## Complex Cobordism and Stable Homotopy Groups of Spheres

Complex cobordism and stable homotopy groups of spheres, also known as the green book.

The second edition is now (December, 2003) available and is part of the AMS Chelsea Series.   The new cover is not green, but dark red. An online edition is available below.  The first edition, published in 1986 by Academic Press, is now out of print.

### Online edition

The book has been republished by the AMS and an updated online version of the book is available here. This version is in the process of being revised. Comments and corrections are welcome.

The entire book, including all errata, figures and tables, is now available as a searchable hyperlinked pdf file (under 5MB) here, uploaded June 21, 2015 and most recently revised on March 10, 2017.

If you are unable to see and use the hyperlinks in this copy, the problem most likely lies with your browser or pdf viewer, not with the file. As of June 2017, I am unaware of a way to make such links work while viewing a pdf file with the Chrome, my browser of choice. If you know of a way to do this, please let me know.

The book was reviewed by Peter Landweber in the AMS Bulletin in 1988, and by Joseph Neisendorfer in the AMS Math Reviews .

I have a list of errata for the first edition. As typos are found in the second edition, corrections will be made in the online edition. If you find additional misprints, please email them to me at doug@removethis.math.&this.rochester.edu .

For charts of the stable homotopy groups of sphere's computed and tabulated there, see Allen Hatcher's home page .  Toda's tables are reproduced on Jie Wu's home page

For charts of Ext groups, see Christian Nassau's Cohomology charts and Bob Bruner's Cohomology of modules over the mod 2 Steenrod algebra.

### Neisendorfer's review of the first edition:

In the preface, the author states: "The purpose of this book is threefold; (i) to make BP-theory and the Adams-Novikov spectral sequence more accessible to nonexperts, (ii) to provide a convenient reference for workers in the field, and (iii) to demonstrate the computational potential of the indicated machinery for determining stable homotopy groups of spheres." He has succeeded in doing this and more. This book provides a substantial introduction to many of the current problems, techniques, and points of view in homotopy theory.

One of the nice features of this book is Chapter 1, "An introduction to the homotopy groups of spheres". It begins with a quick historical survey, starting with the Hurewicz and Freudenthal theorems and leading, via the Hopf map, to the Serre finiteness theorem, the Nishida nilpotence theorem, and the exponent theorem of Cohen, Moore, and the reviewer. Then results relating to the special orthogonal group are described, for example, Bott periodicity and the image of $J$. The history of computing homotopy groups is illustrated by a brief discussion of the Cartan-Serre method of killing homotopy groups and of its descendent, the classical Adams spectral sequence. Some of the triumphs of this spectral sequence, or, more precisely, of the secondary cohomology operations related to it, are indicated; for example, the solutions to the classical and $\text{mod}\,p$ Hopf invariant one problems. At this point, the author makes the transition to the main subject matter of this book by describing the complex cobordism ring, formal group laws, and the Adams-Novikov spectral sequence. The applications of this and related techniques to the existence of infinite families of elements in the stable homotopy groups of spheres are then indicated. Next, the author replaces cobordism by the more tractable BP-theory and introduces the chromatic spectral sequence. Chapter 1 closes with a discussion of the way in which the unstable homotopy groups of spheres relate to the vector field theorem, the Kervaire invariant, and the Segal conjecture. Present in this discussion are James periodicity, the $EHP$ sequences of James and Toda, and the Kahn-Priddy theorem. The description of Mahowald's work on the stable $EHP$ spectral sequence is likely to be of special value to the experts. It should be clear that a reader of Chapter 1 can come away with some understanding of a substantial portion of current homotopy theory.

Chapter 2 gives a quick description of how to set up an Adams spectral sequence, first in the classical case where it is based on the $\text{mod}\,p$ Eilenberg-Mac Lane spectrum and then for a more general spectrum. Convergence and products are given a good discussion. All of this treatment follows Adams and is done in homology.

Chapter 3, "The classical Adams spectral sequence", is a good indicator of the general utility of this book to students of homotopy theory. Following Milnor and Novikov, it applies the Adams spectral sequence to compute the homotopy of $M\text{U}$. In this case and in Bott's computation of the homotopy of $b\text{o}$, the $E\sb 2$ term is rather nice and the spectral sequence collapses. The computations for the homotopy of spheres are more difficult and useful techniques such as the May spectral sequence and the lambda algebra are introduced. Along the way, one computes differentials and observes James periodicity, the Adams vanishing line, and Adams periodicity. This chapter can be used independently as a good introduction to the classical Adams spectral sequence.

Chapter 4, "BP-theory and the Adams-Novikov spectral sequence", begins the detailed study of the main topics of this book. Quillen's theorem that the complex cobordism ring is isomorphic to the Lazard ring is proved and Quillen's method of constructing the BP spectrum by means of an idempotent is given. The BP-theoretic analogue of the dual of the Steenrod algebra is described and then used to make computations of the stable homotopy groups of spheres in a range which is impressive at this stage of the book. A nice survey of BP-theory is also included in this chapter.

Chapter 5 discusses the chromatic spectral sequence and its applications to the Hopf invariant one problem, the image of $J$, and the existence of periodic families. Chapter 6 is devoted to Morava stabilizer algebras and gives as an application a solution to the odd primary version of the Kervaire invariant problem. Taken together, these two chapters are impressive proof of the effectiveness of the Adams-Novikov spectral sequence.

In Chapter 7, the 3-primary and the 5-primary components of the stable homotopy groups of spheres are computed in very extensive ranges. For the 5-primary component, the computations go up to the one thousand stem, which is a new record.

The book closes with three appendices which provide background for the rest of the book but which are also valuable references in themselves. The first appendix deals with Hopf algebras and their generalizations, Hopf algebroids. Among the topics covered in this appendix are the change of rings theorem, Massey products following May's treatment, and algebraic Steenrod operations, including the Kudo transgression theorem. The second appendix contains an account of the theory of commutative one-dimensional formal group laws. The third appendix contains tables of the homotopy groups of spheres.

The book has an extensive bibliography.

In conclusion, this book 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. In addition, the material covered is related to conjectures made by its author concerning the global properties of stable homotopy theory. Even though these conjectures are absent from this book, their recent solution gives added meaning to the mathematics in this fine exposition.