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Differential Topology (AMS Chelsea Publishing) Reprint Edition
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Top Customer Reviews
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 ...Read more ›
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).Read more ›
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.
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.
Most Recent Customer Reviews
The approach taken in this book is a little dated, but with G+P's witty commentary and valuable insights, there is still much to love. Read morePublished 8 months ago by Taylor
It's a perfectly serviceable book but it's certainly an introduction. If you want anything with depth then you'll need a more advanced book. Read morePublished 12 months ago by Dan
Back in the day there must have been a movement towards thin sleek books. Of course, there's tradeoffs. Read morePublished on March 27, 2010 by Mitchell Chan
I agree with the reviewer who is not a "higher mathematician". Neither am I; in fact, I repeatedly found that both Milnor and Hirsch became remarkably clearer after reading the... Read morePublished on December 22, 2009 by V. Nagar
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... Read morePublished on August 13, 2009 by Mayer A. Landau
The book is very good, it is concise but I really like how the authors describe conceptual issues, it is easy to see the motivation behind most of the material we have covered so... Read morePublished on February 7, 2009 by Dan Barkley
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. Read morePublished on July 30, 2008 by Justin Hilburn
This text is really great, once you know the material. It's a bit hard to learn from, but it has so much knowledge in it for people to look up. Read morePublished on December 7, 2007 by Tyler Lee Kelly