This volume, written by someone who has made many significant contributions to mathematical physics, not least to the present dialogue between mathematicians and physicists, aims to present some of the basic material in algebraic topology at the level of a fairly sophisticated theoretical physics graduate student.

This book deals with the cohomology of groups, particularly finite ones. His- torically, the subject has been one of significant interaction between algebra and topology and has directly led to the creation of such important areas of mathematics as homological algebra and algebraic K-theory. It arose primar- ily in the 1920's and 1930's independently in number theory and topology. In topology the main focus was on the work of H. Hopf, but B. Eckmann, S. Eilenberg, and S. MacLane (among others) made significant contributions. The main thrust of the early work here was to try to understand the mean- ings of the low dimensional homology groups of aspace X.

This work examines the beautiful and important physical concept known as the 'geometric phase,' bringing together different physical phenomena under a unified mathematical and physical scheme.