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Characteristic Classes. (AM-76) First Edition Edition

4.4 out of 5 stars 9 customer reviews
ISBN-13: 978-0691081229
ISBN-10: 0691081220
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Frequently Bought Together

  • Characteristic Classes. (AM-76)
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  • Morse Theory (Annals of Mathematic Studies AM-51)
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  • Differential Forms in Algebraic Topology (Graduate Texts in Mathematics)
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Editorial Reviews

From the Back Cover

The theory of characteristic classes began in the year 1935 with almost simultaneous work by Hassler Whitney in the United States and Eduard Stiefel in Switzerland. Stiefel's thesis, written under the direction of Heinz Hopf, introduced and studied certain 'characteristic' homology classes determined by the tangent bundle of a smooth manifold.
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Product Details

  • Series: Annals of Mathematics Studies (Book 76)
  • Paperback: 340 pages
  • Publisher: Princeton University Press; First Edition edition (August 1, 1974)
  • Language: English
  • ISBN-10: 0691081220
  • ISBN-13: 978-0691081229
  • Product Dimensions: 6 x 0.8 x 9.2 inches
  • Shipping Weight: 1.4 pounds (View shipping rates and policies)
  • Average Customer Review: 4.4 out of 5 stars  See all reviews (9 customer reviews)
  • Amazon Best Sellers Rank: #413,738 in Books (See Top 100 in Books)

Customer Reviews

Top Customer Reviews

By Alex on June 6, 2002
Format: Paperback
The point to be made here is that M&S and books comparable to it ( I can think of those by Morita, off hand) are written in a style amenable to mathematicians. The purely formal, albeit axiomatic, approach survives as it appeals more to purists than to physicists. There's really nothing lacking in MS: although dated, it's very readable, requiring only a minimum of prerequisites, and this is what makes it attractive, with an appeal to a wide spectrum of audiences. Physical applications are bountiful, and they can be sought elsewhere in the literature.
On the history of algebraic topology, have a look at the monographs of Dieudonne.
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Format: Paperback
I read this page to see how Milnor's book could average only 4 stars. I found two unfair, but intelligent, 2/3 star reviews apparently by physicists, miffed that a masterful 50 year old book on the foundations of the topic, does not provide a history of their invention, and a survey of their current use in physics. If a book serves its own and its author's purpose wonderfully well, but does not serve yours, that is your fault, not the author's. As a hint, it is in the Annals of Math series, not the Handbook for Physicists series. Please read the chapter on "obstructions", p. 139. There Milnor succintly provides exactly the interpretation requested, and a reference to the classic text by Steenrod.

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.
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By A Customer on April 26, 2000
Format: Paperback
This is a wonderful book, both for the student and for the researcher. Whether you're an aspiring topologist or string theorist, you need this book. If you're a physicist, I would recommend Nakahara's book to supplement some of the discussions. This book progresses very nicely, gently easing you into things with some elementary topics early on, and then building up motivation and machinery as the more advanced topics are reached. You will almost definitely not be left confused or lost. Simply put, this is a classic that should reside on everyone's shelf.
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Format: Paperback
Milnor & Stasheff's "Characteristic Classes" is the standard reference for them. It includes a number of different, but equivalent, definitions and properties of the Stiefel-Whitney, Chern, Euler, and Pontrjagin classes, with a formal, heavily algebraic topological flavor. The material presented is all easy to follow and well explained, and there are problems included in every chapter, some of which are challenging, earning this work its high reputation. However, there is a dearth of geometric intuition and many important, basic facts about the classes are absent, so while a student with only a standard background in algebraic topology and differential topology could read and understand the book, to really gain a practical knowledge of characteristic classes as one would need in studying, say, low-dimensional topology (let alone physics), the book must be supplemented with other book(s) on the subject, such as Bott & Tu's Differential Forms in Algebraic Topology.

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).
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Format: Paperback
I'm tempted to say this book is indispensible for the student who wants to learn about characteristic classes. But it's sufficient to simply say that I have never seen a better book on the subject. The work is not the latest to be sure, but the exposition and the material covered make this an excellent introductory text as well as a reference.
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