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From Calculus to Cohomology: De Rham Cohomology and Characteristic Classes

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De Rham cohomology and the theory of characteristic classes are not only two of the most important topics in mathematics, but also in theoretical physics. Indeed, an understanding of the geometry and topology of fiber bundles requires a mastery of these topics, and, if one is to make sense of topological phenomena in quantum field theory, one must understand how to perform the calculation of characteristic classes. This book gives a fairly good start in meeting these goals, and, the authors say, is written for upper-level undergraduates with no background in topology or differential geometry. However topological spaces are not defined in the book, but the authors use them as though the reader has had prior exposure.

De Rham cohomology is introduced very early in the book (p. 15), with a differential p-form defined as a smooth map from an open set in n-dimensional Euclidean space to the space of alternating forms. The authors do motivate the definition through the consideration of ordinary vector calculus, which serves to ease the transition to the more formal theory. Concepts from algebraic topology immediately follow, these being chain complexes and their corresponding homological algebra. The foremost strategy for the calculation of the De Rham cohomology, the Mayer-Vietoris sequence is given, the treatment emphasizing the role of the Poincare lemma. Considerations from homotopy are used to calculate the de Rham cohomology of punctured Euclidean space. The De Rham theory is then used to prove the Brouwer fixed point theorem. The famous theorem of J.F. Adams on the maximal number of linearly independent vector fields on the n-dimensional sphere is stated but not proved. No doubt the proof was omitted due to the advanced techniques that must be used to prove it.

Differential forms on smooth manifolds are discussed also, along with the accompanying topics of curvature and integration on smooth manifolds. Stokes' theorem is proved in detail. A very detailed study of the concept of degree, linking numbers, and indexes of vector fields is given, as preparation for later discussions on Morse theory and the Poincare-Hopf theorem. The physicist reader will definitely want to pay attention to this discussion because of its importance in applications.This discussion also marks the beginning of the more advanced topics in the book, which continues to its end. Readers will definitely have to pay attention to more of the details here, and the authors replace geometric intuition by more formal, algebraic considerations.

The theory of fiber bundles and vector bundles are given fair treatment in the book too, but the authors should have motivated the subject with some examples of elementary bundles, such as the Mobius strip. They do however prove that a vector bundle over a compact base space has an inner product and they do this with the help of partitions of unity. Partitions of unity are one of most useful concepts to illustrate how the different fibers of a bundle can be joined together. Also, they show how vector bundles over a compact base can be trivialized by taking the direct sum with a suitable bundle, called its complement. This motivates the definition of an Abelian semigroup of isomorphism classes of vector bundles over compact bases. This semigroup can, and the authors show this, be completed to an Abelian group via the Grothendieck construction. These considerations are the origin of the famous K-theory of vector bundles. Along these same lines the authors show that there is a homotopy classification of vector bundles by using the notion of a "pull-back" of vector bundles (the pull-back of a vector bundle "dilutes" the bundle, i.e. makes it less "twisted").

The way the authors present the theory of characteristic classes is much too formal, and does not give the reader an appreciation of their origins and why they work as well as they do. Readers at this level need to be given a lot more motivation to the underlying intuition behind characteristic classes. Physicists in particular, who are faced with these objects in many applications, need a more in-depth discussion. Indeed, the authors really take off in their proof of the Thom isomorphism theorem. They do not discuss why this result is so important nor give concrete examples of its utilization.

A fairly long list of exercises is given in the back of the book, and the reader should work most of these in order to be able to understand the results in the book. It will also help to go back to some of the original papers on vector and fiber bundles, even ones published in the 1930s, to gain more of an appreciation of the concepts in the book.

Rigor is of upmost importance in mathematics, but so is understanding.

This book is true to his objectives: teaching cohomology and characteristic classes starting from calculus in several variables, in the sense that the background needed is more or less just about this start (along with some linear algebra). However, the mathematical maturity needed to fully understand the topics is a great deal bigger than that. The book can get quite esoteric very quickly, and I feel somehow that it could have been more natural to insert the example of cohomology from calculus given in the first chapter after differential forms, for example.

