Differential Topology

by Victor W. Guillemin, Alan Pollack

This text fits any course with the word "Manifold" in the title. It is a graduate level book.

Genres: Mathematics, Technical

222 pages, Hardcover

First published August 14, 1974

About the author

Victor William Guillemin mathematician

Reviews

[Moetanabbi · Rating 4 out of 5]

This is one of the books in a great American tradition of textbooks which are clear, well presented, aimed at helping the student understand, rather than at showing the scholarly knowledge of the authors. I first read the book in my twenties and reread it in my fifties, always a risky thing to do if you are enthusiastic about it in the beginning. The experiment was certainly a success.
The assumption that all manifolds are a priori embedded in a large Euclidean space may avoid a lot of technicality, but it is sometimes confusing, as a lot of theorems depend on a local parametrisation anyway, i.e. the classical approach.
On the other hand, proving Hopf's degree theorem with such a small technical apparatus is really a tour de force. I enjoyed chapter 4 most and will try to discover some of the titles mentioned in the bibliography to explore the items in it further.
A must read if you are relatively new to topology and want to understand why people are enthusiastic about it.

[Daniel]

a real thriller!

[Brian Powell · Rating 4 out of 5]

This is a highly accessible book for first-year grad students who have had some analysis, linear algebra, and know some basic topology. The text is organized around a collection of classic theorems, arrived at and appreciated using the smoothness properties of manifolds (the differential in differential topology). These include the theorems of Borsuk-Ulam (aka the balloon-animal theorem), Poincare-Hopf, Jordan-Brouwer separation, and Gauss-Bonnet, among others.

We come to behold the deep, topological (global) content of these theorems as a explicable ultimately from the differential (local!) properties of mappings; that is, we can understand what it means to be a sphere topologically (in terms of say the Euler characteristic) by seeing how various maps behave on the surface. It's like guessing the identity a person by the clothes they wear. The central idea underlying this view is the concept of transversality of manifolds: two transverse submanifolds essentially 'fill out' the space of which they are a part (the x and y axes are transversal in R^2, for example). The concept allows us to define a more generic notion of 'intersection', and serves as the basis of a powerful set of ideas known as intersection theory.

The text could be considered a more readable, expansive, and verbose version of Milnor's classic text, and serves as an appropriate first look at the subject.

[Ella · Rating 5 out of 5]

This book is my bible. So many fun passages and pictures:
• Cover image as an application of the Jordan curve theorem
• Snail
• Squid
• Pants
• Chocolate fudge
• "Without transversality, X ∩ Z may be some frowzy, useless conglomeration."
• "If our propaganda has not yet made you a true believer in forms, we invite you to try defining the integral of a function." 😎

[Chris · Rating 2 out of 5]

There's a great picture in here that every high school or college math student wondering what continuity means should see. The book is well written. However, differential topology is a boring topic.

[Jeff · Rating 5 out of 5]

I'm kinda upset that the cover my copy isn't as cool as the one pictured here but...

This book is phenomenal. Reasons to buy it:

1) Concise. You'll get so much information in such a short, well-written book.
2) Intersection theory mod2. I haven't found a comparable treatment of this idea anywhere.
3) Transversality. Excellent treatment of transversality.

[Henry · Rating 5 out of 5]

One of the most well-written math book I've ever read

[Lulu]

Geometry and topology, straight up