Topology from the Differentiable Viewpoint by John Willard Milnor (PDF)

Description

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.

User’s Reviews

Editorial Reviews: Review “John Milnor, Winner of the 2011 Abel Prize from the Norwegian Academy of Science and Letters””John Willard Milnor, Winner of the 2011 Leroy P. Steele Prize for Lifetime Achievement, American Mathematical Society”

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐A short booklet focus on the intuition behind manifold. Concise and illuminating.

⭐An excellent accompaniment to the vid.eo lectures available on linbe . To find them simply Google: “John Milnpr, diffddrential gopology lectures”, a sdries of three 60-90 minute lectures.

⭐This is a fantastic book on intro to differential topology. Milnor is amazing.

⭐My boyfriend bought this book for his favorite class in college. The condition of the book is great, (Of course, it’s brand new)…and I think he really love the contents and I think it’s interesting too, through his explanation.

⭐Very interesting little book.

⭐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

⭐or Hirsch’s

⭐, 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.

⭐Despite the lovely subject matter covered in this book, it more importanty gives one a taste of Mathematics as an intellectual discipline. It in outline shows how a mathematical theory – in this case Differential Topology – is constructed and consquently what mathematicians actually do and think about.Anyone who would like to appreciate Mathematics as a field of study rather than just learn some math should open this book.Better still, the prerequisite is only multivariate calculus!I have long thought this book should be the third year of calculus rather than differential equations or complex analysis.Additionally, for the novice it is the only entry I know of into the mysteries of high dimensional geometry, that amazing almost unbelieveable accomplishment of the human mind.There is a Star Trek episode in which a blind woman wears a dress of sensors which enable her to know more about her environment than a person can know from seeing. She knows exact distances and dimensions, can detect minute movements, can process the complete spectrum of light. In some sense she sees better. Modern topology and geometry are like that sensor dress for seeing higher dimensions. While we can not visualize the sphere in 5 dimensions, we know more about it from these mathematical theories than a five dimensionally sighted being ever could.Today, mathematics is often considered to be just a practical tool – like a spread sheet – or a toaster oven. We forget its power to widen our imagination, to frame the unimaginable. This book reminds us of this and shows why Mathematics is the Queen of Sciences.

⭐This book packs a lot of interesting material into a small volume. E.g., I picked up another book recently that started talking about cobordisms right off the bat; despite my having a couple of shelves full of well-known Dover, Springer, Cambridge UP etc. books on topology, differential geometry, mathematical physics, etc., Milnor’s tiny book was the only one I found that could help me understand what cobordisms are right away. The book also uses many illustrations to help understanding.I demote this to 4 stars only because Princeton UP’s price is a bit high; many years ago I was lucky enough to find a used copy of the old U. Virginia edition, and paid much less.

⭐Milnors Werke zählen zu den Klassikern. Sein Schreibstil ist prägnat und äußerst elegant. Dies Buch ersetzt zwar kein vollständiges Buch über Differentialgeometrie, siehe De Carmo, Pollak. Milnor nimmt in seinem Buch nur den Idealfall an, so gelingen ihm sehr kurze Beweise einiger wichtiger Theoreme der Topology. Es lohnt sich, dieses Büchlein immer Griffbereit zu haben. Milnor ist einer der Großmeister seines Fachs. Milnor konnte prägende Impulse in der Topology setzen. Dieses Büchlein ist ein Muss haben Buch. Zum Glück sind auch andere Werke von Milnor noch greifbar, wie das Buch “Morse Theory” oder “K-Theory”.perfecto, es un libro impresionante por su sencillez y profundidad, escrito por un matemático de primerísima fila, un auténtico genio.ok

⭐前半はサードの定理、後半は枠付きコボルディズムを中心に書かれている入門書です。これらは力学系や結び目などにおいてよく使用される項目である。本書は数学書であるが、これらの項目を理解したいと考えている方にも有用だと思います。分量は多くありませんが、簡潔に書かれており読みやすいと思います。 また、本書の邦訳「微分トポロジー講義」も出版されているので、日本語でじっくりと読み直すことも可能です。理論物理専攻の方も一読の価値ありです。

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