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Spectral sequences are among the most elegant and powerful methods of computation in mathematics. This book describes some of the most important examples of spectral sequences and some of their most spectacular applications. The first part treats the algebraic foundations for this sort of homological algebra, starting from informal calculations. The heart of the text is an exposition of the classical examples from homotopy theory, with chapters on the Leray-Serre spectral sequence, the Eilenberg-Moore spectral sequence, the Adams spectral sequence, and, in this new edition, the Bockstein spectral sequence. The last part of the book treats applications throughout mathematics, including the theory of knots and links, algebraic geometry, differential geometry and algebra. This is an excellent reference for students and researchers in geometry, topology, and algebra.
- Sales Rank: #1988089 in Books
- Brand: Brand: Cambridge University Press
- Published on: 2000-11-27
- Original language: English
- Number of items: 1
- Dimensions: 8.98" h x 1.30" w x 5.98" l, 1.69 pounds
- Binding: Paperback
- 578 pages
- Used Book in Good Condition
Review
From reviews of the first edition: 'McCleary has undertaken and completed a daunting task; few algebraic topologists would have the courage to even try to write a book such as this. The mathematical community is indebted to him for this achievement!' Bulletin of the AMS
'... this guide is a treasure trove ...'. Niew Archief voor Wiskunde
About the Author
John McCleary is Professor of Mathematics at Vassar College on the Elizabeth Stillman Williams Chair. His research interests lie at the boundary between geometry and topology, especially where algebraic topology plays a role. His papers on topology have appeared in Inventiones Mathematicae, the American Journal of Mathematics and other journals, and he has written expository papers that have appeared in American Mathematical Monthly. He is also interested in the history of mathematics, especially the history of geometry in the nineteenth century and of topology in the twentieth century. He is the author of Geometry from a Differentiable Viewpoint and A First Course in Topology: Continuity and Dimension and he has edited proceedings in topology and in history, as well as a volume of the collected works of John Milnor. He has been a visitor to the mathematics institutes in Goettingen, Strasbourg and Cambridge, and to MSRI in Berkeley.
Most helpful customer reviews
16 of 18 people found the following review helpful.
A superb overview
By Dr. Lee D. Carlson
Spectral sequences have generally been thought of as being complicated, esoteric constructions, due mainly to the way they are presented in the mathematical literature. This book is very unusual, in that it attempts to explain the need for spectral sequences and give insight into how they arise and in what contexts. Anyone who is curious about spectral sequences will find an exceptionally well-written book here. This goes especially for the physicist reader, who if involved in fields such as string theory or quantum field theory, is faced with a daunting task of learning both the physics and mathematics behind these theories, formidable as both of these are. Chapter one, entitled `An Informal Introduction', is one of the best introductions to spectral sequences in print, in both books and research papers. The intuition gained by the reading of this chapter is invaluable for the chapters that follow, since the author motivates the construction of spectral sequences exceedingly well, with many examples given.
The author introduces spectral sequences as a tool for computing the homology or cohomology (which he labels as H*) of a space or an algebraic invariant assigned to a space or algebraic object. In order to obtain a more tractable problem and to motivate the calculation of H* using spectral sequences, the author assumes at first that H* is `filtered', in particular that H* is a graded vector space. As a first approximation to H*, one uses the associated graded vector space to some filtration of H*, which is the "target" of the spectral sequence. The "two-index" property of spectral sequences in this case arises from the fact that the associated graded vector space to the filtered graded vector space is in fact `bigraded'. One of the indices is called the `complementary degree' while the other is called the `filtration degree.' More formally, the spectral sequence is a sequence of differential bigraded vector spaces, where each bigraded vector space in the sequence is equipped with a linear mapping that is also a differential. The goal is then to find the conditions under which the spectral sequence will `converge' to H*. In the introductory chapter, the author outlines various situations that allow one to compute with a spectral sequence. Some familiar constructions appear, such as the Gysin sequence, known from homological algebra and differential geometry, and the exterior algebra, also from differential geometry.
With the motivation for spectral sequences established in the introduction, the author proceeds to more formal constructions in the next chapter. Spectral sequences arise as a collection of differential bigraded R-modules between which are defined differentials. The author shows in detail how to build spectral sequences using a filtered differential module and using an exact couple. As per the historical development, he also constructs spectral sequences of algebras using tensor products of differential graded modules. After these constructions are made, the author turns his attention to how well the spectral sequence can approximate its target. This entails, as expected, a rigorous notion of limits. The author in fact defines limits and colimits of modules and the notion of a morphism between spectral sequences. For filtered differential graded modules, he shows how conditions on the filtration will ensure the associated spectral sequence converges uniquely to its target. For exact couples, the convergence can be shown but certain properties such as the Hausdorff property for the filtration must be satisfied.
