STUDY ON THE GROUPS OF DIFFEOMORPHISMS

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We studied the groups of diffeomorphisms of the circle and the disk. In particular, we found an important relationship between the group of diffeomorphisms of the circle and the group of the area preserving diffeomorphisms of the disk. Namely using tha fact that any diffeomorphism of the boundary circle of the disk has an extention to an area preserving diffeomorphism of the disk, we wrote down the relationship between the Euler class for the group of diffeomorphisms of the circle and the Calabi invariant for the group of the area preserving diffeomorphisms of the disk. Moreover we showed the sane result for the groups of Lipschitz homeomorphisms.We studied similar relationship between the group of diffeomorphisms of 2-sphere and that of 3-ball, between the group of diffeomorphisms of 3-sphere and that of 4-ball, or more generally, the group of diffeonorphisms of the boundary of a compact manifold and that of the manifold. We also studied the group of Lipschitz homeomorphisms and we obtained a new result on the perfectness of such groups.Relating to the contact structure on the 3-manifolds, we investigated the generalization of the notion of the projectively Anosov flows in higher dimensions. We studied the diffeomorphism classes of such objects. We look at the algebraic models in detail. We also studied Anosov flows and found a classification for regular projectively Anosov flows without compact leaves. We also studied complex vector fields and complex contact structures.We studied the characteristic classes for foliations and the SC^-*S algebra for the foliations. In particular we investigated the case where the foliation has transversely piecewise linear structure. We also looked at the finitely presented simple groups.