Topological study on the structure of the group of homeomorphisms

1.我们考虑了Lipschitz流形上的Lipschitz同胚群，并证明了这个群是局部可收缩的和完备的。作为应用，我们还证明了当G是紧李群时，主G-流形的等变Lipschitz同胚群是完备的。此外，我们还证明了R^n的原点不动的Lipschitz同胚群是完备的。作为应用，我们证明了Lipschitz轨道流形的Lipschitz同胚群和紧致Hausdorff C^1-叶流形的保叶Lipschitz同胚群是完备的.已知主G-流形M的等变同同态群是完备的。如果M至少有两种轨道类型，则它不为真。当M是余维为1轨道的G-流形时，我们确定了M的等变同同态群的第一同调群.我们考虑了叶化流形的保Lipschitz同胚的叶化群，并计算了余维为1的C^2-叶化群的第一同调。我们证明了如果叶理没有D型分支，只有有限个R型分支，则群是完全群。进一步证明了如果叶理有D型分支且线性化映射是C^1-单同态，则群不是完美群。但我们证明了如果叶理有D型分量且线性化映射不是绝对连续的，则群是完美的。这一现象与拓扑情况下的现象不同。

1. We considered the group of Lipschitz homeomorphisms of a Lipschitz manifold and showed that the group is locally contractible and perfect. As its application, we also showed that the group of equivariant Lipschitz homeomorphisms of a principal G-manifold is perfect when G is a compact Lie group. Furthermore we showed that the group of Lipschitz homeomorphisms of R^n leaving the origin fixed is perfect. As its application, we can show that the groups of Lipschitz homeomorphisms of a Lipschitz orbifold and of foliation preserving Lipschitz homeomorphisms of a compact Hausdorff C^1-foliated manifold are perfect.2. It is known that the equivariant diffeomorphism group of a principal G-manifold M is perfect. If M has at least two orbit types, then it is not true. We determined the first homology group of the equivariant diffeomorphism group of M when M is a G-manifold with codimension one orbit.3. We considered the group of foliation preserving Lipschitz homeomorphisms of a foliated manifold and computed the first homologies of the groups for codimension one C^2-foliations. We showed that if the foliation has no type D components and has only a finite number of type R components, then the group is perfect. Furthermore we showed that if the foliation has a type D component and the linearization map is a C^1-diffeomorphism, then the group is not perfect. But we showed that if the foliation has a type D component and the linearization map is not absolutely continuous, then the group is perfect. This phenomenon is different from that in topological case.