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

Research on stable homotopy groups of finite complexes

Research Project

Project/Area Number 10640082
Research Category

Grant-in-Aid for Scientific Research (C)

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

Principal Investigator

SHIMURA Katsumi  Faculty of Science, KOCHI UNIVERSITY, Associate Professor, 理学部, 助教授 (30206247)

Co-Investigator(Kenkyū-buntansha) YAGITA Nobuaki  Ibaraki University Faculty of Education, Professor, 教育学部, 教授 (20130768)
YOSIMURA Zen-ichi  Nagoya Institute of Technology, Professor, 工学部, 教授 (70047330)
HEMMI Yutaka  Faculty of Science, KOCHI UNIVERSITY, Professor, 理学部, 教授 (70181477)
小林 貞一  高知大学, 理学部, 教授 (30033806)
Project Period (FY) 1998 – 1999
Project Status Completed (Fiscal Year 1999)
Budget Amount *help
¥3,300,000 (Direct Cost: ¥3,300,000)
Fiscal Year 1999: ¥1,500,000 (Direct Cost: ¥1,500,000)
Fiscal Year 1998: ¥1,800,000 (Direct Cost: ¥1,800,000)
Keywordshomotopy groups / spheres / finite complexes / spectra / Morava K-theories / Bousfield localization / Johnson-Wilson spectrum / Adams-Novikov spectral sequence / Adams-Novikov スペクトル系列 / Bousfield 局所化 / Jonson-Wilson スペクトラム / Morava K 理論
Research Abstract

In this research, we aimed two subjects. One is to make a more deep understanding of the Bousfield localization of finite complexes with respect to the Morava K-theories, and the other is to determine the homotopy groups πィイD2*ィエD2(LィイD22ィエD2M) of the Bousfield localized Moore spectrum LィイD22ィエD2M with respect to K(2).
For the first one, we obtain some information on the localization with respect to K(1) from the view point of KOィイD2*ィエD2-quasi equivalence. We understand that the K(n)ィイD2*ィエD2 homologies reflect many properties of the space itself by observing Lie groups and H-spaces. Yagita and his coauthors show that some properties obtained from the structure of Morava K(n)ィイD2*ィエD2-homology imply the similar properties obtained from the one of BPィイD2*ィエD2.-homology, whose converse is the standard philosophy when we study this kind of homotopy theory.
For the second, we determined the homotopy groups πィイD2*ィエD2(LィイD22ィエD2M) in the first year. In the second year we concentrate to determine the homotopy groups πィイD2*ィエD2(LィイD22ィエD2SィイD10ィエD1) at the prime 3 and succeeded. Note that πィイD2*ィエD2( LィイD22ィエD2SィイD10ィエD1) has already determined at a prime > 3. The results made Hopkins set up the chromatic splitting conjecture, and the result at the prime 3 gives a counter example of it. This shows that the structure of the homotopy groups πィイD2*ィエD2(LィイD22ィエD2SィイD10ィエD1) of the localized sphere spectrum at the prime 3 is different from the one at a prime > 3. At the prime 2, the structure seems very complex and we determine only the EィイD22ィエD2-term of the Adams-Novikov spectral sequence converging to πィイD2*ィエD2(LィイD22ィエD2M).

Report

(3 results)
  • 1999 Annual Research Report   Final Research Report Summary
  • 1998 Annual Research Report
  • Research Products

    (24 results)

All Other

All Publications (24 results)

  • [Publications] K.Shimomura: "The Adams-Novikov E_2-term for computing π_*(L_2V(0)) at the prime 2"Topology and its Applications. 96. 133-152 (1999)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] K.Shimomura: "The homotopy groups of the L_2-localized mod 3 Moore spectrum"J.Math.Soc.of Japan. 52. 65-90 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] K.Shimomura: "On the action of β_1 in the stable homotopy of spheres at the prime 3"Hiroshima Math.J.. (印刷中).

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] J.Lin and Y.Hemmi: "Odd Generators of the mod 3 Cohomology of FiniteH-spaces"J.Math.Kyoto Univ. (印刷中).

