• Search Research Projects
  • Search Researchers
  • How to Use
  1. Back to previous page

Study on unstable homotopy groups of finite complexes

Research Project

Project/Area Number 15540067
Research Category

Grant-in-Aid for Scientific Research (C)

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

Principal Investigator

MUKAI Juno  Shinshu University, Faculty of Science, Professor, 理学部, 教授 (50029675)

Co-Investigator(Kenkyū-buntansha) KACHI Hideyuki  Shinshu University, Faculty of Science, Professor, 理学部, 教授 (50020657)
ABE Kojun  Shinshu University, Faculty of Science, Professor, 理学部, 教授 (30021231)
MATSUDA Toshimitsu  Shinshu University, Faculty of Science, Associate Professor, 理学部, 助教授 (70020667)
KAMIYA Hisao (GOLASINSKI Marek)  Shinshu University, Faculty of Science, Lecturer, 理学部, 講師 (80020676)
Project Period (FY) 2003 – 2004
Project Status Completed (Fiscal Year 2004)
Budget Amount *help
¥1,400,000 (Direct Cost: ¥1,400,000)
Fiscal Year 2004: ¥600,000 (Direct Cost: ¥600,000)
Fiscal Year 2003: ¥800,000 (Direct Cost: ¥800,000)
Keywordsprojective space / suspension order / homotopy type / Hopf construction / Hopf homomorphism / Toda bracket / Whitehead product / m-twisted / Adams写像 / ホモトピー群 / Hopf準同形写像 / 実射影空間 / Moore空間 / 戸田の積
Research Abstract

Summary of research results is the following.
1.Concerning the formula about the Hopf invariant of some Toda bracket, there exist some restrictions. Inoue and I generalize the formula applicable for the case of finite complexes.
2.Let denote by P^n the real n dimensional projective space and by γ_{n-1}:S^{n-1}→P^{n-1} the attaching map of the top n cell. In 2000, some homotopy groups of a suspension EP^3 were determined and a suspension map Eγ_4 was represented as the composite of three maps. This time, the fourth multiple element of the identity class of EP^6 is takes as a representative of some Toda bracket, and this relates with the composite of Eγ_5 and the Hopf map. And these procedure gave an affirmative solution to the suspension order conjecture of P^{2n}.
3.We Set M^n=E^{n-2}P^2, which is called a mod 2 Moore space. Let i_n : S^{n-1}→M^n be the inclusion. The problem whether the Whitehead product of i_{n+1} and itself generates the direct summand Z/2Z in the homotopy group π_{2n-1}(M^{n+1}) was proposed by Spokenkov at Suzue University. We gave a partial answer.
4.By use of the method solving the suspension order conjecture of P^{2n}, Miyauchi and I determined the group of self-homotopies of a double suspension of P^6.
5.By the jointed work with K. Yamaguchi, the homotopy type of the complex 4 dimension m twisted projective space was determined according as the case m=0, m : odd and m is a multiple of 8. Furthermore, it was proven that there exists no homotopy type in the case m is even and m is not divisible by 8.
6.Inoue and I proved that ε_3 and μ_3 are lifted to the Moore Spaces M^{11} and M^{12}, respectively.
7.To develop the joint work with Golasinski, I tried determining the orders of the Whitehead products of iota_n with alphainpi^n_{n+k} for n>k+1, k<25.
Investigators, except for Abe and Golasinski, could obtain no result.

Report

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

    (21 results)

All 2005 2004 2003 Other

All Journal Article (17 results) Publications (4 results)

  • [Journal Article] Self-homotopy of the double suspension of the real 6-projective space2005

    • Author(s)
      Toshiyuki Miyauchi, Juno Mukai
    • Journal Title

      Kyushu J.Math. 59

      Pages: 101-116

    • NAID

      110001263493

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Homotopy classification of twisted complex projective spaces2005

    • Author(s)
      Juno Mukai, Kohhei Yamaguchi
    • Journal Title

      J.Math.Soc.Japan 57(To appear)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Self-homotopy of the double suspension of the real 6-projective Space2005

    • Author(s)
      Toshiyuki Miyauchi, Juno Mukai
    • Journal Title

      Kyushu J.Math. 58-1

      Pages: 203-209

    • NAID

      110001263493

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Self-homotopy of the double suspension of the real 6-projective Space2005

