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Study on rank filetration of finite H-spaces

Research Project

Project/Area Number 15540074
Research Category

Grant-in-Aid for Scientific Research (C)

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

Principal Investigator

MORISUGI Kaoru  WAKAYAMA UNIVERSITY, Faculty of Education, Professor, 教育学部, 教授 (00031807)

Co-Investigator(Kenkyū-buntansha) OSHIMA Hideaki  Ibaraki University, Faculty of Science, Professor, 理学部, 教授 (70047372)
HEMMI Yutaka  Kochi University, Faculty of Science, Professor, 理学部, 教授 (70181477)
WATANABE Takashi  Osaka Prefecture University, Faculty of Liberal Arts and Sciences, Professor, 総合教育研究機構, 教授 (20089957)
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: ¥1,300,000 (Direct Cost: ¥1,300,000)
Fiscal Year 2004: ¥800,000 (Direct Cost: ¥800,000)
Fiscal Year 2003: ¥1,400,000 (Direct Cost: ¥1,400,000)
KeywordsHopf spaces / Lie groups / Whitehead products / homotopy groups of spheres / self maps / nilpotent class / Moore spaces / mod p cohomology / Samelson積 / Homology / Homtopy / rank / filtration / 対称空間 / 交換子
Research Abstract

The summary of research results is as follows.
1 It is important to know what nilpotency the group [X,X] has for a Hopf space X. However another important thing is to study the composition structure of the group [X,X]. Morisugi determined the relationship between the composition structure and the group structure of [X,X] for X=SU(3) and Sp(2). This structure looks like "Square ring" which Baues in Germany studied.
Let M^n be the mod 2 Moore space. For n【greater than or equal】3 it is known that there exists a lift <η_n>^^^^of the suspension of the Hopf map, η_n:S^<n+1>→S^n. We investigated the order of the Whitehead product [<η_n>^^^^,<η_n>^^^^] in π_<2n+1>(M^n).
2 Let G be the simple Lie group of classical type. Oshima showed that the group [G,G] is non-commutative for almost all cases of G mentioned above. And for some cases of G, he determined the nilpotency class of [G,G].
Let X be a Hopf complex. In this case [X,X] is, so called, an algebraic loop, that is, it has a binary operation with both left and right inverses. Oshima also investigated how they differs from each other.
3 Hemmi showed that the possible even dimensional generators of mod 3 cohomology rings of finite Hopf spaces occurs only in dimension 8 or 20. And he almost determined the structure of such mod 3 cohomology rings. He also showed that under some conditions, there is no Hopf space X with H(X;Z/p)≡Λ(x,p^1x,…p^<p-2>x)

Report

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

    (29 results)

All 2006 2005 2004 2003 2002 2001 Other

All Journal Article (24 results) Publications (5 results)

  • [Journal Article] Homotopy classes of self-maps and induced homomorphisms of homotopy groups2006

    • Author(s)
      M.Arkowitz, H.Oshima, J.Strom
    • Journal Title

      J.Math.Soc.Japan 58

    • Related Report
      2005 Annual Research Report
  • [Journal Article] Retractions of H-spaces2005

    • Author(s)
      Y.Hemmi
    • Journal Title

      Hiroshima Math.J 35

      Pages: 159-165

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] A group of self homotopy equivalences of SO(4)2005

    • Author(s)
      H.Oshima
    • Journal Title

      Quart.J.Math.Oxford 56

      Pages: 95-100

    • 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] Samelson products in the exceptional Lie group of rank 22005

    • Author(s)
      H.Oshima
    • Journal Title

      J.Math.Kyoto Univ. 45

      Pages: 411-420

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Retractions of H-spaces2005

    • Author(s)
      Yutaka Hemmi
    • Journal Title

      Hiroshima Math.J. 35

      Pages: 159-165

    • Related Report
      2005 Annual Research Report
  • [Journal Article] Samelson products in the exceptional Lie group of rank 22005

    • Author(s)
      H.Oshima
    • Journal Title

      J.Math.Kyoto.Univ. 45

      Pages: 411-420

    • Related Report
      2005 Annual Research Report
  • [Journal Article] Retractions of H-spaces2005

    • Author(s)
      Y.Hemmi
    • Journal Title

      Math.J 35(to appear in Hiroshima)

    • Related Report
      2004 Annual Research Report
  • [Journal Article] Commutativity of the group of self homotopy classes of Lie groups2004

