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Cohomological Methods in Transformation Groups : Cambridge Studies in Advanced Mathematics - Christopher Allday

Cohomological Methods in Transformation Groups

Cambridge Studies in Advanced Mathematics


Published: 24th April 2007
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In the large and thriving field of compact transformation groups an important role has long been played by cohomological methods. This book aims to give a contemporary account of such methods, in particular the applications of ordinary cohomology theory and rational homotopy theory with principal emphasis on actions of tori and elementary abelian p-groups on finite-dimensional spaces. For example, spectral sequences are not used in Chapter 1, where the approach is by means of cochain complexes; and much of the basic theory of cochain complexes needed for this chapter is outlined in an appendix. For simplicity, emphasis is put on G-CW-complexes; the refinements needed to treat more general finite-dimensional (or finitistic) G-spaces are often discussed separately. Subsequent chapters give systematic treatments of the Localization Theorem, applications of rational homotopy theory, equivariant Tate cohomology and actions on Poincare duality spaces. Many shorter and more specialized topics are included also. Chapter 2 contains a summary of the main definitions and results from Sullivan's version of rational homotopy theory which are used in the book.

"...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

Equivariant cohomology of G-CW-complexes and the Borel construction
Summary of some aspects of rational homotopy theory
General results on torus and p-torus actions
Actions on Poincar duality spaces
commutative algebra
some homotopy theory of differential modules
Index of notation
Table of Contents provided by Publisher. All Rights Reserved.

ISBN: 9780521350228
ISBN-10: 0521350220
Series: Cambridge Studies in Advanced Mathematics
Audience: Professional
Format: Hardcover
Language: English
Number Of Pages: 484
Published: 24th April 2007
Publisher: Cambridge University Press
Country of Publication: GB
Dimensions (cm): 22.8 x 15.2  x 3.2
Weight (kg): 0.88