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Study on phantom maps

Research Project

Project/Area Number 16540076
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field Geometry
Research InstitutionOsaka Prefecture University (2005-2007)
Osaka Women's University (2004)

Principal Investigator

IRIYE Kouyemon  Osaka Prefecture University, Gradate School of Science, Professor (40151691)

Co-Investigator(Kenkyū-buntansha) WATANABE Takashi  Osaka Prefecture University, Faculty of Liberal arts and Sciences, Professor (20089957)
YAMAGUCHI Atsushi  Osaka Prefecture University, Faculty of Liberal arts and Sciences, Professor (80182426)
KATO Kiriko  Osaka Prefecture University, Graduate School of Science, Associate Professor (00347478)
YOSHITOMI Kentaro  Osaka Prefecture University, Faculty of Liberal arts and Sciences, Lecturer (10305609)
加藤 希理子  大阪女子大学, 理学部, 助教授 (10291791)
Project Period (FY) 2004 – 2007
Project Status Completed (Fiscal Year 2007)
Budget Amount *help
¥3,870,000 (Direct Cost: ¥3,600,000、Indirect Cost: ¥270,000)
Fiscal Year 2007: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
Fiscal Year 2006: ¥700,000 (Direct Cost: ¥700,000)
Fiscal Year 2005: ¥700,000 (Direct Cost: ¥700,000)
Fiscal Year 2004: ¥1,300,000 (Direct Cost: ¥1,300,000)
KeywordsPhantom map / Loop space / rational homotopy eauivalence / exceptional Lie groups / symplectic group / stable homotopy / 例外型リー群 / 分解 / Gray指数 / 逆系 / リー群 / スティーロッド作用素 / 局所化
Research Abstract

Through our research we obtained the following three major results.
1. We proved that for every simply connected finite complex X there is a map which induces rational homotopy equivalence from the loop space on X to a product of some odd dimensional spheres and loop spaces on odd dimensional spheres. This result implies that the phantom map out of the loop space on a simply connected finite complex into a finite type space is trivial.
2. If a space X satisfies one of the following three conditions, then the stable suspension order of the universal phantom map out of the space X is infinite.
(1) The fundamental group of X is finite and homotopy groups are vanish in all sufficiently large dimensions.
(2) X is the classifying space of non-trivial Lie group.
(3) X is the loop space on the Lie group whose integral homology group has torsion element.
3. The loop space on symplectic group Sp (n) is stably indecomposable if n is greater than or equal to 2.

Report

(5 results)
  • 2007 Annual Research Report   Final Research Report Summary
  • 2006 Annual Research Report
  • 2005 Annual Research Report
  • 2004 Annual Research Report
  • Research Products

    (38 results)

All 2008 2007 2006 2005 2004 Other

All Journal Article (24 results) (of which Peer Reviewed: 12 results) Presentation (14 results)

  • [Journal Article] Stable indecomposablity of loop spaces on symplectic groups2008

    • Author(s)
      入江 幸右衛門
    • Journal Title

      Proceeclings of the American Mathematical Society 136

      Pages: 727-733

    • 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] Stable indecomposability of loop spaces on symplectic groups2008

    • Author(s)
      K. Iriye
    • Journal Title

      Proceedings of the American Mathematical Society 136

      Pages: 727-733

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Journal Article] Stable indecomposability of loop spaces on symplectic groups2008

    • Author(s)
      入江 幸右衛門
    • Journal Title

      Proceedings of the American mathematical Society 136

      Pages: 727-733

    • Related Report
      2007 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Stable suspension order of universal phantom maps and some stably indecomposable loop spaces2007

    • Author(s)
      入江 幸右衛門
    • Journal Title

      Journal of the Mathematical Society of Japan 59

      Pages: 97-112

    • 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] Real K-cohomology of complex projectives paces2007

    • Author(s)
      山口 睦
    • Journal Title

      Scientiae Mathematicae Japonicas 65

      Pages: 407-422

    • 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] Real K-homology of complex projective spaces2007

    • Author(s)
      山口 睦
    • Journal Title

      Journal of Mathematics of Kyoto University 47

      Pages: 203-222

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Annual Research Report 2007 Final Research Report Summary
    • Peer Reviewed
  • [Journal Article] Morphisms represented by monomorphisms2007

    • Author(s)
      加藤 希理子
    • Journal Title

      Journal of Pure and Applied Algebras 208

      Pages: 261-283

    • 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] Syzygies of modules with positive codimensions2007

    • Author(s)
      加藤 希理子
    • Journal Title

      Journal of Algebra 318

      Pages: 25-36

    • 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] Stable suspension order of universal phantom maps and some stably indecomposable loop spaces2007

    • Author(s)
      K. Iriye
    • Journal Title

      Journal of the Mathematical Society of Japan 59

      Pages: 97-112

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Journal Article] Real K-cohomology of complex projective spaces2007

    • Author(s)
      A. Yamaguchi
    • Journal Title

      Scientiae Mathematicae Japonicae 65

      Pages: 407-422

    • NAID

      10019758803

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Journal Article] Real K-homology of complex projective spaces2007

    • Author(s)
      A. Yamaguchi
    • Journal Title

      Journal of Mathematics of Kyoto University 47

      Pages: 203-222

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Journal Article] Morphisms represented by monomorphisms2007

    • Author(s)
      K. Kato
    • Journal Title

      Journal of Pure and Applied Algebras 208

      Pages: 261-283

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Journal Article] Syzygies of modules with positive codimensions2007

