This book presents an introduction to the theory of transformation groups which will be suitable for all those coming to the subject for the first time. The emphasis is on the study of compact Lie groups acting on manifolds. Throughout, much care is taken to illustrate concepts and results with examples and applications. Numerous exercises are also included to further extend a reader's understanding and knowledge. Prerequisites are a familiarity with algebra and topology as might have been acquired from an undergraduate degree in Mathematics. The author begins by introducing the basic concepts of the subject such as fixed point sets, orbits, and induced transformation groups. Attention then turns to the study of differential manifolds and Lie groups with particular emphasis on fibre bundles and characteristic classes. The latter half of the book is devoted to surveying the main themes of the subject: structure and decomposition theorems, the existence and uniqueness theorems of principal orbits, transfer theorems, and the Lefschetz fixed point theorem.
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Editorial Reviews
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"Introduction to topological transformation groups, accessible at the early graduate level." --The American Mathematical Monthly
"Develops almost every concept under consideration slowly and from scratch. Most claims are proved in detail. Many of the important constructions are explained with examples. Each chapter has a list of problems. The author extracts some of the fundamental concepts from these sources, adds his own ideas, and in this way produces a very readable text." --Mathematical Reviews
Language Notes
Text: English (translation)
Original Language: Japanese
Original Language: Japanese
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4 of 4 people found the following review helpful:
5.0 out of 5 stars
best introduction to equivariant topology,
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This review is from: The Theory of Transformation Groups (Hardcover)
As the title of my review suggests, this IS where one should start learning about the subject. Other books which could then be tackled (in any order, really) are: Bredon's classic, tom Dieck (especially if you're wondering about the connections with representation theory and stable homotopy), Guillemin et al. on equivariant de Rham theory (with a view toward applications), Allday+Puppe and Hsiang on hard-core cohomological methods, among others. Note that Borel's seminar on transformation groups is a little out of date, but still worth looking into. Much more recently, P. May compiled quite a few very interesting articles addressing the equivariant stable homotopy category, attributing them to R. Piacenza.There are many other avenues one can pursue after reading Kawakubo. I've only listed a few. Good luck :o)
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