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Complex Cobordism and Stable Homotopy Groups of Spheres: Second Edition
Douglas C. Ravenel, University of Rochester, NY
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AMS Chelsea Publishing
2004; 395 pp; hardcover
Volume: 347
ISBN-10: 0-8218-2967-X
ISBN-13: 978-0-8218-2967-7
List Price: US$65 Member Price: US$58.50
Order Code: CHEL/347.H

Since the publication of its first edition, this book has served as one of the few available on the classical Adams spectral sequence, and is the best account on the Adams-Novikov spectral sequence. This new edition has been updated in many places, especially the final chapter, which has been completely rewritten with an eye toward future research in the field. It remains the definitive reference on the stable homotopy groups of spheres.

The first three chapters introduce the homotopy groups of spheres and take the reader from the classical results in the field though the computational aspects of the classical Adams spectral sequence and its modifications, which are the main tools topologists have to investigate the homotopy groups of spheres. Nowadays, the most efficient tools are the Brown-Peterson theory, the Adams-Novikov spectral sequence, and the chromatic spectral sequence, a device for analyzing the global structure of the stable homotopy groups of spheres and relating them to the cohomology of the Morava stabilizer groups. These topics are described in detail in Chapters 4 to 6. The revamped Chapter 7 is the computational payoff of the book, yielding a lot of information about the stable homotopy group of spheres. Appendices follow, giving self-contained accounts of the theory of formal group laws and the homological algebra associated with Hopf algebras and Hopf algebroids.

The book is intended for anyone wishing to study computational stable homotopy theory. It is accessible to graduate students with a knowledge of algebraic topology and recommended to anyone wishing to venture into the frontiers of the subject.

Graduate students and research mathematicians interested in algebraic topology.

Reviews

From reviews of the First Edition:

"This book on the Adams and Adams-Novikov spectral sequence and their applications to the computation of the stable homotopy groups of spheres is the first which does not only treat the definition and construction but leads the reader to concrete computations. It contains an overwhelming amount of material, examples, and machinery ... The style of writing is very fluent, pleasant to read and typical for the author, as everyone who has read a paper written by him will recognize ... this is a very welcome book ..."

-- Zentralblatt MATH

"This book provides a substantial introduction to many of the current problems, techniques, and points of view in homotopy theory ... gives a readable and extensive account of methods used to study the stable homotopy groups of spheres. It can be read by an advanced graduate student, but experts will also profit from it as a reference ... fine exposition."

-- Mathematical Reviews

• An introduction to the homotopy groups of spheres
• Setting up the Adams spectral sequence
• The classical Adams spectral sequence
• $$BP$$-theory and the Adams-Novikov spectral sequence
• The chromatic spectral sequence
• Morava stabilizer algebras
• Computing stable homotopy groups with the Adams-Novikov spectral sequence
• Hopf algebras and Hopf algebroids
• Formal group laws
• Tables of homotopy groups of spheres
• Bibliography
• Index