The book covers four main spectral sequences that arise in algebraic topology: the Leray-Serre, Eilenberg-Moore, Adams, and Bockstein spectral sequences. The Leray-Serre spectral sequence arises when studying the homology (and cohomology) of fibrations with path-connected base spaces and connected fibers. The Leray-Serre spectral sequence allows one to compute the cohomology of the total space from knowledge of the cohomology of the base space and the fiber. The author discusses applications in the computation of cohomology of Lie groups. This is accomplished by constructing the fibration resulting from taking quotients by subgroups. Rigorous proofs of all the constructions are given for the interested reader, including a full proof of the theorem that the fourth homotopy group of the two-sphere is the integers modulo two, and the connections with characteristic classes and the Steenrod algebra.
The Eilenberg-Moore spectral sequence also arises in the study of fibrations, when the cohomology of the base space and the cohomology of the total space are known and one wants to compute the cohomology of the fiber. The author studies this case and the dual case of the Eilenberg-Moore spectral sequence for homology. Heavy use is made of differential homological algebra in this study. The reader can see with great clarity the role of torsion in the applications of the Eilenberg-Moore spectral sequences.
The Adams spectral sequence arises in the context of computing the homotopy groups of a nontrivial finite CW-complex. An approximation to the homotopy groups is given by the `stable homotopy groups', and Adams analysis of these groups and his proof that there are no elements of Hopf invariant one led him to construct the spectral sequence that bears his name. The author gives a detailed overview of this spectral sequence, its applications, and its connection with cobordism theory.
The Bockstein spectral sequence arose in the study of Lie groups, and the author gives the details of the construction of this spectral sequence and its application to H-spaces. Bockstein spectral sequences arise from exact couples, the first differential being the Bockstein homomorphism (in the case of homology). The Bockstein spectral sequence can also be constructed for the case of cohomology, wherein the Bockstein homomorphism becomes the stable cohomology operation in the Steenrod algebra. The resulting spectral sequence is in fact a spectral sequence of algebras with the stable cohomology operation being a derivation with respect to the cup product.
2 of 2 people found the following review helpful.
a great place to start
By supermanifold
This text is remarkable in what it sets out to do. I'm especially impressed with its comprehensive coverage. It's a very delightful read, and a good place to start for the uninitiated with a little more than just a nodding acquaintance with topology/homotopy theory. There's no other text quite like it (as far as I know), and it is indeed something of a treasure trove to be cherished.
The author really has done quite a service to the mathematical community in writing a book that best serves the needs of the budding algebraic topologist who wishes to become acquainted with this (at first sight) technically forbidding subject, demonstrating along the way that spectral sequences are not so scary, but very powerful. Perhaps the title is something of a misnomer, but there are a handful of applications to, and outside of, homotopy theory that, I think, justify the author's choice of title to some extent.
Incidentally, homotopy theory (categorical and otherwise) is very much the flavour of the decade.
What I find a little unnerving about Narada's review is that the reviewer thinks it is not particularly user-friendly, and that it is "more enlightening" to read the original papers. Spectral sequences are, by design, not always easy to grapple with, but I did find the text to be written in a style that made the going a little easier (and, as such, user-friendly, in my mind) than the treatments I've seen elsewhere. Also, though it is sometimes instructive to read the original papers, they are no substitute for the (more) modern and accessible treatments of spectral sequences that appear in the literature today.
I should point out that Weibel's book doesn't really go out of his way to flesh out the spectral sequence machinery in the same way McCleary does, so the texts aren't comparable at all (a confusing point that the reviewer made). But then neither does any book ever written on homological algebra for that matter. Finally, a good chunk of Brown's book focuses on the homological algebraic apparatus needed for the cohomology of groups, so to me, Brown's book presents itself as an excellent introduction to homological algebra in disguise.
1 of 6 people found the following review helpful.
Expensive and not especially user friendly
By Narada
The author tried very hard to write a useful book, but succeeded only partially: the book jumps into hairy definitions before giving any example where the spectral sequence technology is useful. The first chapter is an unrelenting onslaught of somewhat unmotivated definitions, and it takes a tough man to make through the thickets of notation through to the interesting examples (there are lots of trivial ones from the beginning). In addition, the book seems to have not changed too much from the first (Publish or Perish) edition, except for the addition of an exorbitant price. The book is still dated, since the primary application is homotopy theory, which is not the flavor of this decade, or the last (of course, it might come back).
It would be much cheaper and more enlightening to read the original papers by Lyndon, Hochschild-Serre, and Serre's thesis. Cohomology of Groups (Graduate Texts in Mathematics, No. 87) is also much better motivated and much cheaper.
EDIT: Another excellent textbook on the subject is An Introduction to Homological Algebra (Cambridge Studies in Advanced Mathematics), in the same series but at 1/3 the price.
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