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33 of 35 people found the following review helpful
5.0 out of 5 stars great introduction to the subject, despite its glaring faults, September 22, 2008
This review is from: Differential Topology (Hardcover)
There are few books really suitable for undergraduates who wish to get a feel for differential topology, and among them Guillemin and Pollack is probably the best. Assuming only multivariate calculus, linear algebra, and some point-set topology (with a typical analysis class covering everything in the first and third categories), G&P presents an intuitive introduction to smooth manifolds with many pictures and simple examples while avoiding much of the formalism. It is most similar to Milnor's Topology from the Differentiable Viewpoint, upon which it was based, but it has additional material, most notably on differential forms and integration.

Books on differential topology (a.k.a. smooth manifolds or differential manifolds) tend to divide neatly into 2 types. Every book begins with basic definitions of smooth manifolds, tangent vectors and spaces, differentials/derivatives, immersions, embeddings, submersions, submanifolds, diffeomorphisms, and partitions of unity. Also the inverse function theorem is at least cited, if not proved (the proof is left to the reader here), as well as Sard's theorem and some sort of embedding theorem, usually Whitney's "easy" one. But beyond that the 2 types of books diverge, with one type treating vector bundles, the Frobenius theorem, differential forms, Stokes's theorem, and de Rham cohomology, and then possibly continuing on to differential geometry or Lie groups, such as in Lee's Introduction to Smooth Manifolds, Lang's Differential and Riemannian Manifolds, or Warner's Foundations of Differentiable Manifolds and Lie Groups, whereas the other type focuses on Morse theory, normal bundles, tubular neighborhoods, transversality, intersection theory, degree, the Hopf degree theorem, the Poincare-Hopf index theorem, and then possibly continues on to surgery, handlebodies, or cobordism, such as in Wallace's Differential Topology: First Steps, Milnor's TFDV, or Hirsch's Differential Topology. The first type of book is most suitable for the analytic aspects of the subject whereas the second for the topological, so comparing Lee to Hirsch is really an apples-to-oranges comparison. Mathematicians must know both, of course, but physicists, for example, usually use the first type more (although Dubrovin et al.'s second book, Modern Geometry. Part 2: The Geometry and Topology of Manifolds,is more of the second type). This book largely falls into the second category, but the final chapter covers differential forms, integration, Stokes's theorem, a little de Rham cohomology, and the Gauss-Bonnet theorem, making it somewhat of a hybrid, like Berger's Differential Geometry: Manifolds, Curves, and Surfaces (which focuses more on differential geometry) and Bredon's Topology and Geometry (which focuses more on algebraic topology).

The book is at its best when explaining concepts such as smoothness, transversality, stability, Whitney's theorem, intersection number, orientation, Lefschetz fixed point theorem, etc., pictorially, discussing the concept for a while before giving the definition or theorem. Many results that also can be proved using algebraic topology, such as the Brouwer fixed-point theorem, the Borsuk-Ulam theorem or the Jordan separation theorem, are proved, making the book much more interesting than those like Lee or Lang that just focus on machinery.

The main drawback of the book is its carelessness in definitions, particularly at the beginning. As some other reviewers have noted, manifolds are defined as being subsets of some Euclidean space, and diffeomorphisms are defined as being (a type of) maps between open sets in Euclidean spaces, which obviates an explanation of transition functions, but then makes awkward the many places later where references are made to "arbitrary" manifolds, which are never defined. But beyond the introduction, this embedding in some R^N is never used in the proofs, which use only local coordinate neighborhoods, so the results hold more generally (of course, every manifold does embed in some R^N, but one cannot use the proof of Whitney's theorem given here since manifolds were defined as subsets of Euclidean space to begin with).

The other notable feature of this book is its exercises, of which there are many, most being rather easy, but some being important theorems (the Whitney immersion theorem, smooth Urysohn theorem, tubular neighborhood theorem, etc.), with frequent hints provided. Later in the book, some proofs are rendered as extended problem sets, with the proof broken down into steps and each step treated as a separate exercise. This allows the reader to build up the ability to derive these results on his/her own, as well as forcing the reader to actually practice these techniques and thus truly learn the subject matter.
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Initial post: Dec 17, 2008 12:57:56 AM PST
Last edited by the author on Dec 17, 2008 12:58:16 AM PST
Malcolm says:
A copy of this book is also available online as a djvu file on a Russian website. I'll add the details of where to find it to one of my Listmania lists of free online math books later (probably Part 15 or so - I have a large backlog of books to add to my lists).
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