Cohomological Methods in Transformation Groups: (Series Number 32 Cambridge Studies in Advanced Mathematics)

This is an account of the theory of certain types of compact transformation groups, namely those that are susceptible to study using ordinary cohomology theory and rational homotopy theory, which in practice means the torus groups and elementary abelian p-groups. The efforts of many mathematicians have combined to bring a depth of understanding to this area. However to make it reasonably accessible to a wide audience, the authors have streamlined the presentation, referring the reader to the literature for purely technical results and working in a simplified setting where possible. In this way the reader with a relatively modest background in algebraic topology and homology theory can penetrate rather deeply into the subject, whilst the book at the same time makes a useful reference for the more specialised reader.

Table of Contents:
Preface; 1. Equivalent cohomology of G-CW-complexes and the Borel construction; 2. Summary of some aspects of rational homotopy theory; 3. Localisation; 4. General results on torus and p-torus actions; 5. Actions on Poincaré duality spaces; Appendices; References; Indexes.

Review :
"...a clear, beautifully written presentation of some of the central developments in topology in the last thirty-odd years, centering on a subject which we dare predict will never cease to surprise, namely, the action of groups on topological spaces. May it be the forerunner of several other such expositions." Gian-Carlo Rota, The Bulletin of Mathematical Books "...written in a lucid and careful style. All the areas previously mentioned are discussed (as well as many more), paying special attention to the key elements involved in the proofs. Alternate approaches are often discussed, and many interesting examples are provided. The authors have done an admirable job of explaining this area of mathematics. Thoughtful remarks are included in several places, there are exercises at the end of each chapter, and the references are abundant. Moreover, there are two appendices which provide much of the necessary background in commutative and differential algebra....[I]t should prove useful to a broad spectrum of mathematicians." Alejandro Adem, Bulletin of the American Mathematical Society