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Title: Differential Topology by Victor Guillemin, Alan Pollack ISBN: 0-13-212605-2 Publisher: Prentice Hall Pub. Date: 14 August, 1974 Format: Hardcover Volumes: 1 List Price(USD): $76.00 |
Average Customer Rating: 3.17 (6 reviews)
Rating: 2
Summary: Lightweight and overpriced
Comment: 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.
Rating: 4
Summary: A good start....
Comment: Differential topology has influenced many areas of mathematics, and also has many applications in physics, engineering, comptuer graphics, network engineering, and economics. The authors, well-known contributors to the field, have written a nice introduction in this book, which is suitable for readers having a background in linear algebra and advanced calculus.
The authors begin the book with a general overview of manifolds and smooth maps between them. The local behavior of smooth maps is studied first, such behavior determined by the derivative (modulo a diffeomorphism). The inverse function theorem is stated but not proved, the authors encouraging the reader to do the proof using local parametrizations. Generalizations of local diffeomorphisms, the immersions, are discussed, and the local immersion theorem proved. Immersions that are injective and proper, the embeddings, are then discussed. When the dimension of the target manifold is less than or equal to the domain manifold, the surjectivity of the derivative of the map at every point leads to the map being what is called a submersion. Submersions can be viewed as generalizations of projection maps of standard Euclidean space to one of equal or lower dimension. The authors prove this as the local submersion theorem. Regular values of smooth maps are defined and the authors show that the premiage of regular values are submanifolds. A brief discussion of Lie groups is given as an application of the preimage theorem.
The main theme of the book, a generalization of the notion of regularity, called transversality, is also introduced in this chapter. The concept of transversality is fully elaborated on in chapter 2, in the context of manifolds with boundary. Sard's theorem is proved for manifolds with and without boundary. The Transversality Theorem for families of smooth maps is proven in detail, showing that transversal maps are generic when the target manifold is Euclidean space. The authors give an excellent discussion of intersection theory modulo two, along with the famous Jordan-Brouwer separation theorem and Borsuk-Ulam theorem.
In order to make intersection numbers an invariant of homotopy, intersection theory for oriented manifolds is considered in chapter 3. It is shown that homotopic maps always have the same intersection numbers. This brings up naturally the subject of Lefschetz fixed-point theory and the authors give a very clear overview of this. Index theory and the famous Poincare-Hopf index theorem are discussed in this chapter also, along with the Hopf degree theorem. And, thankfully, the authors do not hesitate to employ a myriad of diagrams to illustrate the main points and develop intuition.
The last chapter of the book is more formal than the rest of the book, and covers integration on manifolds. A familiar subject in courses on advanced calculus, the authors do a good job of discussing exterior algebra, differential forms, how to integrate these on manifolds via suitable partitions of unity, and how to differentiate these on manifolds. A very brief introduction to de Rham cohomology is given. The famous Gauss-Bonnet theorem is shown to follow from the Poincare-Hopf theorem.
Rating: 1
Summary: Substitutes rigor for confusion
Comment: 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.
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Title: Topology from the Differentiable Viewpoint by John Willard Milnor ISBN: 0691048339 Publisher: Princeton Univ Pr Pub. Date: 24 November, 1997 List Price(USD): $17.95 |
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Title: Calculus on Manifolds: A Modern Approach to Classical Theorems of Advanced Calculus by Michael Spivak ISBN: 0805390219 Publisher: Westview Press Pub. Date: June, 1965 List Price(USD): $44.00 |
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Title: Functional Analysis by Peter D. Lax ISBN: 0471556041 Publisher: John Wiley & Sons Pub. Date: 22 March, 2002 List Price(USD): $94.95 |
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Title: Topology (2nd Edition) by James Munkres ISBN: 0131816292 Publisher: Prentice Hall Pub. Date: 28 December, 1999 List Price(USD): $101.33 |
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Title: Counterexamples in Analysis by Bernard R. Gelbaum, John M. H. Olmsted ISBN: 0486428753 Publisher: Dover Pubns Pub. Date: 04 June, 2003 List Price(USD): $14.95 |
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