7 of 7 people found the following review helpful:
4.0 out of 5 stars
Pretty good overview, July 6, 2001
This review is from: Loop Spaces, Characteristic Classes and Geometric Quantization (Progress in Mathematics) (Hardcover)
Characteristic classes, one of the most abstract and most difficult subjects to teach, are treated in this book at a level that is fairly understandable. The author endeavors to explain how characteristic classes "do their jobs" in the areas in which they are employed, and, even though he does not give an understanding of the foundations of the subject, a reading of the book will give one some helpful guidance in the gaining of such an understanding. In the introduction in particular, the author gives an excellent overview of the history of characteristic classes and explains how the arise in different areas of mathematics. The book is written for the mathematician in mind, but readers interested in applying the theory of characteristic classes, such as high energy physicists, could gain a great deal from the reading of this book.
In chapter 1, the author overviews the language of sheaf theory and how to construct complexes of sheaves. Although the presentation is somewhat abstract, the author does give some examples of the constructions, such as the exponential exact sequence of sheaves. Using an injective resolution of a sheaf, the sheaf cohomology groups are defined and then shown to be independent of the injective resolution. Using the idea of a double complex, spectral sequences are introduced, along with the concept of sheaf hypercohomology. The later is constructed using an injective resolution corresponding to a sheaf complex. Most interestingly, the author shows how the hypercohomology of sheaves is related to the Cech cohomology. The later is more concrete from an applications point of view, and is one that can be more readily understood by physicists, as well as de Rham cohomology that is introduced later, and is shown to be a resolution of the constant sheaf of a smooth manifold. The Cech cohomology groups are shown to be canonically isomorphic to the de Rham cohomology groups.
A cohomology theory not so familiar to most is the Deligne cohomology, which is also introduced in chapter 1. This is also called Cheeger-Simons cohomology by some, and has applications in conformal field theory. The presentation here is actually quite good, as the author shows how Deligne cohomology is related to ordinary cohomology via a few examples, and how Deligne cohomology can be used to compare Cech cohomology classes with de Rham cohomology classes. The chapter ends with an overview of the famous Leray spectral sequence.
In chapter 2, the author goes into the classification of line bundles, basically using the Weil-Kostant theory. When the line bundle has a connection, the author shows that the isomorphism classes of line bundles with connections is related to the second Deligne cohomology group. The Kostant central extensions of the group of symplectic diffeomorphims is also considered, and the author shows how this acts on sections of line bundles. In chapter 3, the author considers first the topology on the space of singular knots in a smooth three-dimensional manifold, which is shown to great surprise to be a Kahler manifold. Not only that, the author further shows it to have a symplectic, complex, and a Riemannian structure.
The discussion gets considerably more interesting in chapter 4, wherein the author discusses how to generalize the classical result that the second integral cohomology group of a manifold is the group of isomorphism classes of line bundles over the manifold. The goal is to characterize the third integral cohomology group, and the author does this by using the theory of C*-algebras. The result of Dixmier-Douady relating the algebra of compact operators on a separable Hilbert space is shown to give the geometric description of the third integral cohomology group. The section on connections and curvature in this chapter is especially well written because the author explains and motivates well the eventual identification of the Hilbert space as the space of infinitely differentiable functions on the unit circle.
In chapter 5, things get more complicated, where the Dixmier-Douady theory of sheaves of groupoids is related to the third integral cohomology group. Torsors are introduced as a generalization of principal bundles. Algebraic geometers frequently refer to sheaves of groupoids as "stacks" and the author discusses these and the idea of a gerbe from the standpoint of category theory. The sheaf of groupoids is shown to represent the third integral cohomology group and the author constructs a cohomology class of the sheaf of groupoids using differential-geometric constructions.
Chapter 6 considers line bundles over loop spaces, with the holonomy of line bundles initiating the discussion. Interestingly, the author shows that Deligne cohomology again plays a role here, in that the holonomy of a line bundle with a connection can be expressed in terms of a transgression map in Deligne cohomology. The line bundle over the loop space of a smooth manifold is constructed using the sheaf of groupoids over the manifold, and is called the anomaly line bundle associated to the sheaf of groupoids. When the (free) loop space is generalized to the space of oriented singular knots, and the manifold is 3-dimensional, the author shows how to obtain a bundle over this space, and shows the relation to geometric quantization. The central extension of the loop group is considered, interestingly, in terms of action functionals, an approach which has its origins in physics.
The author ends the book with a discussion of the Dirac monopole. This object has been studied in great detail in the literature, but here the author gives an interesting twist wherein he relates the monopole to the Dixmier-Douady sheaf of groupoids over the three-dimensional sphere, and gives an explicit generator of the third integral cohomology group of this sphere. The classical quantization condition then follows naturally.
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4 of 4 people found the following review helpful:
5.0 out of 5 stars
Very interesting, self-contained textbook., April 10, 2000
By A Customer
This review is from: Loop Spaces, Characteristic Classes and Geometric Quantization (Progress in Mathematics) (Hardcover)
Quite clear cohomological approach to line bundles and geometric pre-quantization. Well-developed theory of gerbes. Very interesting chapters on Kaehler geometry of the space of knots and line bundles on loop spaces. The book is self-contained; it starts with definitions and basic theory of complexes of sheaves, hypercohomology, and a concise introduction to the Deligne and Cheeger-Simons cohomologies. My personal favorite is chapter 5, most likely the only readable text on gerbes. I would recommend this book as a textbook for relevant grad courses. The book is also accesible for advanced math and physics majors.
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