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2005 Fiscal Year Final Research Report Summary

Classifying spaces of compact Lie groups topology of p-compact groups

Research Project

Project/Area Number 16540088
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 University, Faculty of Science, Professor, 理学部, 教授 (00268971)

Co-Investigator(Kenkyū-buntansha) ODA Nobuyuki  Fukuoka University, Faculty of Science, Professor, 理学部, 教授 (80112283)
KUROSE Takashi  Fukuoka University, Faculty of Science, Associate Professor, 理学部, 助教授 (30215107)
TORII Takeshi  Fukuoka University, Faculty of Science, Assistant, 理学部, 助手 (30341407)
Project Period (FY) 2004 – 2005
KeywordsClassifying spaces / Homotopy / p-compact groups / Compact Lie groups / Homotopy fixed point set / Invariant theory / Weyl群
Research Abstract

The research on the classifying spaces of compact Lie groups is one of the major area in Homotopy Theory. Our results obtained during 2004 through 2005 are basically concerned with maps between classifying spaces and their applications, as well as the invariant theory. Dwyer--Wilkerson introduced the notion of p-compact group and studied its properties. The purely homotopy theoretic 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 written by Moeller in the AMS Bulletin summarizes their work. Further progress on the homotopy theory of p-compact groups are being made.
We state here our main results. First, we considered a further generalization of a result of Dror Farjoun and Zabrodsky on the relationship between fixed point sets and homotopy fixed point sets, which is related to the generalized Sullivan Conjecture : Suppose a-space X with an action of a finite p-group is p-complete and the cohomological dimension is finite. Then the fixed point set is an empty set if and only if the homotopy fixed point set is empty. Next, we consider a problem on the conditions of a compact Lie group G that its loop space of the p-completed classifying space be a p-compact group for a set of primes. In particular, we discuss the classifying spaces BG that are p-compact for all primes when the groups are certain subgroups of simple Lie groups. Finally, we discuss the invariant theory and the cohomology of classifying spaces. The cohomology can be expressed as an invariant ring under the action of the Weyl group. All Weyl groups are reflection groups. We obtained some results on certain reflection groups.

  • Research Products

    (32 results)

All 2005 2004 Other

All Journal Article (32 results)

  • [Journal Article] Splitting off rational parts in homotopy types2005

    • Author(s)
      N.Iwase, N.Oda
    • Journal Title

      Top. and Appl. 153

      Pages: 133-140

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Maximal gamma-open sets2005

    • Author(s)
      F.Nakaoka, N.Oda
    • Journal Title

      Proceedings of the 10th Meetings on Topological Spaces Theory and its Applications

      Pages: 21-25

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] On relations between 1-lines of Adams-Novikov spectral sequences modulo invariant prime ideals2005

    • Author(s)
      T.Torii
    • Journal Title

      Topology and Its Applications 150

      Pages: 33-57

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Splitting off rational parts in homotopy types2005

    • Author(s)
      N.Iwase, N.Oda
    • Journal Title

      Top.and Appl. no.153

      Pages: 133-140

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] Maximal gamma-open sets2005

    • Author(s)
      F.Nakaoka, N.Oda
    • Journal Title

      Proceedings of the 10th Meetings on Topological Spaces, Theory and its Applications, Fukuoka University Seminar House

      Pages: 21-25

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] On relations between 1-lines of Adams-Novikov spectral sequences modulo invariant prime ideals2005

    • Author(s)
      T.Torii
    • Journal Title

      Topology and Its Applications vol 150(1)--(3)

      Pages: 33-57

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] Homotopy fixed point sets and actions on homogeneous spaces of p-compact groups2004

    • Author(s)
      K.Ishiguro, H.-S.Lee
    • Journal Title

      J.Korean Math. Soc. 41

      Pages: 1101-1114

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Triple brsckets and lax morphism categories2004

    • Author(s)
      K.A.Hardie, K.H.Kamps, H.J.Marcum, N.Oda
    • Journal Title

      Homotopy theory Appl. Categ. Structures 12

      Pages: 3-27

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] On decomposition theorems2004

    • Author(s)
      N.Oda
    • Journal Title

      Trends in Topology 1

      Pages: 38-56

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Minimal open sets and minimal closed sets2004