    • Author(s)
      A.Kono, H.Oshima
    • Journal Title

      Bull.London Math.Soc. 36

      Pages: 37-52

    • Related Report
      2004 Annual Research Report
  • [Journal Article] Higher homotopy commutativity of H-spaces and the permuto-associahedra2004

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

      Trans.Amer.Math.Soc. 356

      Pages: 3823-3839

    • Related Report
      2004 Annual Research Report
  • [Journal Article] On the Bott suspenssion for non-compact Lie groups2003

    • Author(s)
      T.Watanabe
    • Journal Title

      J.Math.Kyoto Univ. 43-4

      Pages: 689-698

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] The group of self homotopy classes of SO (4)2003

    • Author(s)
      H.Oshima
    • Journal Title

      J.Pure Appl.Algebra 108

      Pages: 193-205

    • Related Report
      2004 Annual Research Report
  • [Journal Article] H-spaces as direct product factors of loop spaces2003

    • Author(s)
      Y.Hemmi
    • Journal Title

      Topology Appl. 132

      Pages: 37-47

    • Related Report
      2004 Annual Research Report
  • [Journal Article] On the Bott suspension map for non-compact Lie groups2003

    • Author(s)
      T.Watanabe
    • Journal Title

      J.Math.Kyoto Univ. 43

      Pages: 689-698

    • Related Report
      2004 Annual Research Report
  • [Journal Article] The Whitehead square of a lift of the Hopf map to a mod 2 Moore space2002

    • Author(s)
      K.Morisugi, J.Mukai
    • Journal Title

      J.Math.Kyoto Univ. Vol.42-2

      Pages: 331-336

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] The inverses of an H-space2002

    • Author(s)
      M.Arkowitz, H.Oshima, J.Strom
    • Journal Title

      Manuscripta Math. Vol.108

      Pages: 399-408

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Noncommutativity of the group of self homotopy classes of classical simple Lie groups2002

    • Author(s)
      M.Arkowitz, H.Oshima, J.Strom
    • Journal Title

      Topology Appl. Vol.125

      Pages: 87-96

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Cohomology Rings of 3-local Finite H-spaces2002

    • Author(s)
      Y.Hemmi, J.Lin
    • Journal Title

      J.Pure Appl.Algebra Vol.167

      Pages: 1-14

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Unstable p-th order operation and H-spaces2002

    • Author(s)
      Y.Hemmi
    • Journal Title

      Contemporary Math. Vol 293

      Pages: 75-88

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Composition structure of the self maps of SU(3) and Sp(2)2001

    • Author(s)
      K.Morisugi
    • Journal Title

      Contemporary Math. Vol.274

      Pages: 233-240

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Noncommutativity of self homotopy groups2001

    • Author(s)
      H.Oshima, Nobuaki Yagita
    • Journal Title

      Kodai Math.J. Vol.24

      Pages: 15-25

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Multiplications on a localized Hopf space2001

    • Author(s)
      H.Oshima
    • Journal Title

      Math.J.Ibaraki Univ. Vol.33

      Pages: 17-21

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Mod p decompositions of mod p finite H-spaces2001

    • Author(s)
      Y.Hemmi
    • Journal Title

      Mem.Fac.Sci.Kochi Univ.Ser.A (Math.) Vol.22

      Pages: 59-65

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Higher homotopy commutativity and cohomology of finite H-spaces

    • Author(s)
      Yutaka Hemmi, Yusuke Kawamoto
    • Journal Title

      Geometry and Topology Monographs (to appear)

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

    • Author(s)
      Yutaka Hemmi
    • Journal Title

      Progress in Algebraic Topology Research, Nova Science (to appear)

    • Related Report
      2005 Annual Research Report
  • [Publications] K.Morisugi, J.Mukai: "The Whitehead square of a lift of the Hopf map to a mod 2 Moore space"J.Math.Kyoto Univ.. 42-2. 331-336 (2002)

    • Related Report
      2003 Annual Research Report
  • [Publications] H.Oshima: "The group of self homotopy classes of SO(4)"J.Pure Appl.Algebra. 108. 193-205 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] Y.Hemmi: "H-spaces as direct product factors of loop spaces"Topology Appl.. 132. 37-47 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] Y.Hemmi, Y.Kawamoto: "Higher homotopy commutativity of H-spaces and the permuto-associahedra"Trans.Amer.Math.Soc.. (to appear). (2004)

    • Related Report
      2003 Annual Research Report
  • [Publications] T.Watanabe: "On the Bott suspension map for non-compact Lie groups"J.Math.Kyoto Univ.. 43(to appear). 689-698 (2003)

    • Related Report
      2003 Annual Research Report

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

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