*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.

It was reviewed by Peter Landweber in the AMS Bulletin in 1987,
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.

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.

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 and should not be quoted
publicly. Comments and corrections are welcome.

### Return to Ravenel's home page .

* 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.*