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Research on higher dimensional polyhedra and operads

Research Project

Project/Area Number 13640080
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field Geometry
Research InstitutionKOCHI UNIVERSITY

Principal Investigator

HEMMI Yutaka  Faculty of Science, Professor, 理学部, 教授 (70181477)

Co-Investigator(Kenkyū-buntansha) MORISUGI Kaoru  Wakayama University, Faculty of Education, Professor, 教育学部, 教授 (00031807)
KOMATSU Kazushi  Faculty of Science, Associate Professor, 理学部, 助教授 (00253336)
SHIMOMURA Katsumi  Faculty of Science, Professor, 理学部, 教授 (30206247)
KAWAMOTO Yusuke  National Defense Academy in Japan, School of Liberal Arts and General Education, Research Associate, 総合教育学群, 助手
TSUKIYAMA Kouzou  Shimane University, Faculty of Education, Professor, 教育学部, 教授 (20093651)
Project Period (FY) 2001 – 2002
Project Status Completed (Fiscal Year 2002)
Budget Amount *help
¥3,400,000 (Direct Cost: ¥3,400,000)
Fiscal Year 2002: ¥1,700,000 (Direct Cost: ¥1,700,000)
Fiscal Year 2001: ¥1,700,000 (Direct Cost: ¥1,700,000)
KeywordsHopf space / H-map / A_n-space / A_n-map / projective space / associahedron / homotopy group / Whitehead product / ポップ空間 / quasi p-regular / unstable higher order operation / 高位ヒモトピー可換性 / 準Cp-空間 / mod p torus theorem / 球面のホモトピー群 / タイリング空間
Research Abstract

The summary of reserch results is as follows.
1. We study conditions for mod p finite H-spaces to be quasi p-regular, and we give a generalization to the theorem of Kumpel on p-regulraity. Our result includes results by Harper, McClearly and Wilkerson.
2. We define unstable p-th order mod p cohomology operations for odd primes p. Then, by using the operations we show that there are no H-spaces with certain cohomology.
3. Williams gave a definition of higher order homotopy commutativity of the multiplication on associative H-spaces. We generalize his definition to higher homotopy assocative H-spaces. Then, by using this definition we give the mod p torus theorem for H-spaces with finitely generated cohomology.
4. We show that there is no even degree generators except for degree eight and twenty in the mod 3 cohomology of finite H-spaces. We also determine the cohomology strucre of them as algebras.
5. We consider the inbedding of an An-space X to its loop space of the projective n-1-space ΩP_<n-1>X. Then we study relations between an A_<n-1>-cohomology class x∈H^*(X;F) and its lift x^^^∈H^*(ΩP_<n-1>X;F), where F is a field.
6. It is well known that if a map f: X → Y between H-spaces is an H-map then the map induces a map between their projective planes. Moreover, it is also known that if Y is homotopy associative then the converse holds. We show that the assumption of homotopy associativity is necessary in showing the converse. We also generalize the result to the case of A_n-spaces.
7. It is proved that the moduli space of a punctured Rieman sphere has cellular decomposition. We consider a decomposition of the associahedron by using a homotopy theoretical property, and then by using the decomposition we construct a handle decomposition of the moduli space which is a dual decomposition of the original decomposition.

Report

(3 results)
  • 2002 Annual Research Report   Final Research Report Summary
  • 2001 Annual Research Report
  • Research Products

    (36 results)

All Other

All Publications (36 results)

  • [Publications] Y.Hemmi: "Mod p decompositions of mod p finite H-spaces"Mem. Fac. Sci. Kochi Univ. Ser. A (Math.). 22. 59-65 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] Y.Hemmi, J.Lin: "Cohomology Rings of 3-local Finite H-spaces"J. Pure Appl. Algebra. 167. 1-14 (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] Y.Hemmi: "Unstable p-th order operation and H-spaces"Contemporary Math.. 293. 75-88 (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] Y.Hemmi: "H-spaces as direct product factors of loop spaces"Topology and its Applications. (印刷中).

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] K.Morisugi, J.Mukai: "The Whitehead square of a lift of the Hopf map to a mod 2 Moore space"J. Math. Kyoto Univ.. 42. 331-336 (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] Y.Kawamoto: "Higher homotopy commutativity of H-spaces with finitely generated co-homology"Pacific J. Math.. 204. 145-161 (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] Y. Hemmi: "Mod p decompositions of mod p finite H-spaces"Mem. Fac. Sci. Kochi Univ. Ser. A (Math.). Vol.22. 59-65 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] Y. Hemmi and J. Lin: "Cohomology Rings of 3-local Finite H-spaces"J. Pure Appl. Algebra. Vol. 167. 1-14 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] Y. Hemmi: "Unstable p-th order operation and H-spaces"Contemporary Math.. Vol. 293. 75-88 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] Y. Hemmi: "H-spaces as direct product factors of loop spaces"Topology and its Applications. (to appear).

