Study on the structure of the group of homeomorphisms

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1. We considered the group of foliation preserving homeomorphisms of a foliated manifold and computed the first homologies of the groups for condimension one 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. Especially the group for the Reeb foliation on the 3-sphere is perfect. Furthermore we showed than if the foliation preserving Lipschitz homeomorphisms of a Lipschitz foliated manifold and computed the first homology of the group of foliation preserving Lipschitz homeomorphisms of a codimension one CィイD11ィエD1-foliated manifold. Then we have a phenomenon different from that in topological case.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 stability of compact leaves. Then we showed that all compact Hausdorff CィイD1rィエD1-foliations of 4-manifolds by hyperbolic surfaces are not stable (1 【less than or equal】 r【less than or equal】∞).