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Study of the higher homotopy commutative Hopf spaees and its application to the higher category

Research Project

Project/Area Number 17540083
Research Category

Grant-in-Aid for Scientific Research (C)

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

Principal Investigator

HEMMI Yutaka  Kochi University, Faculty of Science, Professor (70181477)

Co-Investigator(Kenkyū-buntansha) SHIMOMURA Katsumi  Kochi University, Faculty of Science, Professor (30206247)
MORISUGI Kaoru  Wakayama University, Faculty of Education, Professor (00031807)
TSUKIYAMA Kouzou  Shimane University, Faculty of Education, Professor (20093651)
小松 和志  高知大学, 理学部, 助教授 (00253336)
Project Period (FY) 2005 – 2007
Project Status Completed (Fiscal Year 2007)
Budget Amount *help
¥3,560,000 (Direct Cost: ¥3,200,000、Indirect Cost: ¥360,000)
Fiscal Year 2007: ¥1,560,000 (Direct Cost: ¥1,200,000、Indirect Cost: ¥360,000)
Fiscal Year 2006: ¥900,000 (Direct Cost: ¥900,000)
Fiscal Year 2005: ¥1,100,000 (Direct Cost: ¥1,100,000)
Keywordsp-compact group / atomic space / topological monoid / higher homotopv commutativity / shuffle polytopes / A_n space / reduced power operation / resulthedra / ホモトピー可換性 / 実射影空間 / 安定拡張性 / 安定分解問題 / H空間 / 高位ホトモトピー可換性 / AC_n構造 / 法束 / retraction / 高位ホトモトピー / projective space / Clark-Ewing space / cohomology operation
Research Abstract

We have the following result during the period.
1. We study various types of homotopy commutativity related to the C_n-structure given by Williams, and we consider the polytopes representing them as subsets of the permutohedron. In particular, we show that the polytopes representing the higher homotopy commutativity given by the subset of n-th symmetric group consists of the inverse of some fixed shuffles are shuffle polytopes.
2. We introduce the concept of AC_n-structure on A_n-spaces, which is a higher homotopy commutativity. Moreover, we study that the action of the reduced power operation on the mod p cohomology finite A_p-space with AC_n-structure for n>(p-1)/2. The result is published in Geometry and Topology Monographs.
3. We determine the action of the reduced power operation on the simply connected finite p-compact groups with no torsion in homology for odd primes p. We also determine the decomposition of the spaces as products of atomic spaces. The result is a generalization of the fact on the simply connected simple Lie groups given by Mimura-Nishida-Toda.
4. We introduced a new concept on higher homotopy commutativity of topological monoids. To do so we used resulthedra which are constructed by Gelfand, Kapranov and Zelevinsky. Our study has a deep relation ship with the concepts given by Felix-Tanre and Aguade.

Report

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

    (21 results)

All 2008 2007 2006 2005 Other

All Journal Article (13 results) (of which Peer Reviewed: 4 results) Presentation (8 results)

  • [Journal Article] Stable extendibility of normal bundles associated to immersions of RP^n in R^{2n}2008

    • Author(s)
      Y. Hemmi, T. Kobayashi and Min Lwin Oo
    • Journal Title

      Kochi J. Math. 3

      Pages: 167-180

    • 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 extendibility of normal bundles associated to immersions of RP^n in R^{2n}2008

    • Author(s)
      Y. Hemmi, T. Kobayashi, Min Lwin, Oo
    • Journal Title

      Kochi J. Math 3

      Pages: 167-180

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Journal Article] Stable extendibility of normal bundles associated to immersions of RP^n in R^{2n}2008

    • Author(s)
      Y.Hemmi, T.Kobayashi and Min Lwin Oo
    • Journal Title

      Kochi J.Math 3

      Pages: 167-180

    • Related Report
      2007 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Higher homotopy commutativity and cohomology of finite H-spaces2007

    • Author(s)
      Y. Hemmi and Y. Kawamoto
    • Journal Title

      Geometry and Topology Monographs 10

      Pages: 167-186

    • 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] The power of the normal bundle associated to an immersion of RP^n, its complexification and extendibility2007

    • Author(s)
      Y. Hemmi, T. Kobayashi and Min Lwin Oo
    • Journal Title

      Hiroshima Math. J. 37

      Pages: 101-109

    • 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] Higher homotopy coimnutativity and cohomology of finite H-spaces2007

    • Author(s)
      Y. Hemmi, Y. Kawamoto
    • Journal Title

      Geometry and Topology Monographs 10

      Pages: 167-186

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Journal Article] The power of the normal bundle associated to an immersion of RP^n, its complexification and extendibility2007

