Study on Diffeomorphism Groups of Manifolds with Geometric Structures

从以下四个方面研究了具有几何结构的流形及其子群的自同态群的代数拓扑结构.研究Lipschitz映射的一个拓扑性质和Lipschitz同胚群的一个代数结构。我们证明了一个所谓的反函数定理在Lipschitz范畴中成立。考虑了具有标准U（n）-作用的复n空间C^n，证明了在紧开拓扑下，具有紧支撑的等变Lipschitz同胚群的单位分支的第一同调不为零，且允许连续模2.等变双同态群的研究。考虑了具有有限群作用的真实的n空间R^n，确定了R^n的具有紧支集的等变同同态群的单位分支的第一同调.作为推论，我们可以确定轨道折叠、具有紧Hausdorff叶理的流形和具有局部自由S^1作用的3-流形的自同构群的第一同调。具有奇异性的叶状流形的叶状保持自同态群的研究。研究了具有莫尔斯型奇点的叶状流形，确定了该类流形的保同态叶状流形群的单位分支的第一同调.关于保子流形和换位子长的自同态群的研究。我们考虑了一个流形及其子流形，证明了如果子流形的维度大于0，则该流形的微分同胚群中保持子流形的单位分量是完美的。此外，我们还讨论了单位元附近的复同态的换位子长。

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.