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Characteristic Classes. (AM-76) Paperback – August 1, 1974

ISBN-13: 978-0691081229 ISBN-10: 0691081220 Edition: First Edition

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Characteristic Classes. (AM-76) + Morse Theory (Annals of Mathematic Studies AM-51)
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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: 0.8 x 6.1 x 9 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: #470,621 in Books (See Top 100 in Books)

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Please read the chapter on "obstructions", p. 139.
mathwonk
For Euler and S-W classes, the classes are intimately connected to the Thom isomorphism and Thom class of a vector bundle, which is one of the emphases of the book.
Malcolm
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.
Jamey Bass

Most Helpful Customer Reviews

16 of 18 people found the following review helpful 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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27 of 34 people found the following review helpful By mathwonk on April 12, 2005
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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12 of 14 people found the following review helpful 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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7 of 8 people found the following review helpful By Malcolm on May 7, 2010
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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11 of 15 people found the following review helpful By Jamey Bass on June 27, 2001
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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