Complex Cobordism and Stable Homotopy Groups of Spheres: Second Edition
About this Title
Douglas C. Ravenel, University of Rochester, Rochester, NY
Publication: AMS Chelsea Publishing
Publication Year: 2004; Volume 347
ISBNs: 978-0-8218-2967-7 (print); 978-1-4704-2998-0 (online)
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.
Table of Contents
- Chapter 1. An introduction to the homotopy groups of spheres
- Chapter 2. Setting up the Adams spectral sequence
- Chapter 3. The classical Adams spectral sequence
- Chapter 4. $BP$-theory and the Adams-Novikov spectral sequence
- Chapter 5. The chromatic spectral sequence
- Chapter 6. Morava stabilizer algebras
- Chapter 7. Computing stable homotopy groups with the Adams-Novikov spectral sequence
- Appendix A1. Hopf algebras and Hopf algebroids
- Appendix A2. Formal group laws
- Appendix A3. Tables of homotopy groups of spheres