Differential Topology 1St Edition Edition

20 customer reviews
ISBN-13: 978-0132126052
ISBN-10: 0132126052
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From the Publisher

This text fits any course with the word "Manifold" in the title. It is a graduate level book.
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Product Details

  • Hardcover: 222 pages
  • Publisher: Prentice Hall; 1St Edition edition (August 24, 1974)
  • Language: English
  • ISBN-10: 0132126052
  • ISBN-13: 978-0132126052
  • Product Dimensions: 6.2 x 0.7 x 9 inches
  • Shipping Weight: 1.1 pounds
  • Average Customer Review: 3.9 out of 5 stars  See all reviews (20 customer reviews)
  • Amazon Best Sellers Rank: #1,583,132 in Books (See Top 100 in Books)

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Most Helpful Customer Reviews

36 of 38 people found the following review helpful By Malcolm on September 22, 2008
Format: 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
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32 of 37 people found the following review helpful By A Customer on September 3, 2002
Format: Hardcover
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.
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23 of 26 people found the following review helpful By Adrian Jenkins on July 13, 2004
Format: Hardcover
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).
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11 of 11 people found the following review helpful By Lucius Schoenbaum on October 29, 2010
Format: Hardcover Verified Purchase
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.
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7 of 7 people found the following review helpful By Mayer A. Landau on August 13, 2009
Format: Hardcover
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.
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