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Bounded cohomology and 3-dimensional hyperbolic geometry

Research Project

Project/Area Number 07640140
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field Geometry
Research InstitutionTokyo Denki University

Principal Investigator

SOMA Teruhiko  Tokyo Denki University College of Science and Engineering, Professor, 理工学部, 教授 (50154688)

Project Period (FY) 1995 – 1997
Project Status Completed (Fiscal Year 1997)
Budget Amount *help
¥1,900,000 (Direct Cost: ¥1,900,000)
Fiscal Year 1997: ¥400,000 (Direct Cost: ¥400,000)
Fiscal Year 1996: ¥800,000 (Direct Cost: ¥800,000)
Fiscal Year 1995: ¥700,000 (Direct Cost: ¥700,000)
Keywordsbounded cohomology / huperbolic geometry / hyperbolic 3-manifold / pseudonorm / zeronorm subspace / hyperbolic metric / 双曲3-単体 / マイクロチップ分解 / 双曲的計量 / ユークリッド的計量 / 擬Anosov自己同型 / クライン群 / 双曲多様体 / 基本コホモロジー類 / 双曲構造
Research Abstract

Let H^3_ (SIGMA ; R) be the third bounded cohomology of a closed, orientable surface SIGMA of genus g>1. The head investigator proved that the pseudonorm ||・|| on H^3_ (SIGMA ; R) is not a norm by relying on the results in S.Matsumoto-S.Morita (1985). Moreover, by using a similar argument, we construct examples of the n-th bounded cohomology whose pseudonorm is not a norm for any n <greater than or equal> 5. They are the first examples showing that there exist bounded cohomologies without norm.
For a topological space X,the subspace consisting of elements alpha of the k-th bounded cohomology H^k_ (X ; R) with ||alpha||=0 is called the zero-norm subspace of H^k_ (X ; R) and denoted by N^k (X). In this research, we investigated the third zero-norm subspace N^3 (SIGMA). The head investigator constructed non-trivial elements of N^3 (SIGMA) practically by using both a hyperbolic metric and a singular euclidean metric on SIGMA*R,where the euclidean metric is defined by using a measured lamination associated to a pseudo-Anosov automorphism of SIGMA. As an application of this practical construction, it was shown that the dimension of R-vector space N^3 (SIGMA) is the cardinality of continuum.
Throughout the research of bounded cohomology, the head investigator obtained the notion of microchip decompositions on complexes consisting of hyperbolic 3-simplices. Later, it was turned out that the notion is useful also in investigating non-zero degree maps between 3-manifolds. In particular, if a non-zero degree map f : M*N from a closed 3-manifold to a hyperbolic 3-manifolds is given, one can define the structurc of a complex on M consisting of hyperbolic 3-simplices by using the hyperbolic structure on N.By using microchip decompositions on such complexes, it was proved that the number ofhyperbolic 3-manifolds admitting non-zero degree maps from a fixed M is finite.

Report

(4 results)
  • 1997 Annual Research Report   Final Research Report Summary
  • 1996 Annual Research Report
  • 1995 Annual Research Report
  • Research Products

    (24 results)

All Other

All Publications (24 results)

  • [Publications] Teruhiko Soma: "Bounded cohomology of closed surfaces" Topology. 36. 1221-1246 (1997)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1997 Final Research Report Summary
  • [Publications] Teruhiko Soma: "Bounded cohomology and topologically tame Kleinian groups" Duke Math.J.88. 357-370 (1997)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1997 Final Research Report Summary
  • [Publications] Teruhiko Soma: "The zero-norm subspace of bounded cohomology" Comment.Math.Helv.72. 582-592 (1997)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1997 Final Research Report Summary
  • [Publications] Michihiko Fujii, Teruhiko Soma: "Totally geodesic boundaries are dense in the moduli space" J.Math.Soc.Japan. 49. 589-601 (1997)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1997 Final Research Report Summary
  • [Publications] Teruhiko Soma: "Existence of non-Banach bounded cohomology" Topology. 37. 179-193 (1998)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1997 Final Research Report Summary
  • [Publications] Teruhiko Soma: "Spatial-graph isotopy and the rearrangement theorem" Osaka J.Math.(印刷中).

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1997 Final Research Report Summary
  • [Publications] Teruhiko Soma: "Bounded cohomology of closed surfaces" Topology. 36. 1221-1246 (1997)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1997 Final Research Report Summary
  • [Publications] Teruhiko Soma: "Bounded cohomology and topologically tame Kleinian groups" Duke Math.J.88. 357-370 (1997)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1997 Final Research Report Summary
  • [Publications] Teruhiko Soma: "The zero-norm subspace of bounded cohomology" Comment.Math.Helv.72. 582-592 (1997)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1997 Final Research Report Summary
  • [Publications] Michihiko Fujii and Teruhiko Soma: "Totally geodesic boundaries are dense in the moduli space" J.Math.Soc.Japan. 49. 589-601 (1997)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1997 Final Research Report Summary
  • [Publications] Teruhiko Soma: "Existence of non-Banach Bounded cohomology" Topology. 37. 179-193 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1997 Final Research Report Summary
  • [Publications] Teruhiko Soma: "Spatial-graph isotopy and the rearrangement theorem" Osaka J.Math.(in press).

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1997 Final Research Report Summary
  • [Publications] Teruhiko Soma: "Bounded cohomology of closed surfaces" Topology. 36・6. 1221-1246 (1997)

    • Related Report
      1997 Annual Research Report
  • [Publications] Teruhiko Soma: "Bounded cohomology and topologically tame kleinlan groups" Duke Math.J.88・2. 357-370 (1997)

    • Related Report
      1997 Annual Research Report
  • [Publications] Teruhiko Soma: "The zero-norm subspace of bounded cohomology" Comment.Math.Helv.72・4. 582-592 (1997)

    • Related Report
      1997 Annual Research Report
  • [Publications] Michihiko Fujii, Teruhiko Soma: "Totally geodesic boundaries are dense in the moduli space" J.Math.Soc.Japan. 49・3. 589-601 (1997)

    • Related Report
      1997 Annual Research Report
  • [Publications] Teruhiko Soma: "Existence of non-Banach bounded cohomology" Topology. 37・1. 179-193 (1998)

    • Related Report
      1997 Annual Research Report
  • [Publications] Teruhiko Soma: "Spatial-graph isotopy and the vearrangement theorem" Osaka J.Math.(発表予定).

    • Related Report
      1997 Annual Research Report
  • [Publications] Teruhiko Soma: "Bounded cohomology of closed surfaces" Topology. (発表予定).

    • Related Report
      1996 Annual Research Report
  • [Publications] Teruhiko Soma: "Bounded cohomology and topologically tame Kleinian groups" Duke Math.J.(発表予定).

    • Related Report
      1996 Annual Research Report
  • [Publications] Teruhiko Soma: "Existence of non-Banach bounded cohomology" Topology. (発表予定).

    • Related Report
      1996 Annual Research Report
  • [Publications] Michihiko Fujii,Teruhiko Soma: "Totally geodesic boundaries are dense in the moduli space" J.Math.Soc.Japan. (発表予定).

    • Related Report
      1996 Annual Research Report
  • [Publications] Teruhiko Soma,: "Disk/band surfaces of spatial graphs" Tokyo J. Math.(発表予定).

    • Related Report
      1995 Annual Research Report
  • [Publications] Teruhiko Soma: "Spatial-graph isotopy for trivalent graphs and minimally Knotled embeddings." Topology and Appl.(発表予定).

    • Related Report
      1995 Annual Research Report

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Published: 1995-04-01   Modified: 2016-04-21  

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