Bounded cohomology and 3dimensional hyperbolic geometry
Project/Area Number  07640140 
Research Category 
GrantinAid for Scientific Research (C)

Section  一般 
Research Field 
Geometry

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

Project Fiscal Year 
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)

Keywords  bounded cohomology / huperbolic geometry / hyperbolic 3manifold / pseudonorm / zeronorm subspace / hyperbolic metric / 有界コホモロジー / 双曲幾何学 / 双曲3次元多様体 / 擬ノルム / 零ノルム空間 / 双曲計量 / 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.MatsumotoS.Morita (1985). Moreover, by using a similar argument, we construct examples of the nth 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 kth bounded cohomology H^k_ (X ; R) with alpha=0 is called the zeronorm subspace of H^k_ (X ; R) and denoted by N^k (X). In this research, we investigated the third zeronorm subspace N^3 (SIGMA). The head investigator constructed nontrivial 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 pseudoAnosov automorphism of SIGMA. As an application of this practical construction, it was shown that the dimension of Rvector 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 3simplices. Later, it was turned out that the notion is useful also in investigating nonzero degree maps between 3manifolds. In particular, if a nonzero degree map f : M*N from a closed 3manifold to a hyperbolic 3manifolds is given, one can define the structurc of a complex on M consisting of hyperbolic 3simplices by using the hyperbolic structure on N.By using microchip decompositions on such complexes, it was proved that the number ofhyperbolic 3manifolds admitting nonzero degree maps from a fixed M is finite.

Report
(5results)
Research Output
(24results)