Topology from the Differentiable Viewpoint [Paperback]

John Willard Milnor (Author)

Book Description

Publication Date: November 24, 1997 | ISBN-10: 0691048339 | ISBN-13: 978-0691048338 | Edition: Revised

This elegant book by distinguished mathematician John Milnor, provides a clear and succinct introduction to one of the most important subjects in modern mathematics. Beginning with basic concepts such as diffeomorphisms and smooth manifolds, he goes on to examine tangent spaces, oriented manifolds, and vector fields. Key concepts such as homotopy, the index number of a map, and the Pontryagin construction are discussed. The author presents proofs of Sard's theorem and the Hopf theorem.

Product Details

• Paperback: 76 pages
• Publisher: Princeton University Press; Revised edition (November 24, 1997)
• Language: English
• ISBN-10: 0691048339
• ISBN-13: 978-0691048338
• Product Dimensions: 8.8 x 5.9 x 0.4 inches
• Shipping Weight: 4.2 ounces (View shipping rates and policies)
• Average Customer Review: 4.9 out of 5 stars  See all reviews (9 customer reviews)
• Amazon Best Sellers Rank: #656,272 in Books (See Top 100 in Books)

Customer Reviews

Most Helpful Customer Reviews

16 of 16 people found the following review helpful:

5.0 out of 5 stars Excellent starting point for differential topology, July 24, 2000

By 

Steve P. (CA USA) - See all my reviews

This review is from: Topology from the Differentiable Viewpoint (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.

26 of 29 people found the following review helpful:

5.0 out of 5 stars An excellent introduction to topology of manifolds., September 30, 1998

By 

Dev Sinha ([e-mail]) (Providence, RI, USA) - See all my reviews

This review is from: Topology from the Differentiable Viewpoint (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.

9 of 9 people found the following review helpful:

5.0 out of 5 stars a must-read supplement for topology students, September 27, 2008

By 

Malcolm (Tokyo) - See all my reviews

This review is from: Topology from the Differentiable Viewpoint (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. Milnor really emphasizes the topology of the subject, giving applications such as the fundamental theorem of algebra, Brouwer's fixed point theorem, the hairy ball theorem, the Poincare-Hopf theorem, and Hopf's theorem. Most of the book focuses on degree theory, but there is also a nice introduction to framed cobordism, which is rare for an elementary book. Guillemin & Pollack's book was based in large part on this one, and could be read together, with G&P giving more elementary explanations and additional topics, while Milnor's book provides a proof of the Sard theorem and the Pontrjagin-Thom construction. The exercises, though not particularly difficult, do provide a good opportunity to practice proving theorems in the subject, as there are no hints for them, as one would find in many other differential topology books, and they are not separated by chapter.