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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Replying to @divbyzero
Do they already know at least some combinatorial topology? If not, then I would strongly suggest supplementing Guillemin & Pollack with Henle's book, which (Hallelujah!) is available in Dover paperback (and is a masterpiece.) Maybe also consider using Bishop & Goldberg?
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Replying to @MathPrinceps @divbyzero
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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Guillemin & Pollack, as well as Milnor, try to skirt the issue of saying what a manifold is by speaking only of submanifolds of R^n. Seems like a shrewd strategy, and in a sense it "works." But it's wildly ahistorical, and separates the subject from its most natural applications.
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