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Topology from the Differentiable Viewpoint Rev ed. Edition

4.9 out of 5 stars 11 customer reviews
ISBN-13: 978-0691048338
ISBN-10: 0691048339
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Editorial Reviews

About the Author

John Milnor is Professor of Mathematics and Co-Director of the Institute for Mathematical Sciences at SUNY, Stony Brook. He is the author of "Topology from the Differential Viewpoint, Singular Points of Complex Hypersurfaces, Morse Theory, Introduction to Algebraic K-Theory, Characteristic Classes" (with James Stasheff), and "Lectures on the H-Cobordism Theorem" (Princeton).
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Product Details

  • Series: Princeton Landmarks in Mathematics and Physics
  • Paperback: 80 pages
  • Publisher: Princeton University Press; Rev ed. edition (November 24, 1997)
  • Language: English
  • ISBN-10: 0691048339
  • ISBN-13: 978-0691048338
  • Product Dimensions: 6 x 0.2 x 9 inches
  • Shipping Weight: 4.2 ounces (View shipping rates and policies)
  • Average Customer Review: 4.9 out of 5 stars  See all reviews (11 customer reviews)
  • Amazon Best Sellers Rank: #568,335 in Books (See Top 100 in Books)

Customer Reviews

Top Customer Reviews

Format: Paperback
Milnor's "Topology from the Differentiable Viewpoint" is a brief sketch of differential topology, well written, as are all books by Milnor, with clear, concise explanations. For students who wish to learn the subject, it should be read as a companion to a more substantive text, such as Guillemin & Pollack's Differential Topology or Hirsch's Differential Topology, as too much of the material is left out for this to be adequate as a textbook. OTOH, it does make for good bedtime reading.

While this book is highly regarded among mathematicians, it is not without its faults, namely,
- it fails to cover many topics of importance, such as transversality (only mentioned in an exercise), embeddings, differential forms, integration, Morse theory, and the intersection form;
- it only cites some theorems without proving them, or it leaves the proofs to the reader;
- it offers proofs of many theorems that are really only sketches without all the details;
- manifolds are only defined as subsets of Euclidean spaces;
- there is only 1 collection of 17 problems at the end of the book, which are used to introduce important concepts; and
- it probably moves too quickly for true beginners, packing a lot into only 51 pages.

So don't buy this as your only, or even first, book on differential topology. Oddly, many of the faults that I listed above are simultaneously strengths, in that it can be read very quickly, with relatively little effort and a high rate of retention.
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Format: Paperback
One of the best points of this little book is its brevity and clear exposition of the basic ideas. It makes a great reference guide because it's so short and well-organized. Written by a distinguished mathematician, it's no wonder that other graduate-level texts such as Guillemin & Pollacks "Differential Topology" highly recommend reading it alongside their book. Milnor's booklet is a classic, whose style and ideas surely pervade other texts.
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Format: Paperback
Perfect for a first-year graduate or advanced undergraduate course, Milnor takes us on a brief stroll through elementary differential topology. Elegant and self-contained, this book serves as an excellent first taste of the subject. Milnor is a master expositor, and is at his best in this book.
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Format: Paperback
Do the exercises. Many were Ph.D. dissertation-level problems in the 1960s; today, they're aptly described as "elementary"- because Milnor MADE them elementary.
This book forms part of the toolkit you will need to fully explore the more modern work in dynamics, complexity, and applications (e.g., economics, physics).
The clarity of the exposition also forms an ideal example of how to communicate mathematics powerfully and simply.
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Format: Paperback
I would suggest to use this book as a companion to more serious books on topology. Weighing in at a mere 51 pages, this book accomplishes what it needs to: a brief, succinct introduction to topology mostly based on the work of Brouwer. There is a nice mixture of topics, ranging from Sard's theorem to Poincare-Hopf theorem. The proofs and ideas are not fully rigorous or developed, but that would be quite a bit to expect from such a short exposition.
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Format: Paperback
This book is exceptionally well written and contains almost no errors. In this book, Milnor makes the decision to only treat the ideal cases. He does not delve into technical refinements. As a result, Milnor is able to prove a major theorem on almost every page. This also makes the book an ideal introduction for someone who is encountering differential topology for the first time since the reader is able to digest the key ideas in differential topology without having to go through extra technical baggage, which is required to extend the methods to more general cases. Despite the fact it is less than 80 pages, the book covers a significant amount of material, which allows the reader to learn a considerable amount per unit time.
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