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Homotoy Theory of Classifying Spaces

Research Project

Project/Area Number 09640138
Research Category

Grant-in-Aid for Scientific Research (C)

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

Principal Investigator

ISHIGURO Kenshi  Fukuoka Univ., Fac.of Science, Assoc.Prof., 理学部, 助教授 (00268971)

Co-Investigator(Kenkyū-buntansha) AKITA Toshiyuki  Fukuoka Univ., Fac.of Science, Assistant, 理学部, 助手 (30279252)
KUROSE Takashi  Fukuoka Univ., Fac.of Science, Asso.Prof., 理学部, 助教授 (30215107)
ODA Nobuyuki  Fukuoka Univ., Fac.of Science, Professor, 理学部, 教授 (80112283)
Project Period (FY) 1997 – 1998
Project Status Completed (Fiscal Year 1998)
Budget Amount *help
¥2,300,000 (Direct Cost: ¥2,300,000)
Fiscal Year 1998: ¥1,000,000 (Direct Cost: ¥1,000,000)
Fiscal Year 1997: ¥1,300,000 (Direct Cost: ¥1,300,000)
KeywordsClassifying spaces / Homotopy / p-compact groups / Compact Lie groups / Pairing / Admissible maps / Whitehead積 / Coxter群
Research Abstract

The research on the classifying spaces of compact Lie groups has been one of the major area in Homotoy Theory. Our results obtained during 1997 through 1998 are basically concerned with maps between classifying spaces and their applications. Dwyer-Wilkerson defined ap-compact group and studied its properties. The purely homotopy theoritic object appears to be a good generalization of a compact Lie group. A p-compact group has rich structure, such as a maximal torus, a Weyl group, etc. A note wrtten by Moeller in the AMS Bulletin summarizes their work. Further progress on the homotopy theory of the classifying spaces of p-compact groups are being made.
We state here our main results. First, we consider the maps of p-compact groups of the form BX * BY*BZ.The main theorem shows that if the restriction map on BY is a weak epimorphism, then the restriction on BX should factor through the classifying spaces of the center of the p-compact group Z.Next, for G =S^3 * .. * S^3, let X be a genus of BG.We investigate the monoid of rational equivalences of X, denoted by epsilon(X). It is shown that a submonoid of epsilon_0(X), denoted by delta_00(X), determines the decomposability of the space X.We also show converses to some known results for the classifying spaces of p-toral groups or p-compact toral group. Suppose G is a compact Lie group. The following results are obtained. If there is a positive integer k such that the n-th homotopy groups of the p-completion of BG are zero for all n <greater than or equal> k then the loop space of this space is a p-compact toral group. If the canonical map Rep(G, K)*[BG, BK] is bijective for any compact connected Lie group K, then G is a p-toral group. in addition, our work containesa research on the conditions of a compact Lie group that its loop space of the p-completed classifying space be a p-compact group.

Report

(3 results)
  • 1998 Annual Research Report   Final Research Report Summary
  • 1997 Annual Research Report
  • Research Products

    (37 results)

All Other

All Publications (37 results)

  • [Publications] K.Ishiguuro and Maki Tokuzawa: "Unitary groups and pairings of classifying spaces" Hiroshima Mathematical Journal. to appear.

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] K.Ishiguro, Jesper Moller and Dietrich Notbohm: "Rational self-equivaences of spaces in the genus of a product of quaternionic projective spaces" Journal of the Math.Soc.of Japan. 51(1). 45-61 (1999)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] K.Ishiguro: "Pairings of p-compact groups and H-structures on the classifying spaces of finite loop spaces" Publications of Research Institute for Mathematical Sciences. 34(6). 567-578 (1998)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] K.Ishiguro: "Projective unitary groups and K-theory of classifying s paces" Fukuoka University Science Reports. 28(1). 1-6 (1998)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] K.Ishiguro: "Classifying spaces and finite loop spaces (expository)" The proceedings of Workshops in Pure Mathematics 1996, Korean Academic Council. 1-13 (1997)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] N.Oda: "A generalization of the Whitehead product" Math.J.Okayama Univ.to appear.

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] N.Oda and T.Shimizu: "Smash product with Γ-Whitehead product and its dual." Far East J.Math.Sci.6(5). 707-726 (1998)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] N.Oda and T.Shimizu: "Transformation between pairings" Fukuoka Univ.Sci.Reports. 28(2). 53-64 (1998)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] T.Akita: "Cohomology and Euler characteristics of Coxeter groups" Science Bulletin of Josai University, Special Issue. 2. 3-16 (1997)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] T.Akita: "On the Euler characteristic of the orbit space of a proper Γ-complex" Osaka Journal of Mathematics. to appear.

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] T.Akita: "On the homology of Torelli groups and Torelli spaces" Osaka Journal of Mathematics, Proc.Japan Acad.Ser.A. to appear.

