STUDY ON THE GROUPS OF DIFFEOMORPHISMS

我们研究了圆和盘的微分同态群。特别地，我们发现了圆的微同构群与盘的保面积微同构群之间的重要关系。即利用圆盘边界圆的任何微分同构都可以推广到圆盘的保面积微分同构这一事实，我们给出了圆的保面积微分同构群的欧拉类与圆盘的保面积微分同构群的卡拉比不变量之间的关系。此外，我们还对Lipschitz同胚群给出了相同的结果。研究了2球的微分同构群与3球的微分同构群，3球的微分同构群与4球的微分同构群，更一般地说，研究了紧流形边界的微分同构群与流形边界的相似关系。我们还研究了Lipschitz同胚群，得到了该类群完备性的一个新结果。针对三维流形上的接触结构，研究了射影Anosov流概念在高维上的推广。我们研究了这类对象的微分同胚类。我们将详细研究代数模型。我们还研究了Anosov流，并发现了一种无紧叶正射影Anosov流的分类方法。我们还研究了复向量场和复接触结构。研究了叶形的特征类和叶形的SC^-*S代数。我们特别研究了叶理具有横向分段线性结构的情况。我们还研究了有限表示的简单组。

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.