    • Author(s)
      F.Nakaoka, N.Oda
    • Journal Title

      Proceedings of the 9th Meetings on Topological Spaces Theory and its Applications

      Pages: 5-8

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Topology of the moduli of representations with Borel mold2004

    • Author(s)
      K.Nakamoto, T.Torii
    • Journal Title

      Pacific J. Math. 213(2)

      Pages: 365-387

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Homotopy fixed point sets and actions on homogeneous spaces of p-compact groups2004

    • Author(s)
      K.Ishiguro, Hyang-Sook Lee
    • Journal Title

      J.Korean Math.Soc. 41, no.6

      Pages: 1101-1114

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] Triple brackets and lax morphism categories2004

    • Author(s)
      K.A.Hardie, K.H.Kamps, H.J.Marcum, N.Oda
    • Journal Title

      Homotopy theory Appl.Categ.Structures no.12

      Pages: 3-27

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] On decomposition theorems2004

    • Author(s)
      N.Oda
    • Journal Title

      Trends in Topology no.1

      Pages: 38-56

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] Minimal open sets and minimal closed sets2004

    • Author(s)
      F.Nakaoka, N.Oda
    • Journal Title

      Proceedings of the 9th Meeting on Topological Spaces Theory and its Applications, Fukuoka University Seminar House

      Pages: 5-8

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] Topology of the moduli of representations with Borel mold2004

    • Author(s)
      K.Nakamoto, T.Torii
    • Journal Title

      Pacific J.Math. vol 213(2)

      Pages: 365-387

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] Classifying spaces of compact Lie groups that are p-compact for all prime numbers.

    • Author(s)
      K.Ishiguro
    • Journal Title

      the Kinosaki proceedings (to appear in)

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Pairings of function spaces

    • Author(s)
      Y.Hirashima, N.Oda
    • Journal Title

      preprint

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Hadamard-type theorems for hypersurfaces in hyperbolic spaces

    • Author(s)
      L.~J.~Alias, T.Kurose, G.Solanes
    • Journal Title

      Differential Geometry and its Applications (to appear in)

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Milnor operations and the generalized Chern character

    • Author(s)
      T.Torii
    • Journal Title

      Geometry and Topology Monographs (to appear in)

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Algebraic vector bundles on SL(3, C)

    • Author(s)
      K.Nakamoto, T.Torii
    • Journal Title

      Rocky Mountain Journal of Mathematics (to appear in)

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Equivariance of generalized Chern characters

    • Author(s)
      T.Torii
    • Journal Title

      Preprint

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] On additive unstable operations for cohomology theories over P(n)

    • Author(s)
      T.Torii
    • Journal Title

      Preprint

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Rational homotopy type of the moduli of representations with Borel mold

    • Author(s)
      K.Nakamoto, T.Torii
    • Journal Title

      Preprint

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Classifying spaces of compact Lie groups that are $p$--compact for all prime numbers

    • Author(s)
      K.Ishiguro
    • Journal Title

      the Kinosaki proceeding (to appear)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] Pairings of function spaces

    • Author(s)
      Y.Hirashima, N.Oda
    • Journal Title

      (Preprint)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] Hadamard-type theorems for hypersurfaces in hyperbolic spaces

    • Author(s)
      L.~J.~Al'ias, T.Kurose, G.Solanes
    • Journal Title

      Differential Geometry and its Applications (to appear)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] Milnor operations and the generalized Chern character

    • Author(s)
      T.Torii
    • Journal Title

      Geometry and Topology Monographs (to appear)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] Algebraic vector bundles on SL(3,C)

    • Author(s)
      K.Nakamoto, T.Torii
    • Journal Title

      Rocky Mountain Journal of Mathematics (to appear)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] Equivariance of generalized Chern characters

    • Author(s)
      T.Torii
    • Journal Title

      (Preprint)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] On additive unstable operations for cohomology theories over P(n)

    • Author(s)
      T.Torii
    • Journal Title

      (Preprint)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] Rational homotopy type of the moduli of representations with Borel mold

    • Author(s)
      K.Nakamoto, T.Torii
    • Journal Title

      (Preprint)

    • Description
      「研究成果報告書概要(欧文)」より

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Published: 2007-12-13  

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