Bounded cohomology and 3-dimensional hyperbolic geometry

Let H^3_（SIGMA ; R） be the third bounded cohomology of a closed， orientable surface SIGMA of genus g>1。首席调查员提出了一个假名词。||·||关于H^3_（SIGMA ; R）不是根据S. Matsumoto-S. Morita的结果来解释的正常（1985年）。Moreover，通过使用一个类似的参数，我们构建了n-边界同源学的示例，无论是假的，都不是任何n5的标准<greater than or equal>。他们是第一个示例，展示了存在着没有正常边界的同源性：拓扑空间X，与K-th边界同源性H^k_（X ; R）的子空间稳定||Alpha（Alpha）||=0被称为H^k_（X ; R）的零规范子空间，被N^k （X）拒绝。在这项研究中，我们调查了第三个零规范子空间N^3 （SIGMA）。首席调查员构建的N^3 （SIGMA）的非琐事元素实际上是通过在SIGMA*R上同时使用超声波度量和在SIGMA*R上使用单一的euclidean度量来定义的，在哪里可以使用与SIGMA的模拟-Anosov自动化有关的euclidean度量。作为这个实际构造的一个应用，它显示了R-矢量空间N^3 （SIGMA）的尺寸是连续性的本质，研究了边界同源学的研究，首席研究员发现了一个微型芯片分解的运动，使超玻色3简单件的复杂性。后来，它被改变了，因为运动也是一个有用的，可以在3个轮廓之间进行调查的非零度地图。在具体情况下，如果一个非零度映射f ： M*N从一个封闭的3块到一个超轴3块被赋予，一个可以定义一个超轴3简单的结构，通过在N上使用超轴结构上的微芯片分解，它就证明了从一个固定的M上获得的非零度映射的数目的超轴3块是有限的。

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.