Topology from the Differentiable Viewpoint

by John Milnor

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.

Genres: Mathematics, Textbooks, Geometry, Stem, Technical, Science, Nonfiction

80 pages, Paperback

First published November 24, 1997

Reviews

[Woflmao · Rating 4 out of 5]

As usual with Milnor, this book is written in a very accessible and unpretentious way. It focusses only on the bare essentials of the theory, and it in some parts it is quite sketchy.

[Tom · Rating 5 out of 5]

A great book, but you have to know when there are gaps, and how to fill them in.

[Dow · Rating 5 out of 5]

This book fucking bangs. Milnor is the king.

[Simona · Rating 5 out of 5]

A first introduction to differential topology, very readable, I studied from it in my basic differential topology class. It is much less frightening to start with then alternatives because of length and the number of pictures.

[You Chi · Rating 5 out of 5]

This text is the most elementary one among John Milnor’s introductory writings on differential topology. Especially, I would recommend to those who have finished undergraduate analysis course to read this one. As an prominent figure in differential topology, Milnor’s exposition is clear, and his choice of topics is superb. He fully illustrated how to use techniques in multivariable calculus such as implicit/inverse function theorem to obtain global topological results on differential manifolds. For instance, using simple arguments based on regular value, he gives a simple proof on fundamental theorem of algebra.

One of highlight in the book is Sard’s theorem in chapter 2 and 3, and it’s proof is one the most technical parts in the book. Yet it worths the price. With Sard’s theorem, he is able to prove Brouwer fixed point theorem, which assets that continuous map on unit closed ball in Euclidean space must have a fixed point, without using concepts and machinery in algebraic topology.

Another highlight is that he defines the concepts of degree of a smooth map between smooth manifolds in both non-oriented and oriented case, and he proves that it is a smooth homotopy invariant. The application of this is that he gives a (yet incomplete) proof on Poincaré-Hopf theorem, which relates the local analytical invariant (index of vector field) and global topological invariant (Euler characteristic). This is a prototype of many important theorems in geometry of this kinds, such as Gauss-Bonnet-Chern theorem and the last Atiyah-Singer index theorem.

I read this book in my junior year as undergrad., and the my experience reading this text is unforgettable. It also triggers me to further pursue more advanced text on differential topology such as Bott and Tu’s “Differential Forms in Algebraic Topology” and the proof of Atiyah-Singer index theorem. I strongly recommend everyone who wants to do research in geometry or topology seriously to read this one.

[Lucas Chen · Rating 5 out of 5]

truly masterful. restores your faith in mathematics. makes you want to study topology all summer in the sweltering heat and stay at school while your friends make real memories with their lives.

serious disclaimers: go through every proof multiple times because milnor makes you mistakenly think you understand his proofs really easily, but he also does place the tools for really trying to get each detail right inside the text. read a real manifolds book after and understand atlases better. there are a lot of really terribly typeset versions of this book (professor may, if you ever read this please put a better version on your website). lastly, if you really want to feel significant because you're doing math for the sake of being an egotistical bastard, go watch milnor's original lectures on youtube because they're also so so brilliant.

[Yuxi Liu · Rating 5 out of 5]

Elegant little book that covers some of the basics of differential topology.

About the level of a first year graduate.

A good follow-up book would be Introduction to Smooth Manifolds (John Lee).

[Christopher Waller]

I enjoyed it for the most part, but I think Milnor tries so hard to streamline that he leaves out quite a few intermediary details the reader has to fill in for themselves or risk being completely lost.

[Marcos Vinicius Strieder · Rating 5 out of 5]

Great great book, Milnor is awesome