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Homotopy Methods in a Generalization of Lie Group Theory

Research Project

Project/Area Number 14540096
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) TORII Takeshi  Fukuoka University, Faculty of Science, Assistant, 理学部, 助手 (30341407)
KUROSE Takashi  Fukuoka University, Faculty of Science, Assistant Professor, 理学部, 助教授 (30215107)
ODA Nobuyuki  Fukuoka University, Faculty of Science, Professor, 理学部, 教授 (80112283)
Project Period (FY) 2002 – 2003
Project Status Completed (Fiscal Year 2003)
Budget Amount *help
¥2,800,000 (Direct Cost: ¥2,800,000)
Fiscal Year 2003: ¥1,400,000 (Direct Cost: ¥1,400,000)
Fiscal Year 2002: ¥1,400,000 (Direct Cost: ¥1,400,000)
KeywordsClassifying spaces / Homotopy / p-compact groups / Compact Lie groups / Homotopy fixed point set / Localizations of spaces / 空間の局所化 / 表現論 / Invariant Theory / コホモロジー
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 2002 through 2003 are basically concerned with maps between classifying spaces and their applications Dwyer-Wilkerson defined a 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 generalize 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 : For a finite p-group Π, suppose a Π -space X is F_p-complete and cd_p(X) is finite. Then X^πis an empty set if and only if the homotopy fixed point set X^<hπ> is empty. As an application, we discuss extension problems considering actions on homogeneous spaces of p-compact groups. 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. A necessary and sufficient condition that BG be p-compact toral for all primes has been obtained. We ask if BH is p-compact for a set of primes when H is a subgroup of a simple Lie group G and obtaine certain results.

Report

(3 results)
  • 2003 Annual Research Report   Final Research Report Summary
  • 2002 Annual Research Report
  • Research Products

    (35 results)

All Other

All Publications (35 results)

  • [Publications] K.Ishiguro: "Classifying spaces of compact Lie groups that are p-compact for allprime numbers"Preprint.

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] K.Ishiguro, Hang-Sook Lee: "Homotopy fixed point sets and actions on homogeneous spaces of p-compact groups"Preprint.

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] K.A.Hardie, H.J.Marcum, N.Oda: "The Whitehead products and powers in W-topology"Proc.Amer.Math.Soc.. 131. 941-951 (2003)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] F.Nakaoka, N.Oda: "Some porperties of maximal open sets"Int.J.Math.and Math.Sci.. 1331-1340 (2003)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] N.Oda: "On decomposition theorems"Preprint. (2003)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] T.Kurose: "Conformal-projective geometry of statistical manifolds"The Interdisciplinary Information Sciecnces. 8. 89-100 (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] Hitoshi Furuhata, T.kurose: "Self-dual centroaffine surfaces of codimension two with constant affine mean curvature"Bulletin of the Belgian Mathematical Society. Simon Stevin. 9. 573-583 (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] Kazunori Nakamoto, T.Torii: "Topology of the moduli of representations with Borel mold"Pacific J.Math.. 213. 365-387 (2004)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] T.Torii: "On degeneration of one-dimensional formal group laws and applications to stable homotopy theory"Amer.J.Math.. 125. 1037-1077 (2003)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] T.Torii: "The geometric fixed point spectrum of (Z/p)^k Borel cohomology for En and its completion"Recent progress in homotopy theory (Baltimore, MD, 2000) Contemp.Math.. 293. 343-369 (2003)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] T.Torii: "On relations between 1-lines of Adams-Novikov spectral sequences modulo invariant prime ideals"Preprint.

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] T.Torii: "Equivariance of generalized Chern characters"Preprint.

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] Kazunori Nakamoto, T.Torii: "Preprint"Algebraic vector bundles on SL(3,C).

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] K.Ishiguro: "Classifying spaces of compact Lie groups that are p-compact for allprime numbers"(Preprint).

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] K.Ishiguro, Hyang-Sook Lee: "Homotopy fixed point sets and actions on homogeneous spaces of p-compact groups"(Preprint).

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] K.A.Hardie, H.J.Marcum, N.Oda: "The Whitehead products andpowers in W-topology"Proc.Amer.Math.Soc.. 131. 941-951 (2003)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] F.Nakaoka, N.Oda: "Some properties of maximal open sets"Int.J.Math. and Math.Sci.. 1331-1340 (2003)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] N.Oda: "On decomposition theorems"(Preprint).

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] T.Kurose: "Conformal-projective geometry of statistical"The Interdisciplinary Information Sciences. 8. 89-100 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] Hitoshi Furuhata, T.kurose: "Self-dual centroaffine surfaces of codimension two with constant affine mean curvature"Bulletin of the Belgian Mathematical Society. Simon Stevin. 9. 573-587 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] Kazunori Nakamoto, T.Torii: "Topology of the moduli of representations with Borel mold"Pacific J.Math.. 213 no.2. 365-387 (2004)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] T.Torii: "On degeneration of one-dimensional formal group laws and applications to stable homotopy theory"Amer.J.Math.. 125 no.5. 1037-1077 (2003)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] T.Torii: "The geometric fixed point spectrum of (Z/p)^k cohomology for En and its completion"Contemp.Math.,293,Amer.Math.Soc., Borel Providence, RI. 343-369 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] Kazunori Nakamoto, T.Torii: "Algebraic vector bundles on SL(3,C)"(Preprint).

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] T.Torii: "On relations between 1-lines of Adams-Novikov spectral sequences modulo invariant prime ideals."(Preprint).

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] T.Torii: "Equivariance of generalized Chern characters"(Preprint).

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] Kazunori Nakamoto, Takeshi Torii: "Topology of the module of representations with Borel mold"Pacific J.Math.. 213・2. 365-387 (2004)

    • Related Report
      2003 Annual Research Report
  • [Publications] F.Nakaoka, N.Oda: "Some porperties of maximal open sets"Int.J.Math. and Math.Sci.. 21. 1331-1340 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] Takeshi Torii: "On degeneration of one-dimensional formal group laws and applications to stable homotopy theory"Amer.J.Math. 125(2003). 125・5. 1037-1077 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] K.A.Hardie, H.J.Marcum, N.Oda: "The Whitehead products and powers in W-topology"Proc.Amer.Math.Soc.. 131. 941-951 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] K.A.Hardie, H.J.Marcum, N.Oda: "The Whitehead products and powers in W-topology"Proc. Amer. Math. Soc.. 131・3. 941-951 (2003)

    • Related Report
      2002 Annual Research Report
  • [Publications] F.Nakaoka, N.Oda: "Some porperties of maximal open sets"International Journal of Mathematics and Mathematical Sciences. (to appear).

    • Related Report
      2002 Annual Research Report
  • [Publications] Takashi Kurose: "Conformal-projective geometry of statistical manifolds"The Interdisciplinary Information Sciecnces. 8. 89-100 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] Hitoshi Furuhata, Takashi Kurose: "Self-dual centroaffine surfaces of codimension two with constant affine mean curvature"Bulletin of the Belgian Mathematical Society-Simon Stevin. (to appear).

    • Related Report
      2002 Annual Research Report
  • [Publications] Takeshi Torii: "The geometric fixed point spectrum of (Z/p)^k Borel cohomology for E_n and its completion. Recent progress in homotopy theory (Baltimore, MD,2000)"Contemp. Math., Amer. Math. Soc., Providence, RI, 2002. 293. 343-369 (2002)

    • Related Report
      2002 Annual Research Report

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Published: 2002-04-01   Modified: 2016-04-21  

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