERSITY, Graduate School of Environmental Science and Technology, Lecturer, 大学院・環境学研究科, 講師 (20260664)
Project Period (FY) 2003 – 2005
Project Status Completed (Fiscal Year 2005)
Budget Amount *help
¥3,500,000 (Direct Cost: ¥3,500,000)
Fiscal Year 2005: ¥1,000,000 (Direct Cost: ¥1,000,000)
Fiscal Year 2004: ¥1,200,000 (Direct Cost: ¥1,200,000)
Fiscal Year 2003: ¥1,300,000 (Direct Cost: ¥1,300,000)
Keywordssurgery theory / homology / quivariant manifold / transformation groups / sphere / fixed point set / equivariant connected sum / ホモロジー同値 / 多様体 / 被覆空間 / 群作用 / Mackey函手 / Burnside環
Research Abstract

(1)Let G be a finite group, Y a homology disk with a smooth G-action, f : X → Y a G-framed map, H a subgroup of G. Suppose that Y has a G-fixed point and a point of isotropy subgroup H. Then we constructed G-connected sum f#_G G x_H D(f) : X #_G G x_H D(X) → Y.
(2)We generalized the notion of quadratic forms of Cappell-Shaneson and their group. We proved the generalized group is isomorphic to the original one. Using this new notion, we proved a sum formula of surgery obstructions for the G-connected sum above.
(3)For Z_{(p)} where p is a prime such that the order of fundamental group of Y, we proved that the equivariant homology obstruction group is isomorphic to the Wall group. So, we could develop induction-restriction theory of equivariant homology surgery obstruction groups.
(4)Using above results, we proved a deleting-inserting theorem of G-fixed point sets for gap Oliver group G.
(5)We decided manifolds appearing as the G-fixed point sets of smooth G-actions on spheres where G was a nilpotent Oliver group or a nontrivial perfect group.

Report

(4 results)
  • 2005 Annual Research Report   Final Research Report Summary
  • 2004 Annual Research Report
  • 2003 Annual Research Report
  • Research Products

    (22 results)

All 2005 2004 2003 Other

All Journal Article (19 results) Publications (3 results)

  • [Journal Article] ホモロジー同値写像を得るための同変手術理論とその応用2005

    • Author(s)
      森本 雅治
    • Journal Title

      第52回トポロジーシンポジウム講演集

      Pages: 1-12

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] 同変手術理論と球面上の滑らかな作用の不動点集合2005

    • Author(s)
      森本 雅治
    • Journal Title

      数理解析研究所講究録 1449

      Pages: 109-118

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Annual Research Report 2005 Final Research Report Summary
  • [Journal Article] The dimension of spheres with smooth one fixed point actions2005

    • Author(s)
      Anthony Bak, Masaharu Morimoto
    • Journal Title

      Forum Mathematicum 17

      Pages: 199-216

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Annual Research Report 2005 Final Research Report Summary
  • [Journal Article] Equivariant surgery theory to obtain homology equivalences and its applications2005

    • Author(s)
      Masaharu Morimoto
    • Journal Title

      Abstracts of 52nd Topology Symposium

      Pages: 1-12

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Equivariant surgery theory and the fixed point sets of smooth actions on spheres2005

    • Author(s)
      Masaharu Morimoto
    • Journal Title

      RIMS Kokyuroku 1449

      Pages: 109-118

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] The dimension of spheres with smooth one fixed point actions2005

    • Author(s)
      Anthony Bak, Masaharu Morimoto
    • Journal Title

      Forum Mathematicum (to appear)(印刷中)

    • Related Report
      2004 Annual Research Report
  • [Journal Article] An exact sequence of Grothedieck-Witt rigns2004

    • Author(s)
      Anthony Bak, Masaharu Morimoto
    • Journal Title

      数理解析研究所講究録 1393

      Pages: 58-63

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] An exact sequence of Grothendieck-Witt rings2004

    • Author(s)
      Anthony Bak, Masaharu Morimoto
    • Journal Title

      RIMS Kokyuroku 1393

      Pages: 58-63

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] An exact sequence of Grothendieck-Witt rings2004

    • Author(s)
      Anthony Bak, Masaharu Morimoto
    • Journal Title

      数理解析研究所講究録 1393

      Pages: 58-63

    • Related Report
      2004 Annual Research Report
  • [Journal Article] Deleting and inserting fixed point sets ondisks under the strong gap condition2004

    • Author(s)
      Masaharu Morimoto
    • Journal Title

      K-theory Preprint Archives 707

      Pages: 1-28

    • Related Report
      2004 Annual Research Report
  • [Journal Article] Mackey and Frobenius stuructures on odd dimensional surgery obstruction groups2003

    • Author(s)
      XianMeng Ju, Katsuhiko Mathuzaki, Masaharu Morimoto
    • Journal Title

      K-Theory 29

      Pages: 285-312

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Induction theory of equivariant-surgery-obstruction groups2003

    • Author(s)
      XianMeng Ju, Katsuhiko Mathuzaki, Masaharu Morimoto
    • Journal Title

      数理解析研究所講究録 1343

      Pages: 129-143

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Mackey and Frobenius structures on odd dimensional Surgery obstruction groups2003

    • Author(s)
      XianMeng Ju, Katsuhiko Matsuzaki, Masaharu Morimoto
    • Journal Title

      K-Theory 29

      Pages: 285-312

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Induction theory of equivariant-surgery-obstruction groups2003

    • Author(s)
      XianMeng Ju, Katsuhiko Matsuzaki, Masaharu Morimoto
    • Journal Title

      RIMS Kokyuroku 1343

      Pages: 129-143

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Equivariant surgery theory for homology equivalences under the gap condition

    • Author(s)
      Masaharu Morimoto
    • Journal Title

      Publ. Res. Inst. Kyoto Univ. (to appear)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] 「研究成果報告書概要(和文)」より

    • Author(s)
      原稿の記載なし
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Equivariant surgery theory for homology equivalences under the gap condition

    • Author(s)
      Masaharu Morimoto
    • Journal Title

      Publ.Res.Inst.Math.Sci. (to appear)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Fixed-point sets of smooth actions on spheres

    • Author(s)
      Masaharu Morimoto
    • Journal Title

      K-Theory (to appear)

    • Related Report
      2005 Annual Research Report
  • [Journal Article] Equivariant surgery theory for homology equivalences under the gap condition

    • Author(s)
      Masaharu Morimoto
    • Journal Title

      Publ.Res.Inst.Kyoto Univ. (to appear)

    • Related Report
      2005 Annual Research Report
  • [Publications] A.Bak, M.Morimoto: "The dimension of spheres with smooth one fixed point actions"Forum Mathematicum. (in press).

    • Related Report
      2003 Annual Research Report
  • [Publications] X.M.Ju, K.Matsuzaki, M.Morimoto: "Mackey and Frobenius structures on odd dimensional surgery obstruction groups"K-Theory. 29・4. 285-312 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] X.M.Ju, K.Matsuzaki, M.Morimoto: "Induction theory of equivariant-surgery-obstruction groups"京都大学 数理解析研究所講究録. 1343. 129-143 (2003)

    • Related Report
      2003 Annual Research Report

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

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