A User's Guide to Spectral Sequences (Cambridge Studies in Advanced Mathematics)

Rating:
Summary: A superb overview
Review: 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.