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Research of the Localization with respect to K-Theory and its Picard Group

Research Project

Project/Area Number 15540068
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field Geometry
Research InstitutionNagoya Institute of Technology

Principal Investigator

YOSHIMURA Zen-ichi  Nagoya Institute of Technology, Technology, Professor, 工学研究科, 教授 (70047330)

Co-Investigator(Kenkyū-buntansha) MINAMI Norihiko  Nagoya Institute of Technology, Technology, Professor, 工学研究科, 教授 (80166090)
NAKAI Hirofumi  Nagoya Institute of Technology, Technology, Lecture, 講師 (80343739)
Project Period (FY) 2003 – 2005
Project Status Completed (Fiscal Year 2005)
Budget Amount *help
¥2,200,000 (Direct Cost: ¥2,200,000)
Fiscal Year 2005: ¥700,000 (Direct Cost: ¥700,000)
Fiscal Year 2004: ¥700,000 (Direct Cost: ¥700,000)
Fiscal Year 2003: ¥800,000 (Direct Cost: ¥800,000)
KeywordsReal K-Theory / Complex K-Theory / K-Localization Theory / KO-Homology Equivalence / Conlugation / Adams Operation / CW-Spectrum / Bousfield's Category / 局所化理論 / CW複体 / 共役作用素 / ねじれ群 / ピカール群
Research Abstract

The head investigator introduced the concept of "quasi KO_*-equivalence" in 1990, in order to give a certain classification of CW-complexes or manifolds, which is weaker than the classification based on the K_*-localization. Since then, he has continued to research mainly the several subjects concerned with the quasi KO_*-equivalence and the K_*-localization. He has already obtained some satisfactory results, and moreover obtained a new result mentioned below during the period of Scientific Research Project in 2003-2005.
The subject is to classify CW-complexes (or CW-spectra) X by the K_*-localization when once X has been classified by the quasi KO_*-equivalence. For our purpose it is necessary to investigate the behavior of the Adams operation on KO_*X. However it is never easy to determine their K_*-local types because the behavior of the Adams operation on KO_*X is very complicated. In 2001-2002 he determined the K_*-local type of X when KU_*X is isomorphic to Z Z the free abelian gr … More oup of rank 2.
During the period of Scientific Research Project in 2003-2005 he moreover established to determine the K_*-local type of X when KU_* X is isomorphic to Z Z/2^m the direct sum of the free abelian group of rank 1 and a 2-cyclic group.
Our result is mentioned below :
In order to determine the K_*-local type of X with KU_*X Z Z/2^m we may treat CW-spectra X under the restriction that KU_0X Z Z/2^m and KU_1X 0 by virtue of the duality theorem. Such a CW-spectrum X is quasi K0_*-equivalent to one of the following 7 kinds of spectra Σ^<2i>∨SZ/2^m,Σ^<2i>∨V_m, Σ^<2i>∨W_m,M_m,N_m,Q_m,R_m (0【less than or equal】i【less than or equal】3). When a CW-spectrum X is quasi KO_*-equivalent to each of these small spectra, it has one or two kinds of the Adams complex operation on KU_*X following its quasi KO_*-equivalent type. The Adams operation on its KO-homology groups KO_*X are completely determined by the behavior of the Adams complex operation on KU_*X. By these observation of the Adams operation and construction of several small cells complexes we can completely determine the K_*-local type of CW-spectra X with KU_0X Z Z/2^m and KU_1X 0. Consequently, a CW-spectrum X has essentially 31,33,1,4,2,3,12 kinds of K_*-local types if it is quasi KO_*- equivalent to Σ^<2i>∨SZ/2^m, Σ^<2i>∨V_m,Σ^<2i>∨W_m,M_m,N_m,Q_m,R_m respectively. Less

Report

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

    (19 results)

All 2006 2005 2003 Other

All Journal Article (15 results) Book (3 results) Publications (1 results)

  • [Journal Article] Nilpotency of the Bauer-Furuta stable homotopy Seiberg-Witten invariants2006

    • Author(s)
      M.Furuta, Y.Kametani, N.Minami
    • Journal Title

      Geometry and Topology Monographs (印刷中)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Homotopy theorical considerations of the Bauer-Furuta stable homotopy Seiberg-Witten invariants2006