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] Z.Yosimura: "Quasi KO_*-types of CW-spectra X with KU_*X=^^〜Free【symmetry】 Z/2^m"Osaka J.Math.. 36. 1-19 (1999)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] D.Ravenel,S.Wilson and N.Yagita: "Brown-Peterson cohomology from Morava K-theory"K-theory. 15. 147-199 (1998)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] K. Shimomura: "The Adams-Novikov EィイD22ィエD2-term for computing πィイD2*ィエD2(LィイD22ィエD2V(0)) at the prime 2"Topology and its Applications. Vol. 96. 133-152 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] K. Shimomura: "The homotopy groups of the LィイD22ィエD2-localized mod 3 Moore spectrum"J. Math. Soc. of Japan. Vol. 52. 65-90 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] K. Shimomura: "On the action of βィイD21ィエD2 in the stable homotopy of spheres at the rime 3"Hiroshima Math. J.. in press.

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] J. Lin and Y. Hemmi: "Odd Generators of the mod 3 Cohomology of Finite H-spaces"J. Math. Kyoto Univ.. in press.

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] Z. Yosimura: "Quasi KOィイD2*ィエD2-types of CW-spectra X with KUィイD2*ィエD2X【similar or equal】Free【symmetry】 Z/2ィイD1mィエD1"Osaka J. Math. Vol. 36. 1-19 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] D. Ravenel, S. Wilson and N. Yagita: "Brown-Peterson cohomology from Morava K-theory"K-Theory. Vol. 15. 147-199 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] K.Shimomura: "The Adams-Novikov E_2-term for computing π_*(L_2V(0)) at the prime 2"Topology and its Applications. 96. 133-152 (1999)

    • Related Report
      1999 Annual Research Report
  • [Publications] K.Shimomura: "The homotopy groups of the L_2-localized mod 3 Moore spectrum"J.Math.Soc.of Japan. 52. 65-90 (2000)

    • Related Report
      1999 Annual Research Report
  • [Publications] K.Shimomura: "On the action of β_1 in the stable homotopy of spheres at the prime 3"Hiroshima Math.J.. (印刷中).

    • Related Report
      1999 Annual Research Report
  • [Publications] J.Lin and Y.Hemmi: "Odd Generators of the mod 3 Cohomology of FiniteH-spaces"J.Math.Kyoto Univ. (印刷中).

    • Related Report
      1999 Annual Research Report
  • [Publications] Z.Yosimura: "Quaki KO_*-types of CW-spectra X with KU_* X =^^〜Free 【cross product】 Z/2^m"Osaka J.Math.. 36. 1-19 (1999)

    • Related Report
      1999 Annual Research Report
  • [Publications] D.Ravenel,S.Wilson and N.Yagita: "Brown-Peterson cohomology from Morava K-theory"K-Theory. 15. 147-199 (1998)

    • Related Report
      1999 Annual Research Report
  • [Publications] K.Shimomura: "The homotopy groups of an L_2-localized type one finite spectrum at the prime 2" Hiroshima Mathematical Journal. 28. 113-127 (1998)

    • Related Report
      1998 Annual Research Report
  • [Publications] K.Shimomura: "The Adams-Novikov E_2-term for computing π_*(L_2V(0)) at the prime 2" Topology and its Applications. 印刷中. 印刷中 (1999)

    • Related Report
      1998 Annual Research Report
  • [Publications] K.Shimomura: "The homotopy groups of the L_2-localized mod 3 Moore spectrum" Journal of the Mathematical Society of Japan. 印刷中. 印刷中 (1999)

    • Related Report
      1998 Annual Research Report
  • [Publications] T.Kobayashi: "Immersions of lens spaces in complex projective spaces" Memoirs of the Faculty of Science Kochi Univ.Ser.A. 20. 57-65 (1999)

    • Related Report
      1998 Annual Research Report
  • [Publications] Y.Nishimura and Z.Yosimura: "The quasi KO^*-types of weighted mod 4 lens spaces" Osaka Journal of Mathematics. 35. 895-914 (1998)

    • Related Report
      1998 Annual Research Report
  • [Publications] N.Yagita: "Localization of the spectral sequence converging to the cohomology ofan extra special p-group for odd prime" Osaka Journal of Mathematics. 35. 83-116 (1998)

    • Related Report
      1998 Annual Research Report

URL: 

Published: 1998-04-01   Modified: 2016-04-21  

Information User Guide FAQ News Terms of Use Attribution of KAKENHI

Powered by NII kakenhi