Customer Reviews: Read 6 more reviews...
 A wonderful introduction to differential topology July 14, 2004
M. A Jenkins (West Lafayette, IN)
10 out of 10 found this review helpful
First, I must comment about the reviewer below (who is obviously a greater mathematician than I) - I wouldn't recommend Bredon's book to anyone who wants to study differential topology. Man, I fought through a year of algebraic topology with that book, and I'm not sure I got a darn thing out of it! Being of a more analytic, geometric mindset, however, Guillemin and Pollack's book was right up my alley.
First, the authors make the wonderful assumption in the beginning that all manifolds live in R^n for some large enough n. This made study a great deal easier for me, as fighting through charts and atlases may not be the best place to start manifold theory (I don't mean to shortchange other important methods for working with differentiable manifolds, but rather I want to emphasize that many students might get lost in the machinery before learning anything of the theory). The book moves casually along (as the authors suggest, this book is nice for a smell-the-flowers two semester grad school class; we finished in Wisconsin in about a semester and a half before moving on to other pastures). The authors' reluctance to mention functors is also quite nice (I have asked many an algebraic topologist to describe these little guys, and the best answer I've heard is "A functor is an arrow"). A bit of analysis knowledge is nice, particularly in chapter four, and linear algebra (which seems to be a lost art, at least over here in the states) is absolutely critical.
For those of you out there who want to learn a little of this vast and incredibly interesting subject, I would highly recommend this book (even over Milnor's "Topology from the Differential Viewpoint", although the price of Milnor is much nicer). I must agree that this book is outrageously overpriced, but I ended up sucking it in for a month to spare the change for it. If Bredon is your cup of tea, so be it, but I think that most will find this book much more to their liking. One caveat, however: you MUST do some exercises. The authors leave important theorems entirely to exercises (some that come to mind are the "Stack of Records" theorem, the Jordan curve theorem, the Hopf degree theorem, the Cauchy integral formula, etc.).
Solid Introduction April 4, 2000
7 out of 8 found this review helpful
This book confers the intuition and understanding necessary for a solid foundation in Differential Topology. The exercises are well crafted and relevant too. What's more is that this book is actually readable: Some books are great once one already knows the topic whilst exceedingly difficult to the uninitiated. This book, however, is a reference as well as a solid introduction. And as an added plus the authors stray from the standard clinical textbook writing styles. Brought together with Munkres' Elements of Algebraic Topology and perhaps Massey's Algebraic Topology: An Introduction, you've got the foundations of a graduate career in Topology.
Guillemin & Pollack -- review February 23, 2001
7 out of 9 found this review helpful
This is an excellent introduction to manifold theory. The geometric intuition is as clean as it can be. This is a perfect book for a first course in manifold theory, provided the student has studied basic multivariable calculus and the differential geometry of curves and surfaces in 3-space. The prerequisites are kept lean by ignoring the general technology of abstract manifolds and dealing only with submanifolds of euclidean space.
Great, but it can't be your only book on the subject July 31, 2008
Justin Hilburn (Austin, TX)
As someone who normally likes mathematical machinery I was inclined to dislike this book, but it quickly won me over. Unlike many other books on the subject G+P covers a myriad of advanced topics like degree theory, the Cauchy integral formula, the Poincare-Theorem and the Jordan curve theorem from an intuitive geometric viewpoint. The exercises in this book are its greatest strength. I was impressed how many nontrivial theorems are left as exercises to the reader (with copious hints of course).
The fact that the book is so concrete and geometric is also it's greatest weakness. As opposed to Lee's book (which I noticed another reviewer mentioned), G+P avoids introducing much of the machinery of abstract manifolds in order to get to more interesting theorems more quickly. I think this was a good choice because even after 500+ pages Lee really never really gets beyond introducing machinery (even though I do agree his book is very clear and well written).
I would recommend that you buy G+P to develop your geometric intuition along with another book like Bredon or Lee in order to introduce the modern machinery of topology. If you need further reading I think that Baez's book and Morita's book are both very good.
great introduction to the subject, despite its glaring faults September 23, 2008
Malcolm (Tokyo)
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 (aka 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 theroem, 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-orange 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 sownwhat 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 geometry).
The book is at its best when explaining concepts such as smoothness, transversality, stability, Whiteny'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 drawbook 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 Whiteny'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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