Customer Reviews

Differential Topology

5 star:		(9)
4 star:		(2)
3 star:		(1)
2 star:		(3)
1 star:		(2)

 

 

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.

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.).

I had to study this for my degree. It was one of those books that one person bought and was passed around mainly due to it's outrageous cost. It has a lack of rigour that is not made up by being more intuitive or giving the reader insight into why differential topology is such a great subject.

Transversality is rightly given prominence, but you don't really walk away with a good feel for it's importance or power. Degrees, linking numbers etc I got for 10 GBP with Milnor's Topology from a Differential Point of View.

As an introduction to differential topology - with a little point set and alot of algebraic throw in - Bredon's Geometry and Topology sets the gold standard, with Darling's Differential Forms and Connections doing a good job on the differential geometry front and Milnor's book above providing bedtime reading beforehand. You can buy all three together for around the same price as this book.

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.

the AMS Chelsea edition appears to be a digital facsimile of the original with pixillated letters. the typeface is visibly deteriorated - a cleaner image comes from an ordinary laser printer. It's distracting when reading what I think is a very nice book.

Back in the day there must have been a movement towards thin sleek books. Of course, there's tradeoffs. On the downside there's a lack of narration and context - the usual what, why, and where we're going type of stuff. The upside is what mathematicians call 'elegance'. For the layman this can be summarized as describing an object or thought in its most minimal form, whatever that may be.

My rating system of five stars is based on how successful the two authors succeed in the thin book paradigm. That is, I think there's enough there to "get it" and what's there is correct (or with a minimum of erratas). I agree with all the reviewers that gave this book 5 stars and 1-2 stars (it sucks). With respect to the 5 star folks, I agree that the authors met their objectives stated in the preface. They succeeded of creating an engineering schematic or wiring diagram to get from A to B. With respect with the 1-2 star folks I concur with their opinion. Yes, this book would not pass an editorial board standards based on modern publishing criteria. Nowadays it's more than just the handwaving. For example, reading chapter 1 section 1 I don't think there's enough there to explain to your grandmother what a manifold is and why should we care about them - a true test of mastery.

My advice to the 1-2 star folks that's not used to reading these thin sleek books is a technique I call pre-reading the book. It's a three-fold process. If you're forced into a quarter sytem (12-14 week) math class covering the whole book I highly recommend doing an 80-hour crash study session prior to the first class. Otherwise it'll seem like you just walked into a middle of a movie. First, read the preface to see where the authors 'think' they're going. Second, map out the key chapters and problems associated with each topical goal. Third, starting from the end of the book and going all the way to the start, build a dependency outline linking the 'big' result at the end with all the preceding. You'll be surprised how quickly you can cover what initially seemed advanced book. Most of the 'filler' can be gotten off the web. There's alot out there since this book has been used for the last 30 plus years.

My excitement from reading this book is how the authors bridge the study of smooth compact objects to topics that previously were in the realm of algebraic topology. These objects can be classified quickly by looking at global properties such as how many holes, described by the genus, how much curvature they have, described by the euclidean characteristic, how many vector zeros on the surfaces, described by how many bald spots would appear if hair sprouted on the surface and someone wanted to comb it and how many times the surface wraps around the interior, described by its degree. The authors do a good job of pretending like you don't have to know anything about algebraic topology but like I stated in the previous paragraph I couldn't resist googling because without getting some precursory knowledge it felt like being in the middle of a movie.

I took differential topology as an undergraduate. We used this text. Neither I, nor any of the other students, had had any prior introduction to topological manifolds prior to taking this course. At that level, the problems with this book are immediate. The authors never define exactly what a manifold is. This is true of most mathematical objects introduced in the first and second chapters of the book. They are never precisely defined. Many of the exercises are very simple, testing your understanding of the definitions. But, without proper definitions, you are never sure what is being asked. You are also not sure what constitutes a correct answer. My feeling was that no one finished the class with any real understanding. A month after the class was over a professor asked me what a manifold was, and I couldn't answer him. I then took a course using Spivak's first volume differential geometry and a course in algebraic topology using Massey's book. With that background, I returned to this book and found it a delightful read up to the middle of chapter 3. Towards the end of chapter 3 the authors get totally lazy and make everything an exercise. That pretty much sums it up. If you have the mathematical background to consult other books for details, then you'll be able to get past the initially poor exposition and you'll find this book fun and more interesting as you get further into it. But, past a certain point, you'll realize that the textbook has become one long exercise or workbook, and not a textbook at all.

The treatment of the material is oversimplified. From the outset, there is no discussion of atlases or charts, and while I applaud attempts at simplification in the name of developing intuition and understanding, this book just sacrifices too much depth and rigor to be of any use to the serious mathematics student. What material is covered suffers from hand-waving, inprecise definitions, and awkward notation; unfortunate for a subject that requires the development of a large arsenal of definitions.

It seems that books on differential topology are either extremely complicated (see Serge Lang, Fundamentals of Differential Geometry) or extremely simplified (like this book). Recommended substitute: _Introduction to Smooth Manifolds_, by John M. Lee (University of Washington). I don't believe this book has been published yet, but you can find preliminary copies online. I've also heard that Hirsch's _Differential Topology_ is good, but I personally haven't read it.

I loved to study this book several years ago as an undegraduate. Now I have to teach those subjects, so I decided to buy a copy for myself. I received the book, admired the beautiful hardcover, but when I opened it I was immediately shocked by the crude quality of the printing. The problem is that this is a poorly scanned version of the old edition (which I took from the library to compare). I fear I'll be dizzy if I start reading this; I guess that if I try to scan a page of the original with my home scanner and print it on a laser printer I'll get a better result. I'm very surprised that AMS published this. Now, for the first time in many years as a costumer, I'll try to return this book to Amazon.

PS: Another reviewer (Lucius Schoenbaum) had similar complaints as me, but for some reason he gave a 5 star rating. My single star refers not to the text itself, but to the quality of this printing and to the value of the purchase.

As someone who normally likes to see things developed in more generality 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 and intersection theory, the Cauchy integral formula, the Poincare-Hopf 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 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 Spivak, or Lee in order to introduce the modern machinery of differential topology. If you want further reading Bott and Tu, Bredon, and Baez are all fantastic.