So, I'm currently at the point of learning about derivatives and integrals in Elementary Real Analysis, I'm also learning Point set topology and I've already gone over the main definition of topology, basis, subspace topology, quotient topology, product topology up to continuous functions thus far. I also understand elementary linear algebra and know enough Abstract Algebra.

My question is whether or not I can start reading Differential Topology by Guillemin and Pollack. It states a full year of analysis and semester of linear algebra but I'm not exactly sure if my current level understanding is sufficient to start studying differential topology.

If not should I start going over Hatcher's book on Algebraic Topology? I've read that knowing algebraic topology before differential topology is a good idea.

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    $\begingroup$ I'd say Guillemin and Pollack is a good gentle start; very concrete. If you feel more ambitious John Lee's smooth manifolds book is also okay. I don't think alg top is a requisit for dif top; really one learns one of them picks up on the other as one goes. $\endgroup$ – chriseur Mar 24 '17 at 1:52
  • $\begingroup$ Just a quick side question, I saw John Lee's Riemannian Manifolds and wondered if I really need to go through his smooth manifolds to understand basic Riemannian Manifolds? $\endgroup$ – Alexander King Mar 24 '17 at 2:08
  • $\begingroup$ Not the whole book for sure. Basic chapters (first four or so I think?) + vector bundles would probably suffice, depending on how thoroughly one wants to understand it. $\endgroup$ – chriseur Mar 24 '17 at 2:13
  • $\begingroup$ Thanks. I'm just trying to see what kind of prerequisites are really needed to study manifolds and even algebraic topology. $\endgroup$ – Alexander King Mar 24 '17 at 2:15

I would say no. In my experience, in order to really study differential topology you need to have a firm ground in multivariable calculus. In particular, things like understanding the derivative is a linear map (best linear approximation), implicit function theorem, inverse function theorem, etc.

Ted Shifrin has a really awesome book that will give you the relevant linear algebra and multi- calculus to begin studying differential topology here. I would also recommend Tu's introduction to manifolds, or Lee who develop a bit more geometric understanding of tangent space (than G & P) which is the first topic to digest. A lot of users may not like this but I think the order of things follows best if you go,

$$\textbf{linear algebra + multivariable calculus} \to \textbf{differential geometry} \to \textbf{differential topology}$$

  • $\begingroup$ That's what I thought. Guess I'll stick with the route suggested above since it seems a lot wiser. $\endgroup$ – Alexander King Mar 25 '17 at 0:43
  • $\begingroup$ If you are pleased with this answer, please accept to close the question. $\endgroup$ – Faraad Armwood Mar 26 '17 at 6:04
  • $\begingroup$ Yes, I'm satisfied with the answer. $\endgroup$ – Alexander King Mar 26 '17 at 16:52
  • $\begingroup$ You can accept it on your end. $\endgroup$ – Faraad Armwood Mar 26 '17 at 16:56
  • $\begingroup$ Can the downvoter please explain? The question of the OP is clearly one which leaves room for a wide range of responses. I truly believe the outline above is the way to since that was how it all worked for me, what else do I have to go off of? The OP is also pleased with the answer and even more so, agreed that this was also his/her thoughts as well. The fact that you (downvoter) feels differently is irrevelvant if the OP already has his/her mindset on how they wish to learn the subject, which agrees with my way. $\endgroup$ – Faraad Armwood Mar 27 '17 at 12:32

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