This version is more screen friendly than its predecessors. Like them it is in the process of being revised. Comments and corrections are welcome. If you find additional misprints, please email them to me at dcravenel@removethis.gmail.com.

The last revision of the second edition, uploaded April 29, 2026 is here.
I also have a list of errata for the first edition.

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

For charts of the stable homotopy groups of sphere's computed and tabulated there, see Allen Hatcher'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.

Complex cobordism and stable homotopy groups of spheres,

The author has previously written [4, p. 407]: “I am painfully aware of the esoteric nature of this subject and of the difficulties faced by anyone in the past who wanted to become familiar with it.” The subject in question is the topic of the book under review, namely the study of stable homotopy theory by means of the Adams–Novikov spectral sequence. Here, then, is required reading for all who want to enter this field.

The holy grail of stable homotopy theorists is the calculation of the stable homotopy groups of spheres, $\pi_k^S = \pi_{n+k}(S^n)$ for $n > k+1$ (the independence of these groups on $n$ in this range follows from the suspension isomorphism). One finds easily that $\pi_0^S \cong \mathbf{Z}$ and $\pi_k^S = 0$ for $k < 0$. Serre proved in 1953 [5] that, for $k > 0$, $\pi_k^S$ is a finite abelian group. Thus one can fix a prime $p$ and ask to compute the $p$-primary components $\pi_{k,(p)}^S$ for $k > 0$; the problem is usually approached in this form.

Why is the study of the stable homotopy groups of spheres such a central problem? We shall offer some remarks, and in addition urge the reader to consult G. Whitehead’s lovely article [7] for an account of the development of homotopy theory.

The spheres are the most fundamental spaces in algebraic topology. There are many classical theorems about spheres, such as the Borsuk–Ulam theorem and the result of H. Hopf that the degree of a map of the $n$-sphere $S^n$ to itself determines the homotopy class of the map. All graduate students in mathematics learn to compute the homology of $S^n$. Since it is even simpler to define the homotopy groups $\pi_{n+k}(S^n)$, as the homotopy classes of basepoint preserving maps $S^{n+k} \to S^n$, it may come as a surprise to learn that these groups present such a challenge, even for the two- and three-dimensional spheres.

In the face of such a challenge, one looks for patterns. Perhaps the most impressive is that the suspension homomorphism

is an isomorphism for $n > k+1$, proved by Freudenthal in 1937. This brings one to the stable homotopy groups of spheres, where Serre’s result mentioned above is the most striking pattern one sees next. Cell complexes are built from spheres; for a finite cell complex, Serre’s results imply that the groups $\pi_{n+k}(S^n X)$ $(n \gg k)$ are finitely generated with the same rank as $H_k(X;\mathbf{Z})$. It is often straightforward to calculate homology of cell complexes, but an inductive computation of the stable homotopy groups of a cell complex would require a knowledge of the stable homotopy groups of spheres.

Thus there is ample justification for the creation of elaborate machinery to study the stable homotopy groups of spheres. Such an apparatus has been erected, the building blocks being generalized homology theories (especially bordism theory) and various associated spectral sequences.

There are several reasons why people find this an attractive field. Some are keen on homotopy theory (always have been, always will be). Some thrive on a challenge and enjoy computing. And some are devotees of a part of the machinery which can be brought to bear on the problem, much of which (especially generalized homology such as bordism theory) has independent interest.

Thus it is fortunate or unfortunate, depending on one’s outlook, that the machinery used to study the stable homotopy groups of spheres has considerable intricacy. The classical Adams spectral sequence [1] was developed for this purpose. Fixing a prime $p$ and letting $A$ denote the Steenrod algebra of cohomology operations in mod $p$ cohomology, the $E_2$-term of this spectral sequence is

while the limit groups $E_\infty^{s,t}$ for $t - s = k$ form the associated graded group to a filtration of $\pi_{k,(p)}^S$. It is a major task to determine the $E_2$-term, which one has been able to accomplish only in a limited range for each prime. Moreover, one still faces the task of determining the differentials $d_r \colon E_r^{s,t} \to E_r^{s+r,\,t+r-1}$.

The main object of interest in Ravenel’s book is the Adams–Novikov spectral sequence—its origin, its detailed structure, and its applications. Surprisingly, one is asked to replace ordinary homology and cohomology by complex bordism and cobordism theory. The complex bordism groups $MU_n(X)$ of a space $X$ are obtained from maps $f\colon M \to X$ of closed smooth $n$-dimensional manifolds into $X$, $M$ having a complex structure on its stable normal bundle, under the equivalence relation of bordism. The complex bordism groups $MU_*(X)$ form a graded module over $MU_* = MU_*(pt)$, and the latter is a polynomial ring $\mathbf{Z}[x_2, x_4, \ldots]$, where $x_{2n}$ is represented by a suitable $2n$-dimensional manifold (examples: $x_{2(p-1)} = [\mathbf{CP}^{p-1}]$ for each prime $p$). One also replaces the Steenrod algebra by $MU_*(MU)$, the “self-homology of $MU$” which is analogous to the dual of the Steenrod algebra, and obtains a spectral sequence converging to $\pi_*^S$ with

More economical for the study of the $p$-primary component $\pi_{*,(p)}^S$ is the replacement of $MU$ by Brown–Peterson homology $BP$, for which $BP_* = BP_*(pt)$ is a polynomial ring $\mathbf{Z}_{(p)}[v_1, v_2, \ldots, v_n, \ldots]$, $v_n$ of degree $2(p^n - 1)$. Now the $E_2$-term is

and the spectral sequence converges to $\pi_{*,(p)}^S$.

