Study on Diffeomorphism Groups of Manifolds with Geometric Structures

点击翻译按钮获取中文摘要

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.