Differential Forms in Algebraic Topology (Graduate Texts in Mathematics)

That makes this one of my all time favourite mathematics texts.


A first class job   August 18, 2001
Dr. Lee D. Carlson (Saint Louis, Missouri USA)
17 out of 17 found this review helpful

The authors of this book, through clever examples and in-depth discussion, give the reader a rare accounting of some of the important concepts of algebraic topology. The introduction motivates the subject nicely, and the authors succeed in giving the reader an appreciation of where the concepts of algebraic topology come from, how they do their jobs, and their limitations. The de Rham cohomology, which is the main subject of the book, is explained in here in a way that gives the reader an intuitive and geometric understanding, which is sorely needed, especially for physicists who are interested in applications. As an example, they give a neat argument as to why de Rham cohomology cannot detect torsion.

In chapter 1, the authors get down to the task of constructing de Rham cohomology, starting with the de Rham complex on R(n). The de Rham complex is then specialized to the case where only C-infinity functions with compact support are used, giving the de Rham complex with compact supports on R(n). The de Rham complex is then generalized to any differentiable manifold and the de Rham cohomology computed using the Mayer-Vietoris sequence.

The discussion gets a little more involved when the authors characterize the cohomology of a fiber bundle. The all-important Thom isomorphism for vector bundles, is treated in detail. The authors give several good examples of the Poincare duals of submanifolds. The connection to ideas in differential topology is readily apparent in this chapter, namely transversality and the degree of a map. In addition, the first construction of a characteristic class, the Euler class, is done in this chapter.

The Mayer-Vietoris sequence is generalized to the case of countably many open sets in chapter 2, and shown to be isomorphic to the Cech cohomology for a "good" cover of a manifold. Good examples are given for computing the de Rham cohomology from the combinatorics of a good cover. The authors then characterize Cech cohomology groups in more detail, introducing the important concept of a presheaf. Presheaves are usually introduced abstractly in most books, so it is a real treat to see them described here in such an understandable way. Computations of the case of a sphere bundle are given, and the role of orientability and the Euler class in giving the existence of a global form on the total space is detailed. The Thom isomorphism theorem and Poincare duality are generalized to the cases where the manifold does not have a finite good cover and the vector bundle is not orientable. A very concrete introduction to monodromy is given and nice examples of presheaves that are not constant are given.

The authors treat spectral sequences in chapter 4, and as usual with this topic, the level of abstraction can be a stumbling block for the newcomer. The authors though explain that the spectral sequence is nothing other than a generalization of the double complex of differential forms that was considered in chapter 2. The crucial step in the chapter is the transition to cohomology with integer coefficients, which is necessary if one is to study torsion phenomena. The De Rham theory is then extended to singular cohomology and the Mayer-Vietoris sequence studied for singular cochains. The authors show that the singular cohomology of a triangularizable space is isomorphic to its Cech cohomology with the constant presheaf the integers. After a fairly detailed review of homotopy theory (including a discussion of Morse theory) the authors compute the fourth and fifth homotopy groups of S(3). The last section of the chapter discusses the rational homotopy theory of Sullivan as applied to differentiable manifolds. The authors discussion is illuminating, and shows how eliminating any torsion information allows one to prove some interesting results on the homotopy groups of spheres. One such result is Serre's theorem, the other being the computation of some low-dimensional homotopy groups of the wedge product of S(2) with itself.

The last chapter of the book considers the theory of characteristic classes, with Chern classes of complex vector bundles being treated first. The theory of characteristic classes is usually treated formally, and this book is no exception, wherein the authors formulate it using ideas of Grothendieck. They do however give one nice example of the computation of the first Chern class of a tautological bundle over a projective space. The Pontryagin class is defined in terms of a complexification of a real vector bundle and computed for spheres and complex manifolds. A superb discussion is given of the construction of the universal bundle and the representation of any bundle as the pullback map over this bundle.


wonderfully clear, useful book   February 7, 2006
twit
7 out of 7 found this review helpful

I agree with the other reviews, and only wanted to add to one of them that in regard to examples of chern classes, I believe they also use the whitney formula to derive the chern classes of a hypersurface from that of projective space, which really expands the realm of examples significantly.

This was all I needed in writing my notes on the Riemann Roch theorem for hypersurfaces in 3 and 4 space, for instance. I felt I knew little about concrete chern classes, but I was able to take the presentation in this book and use it for my purposes immediately.


a masterpiece of exposition   December 12, 2007
Nim Sudo
2 out of 2 found this review helpful

This is a beautiful book which I have read and re-read with much profit and pleasure over the years. It presents topics in a very unusual order, which minimizes boring technicalities and develops intuition. Everything is very concrete and explicit, with lots of nice pictures and diagrams.

The book begins with a clear and concise treatment of deRham cohomology. If one hasn't seen differential forms before, then it might be a bit too brief and one might need to supplement it. But if one is comfortable with differential forms, then de Rham theory is a setting in which theorems such as Poincare duality can be proved with a minimum of pain. It is also very edifying to see the Poincare dual of a submanifold as a differential form. There is then a natural transition to Cech cohomology and double complexes. With this as a warmup, it is then a small additional step to spectral sequences (although the derived couple approach used here is perhaps not the most elementary possible). This machinery is then used to discuss an assortment of topics in homotopy theory and characteristic classes, which always sticks to the most important points without getting bogged down in technicalities.

It is highly unusual that the definition of singular homology only comes after the introduction of spectral sequences! This book might be best appreciated if one has some familiarity with singular homology and wants to better understand its geometric meaning.

Despite the avoidance of technicalities, the book is carefully written, although there is the occasional sign error. For example, the sign given for the Lefschetz fixed point theorem is wrong for odd-dimensional manifolds; try it for the circle and you will see. (Several other books make the same mistake.)


Painless intro to spectral sequences and algebraic topology   January 29, 2000
Michael Spertus (Chicago, IL USA)
Wonderful book. The real theme of this book is to get the reader to some powerful and compelling applications of algebraic topology and comfort with spectral sequences. The use of differential forms avoids the painful and for the beginner unmotivated homological algebra in algebraic topology. E.g., For example, the wedge product of differential forms allow immediate construction of cup products without digression into acyclic models, simplicial sets, or Eilenberg-Zilber theorem. The authors later come back and do the now-motivated version for singular homology later.

I really like the idea of using spectral sequences from the beginning. It quickly brings in the actual flavor of algebraic topology by introducing today's workhorse tool. Also, because spectral sequences take a long time to become second nature (at least they did for me), the earlier the exposure to them the quicker the reader will be able to do more advanced topics comfortably. Again, spectral sequences are introduced in a painless special case, that of a double complex, and more difficult cases are not treated until the reader feels comfortable with basic spectral sequence calculations.

Finally, the prerequisites are less than or equal to those for other algebraic topology books, making this a nice choice for a first exposure to algebraic topology.