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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.

本文从以下四个方面对具有几何结构的流形及其子群的微分同构群的代数拓扑结构进行了研究。Lipschitz映射的拓扑性质及Lipschitz同胚群的代数结构研究。我们证明了所谓的反函数定理在Lipschitz范畴中成立。考虑正则U(n)作用下的复n空间C^n，证明了紧支持On的等变Lipschitz同纯群在紧开拓扑下的第一同调不消失且允许连续模2。等变微分同胚群的研究。考虑具有有限群作用的实n空间R^n，确定了具有紧支持的R^n的等变微分同态群的恒等分量的第一同调。作为一个推论，我们可以确定轨道形、紧Hausdorff叶形和具有局部自由S^1作用的3-流形的自同构群的第一同调。具有奇点的叶形流形的保叶微分同态群的研究。我们考虑了具有莫尔斯型奇点的叶形流形，并确定了这些叶形流形的保叶微分同态群的恒等分量的第一同调。保留子流形和交换子长度的微分同态群的研究。我们考虑了一个流形及其子流形，证明了当子流形的维数大于0时，保留子流形的流形的微分同态群的恒等分量是完全的。进一步讨论了在恒等附近的微分同胚的对易子长度。