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Study of coupled K-theory and its applications to non-linear actions

Research Project

Project/Area Number 12640072
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, Faculty of Environmental Science and Technology, Professor, 環境理工学部, 教授 (30166441)

Co-Investigator(Kenkyū-buntansha) IKEHATA Shuichi  Okayama University, Faculty of Environmental Science and Technology, Professor, 環境理工学部, 教授 (20116429)
NAKAJIMA Atsushi  Okayama University, Faculty of Environmental Science and Technology, Professor, 環境理工学部, 教授 (30032824)
SHIMAKAWA Kazuhisa  Okayama University, Faculty of Science Professor, 理学部, 教授 (70109081)
野田 隆三郎  岡山大学, 環境理工学部, 教授 (70029726)
Project Period (FY) 2000 – 2002
Project Status Completed (Fiscal Year 2002)
Budget Amount *help
¥1,600,000 (Direct Cost: ¥1,600,000)
Fiscal Year 2002: ¥700,000 (Direct Cost: ¥700,000)
Fiscal Year 2001: ¥900,000 (Direct Cost: ¥900,000)
Keywordssurgery obstruction / quadratic module / intersection form / selfmtersection form / Hermitian module / Burnside ring / Green functor / nonlinear action / Coupled K-Theory / 特異点集合 / Dress型誘導理論 / Mackey functor / w-Mackey functor / K-Theory / 変換 / surgery / 群作用 / 球面
Research Abstract

So fer, surgery obstructions have been defined as certain equivalence classes of quadratic modules M=(K_k(X;Z),λ,μ) or ones with positioning maps a: 〓_o→ K_k(X;Z). Here λ and μ are the intersection form and the selfintersection form, respectively. In this research, we developed a new theory, namely a coupled K-theory, of quadruple (M, M_2, a, a_2) consisting of a Z[G]-quadratic module M, a Z_2[G]-quadratic module M_2, a positioning map a: 〓→ K_k(X;Z), and a positioning map a: 〓_2→ K_k(X;Z_2).
We constructed new equivariant surgery obstructions, define new Lagrangians and metabolic forms, classified them and quadratic modules, studied surgery technique from the view point of geometry, and putting all this together, constructed a new equivariant surgery theory.
We defined coupled Hermitian modules and a new special Grothendieck-Witt group. Furthermore, we studied the group in the respect of a Mackey functor, a module over the Burnside ring, and a Green functor.
Combining the results above with Oliver's results, we obtained various nonlinear smooth actions on disks and spheres. In particular, for nilpotent Oliver groups and perfect groups, we determined simply connected fixed point manifolds of smooth actions on spheres.

Report

(4 results)
  • 2002 Annual Research Report   Final Research Report Summary
  • 2001 Annual Research Report
  • 2000 Annual Research Report
  • Research Products

    (29 results)

All Other

All Publications (29 results)