    • Author(s)
      Toshiyuki Miyauchi, Juno Mukai
    • Journal Title

      Kyushu J.Math. 59-1

      Pages: 1-16

    • NAID

      110001263493

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Homotopy classification of twisted complex projective spaces2005

    • Author(s)
      Juno Mukai, Kohhei Yamaguchi
    • Journal Title

      J.Math.Soc.Japan 57-2 (to appear)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Self-homotopy of the double suspension of the real 6-projective space2005

    • Author(s)
      T.Miyauchi, J.Mukai
    • Journal Title

      Kyushu J.Math. 59・1

      Pages: 101-116

    • NAID

      110001263493

    • Related Report
      2004 Annual Research Report
  • [Journal Article] Homotopy classification of twisted complex projective spaces2005

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

      Japan Math.Soc. 57・2(To appear)

    • Related Report
      2004 Annual Research Report
  • [Journal Article] Orders of Whitehead products of ι_n with α in π ^n_{n+k} (n>k+1,k<25)2005

    • Author(s)
      J, Mukai
    • Journal Title

      Proceeding of the International conference on Homotopy Theory and related topics at Korea University (to appear)

    • Related Report
      2004 Annual Research Report
  • [Journal Article] A direct summand in a homotopy group of the mod 2 Moore space2004

    • Author(s)
      Arcadiy Skopenkov, Juno Mukai
    • Journal Title

      Kyushu J.Math. 58

      Pages: 203-209

    • NAID

      130000063085

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Lifting to the suspension of the real projective plane2004

    • Author(s)
      Tomohisa Inoue, Juno Mukai
    • Journal Title

      Jour.Fac, Sci.Shinshu Univ. 39

      Pages: 1-9

    • NAID

      110004594348

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] A direct summand in a homotopy group of the mod 2 Moore space2004

    • Author(s)
      Arkadiy Spopenkov, Juno Mukai
    • Journal Title

      Kyushu J.Math. 58-1

      Pages: 203-209

    • NAID

      130000063085

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Lifting to the suspension of the real projective plane2004

    • Author(s)
      T.Inoue, J.Mukai
    • Journal Title

      Jour.Fac, Sci.Shinshu Univ. 39

      Pages: 1-9

    • NAID

      110004594348

    • Related Report
      2004 Annual Research Report
  • [Journal Article] A note on the Hopf homomorphism of a Toda bracket and its application2003

    • Author(s)
      Tomohisa Inoue, Juno Mukai
    • Journal Title

      Hiroshima Math.J. 33

      Pages: 379-389

    • NAID

      110000016497

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] The suspension order of the real even dimensional projective space,2003

    • Author(s)
      Juno Mukai
    • Journal Title

      J.Math.Kyoto Univ 43

      Pages: 755-769

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] A note on the Hopf homomorphism of a Toda bracket and its application2003

    • Author(s)
      Tomohisa Inoue, Juno Mukai
    • Journal Title

      Hiroshima Math.J. 33-3

      Pages: 379-389

    • NAID

      110000016497

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] The suspension order of the real even dimensional projective space2003

    • Author(s)
      Juno Mukai
    • Journal Title

      J.Math.Kyoto Univ. 43-4

      Pages: 755-769

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Orders of Whitehead products of ι_n with α

    • Author(s)
      Juno Mukai
    • Journal Title

      Proceeding of the International conference on Homotopy Theory and related topics at Korea University in π^n_{n+k} (n>k+1, k<25) (to appear)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Publications] 井上朋久, 向井純夫: "A note on the Hopf homomorphism of a Toda bracket and its application"Hiroshima Math.J.. 33・3. 397-407 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] 向井純夫: "Suspension order of the real even dimensional projective space"J.Math.Kyoto Univ.. 43・4. 755-769 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] 向井純夫, Arkadiy Skopenkov: "A direct summand in a homotopy group of the mod 2 Moore space"Kyushu J.Math.. 58・1. 203-209 (2004)

    • Related Report
      2003 Annual Research Report
  • [Publications] 阿部孝順, 福井和彦: "On the structure of the group of Lipschitz homeomorphisms and its subgroups II"Jour.Math.Soc.Japan. 55・4. 947-956 (2003)

    • Related Report
      2003 Annual Research Report

URL: 

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

Information User Guide FAQ News Terms of Use Attribution of KAKENHI

Powered by NII kakenhi