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] T.Akita: "On the cohomology of Coxeter groups and their finite parabo lic subgroups. II" Proc.Sympos.Pure Math.63. 1-5 (1998)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] T.Akita: "Euler characteristics of Coxeter groups, PL-triangulation of closed manifolds, and cohomology of subgroups of Artin groups" Journal of the London Mathematical Society. to appear.

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] K.Ishiguro and Maki Tokuzawa: "Unitary groups and pairings of classifying spaces" Hiroshima Mathematical Journal. (to appear).

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] K.Ishiguro, Jesper Moller and Dietrich Notbohm: "Rational selfequivalences of spaces in the genus of a product of quaternionic projective spaces" Journal of the Math.Soc.of Japan. vol 51(1). 45-61 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] K.Ishiguro: "Pairings of p-compact groups and H-structures on the classifying spaces of finite loop spaces" Publications of Research Institute for Mathematical Sciences. vol 34(6). 567-578 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] K.Ishiguro: "Projective unitary groups and K-theory of classifying spaces" Fukuoka University Science Reports. vol 28(1). 1-6 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] K.Ishiguro: "Classifying spaces and finite loop spaces(expository)" The proceedings of Workshops in Pure Mathematics 1996, Korean Academic Council. 1-13 (1997)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] N,Oda: "A generalization of the Whitehead product" Math.J.Okayama Univ.(to appear).

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] N,Oda and T.Shimizu: "Smash product with GAMMA-Whitehead product and its dual" Far East J.Math.Sci.vol 6(5). 707-726 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] N,Oda and T.Shimizu: "Transformation between pairings" Fukuoka Univ.Sci.Reports. vol 28(2). 53-64 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] T.Akita: "Cohomology and Euler characteristics of Coxeter groups" Science Bulletin of Josai University. Special Issue No.2. 3-16 (1997)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] T.Akita: "On the Euler characteristic of the orbit space of a proper GAMMA-complex" Osaka Journal of Mathematics. (to appear).

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] T.Akita: "On the homology of Torelli groups and Torelli spaces" Proc.Japan Acad.Ser.A.(to appear).

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] T.Akita: "On the cohomology of Coxeter groups and their finite parabolic subgroups.II" Group representations : cohomology, group actions and topology (Seattle, WA,1996), 1-5, Proc.Sympos.Pure Math., 63, Amer.Math.Soc., Providence, RI. (1998)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] T.Akita: "Euler characteristics of Coxeter groups, PL-triangulations of closed manifolds, and cohomology of subgroups of Artin groups" Journal of the London Mathematical Society. (to appear).

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] K. Ishiguro and Maki Tokuzawa: "Unitary groups and pairings of classif ying spaces" Hiroshima Mathematical Journal. to appear

    • Related Report
      1998 Annual Research Report
  • [Publications] K. Ishiguro, Jesper Moller and Dietrich Notbohm: "Rational self-equ ivalences of spaces in the genus of a product of quaternionic projec-tive spaces" Journal of the Math. Soc. of Japan. 51(1). 45-61 (1999)

    • Related Report
      1998 Annual Research Report
  • [Publications] K.Ishiguro: "Pairings of p-compact groups and H-structures on the classifying spaces of finite loop spaces" Publications of Research Institute for Mathematical Sciences. 34(6). 567-578 (1998)

    • Related Report
      1998 Annual Research Report
  • [Publications] K.Ishiguro: "Projective unitary groups and K-theory of classifying s paces" Fukuoka University Science Reports. 28(1). 1-6 (1998)

    • Related Report
      1998 Annual Research Report
  • [Publications] N.Oda and T.Shimizu: "Smash product with T-Whitehead product and its dual" Far East J.Math.Sci. 6(5). 707-726 (1998)

    • Related Report
      1998 Annual Research Report
  • [Publications] T.Akita: "On the cohomology of Coxeter groups and their finite parabo lic subgroups. II" Proc.Sympos.Pure Math.63. 1-5 (1998)

    • Related Report
      1998 Annual Research Report
  • [Publications] K.Ishiguro: "Classifying spaces and finite loop spaces(expository)" The proceedings of Workshops in Pure Mathematics 1996,Korean Academic Council. 1-13 (1997)

    • Related Report
      1997 Annual Research Report
  • [Publications] K.Ishiguro: "Projective unitary groups and K-theory of classifying spaces" Fukuoka University Science Reports. (印刷中). (1998)

    • Related Report
      1997 Annual Research Report
  • [Publications] N.Oda: "A generalization of the Whitehead product" Math.J.Okayama Univ.(印刷中). (1998)

    • Related Report
      1997 Annual Research Report
  • [Publications] T.Akita: "Cohomology and Euler characteristics of Coxeter groups" Science Bulletin of Josai University,Special Issue. 2. 3-16 (1997)

    • Related Report
      1997 Annual Research Report
  • [Publications] T.Akita: "On the cohomology of Coxeter groups and their finite parabolic subgroups II" Proceedings of Symposia in Pure Mathematics. 63. 1-5 (1998)

    • Related Report
      1997 Annual Research Report

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Published: 1997-04-01   Modified: 2016-04-21  

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