  • [Publications] Masaharu Morimoto: "Induction theorems of surgery obstruction groups"Trans.Amer.Math.Soc. (electronically published on Feb. 4, 2003). (印刷中). (2003)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] Masaharu Morimoto, Krzysztof Pawalowski: "Smooth actions of Oliver groups on spheres"Topology. 42 no.2. 395-421 (2003)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] Masaharu Morimoto: "Cappell-Shaneson's group and equivariant surgery"数理解析研究所講究録. 1290. 42-47 (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] 島川和久: "Labeled configuration spaces and group-completion"数理解析研究所講究録. 1290. 100-103 (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] Masaharu Morimoto: "The Burnside ring revisited"Current Trends in Transformation Groups, K-Monographs in Mathematics. 7. 129-145 (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] Masaharu Morimoto: "G-surgery on 3-dimensional manifolds for homology equivalences"Publ.Res.Inst.Math.Sci. Kyoto Univ.. 37. 191-220 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] Masaharu Morimoto: "Equivariant surgery with middle dimensional singular sets. II : Equivariant framed cobordism invariance"Trans.Amer.Math.Soc.. 353. 2427-2440 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] Kazuhisa Shimakawa: "Configuration spaces with partially summable labels and homology theories"Math.J. Okayama Univ.. 43. 43-72 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] M.Morimoto, T.Sumi, M.Yanagihara: "Finite groups possessing gap modules"Geometry and Topology : Aarhus, eds. K.Grove, I.Madsen and E.Pedersen, Contemp.Math., Amer.Math.Soc.. 258. 329-342 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] Masaharu Mrimoto: "Induction theorems of surgery obstruction groups"Trans. Amer. Math. Soc. (Artcile electronically published on Feb. 4, 2003). (2003)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] Masaharu Morimoto and Krzysztof Pawalowski: "Smooth actions of Oliver groups on spheres"Topology. 42 no.2. 395-421 (2003)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] Masaharu Morimoto: "Cappell-Shaneson' s group and equivariant surgery"RIMS Kokyuroku. 1290. 42-47 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] Kazuhisa Shimakawa: "Labeled configuration spaces and group-completion"RIMS Kokyuroku. 1290. 100-103 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] Masaharu Morimoto: "The Bumside ring revisited"Current Trends in Transformation Groups, K-Monographs in Mathematics. 7. 129-145 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] Masaharu Morimoto: "G-surgery on 3-dimensional manifolds for homology equivalences"Publ. Res. Inst. Math. Sci. Kyoto Univ.. 37. 391-220 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] Masaharu Morimoto: "Equivariant surgery with Middle dimensional singular sets. II : Equivariant framed cobordim invariance"Trans. Amer. Math. Soc.. 353. 2427-2440 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] Kazuhisa Shimakawa: "Configuration spaces with partially summable labels and homlogy theories"Math. J. Okayama Univ.. 43. 43-72 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] M. Morimoto, T. Sumi and M. Yanagihara: "Finite groups possessing gap modules"Geometry and Topology : Aarhus, eds. K. Grove, I. Madsen and E. Pedersen, Contemp. Math., Amer. Math. Soc. Providence. 258. 329-342 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] Masaharu Morimoto: "Induction theorems of surgery obstruction groups"Trans. Amer. Math. Soc.(Artcile electronically published on Feb. 4, 2003). (2003)

    • Related Report
      2002 Annual Research Report
  • [Publications] Masaharu Morimoto, Krzysztof Pawalowski: "Smooth actions of Oliver groups on spheres"Topology. 42 no.2. 395-421 (2003)

    • Related Report
      2002 Annual Research Report
  • [Publications] Masaharu Morimoto: "The Burnside ring revisited"Current Trends in Transformation Groups, K-Monographs in Mathematics. 7. 129-145 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] Masaharu Morimoto: "The Burnside ring revisited"Trends in Transformation Groups (K-Theory Monographs), Kluwer Academic Publishers. (to appear). (2002)

    • Related Report
      2001 Annual Research Report
  • [Publications] Masaharu Morimoto, Krzysztof Pawalowski: "Smooth actions of finite Oliver groups on spheres"Topology. (to appear). (2002)

    • Related Report
      2001 Annual Research Report
  • [Publications] Masaharu Morimoto: "Equivariant surgery with middle dimensional singular sets. II : Equivariant framed cobordism invariance"Transactions of Amererican Mathematical Society. 353. 2427-2440 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] Masaharu Morimoto: "G-surgery on 3-dimensional manifolds for homology equvalences"Publications of the Research Institute for Mathematical Sciences, Kyoto University. 37・2. 191-220 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] Masaharu Morimoto: "Equivariant surgery with middle dimensional singular sets. II : Equivariant framed cobordism invariance"Transactions of the American Mathematical Society. (in press). (2001)

    • Related Report
      2000 Annual Research Report
  • [Publications] Masaharu Morimoto: "G-surgery on 3-dimensional manifolds for homology equivalences"Publications of Res.Inst.for Math.Sci.Kyoto University. 37巻・2号(in press).

    • Related Report
      2000 Annual Research Report
  • [Publications] Atsushi Nakajima: "On generalized higher derivations"Turkish Journal of Mathematics. 24巻. 295-311 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] Atsushi Nakajima: "Generalized Jordan derivations"Proceedings of International Symposium on Ring Theory (Birkhaeuser, Boston). (in press). 235-243

    • Related Report
      2000 Annual Research Report

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Published: 2001-04-01   Modified: 2016-04-21  

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