Geometry of Characteristic Classes (Translations of Mathematical Monographs)

Characteristic classes are central to the modern study of the topology and geometry of manifolds. They were first introduced in topology, where, for instance, they could be used to define obstructions to the existence of certain fiber bundles. Characteristic classes were later defined (via the Chern-Weil theory) using connections on vector bundles, thus revealing their geometric side. In the late 1960s new theories arose that described still finer structures. Examples of the so-called secondary characteristic classes came from Chern-Simons invariants, Gelfand-Fuks cohomology, and the characteristic classes of flat bundles. The new techniques are particularly useful for the study of fiber bundles whose structure groups are not finite dimensional. The theory of characteristic classes of surface bundles is perhaps the most developed. Here the special geometry of surfaces allows one to connect this theory to the theory of moduli space of Riemann surfaces, i.e., Teichmuller theory. In this book Morita presents an introduction to the modern theories of characteristic classes. Characteristic classes are central to the modern study of the topology and geometry of manifolds. They were first introduced in topology, where, for instance, they could be used to define obstructions to the existence of certain fiber bundles. Characteristic classes were later defined using connections on vector bundles, thus revealing their geometric side. The purpose of this book is to introduce the reader to the three theories of characteristic classes that were developed in the late 1960s. They include characteristic classes of flat bundles, characteristic classes of foliations, and characteristic classes of surface bundles. The book is intended for graduate students and research mathematicians working in various areas of geometry and topology. Characteristic classes are central to the modern study of the topology and geometry of manifolds. They were first introduced in topology, where, for instance, they could be used to define obstructions to the existence of certain fibre bundles. Characteristic classes were later defined (via the Chern-Weil theory) using connections on vector bundles, thus revealing their geometric side. De Rham Homotopy Theory -- Chracteristic Classes Of Flat Bundles -- Characteristic Classes Of Foliations -- Characteristic Classes Of Surface Bundles. Shigeyuki Morita ; [translated From The Japanese By The Author]. Includes Bibliographical References (p. 179-182) And Index.