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I am just about to finish my study of Milnor's book 'Topology from the Differentiable Viewpoint' and I really love the subject. I would like to continue my study of Differential Topology and am looking for some good references. What are the canonical choices of textbooks for this subject ? I am aware of the book by Morris Hirsch but am not sure what are the prerequisites or whether it is a good book. I have almost no knowledge of Algebraic Topology and my knowledge of analysis in limited to undergrad level real and functional analysis. I have a background knowledge of smooth manifolds and differential geometry (Riemannian metrics, curvature, connections etc.) Thanks !

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    $\begingroup$ You may find Bott-Tu useful in bridging the gap between differential topology and algebraic topology. $\endgroup$ – Neal Dec 21 '13 at 18:02
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If you're going to continue in topology, you should learn algebraic topology. Hatcher is a good place to start, and with your background, I would suggest Bott-Tu to bridge the gap between differential and algebraic topoogy.

For other differential topology books, Hirsch is good, as is Guillemin-Pollack. (The latter is standard introductory material; as I have not read Milnor's book, I do not know how strong the overlap is.)

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  • $\begingroup$ Thanks! About Bott & Tu , I wish to ask : is it a good idea to learn basic homological algbera and spectral sequences (say from Weibel ch 1-5) before reading Bott-Tu or is it more logical to read Bott & Tu first and then read homological algbera ? $\endgroup$ – user90041 Dec 21 '13 at 18:07
  • $\begingroup$ And which order would you suggest : learning Algberaic Topology from Hatcher first and then reading Bott-Tu or the other way around ? Thanks ! $\endgroup$ – user90041 Dec 21 '13 at 18:10
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    $\begingroup$ Guillemin and Pollack was inspired by Milnor's book, but there's a good deal of stuff that's quite expanded and somewhat different (including differential forms, Stokes's Theorem, and the Degree Theorem). But you sound like you have a solid analysis (single and multivariable) background, so Hirsch's book is definitely appropriate for you. $\endgroup$ – Ted Shifrin Dec 21 '13 at 18:16
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    $\begingroup$ Hatcher is elementary and is an appropriate textbook for a very first algebraic topology course. More importantly, though, don't get hung up on order -- you should never read things "in order"! Learning is absolutely not a linear process. Read them concurrently and use one to get you unstuck in the other. $\endgroup$ – Neal Dec 21 '13 at 18:18
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    $\begingroup$ Hatcher is a trivial book, real men read spanier!!!! $\endgroup$ – user111072 Dec 22 '13 at 1:53

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