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Characteristic Classes. (AM-76) First Edition Edition
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On the history of algebraic topology, have a look at the monographs of Dieudonne.
The most important construction on a smooth manifold is its tangent bundle, and the basic question is whether smooth never zero vector fields exist. The subject begins with the theorem of Poincare & Hopf: a never zero vector field exists if and only if the topological euler characteristic of the underlying manifold is zero. For a polyhedron, this euler characteristic is the number V-E+F = vertices - edges + faces. Thus the most basic characteristic class is the euler class. Briefly, the others measure existence of sequences of independent vector fields. In 1957 their existence, construction and properties were clouded, and Milnor cleared this away once for all in these notes, published by demand and gratefully received by [almost] everyone. This is a great book, and a 2 star review only serves to rate ones own qualifications to appreciate it. I.e. these reviewers are rating not the book but its suitability for their own narrow interests. For another short introduction try the chapter in the book by Bott & Tu.
Characteristic classes associated to a fiber bundle are cohomology classes of the base manifold that behave naturally under pullbacks (i.e., the characteristic class of the pullback of a bundle is the pullback of the characteristic class of the bundle). In particular, the classic classes that this book deals with are associated to vector bundles, either real (the Stiefel-Whitney (S-W) and Pontrjagin classes), real orientable (the Euler class), or complex (Chern classes).Read more ›
Most Recent Customer Reviews
This was topic for my 2nd oral exam at U of Chicago. Me and a couple of other fellows took turns lecturing on it to each other in preparation for same, to work out a few wrinkles... Read morePublished on October 16, 2010 by email@example.com
Milnor's writing is, as always, exceptionally clear and concise. Foundational material for any geometry student. Every mathematician should own this. Read morePublished on February 22, 2006 by A. Smith
Coming from a physics point of view, I did not find this exposition the most useful. It is rather formal and the ordering isn't that natural for a physicist. Read morePublished on August 14, 2002