Study on Diffeomorphism Groups of Manifolds with Geometric Structures

几何结构流形微分同胚群的研究

基本信息

批准号：
17540098
负责人：
FUKUI Kazuhiko
金额：
$ 2.11万
依托单位：
Kyoto Sangyo University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2005
资助国家：
日本
起止时间：
2005 至 2007
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-17540098/
关键词：
diffeomorphism group geometric structure homology of a group perfect group finite grouo action orbifold foliation with Morse singularity commutator length モース型特異点をもつ葉層同相群の1次元ホモロジー葉層構造レフシェッツ葉層

项目摘要

I researched about an algebraic and topological structure of the diffeomorphism group of a manifold with a geometric structure and its subgroup from the following four viewpoints.1. Study on a topological property of Lipschitz mappings and an algebraic structure of the group of Lipschitz homeomorphisms. We proved that a so-called Inverse Function Theorem holds in the Lipschitz category. We considered the complex n space C^n with canonical U(n)-action and proved that the first homology of the identity component of the group of equivariant Lipschitz homeomorphisms of On with compact support under the compact open topology does not vanish and admits continuous moduli2. Study on the group of equivariant diffeomorphisms. We considered the real n space R^n with finite group action and determined that the first homology of the identity component of the group of equivariant diffeomorphisms of R^n with compact support. As a corollary, we can determine the first homology of the groups of automorphisms of orbifolds, manifolds with compact Hausdorff foliations and 3-manifolds with locally free S^1 action.3. Study on the group of foliation preserving diffeomorphisms of foliated manifolds with singularity. We considered foliated manifolds with singularities of Morse type and determined the first homology of the identity component of the group of foliation preserving diffeomorphisms of the foliated manifolds.4. Study on the group of diffeomorphisms preserving a submanifold and the commutator length. We considered a manifold and its submanifold and proved that the identity component of the group of diffeomorphisms of the manifold preserving the submanifold is perfect if the dimension of the submanifold is greater than 0. Furthermore we discussed the commutator length of diffeomorphisms near the identity.

我研究了具有几何结构的歧义群的代数和拓扑结构，并从以下四个观点中进行了几何结构及其亚组。1。研究Lipschitz映射的拓扑特性以及Lipschitz同构的代数结构。我们证明了所谓的逆函数定理在Lipschitz类别中。我们考虑了具有规范u（n）的复杂n空间c^n，并证明了在紧凑的开放式拓扑下与紧凑的支撑的一组同构同构同构同源的第一同源性并没有消失，并且接纳了连续的模量。对模棱两可的差异性研究。我们考虑了具有有限群体作用的真实n空间r^n，并确定了r^n的dopariant diffefomorlists的第一个同源性具有紧凑的支持。作为推论，我们可以确定Orbifolds自动形态的第一组，具有紧凑的Hausdorff叶子的歧管和具有本地自由S^1动作的3个manifolds。3。对叶状叶片的植物群的研究，以奇异性的叶状歧管的差异性。我们考虑了具有莫尔斯类型的奇异性的叶状流形，并确定了叶面叶子群体的身份组成部分的第一个同源性，从而保留了叶状歧管的差异性。4。对保留子术和换向器长度的差异性的研究。我们考虑了一种歧管及其子序，并证明了保留子策略的歧管的差异性的身份成分，如果子序列的尺寸大于0。