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] K. Morisugi and J. Mukai: "The Whitehead square of a lift of the Hopf map to a mod 2 Moore space"J. Math. Kyoto Univ.. Vol. 42. 331-336 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] Y. Kawamoto: "Higher homotopy commutativity of H-spaces with finitely generated cohomology"Pacific J. Math.. Vol. 204. 145-161 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] Y.Kamiya, K.Shimomura: "The Picard group on E(2)-localized category"Contemporary Math.. (印刷中).

    • Related Report
      2002 Annual Research Report
  • [Publications] Y.Hemmi: "H-spaces as direct product factors of loop spaces"Topology and its Applications. (印刷中).

    • Related Report
      2002 Annual Research Report
  • [Publications] Y.Hemmi: "Unstable p-th order operation and H-spaces"Contemporary Math.. 293. 75-88 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] Y.Hemmi, J.Lim: "Cohomology Rings of 3-local Finite H-spaces"J. Pure Appl. Algebra. 167. 1-14 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] Y.Kamiya, K.Shimomura: "The Picard group on E(2)-localized category"Contemporary Math.. (印刷中).

    • Related Report
      2002 Annual Research Report
  • [Publications] T.Kobayashi, K.Komatsu: "Extendibility and stable extendibility of the square of the normal bundle associated to an immersion of the real projective space"Hiroshima Math. J.. 32. 371-378 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] K.Morisugi, J.Mukai: "The Whitehead square of a lift of the Hopf map to a mod 2 Moore space"J. Math. Kyoto Univ.. 42. 331-336 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] Y.Hemmi: "Mod p decompositions of mod p finite H-spaces"Mem. Fac. Sci. Kochi Univ. Ser. A (Math.). 22. 59-65 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] Y.Hemmi, J.Lin: "Cohomology Rings of 3-local Finite H-spaces"J. Pure Appl. Algebra. (印刷中).

    • Related Report
      2001 Annual Research Report
  • [Publications] Y.Hemmi: "Unstable p-th order operation and H-spaces"Contemporary Math.. (印刷中).

    • Related Report
      2001 Annual Research Report
  • [Publications] Y.Kamiya, K.Shimomura: "The homotopy groups π_*(L_2V(0)∧T(k))"Hiroshima Math. J.. 31. 391-408 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] K.Shimomura, X.Wang: "The homotopy groups π_*(L_2S^0) at the prime 3"Topology. (印刷中).

    • Related Report
      2001 Annual Research Report
  • [Publications] K.Shimomura: "The homotopy groups π_*(L_nT(m)∧V(n-2))"Contemporary Math.. (印刷中).

    • Related Report
      2001 Annual Research Report
  • [Publications] K.Shimomura, X.Wang: "The Adams-Novikov E_2-term for π_*(L_2S^0) at the prime 2"Math. Z.. (印刷中).

    • Related Report
      2001 Annual Research Report
  • [Publications] Y.Kamiya, K.Shimomura: "E_*-homology spheres for a connective spectrum E"Contemporary Math.. (印刷中).

    • Related Report
      2001 Annual Research Report
  • [Publications] T.Kobayashi, K.Komatsu: "Extendibility and stable extendibility of vector bundles over real projective spaces"Hiroshima Math. J.. 31. 99-106 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] K.Komatsu: "Periods of cut-and-project tiling spaces from root lattices"Hiroshima Math. J.. 31. 435-438 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] S.Ishiguro, K.Komatsu: "Symmetries of cut-and-project tilings corresponding to exceptional folding"Mem. Fac. Sci. Koch Univ. (Math.). 22(印刷中). (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] T.Kobayashi, K.Komatsu: "Extendibility and stable extendibility of the square of the normal bundle associated to an immersion of the real projective space"Hiroshima Math. J.. (印刷中).

    • Related Report
      2001 Annual Research Report
  • [Publications] K.Morisugi: "Composition sturctnre of the self maps of SU(3) and Sp(2)"Contemporary Math.. 274. 233-240 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] K.Morisugi: "The Whitehead square of a lift of the Hopf map to a mod 2 Moore space"J. Math. Kyoto Univ.. (印刷中).

    • Related Report
      2001 Annual Research Report
  • [Publications] H.Shiga, K.Tsukiyama, T.Yamaguchi: "Principal S^1-bundles and forgetful maps"Contempprary Math.. 274. 293-297 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] Y.Kawamota, J.Lin: "Homotopy commutativity of H-spaces with finitely generated cohomology"Trans. Amer. Math. Soc.. 353. 4481-4496 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] Y.Kawamota: "Higher homotopy commutativity of H-spaces with finitely generated cohomology"Pacific J. Math.. (印刷中).

    • Related Report
      2001 Annual Research Report

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Published: 2001-04-01   Modified: 2016-04-21  

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