    • Author(s)
      Y. Hemmi, T. Kobayashi, Min Lwin, Oo
    • Journal Title

      Hiroshima Math. J 37

      Pages: 101-109

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Journal Article] Higher homotopy commutativity and cohomology of finite H-spaces2007

    • Author(s)
      Y.Hemmi, Y.Kawamoto
    • Journal Title

      Geometry and Topology Monographs 10

      Pages: 167-186

    • Related Report
      2006 Annual Research Report
  • [Journal Article] The power of the normal bundle associated to an immersion of RP^n, its complexification and extendibility2007

    • Author(s)
      Y.Hemmi, T.Kobayasbi, Min Lwin Oo
    • Journal Title

      Hiroshima Math. J. 37

      Pages: 101-109

    • Related Report
      2006 Annual Research Report
  • [Journal Article] Homotopy groups of a generalized E(2)-local Moore spectrum at the prime three2005

    • Author(s)
      I.Ichigi, K.Shimomura
    • Journal Title

      Hiroshima Math.J. 35

      Pages: 125-142

    • Related Report
      2005 Annual Research Report
  • [Journal Article] Extendibility and stable extendibility of vector bundles over lens spaces mod 32005

    • Author(s)
      T.Kobayashi, K.Komatsu
    • Journal Title

      Hiroshima Math.J. 35

      Pages: 403-412

    • NAID

      110004455825

    • Related Report
      2005 Annual Research Report
  • [Journal Article] Higher homotopy associativity of H-spaces and retractions

    • Author(s)
      Y.Hemmi
    • Journal Title

      Progress in Algebraic Topology Research (印刷中)

    • Related Report
      2005 Annual Research Report
  • [Journal Article] On the homotopy groups of E(n)-local spectra with unusual invariant ideals

    • Author(s)
      K.Shimomura, H.Nakai
    • Journal Title

      The Geometry & Topology Monograph (印刷中)

    • Related Report
      2005 Annual Research Report
  • [Presentation] The power of the normal bundle associated to an immersion of RP^n, its complexification and extendibility2007

    • Author(s)
      Min Lwin Oo, T. Kobayashi and Y. Hemmi
    • Organizer
      日本数学会, 秋季総合分科会
    • Place of Presentation
      東北大学
    • Year and Date
      2007-09-22
    • 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] 高位ホモトピー可換性と終結多面体について2007

    • Author(s)
      河本裕介, 逸見豊
    • Organizer
      日本数学会, 秋季総合分科会
    • Place of Presentation
      東北大学
    • Year and Date
      2007-09-22
    • 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] The power of the normal bundle associated to an immersion of RP^n, its complexification and extendibility2007

    • Author(s)
      Min Lwin, Oo, T. Kobayashi, Y. Hemmi
    • Organizer
      Mathematical Society of Japan meeting
    • Place of Presentation
      Tohoku University
    • Year and Date
      2007-09-22
    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] Higher homotopy commutativity and resulthedra2007

    • Author(s)
      Y. Kawamoto, Y. Hemmi
    • Organizer
      Mathematical Society of Japan meeting
    • Place of Presentation
      Tohoku University
    • Year and Date
      2007-09-22
    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] pコンパクト群の分解とSteenrod作用素2007

    • Author(s)
      逸見 豊, 荻一 士喜, 西信 洋和, 河本 裕介
    • Organizer
      空間の代数的・幾何的モデルとその周辺
    • Place of Presentation
      信州大学
    • Year and Date
      2007-09-07
    • 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] Decomposition Of p-compact groups and the Steenrod operations2007

    • Author(s)
      Y. Hemmi, K. Ogi, H. Nishinobu, Y. Kawamoto
    • Organizer
      Algebraic and geometric models of spaces and related topics
    • Place of Presentation
      Shinshu University
    • Year and Date
      2007-09-07
    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] p-compact群のoohomology上のSteenrod operationの作用について2006

    • Author(s)
      西信 洋和, 逸見 豊
    • Organizer
      日本数学会, 秋季総合分科会
    • Place of Presentation
      大阪市立大学
    • Year and Date
      2006-09-19
    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] On the Steenrod operation on the cohomology of p-compact groups2006

    • Author(s)
      H. Nishinobu, Y. Hemmi
    • Organizer
      Mathematical Society of Japan meeting
    • Place of Presentation
      Osaka City University
    • Year and Date
      2006-09-19
    • 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: 2005-04-01   Modified: 2016-04-21  

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