    • Author(s)
      K. Kato
    • Journal Title

      Journal of Algebra 318

      Pages: 25-36

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Journal Article] On exccss filtration on thc Stccnrod algcbra2007

    • Author(s)
      山口 睦
    • Journal Title

      Geometry & Topology Monographs 10

      Pages: 423-449

    • Related Report
      2007 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Real K-cohomology of complex projective spaces2007

    • Author(s)
      山口 睦
    • Journal Title

      Scientiae Mathematicae Japonicae 65

      Pages: 407-422

    • NAID

      10019758803

    • Related Report
      2007 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Syzygies of modules with positive codimension2007

    • Author(s)
      加藤 希理子
    • Journal Title

      Journal of Algebra 318

      Pages: 25-36

    • Related Report
      2007 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Stable suspension order of universal phantom maps and some stably indecomposable loop spaces2007

    • Author(s)
      入江 幸右衛門
    • Journal Title

      Journal of Mathematical Society of Japan 59

      Pages: 97-112

    • Related Report
      2006 Annual Research Report
  • [Journal Article] Morphisms represented by monomorphisms2007

    • Author(s)
      加藤 希理子
    • Journal Title

      Journal of Pure and Applied Algebra 208

      Pages: 261-283

    • Related Report
      2006 Annual Research Report
  • [Journal Article] Rational equivalence and phantom map out of loop space II2004

    • Author(s)
      入江 幸右衛門
    • Journal Title

      Journal of Mathematics of Kyoto University 44

      Pages: 595-601

    • 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] Rational equivalence and phantom map out of loop space II2004

    • Author(s)
      K. Iriye
    • Journal Title

      Journal of Mathematics of Kyoto University 44

      Pages: 595-601

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Journal Article] Rational equivalence and phantom map out of loop space, II2004

    • Author(s)
      入江幸右衛門
    • Journal Title

      Journal of Mathematics of Kyoto University 44巻3号

      Pages: 595-601

    • Related Report
      2004 Annual Research Report
  • [Journal Article] On excess filtration on the Steenrod algebra

    • Author(s)
      山口 睦
    • 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
      2007 Final Research Report Summary
    • Peer Reviewed
  • [Journal Article] On excess filtration on the Steenrod algebra

    • Author(s)
      A. Yamaguchi
    • Journal Title

      Geometry and Topology Monographs (in press)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Journal Article] Morphisms represented by monomorphisms

    • Author(s)
      加藤 希理子
    • Journal Title

      Journal of Pure and Applied Algebra (印刷中)

    • Related Report
      2005 Annual Research Report
  • [Presentation] Cohomology theory and representation2008

    • Author(s)
      A. Yamaguchi
    • Organizer
      NIT Meeting on Homotopy Theory
    • Place of Presentation
      Nagoya Institute of Technology
    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] コホモロジー論と表現論2007

    • Author(s)
      山口 睦
    • Organizer
      名工大ホモトピー論集会
    • Place of Presentation
      名古屋工業大学
    • Year and Date
      2007-03-25
    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Annual Research Report 2007 Final Research Report Summary
  • [Presentation] Quotient categories of homotopy categories2007

    • Author(s)
      加藤 希理子
    • Organizer
      第29回可換環論シンポジウム
    • Place of Presentation
      名古屋大学
    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] Quotient categories of homotopy categories2007

    • Author(s)
      K. Kato
    • Organizer
      the 29th Conference on commutative algebras
    • Place of Presentation
      Nagaya University
    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] 例外リー群のループ空間の安定圏における非分解性について2006

    • Author(s)
      入江 幸右衛門
    • Organizer
      日本数学会春季総合分科会
    • Place of Presentation
      中央大学
    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] リー群のループ空間の安定圏における非分解性について2006

    • Author(s)
      入江 幸右衛門
    • Organizer
      ホモトピー論シンポジウム
    • Place of Presentation
      愛媛大学
    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] On the stable indecomposability of loop spaces on the exceptional Lie groups2006

    • Author(s)
      K. Iriye
    • Organizer
      Spring MS J Meeting
    • Place of Presentation
      Chuo University
    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] On the stable indecomposability of loop spaces on Lie groups2006

    • Author(s)
      K. Iriye
    • Organizer
      Conference on Homotopy Theory
    • Place of Presentation
      Ehime University
    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] Stable suspension order of universal phantom maps and indecompos ability of loop spaces on exceptional Lie groups2005

    • Author(s)
      入江 幸右衛門
    • Organizer
      International Conference on Algebraic Topology
    • Place of Presentation
      京都大学
    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] Stable suspension order of universal phantom maps and indecomposability of loop spaces on exceptional Lie groups2005

    • Author(s)
      K. Iriye
    • Organizer
      International Conference on Algebraic Topology
    • Place of Presentation
      Kyoto University
    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] Phantom maps out of loop spaces2004

    • Author(s)
      入江 幸右衛門
    • Organizer
      International Conference on Algebraic Topology
    • Place of Presentation
      北京
    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] ループ空間からでる幽霊写像について2004

    • Author(s)
      入江 幸右衛門
    • Organizer
      日本数学会秋季総合分科会
    • Place of Presentation
      北海道大学
    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] Phantom maps out of loop spaces2004

    • Author(s)
      K. Iriye
    • Organizer
      International Conference on Algebraic Topology
    • Place of Presentation
      Beijing
    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] Phantom maps out of loop spaces2004

    • Author(s)
      K. Iriye
    • Organizer
      Autumn MSJ Meeting
    • Place of Presentation
      Hokkaido University
    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary

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Published: 2004-04-01   Modified: 2016-04-21  

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