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Bounded cohomology and 3-dimensional hyperbolic geometry

有界上同调和 3 维双曲几何

基本信息

批准号：
07640140
负责人：
SOMA Teruhiko
金额：
$ 1.22万
依托单位：
Tokyo Denki University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
1995
资助国家：
日本
起止时间：
1995 至 1997
项目状态：
已结题

项目摘要

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.

设H^3_（SIGMA ; R）是亏格g>1的闭可定向曲面SIGMA的第三有界上同调。首席调查员证明，||·||在H ^3_（SIGMA ; R）上的σ_（σ_）不依赖于S.Matsumoto-S.Morita（1985））的结果。此外，通过使用类似的参数，我们构造的例子的n阶有界上同调的，其范数是不是一个规范的任何n <greater than or equal>5。它们是第一个证明存在无范数有界上同调的例子。对于拓扑空间X，由第k个有界上同调H^k_（X ; R）的元素α组成的子空间，||阿尔法||=0称为H^k_（X ; R）的零范数子空间，记为N^k（X）。在本研究中，我们研究了第三个零范数子空间N^3（SIGMA）。首席研究员实际上通过使用SIGMA*R上的双曲度量和奇异欧几里德度量来构造N^3（SIGMA）的非平凡元素，其中欧几里德度量通过使用与SIGMA的伪Anosov自同构相关联的测量层来定义。作为这一实际构造的应用，证明了R-向量空间N^3（SIGMA）的维数是连续统的基数。通过对有界上同调的研究，首席研究员得到了由双曲3-单形组成的复形的微片分解的概念。后来，事实证明，这个概念在研究3-流形之间的非零度映射时也很有用。特别地，如果给定一个从闭3-流形到双曲3-流形的非零度映射f：M*N，则可以利用N上的双曲结构定义M上由双曲3-单形组成的复形的结构.利用这类复形的微片分解，证明了允许来自固定M的非零度映射的双曲3-流形的个数是有限的.