Geometry Of Characteristic Classes
Geometry Of Characteristic Classes
复制标题
特征类的几何
DOI：
10.1090/mmono/199
复制
发表时间：
2001
影响因子：
4.9
通讯作者：
S. Morita
中科院分区：
文献类型：
--
作者：
S. Morita
本书的目的是阐述 20 世纪 60 年代末以来出现的一些关于特征类几何的新理论的基础知识。特征类中，典型的有Stiefel-Whitney类、Euler类或Pontrjagin类和Chern类。这些类是在 1930 年代和 1940 年代引入的，从那时起它们在流形的分类和结构分析中发挥了基础作用。另一方面，从 20 世纪 60 年代末开始，出现了一些比以前更精细地处理流形结构的理论。其中包括Gel'fand-Fuks理论、Chern-Simons理论以及平丛特征类理论，它们彼此密切相关。这些新的特征类是在上述经典特征类（部分）消失时定义的，因此有时将它们称为次要类。除此之外，平丛的特征类已被证明不仅与几何学而且与代数几何和数论有着密切的关系。很可能它们将在未来的数学中发挥越来越重要的作用。纤维丛的特征类理论仍然很大程度上未知，其结构群是无限维群，例如流形的微分同胚群。然而，对于曲面的微分同胚群，已经进行了相当详细的研究。这就是始于 20 世纪 80 年代的面丛特征类理论。正如本书最后“未来研究的方向和问题”中将提到的，对上述新理论的研究正在从不同的角度继续进行。虽然本书的描述仅限于基础部分，但如果读者对这些理论感兴趣，作者将很高兴。九
The purpose of the present volume is to give expositions on the basics of some new theories concerning geometry of characteristic classes which have appeared since the end of the 1960’s. Among characteristic classes, there are Stiefel-Whitney classes, Euler classes or Pontrjagin classes and Chern classes as typical examples. These classes were introduced during the 1930’s and 1940’s, and since then they have played fundamental roles in the classification as well as analysis of the structure of manifolds. On the other hand, from the end of the 1960’s onward, there arose a few theories which treat finer structures of manifolds than before. These include Gel’fand-Fuks theory, Chern-Simons theory and also the theory of characteristic classes of flat bundles, which are closely related to each other. These new characteristic classes are defined when the above mentioned classical characteristic classes vanish (partially) so that sometimes they are called the secondary classes. Among other things, the characteristic classes of flat bundles have been shown to have intimate relations with not only geometry but also algebraic geometry and number theory. It is plausible that they will play more and more crucial roles in future mathematics. The theory of characteristic classes of fiber bundles, whose structure groups are infinite dimensional groups such as the diffeomorphism groups of manifolds, remain largely unknown. However, in the case of diffeomorphism groups of surfaces, quite detailed studies have been made. This is the theory of characteristic classes of surface bundles which began in the 1980’s. As will be mentioned in “Directions and Problems for Future Research” at the end of this book, studies of the above new theories are continuing from various points of view. Although the description of this book is limited to the foundational part, the author would be happy if readers are interested in these theories. ix