Synopses & Reviews
"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. The most important topics, covering spaces, homotopy and homology theory, degree theory fibrations and a little about Lie groups are treated at a brisk pace and informal level. Personally I found the style congenial.(...) extremely useful as background or supplementary material for a graduate course on geometry and physics and would also be useful to those contemplating giving such a course. (...)" Contemporary Physics, A. Schwarz GL 308
Synopsis
In recent years topology has firmly established itself as an important part of the physicist's mathematical arsenal. Topology has profound relevance to quantum field theory-for example, topological nontrivial solutions of the classical equa- tions of motion (solitons and instantons) allow the physicist to leave the frame- work of perturbation theory. The significance of topology has increased even further with the development of string theory, which uses very sharp topologi- cal methods-both in the study of strings, and in the pursuit of the transition to four-dimensional field theories by means of spontaneous compactification. Im- portant applications of topology also occur in other areas of physics: the study of defects in condensed media, of singularities in the excitation spectrum of crystals, of the quantum Hall effect, and so on. Nowadays, a working knowledge of the basic concepts of topology is essential to quantum field theorists; there is no doubt that tomorrow this will also be true for specialists in many other areas of theoretical physics. The amount of topological information used in the physics literature is very large. Most common is homotopy theory. But other subjects also play an important role: homology theory, fibration theory (and characteristic classes in particular), and also branches of mathematics that are not directly a part of topology, but which use topological methods in an essential way: for example, the theory of indices of elliptic operators and the theory of complex manifolds.
Synopsis
In recent years topology has firmly established itself as an important part of the physicist's mathematical arsenal. It has many applications, first of all in quantum field theory, but increasingly also in other areas of physics. This book is devoted to the exposition of topology in a form easily accessible to physicists. It will be also useful to mathematicians who would like to apply topology in their work, without specialising in this discipline. The author, a topologist turned mathematical physicist has contributed many results to quantum field theory using topological methods, and is thus eminently qualified to write a book such as this.
Synopsis
"This is a very interesting book on an important topic both for physics and for mathematics. (...) It starts at the beginning, but is not really for beginners; the physics background develops rapidly, through seven short chapters, and the final eight chapters provide a lightning review of the mathematical topics encountered (...) Part II is the main part of the text, containing a selection of fascinating topics, beautifully presented, to many of which the author has been a significant contributor. The chapters on functional integration, on elliptic operators, their determinants and related index theorems, on calculating instanton contributions and on anomalies are particularly attractive. (...)" Bulletin London Mathematical Society