I have a couple of strong undergraduate students who want to learn differential topology as an independent study with me (they already had a semester of topology). Textbook recommendations? I used Guilleman and Pollack (+ Milnor's book) when I was a student.
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It really boils down to whether you want to try to introduce them to the idea of a manifold, as a collection of (compatible) charts. If they're already used to thinking of topological spaces as a collection of (adjacent) simplices, then that may be a good pedagogical investment.
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Thank you!
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