What has one gained by replacing ordinary cohomology with complex bordism or Brown–Peterson homology? First, there are fewer differentials. The first two figures of the book, on pp. 12 and 15, compare the Adams spectral sequence with the Adams–Novikov spectral sequence for $p = 3$ and $t - s \leq 45$; the reviewer had some fun locating the four differentials inadvertently omitted from the first of these, and suggests that the reader try to spot them. Secondly, one obtains a richer homology theory, built from manifolds and universal among homology theories having a formal group. Formal groups $F(X,Y)$ are power series over a commutative ring with unit element, satisfying the group-like conditions $F(X,0) = X$, $F(X,Y) = F(Y,X)$ and $F(X,F(Y,Z)) = F(F(X,Y),Z)$; they enter into various areas of algebraic geometry and number theory; e.g., elliptic curves have formal groups [6, Chapter IV] expressing the group law near the origin. A great deal of number theory enters into the study of bordism and cobordism via formal groups, and much insight has been gained in the process, notably in working out the ideas of Jack Morava [3].

In recent years, there has been a trend toward less computational themes in stable homotopy theory. This began with the study of infinite families and periodicity phenomena, for example in work by M. Mahowald and by H. Miller, D. Ravenel, and W. S. Wilson in the 70s. More recently, notable success has been obtained in understanding the structure of the stable homotopy category. As an example, Nishida’s nilpotence theorem, asserting that all elements of $\pi_k^S$ with $k > 0$ are nilpotent, has been greatly extended in work of E. Devinatz, M. Hopkins, and J. Smith.

Here now is some advice to users of the book under review. The first chapter is an informal introduction to the field, and can be read for orientation. The next two chapters deal with generalities about Adams spectral sequences, and the use of the classical Adams spectral sequence. The heart of the book is to be found in Chapters 4 and 5, on the Adams–Novikov spectral sequence and the chromatic spectral sequence; the latter serves to organize and make understandable the structure of the former. To get the most out of this part of the book, the reader should be familiar with a bit more about Brown–Peterson homology than is offered here; see §2 of Chapter 4 or Wilson [8] for guidance. The appendices on Hopf algebroids and formal groups also provide essential information. The results in the final section of Chapter 6 are needed as input for the calculations of the final chapter.

There are quite a few figures and tables in the book; the reader might want to compile an index for them, as the reviewer did. It takes a certain amount of youthful zeal to tabulate the 5-primary groups $\pi_{*,(5)}^S$ in dimensions up to 1000, the previous record being 760 by Aubry [2]. In contrast, a smooth picture of $\pi_{*,(p)}^S$ follows from the Adams–Novikov spectral sequence in dimensions up to $2p(p^2-1)$ for odd primes (in dimensions up to 25 for $p = 2$). Thus calculations in dimensions above 25 for $p = 2$, and above 240 for $p = 5$, require a devotion to special techniques such as those presented in the final chapter.

Having spent several very intense weeks with this book I have considerable respect for the hard work and mastery that went into it. There is a good mix of explanation and demonstration of techniques in action. One need not study the entire book to profit from it.

References

1. 1. J. F. Adams, On the structure and applications of the Steenrod algebra, Comment. Math. Helv. 32 (1958), 180–214.
2. 2. M. Aubry, Calculs de groupes d’homotopie stables de la sphère, par la suite spectrale d’Adams–Novikov, Math. Z. 183 (1984), 45–91.
3. 3. J. Morava, Noetherian localisations of categories of cobordism comodules, Ann. of Math. (2) 121 (1985), 1–39.
4. 4. D. Ravenel, A novice’s guide to the Adams–Novikov spectral sequence, Geometric Applications of Homotopy Theory II, Lecture Notes in Math., vol. 658, pp. 404–475, Springer-Verlag, Berlin and New York, 1978.
5. 5. J.-P. Serre, Groupes d’homotopie et classes de groupes abéliens, Ann. of Math. (2) 58 (1953), 258–294.
6. 6. J. H. Silverman, The arithmetic of elliptic curves, Springer-Verlag, Berlin and New York, 1986.
7. 7. G. W. Whitehead, Fifty years of homotopy theory, Bull. Amer. Math. Soc. (N.S.) 8 (1983), 1–29.
8. 8. W. S. Wilson, Brown–Peterson homology, an introduction and sampler, CBMS Regional Conf. Ser. in Math., no. 48, Amer. Math. Soc., Providence, R.I., 1980.

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