This book has its origin in a graduate course delivered at DAMTP in the Lent term of 1989 by one of the authors (J. de A.) on group cohomology. We have aimed at a self-contained presentation. Some of the background sections, however, are provided as review or reference material and occasionally they may not be suitable for first study if the reader is completely unfamiliar with their contents. There are already several good textbooks on differential geometry for physicists and the authors felt that there was no point in reproducing here topics not relevant to the main subject that could be found without much effort in a single source. Thus, a modicum of knowledge of differential manifolds, Cartan calculus and of the theory of connections is assumed. From the physics side, a knowledge of quantum field theory is also required to read certain parts of the book. Nevertheless, almost all the necessary background material is discussed in it, and often repeated in different settings of increasing complexity so that the various concepts can be assimilated more easily.

A comment should be made concerning rigour. The book uses generously the mathematical ‘Theorem, Proposition, etc’ presentation style. We have found this convenient in order to stress certain key ideas or to organize and separate parts of the text that would otherwise look unreasonably long. Nevertheless, the book is primarily intended for physicists wishing to familiarize themselves with certain aspects of the theory of Lie groups, fibre bundles and a few of their physical applications, particularly to problems where cohomology and extension theory play an important rôle.