    • Author(s)
      M.Furuta, Y.Kametani, H.Matsu, N.Minami
    • Journal Title

      Geometry and Topology Monographs (印刷中)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Nilpotency of the Bauer-Furuta stable homotopy Seiberg-Witten invariants2006

    • Author(s)
      M.Furuta, Y.Kemetani, N.Minami
    • Journal Title

      Geometry and Topology Monographs (In press)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Homotopy theoretical considerations of the Bauer-Furuta stable homotopy Seiberg-Witten invariants2006

    • Author(s)
      M.Furuta, Y.Kemetani, H.Matsue, N.Minami
    • Journal Title

      Geometry and Topology Monographs (In press)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Nilpotency of the Bauer-Furuta stable homotopy Seiberg-Witten invariants2006

    • Author(s)
      Mikio Furuta, Yukio Kametani, Norihiko Minami
    • Journal Title

      Geometry and Topology Monographs (印刷中)

    • Related Report
      2005 Annual Research Report
  • [Journal Article] Homotopy theoretical considerations of the Bauer-Furuta stable homotopy Seiberg-Witten invariants2006

    • Author(s)
      Mikio Furuta, Yukio Kanietani, Hirofumi Matsue, Norihiko Minami
    • Journal Title

      Geometry and Topology Monographs (印刷中)

    • Related Report
      2005 Annual Research Report
  • [Journal Article] On the Chromatic Tower spaces2003

    • Author(s)
      N.Minami
    • Journal Title

      The American Journal of Mathematics 125

      Pages: 449-473

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] The structure of general chromatic E_1-term Ext^0_<Γ(2)>(BP_*,M^1_2) at the prime 22003

    • Author(s)
      H.Nakai, D.Yoritomo
    • Journal Title

      Hiroshima Mathematical Journal 33

      Pages: 137-156

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] The first cohomology group of the generalized Morava stabilizer algebra2003

    • Author(s)
      H.Nakai, D.C.Ravenel
    • Journal Title

      Proceedings of American Mathematics Society 13

      Pages: 1629-1639

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] On the Chromatic Tower spaces2003

    • Author(s)
      N.Minami
    • Journal Title

      The American Journal of Mathematics Vol.125

      Pages: 449-473

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] The structure of general chromatic E_1-term Ext^0_<Γ(2)> (BP_*,M^1_2) at the prime 22003

    • Author(s)
      H.Nakai, D.Yoritomi
    • Journal Title

      Hiroshima Mathematical Journal Vol.33

      Pages: 137-156

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] The first cohomology group of the generalized Morava stabilizer algebra2003

    • Author(s)
      H.Nakai, D.C.Ravenel
    • Journal Title

      Proceedings of American Mathematics Society Vol.13

      Pages: 1629-1639

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] On the Chromatic Tower2003

    • Author(s)
      Norihiko Minami
    • Journal Title

      The American Journal of Mathematics 125

      Pages: 449-473

    • Related Report
      2004 Annual Research Report
  • [Journal Article] The structure of general chromatic E1-term Ext0Γ(2)(BP*,M12) at the prime 22003

    • Author(s)
      Hirofumi Nakai, Daichi Yoritomi
    • Journal Title

      Hiroshima Mathematical Journal 33

      Pages: 137-156

    • Related Report
      2004 Annual Research Report
  • [Journal Article] The first cohomology group of the generalized Morava stabilizer algebra2003

    • Author(s)
      Hirofumi Nakai, Douglas C.Ravenel
    • Journal Title

      Proceedings of American Mathematics Society 131

      Pages: 1629-1639

    • Related Report
      2004 Annual Research Report
  • [Book] 工科のための線形代数2005

    • Author(s)
      吉村善一
    • Total Pages
      243
    • Publisher
      数理工学社
    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Book] Linear Algebra for Technology2005

    • Author(s)
      Z.Yoshimura
    • Total Pages
      243
    • Publisher
      Suuri Kougakusha
    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Book] 工科のための 線形代数2005

    • Author(s)
      吉村善一
    • Total Pages
      243
    • Publisher
      数理工学社
    • Related Report
      2005 Annual Research Report
  • [Publications] Norihiko Minami: "On the Chromatic Tower"The American Journal of Mathematics. 125. 449-473 (2003)

    • Related Report
      2003 Annual Research Report

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

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