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Equivariant cohomology on smooth manifolds is the subject of this book which is part of a collection of volumes edited by J. Bruning and V.W. Guillemin. The point of departure are two relatively short but very remarkable papers be Henry Cartan, published in 1950 in the Proceedings of the “Colloque de Topologie”. These papers are reproduced here, together with a modern introduction to the subject, written by two of the leading experts in the field. This “introduction” comes as a textbook of its own, though, presenting the first full treatment of equivariant cohomology in the de Rahm setting. The well known topological approach is linked with the differential form aspect through the equivariant de Rahm theorem. The systematic use of supersymmetry simplifies considerably the ensuing development of the basic technical tools which are then applied to a variety of subjects, leading up to the localization theorems and other very recent results.
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From the reviews: MATHEMATICAL REVIEWS “The authors are very generous to the reader, and explain all the basics in a very clear and efficient manner. The understanding is enhanced by appealing to concepts which developed after Cartan’s seminal work, which also help to place things in a broader context. This approach sheds light on many of Cartan’s motivations, and helps the reader appreciate the beauty and the simplicity of his ideas…There are ‘gifts’ for the more advanced readers as well, in the form of many refreshing modern points of view proposed by the authors…The second part of the book is in my view a very convincing argument for the usefulness and versatility of this theory, and can also serve as a very good invitation to more detailed investigation. I learned a lot from this book, which is rich in new ideas. I liked the style and the respect the authors have for the readers. I also appreciated very much the bibliographical and historical comments at the end of each chapter. To conclude, I believe this book is a must have for any mathematician/physicist remotely interested in this subject.”
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