Nevertheless, I like this book. The authoritative books that treat more or less the same topics (Milnor & Stasheff's "Characteristic Classes", Bott & Tu's "Differential Forms in Algebraic Topology"), although more inspired and clearer (for the initiated), ask for more background and even more maturity than this one. Madsen & Tornehave introduces you to some very powerful machinery of algebraic topology, being at the same time challenging and rewarding. It succeds in the sense that it really teaches the way of thinking "algebro-topologically", a thing that can be invaluable on the study of recent topics in theoretical physics. You also can try to read the classics after reading this book: then you can at the same time understand better the point of view of these authors, and get a better grasp of the topics you've seen before.

It is a bit ambitious to use deRham cohomology as an introduction to differential forms and analysis on manifolds (compare with the easier and clearer 'Analysis on Manifolds' by Munkres). A bit too much for newcomers, too little for graduate students (compare with Bott and Tu). It is good for advanced undergraduates who are able to handle the pace and abstraction. Few examples and computations.

This book studies caharacteristic classes via de Rham cohomology. Compared with Bott and Tu's similar book "Differential Forms in Algebraic Topology", this one is more refined and more sophisticated. Warning: this book contains few applications of characteristic classes. For that matter, Milnor's "Characteritic Classes" is still the best source of information.

This book covers quite a lot of advanced material, in a rigorous fashion. It is quite a demanding introduction to such important concepts as those pertaining to De Rham cohomology, curvature, bundles and characteristic classes. Far more accessible and motivated books exist on the subject for, e.g., the physics student. I suspect that even a well-prepared maths student would have quite a hard time struggling with this book. Returning to the prospective physics student, he would be well advised to have a solid maths undergraduate background in topology, before perusing this book. While this may be desirable, there are plenty other more friendly and less demanding introductions on the same topics.
I would therefore recommend this book only as a second-reading.
A student would then benefit from the sharp, rigorous and quick presentation of most ground notions in algebraic and differential topology.

I give this book two stars simply because the material that is discussed is so inherently interesting. The text itself, however, is a pedagogic nightmare. I would go so far as to say it is effectively unreadable unless you already know the material the author is attempting to communicate. In the preface it is stated that only "standard calculus and linear algebra" are formal prerequisites. That is an extremely doubtful proposition.

The frustrations begin very early on. Consider, for example Theorem 1.4 on page 2: This theorem states, in part, "...any smooth function (defined on a star-shaped domain) that satisfies (2), Question 1.1 has a solution". Ok, that makes perfect sense, right? Who states a theorem like this? Within this review it would be impossible for me to actually state the theorem properly because doing so would involve writing mathematical equations, but basically it provides for the conditions under which a conservative vector field will exist. The first part of the "proof" involves writing down a function that is given by the definite integral of a two term expression. Where did this function come from and what reason did the author have for writing it down? Good question but unfortunately is left unanswered by the author. The next steps involve differentiating this expression which in fact involves differentiating "under the integral". Is this "standard" calculus? Analysis, yes, but not calculus. Lang's Undergraduate Analsis text, for example, addresses the topic on page 499. Even there though, it is not addressed in sufficient generality to cover the situation in which the author here uses it. What would have been helpful at this point is if the author would have stated "Here in step x we used Theorem A.B.C from [Some Text]" Also, nowhere in the proof does the author explicitly state where the "star-shaped" domain hypothesis comes from. In summary, the argument supplied by the author is not a "proof" but, rather, an unmotivated sketch.

That being said, the goals of the book are extremely ambitious and what author is attempting to accomplish is not easy and I give him credit for this attempt. If the book were rewritten to include at least twice the level of detail that it currently has, it could possibly be successful.

In summary, if you can understand this book it is probably because you have learned the material from a more digestible source, such as John Lee's excellent Smooth Manifolds, Tu's Introduction to Manifolds or Bott and Tu's Differential Forms and Algebraic Topology.