Topological study on the structure of the group of homeomorphisms

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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.