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I want to self-study differential topology. I'd like to hear suggestions from you about appropriate books that I could use while studying.

Note: I have not studied differential topology before. I self-studied general topology and some algebraic topology before.

Thank you

Amr
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2 Answers2

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A standard introductory textbook is Differential Topology by Guillemin and Pollack. It was used in my introductory class and I can vouch for its solidity. You might also check out Milnor's Topology from the Differentiable Viewpoint and Morse Theory. (I have not read the first, and I have lightly read the second.)

For other books on topology, Hatcher has a nice list here. You may be interested in books like Bott-Tu or others listed under item III, manifold theory.

Neal
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  • +1 Thanks for your answer. For the first book that you mentioned, am I missing any of its prerequisites ? – Amr Jul 18 '13 at 13:52
  • @Amr Have you studied analysis and measure theory, enough to be familiar with Lebesgue integrals? – Neal Jul 18 '13 at 13:53
  • I did, but I did not go to deep. I studied measure theory from an analysis book. There were 2 chapters on measure theory. I did not study a pure measure theory book. Is this OK ? – Amr Jul 18 '13 at 13:55
  • I am familiar with lesbesgue integrals as well. – Amr Jul 18 '13 at 13:59
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    What you need is a multivariable analysis course -- familiarity with the derivative as a linear map, chain rule, inverse/implicit function theorems. You don't need any abstract measure theory, although the notion of measure zero will show up. – Ted Shifrin Jul 18 '13 at 14:13
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Since you're new to differential topology but have already self-studied general topology and some algebraic topology, you’re in a good position to dive in. Here are some standard and widely respected texts, along with brief comments to help you choose:

1. John M. Lee – Introduction to Smooth Manifolds: This is one of the most comprehensive and pedagogically friendly books for beginners. It starts from the basics (like smooth structures and tangent spaces) and builds up to more advanced topics such as differential forms and de Rham cohomology. It's especially well-suited if you appreciate rigorous development and clear exposition.

2. Victor Guillemin & Alan Pollack – Differential Topology: This book takes a more geometric and intuitive approach, emphasizing transversality, intersection theory, and differential topology proper (e.g., the Whitney embedding and immersion theorems). It's less heavy on differential geometry and more focused on applications of smooth maps.

3. Frank W. Warner – Foundations of Differentiable Manifolds and Lie Groups: This one is a bit more advanced and assumes some mathematical maturity. It’s great for building a bridge to Lie groups and differential geometry but might feel dense if you’re just starting out. Consider tackling it after or alongside Lee’s book for a broader perspective.

You might consider starting with Lee’s Introduction to Smooth Manifolds, and then moving to Guillemin & Pollack once you're comfortable with the basics. Warner’s book is a good reference as you progress deeper, especially if you plan to explore connections with Lie theory.

Kishalay Sarkar
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