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.

令H^3_（Sigma; r）为第三个有界的共同体，是g> 1的封闭，可定向的表面Sigma。首席调查员证明了伪|| ||在H^3_（Sigma; r）上依靠S.matsumoto-s中的结果而不是规范。莫里塔（1985）。此外，通过使用类似的论点，我们构建了第n个有界的共同体的示例，其伪型不是任何n <大于或相等> 5的n <n <n <n <或等于> 5的规范。它们是第一个示例，表明存在有限的共同体，没有规范。对于拓扑空间x，对于拓扑空间x，由k-th bounded cohomology cohomology salpha asspace的子空间= 0 h^k_（x; r）的零 - 基本空间，并用n^k（x）表示。在这项研究中，我们研究了第三个零 - 标准子空间N^3（Sigma）。主管研究人员实际上通过在Sigma*r上使用双曲线度量和奇异欧几里得公制，构建了N^3（Sigma）的非平凡元素，在Sigma*r上，欧几里得度量是通过使用与伪型轴向自动形态相关的测量层压来定义的。作为这种实用结构的应用，结果表明，R-vetor Space N^3（Sigma）的维度是继续进行的基础性。通过有限的共同体研究，负责人研究员获得了由多苯甲部组成的复合物的微芯片分解的概念。后来，事实证明，该概念在研究3个manifolds之间的非零程度图也很有用。特别是，如果给出了从封闭的3个字体到双曲线3个序列的非零程度图f：m*n，则可以通过使用在这种复合物上使用微芯片分解来定义由n.by上的双曲线结构组成的M的M M*N，从而定义了Myrocoloic n.by exporter exparts exparts expards expards axp-manifolds n.by的结构。