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Currently I'm self studying for my own enjoyment topology and algebra (munkres and herstein). Since I start at the university next year everything I'm learning now is for my own enjoyment and I will probably relearn it in the university anyway.

I'm interested in differential topology yet I haven't read a single book on analysis in the level of say, "baby Rudin". Not to say that I'm not familiar with it. I have done a lot of studying in the past on calculus (sequences, differentiation, integration, fourier series, integral transforms etc.) ODE's (and a bit PDE's) and non rigorous complex analysis (visual complex analysis) but my foundation are really far from being solid.

Is it possible to learn differential topology before strengthening my foundations in analysis and leave that to do once i get to the university?

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  • $\begingroup$ I don't know to what extent it is possible, but you could always glance at Milnor's Topology from the differential viewpoint or his book on Morse theory, if only to get a feel for the subject. $\endgroup$ – Olivier Bégassat Dec 20 '13 at 10:03
  • $\begingroup$ here is a nice answer. $\endgroup$ – Mhenni Benghorbal Dec 20 '13 at 10:07
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    $\begingroup$ Yes, it is definitely possible to learn differential topology without any strengthening of your foundations in analysis. In particular the content of "baby Rudin" is essentially irrelevant: rearrangement of series and gamma functions won't help you to understand, say, De Rham cohomology or Morse theory, nor even to compute in the tangent bundle! As an aside, I don't understand the American infatuation for that book, which I find boring, needlessly difficult and concise and without any show of enthusiasm for calculus, one of the most exciting and glorious achievements of mankind. $\endgroup$ – Georges Elencwajg Dec 20 '13 at 10:49
  • $\begingroup$ The important tools you should definitely understand include: integration with respect to curve, Picard-Lindelof theorem, inverse and implicit function theorems, bump functions... if these are fairly familiar, you shouldn't have too much trouble with the basics, as far as I can recall. $\endgroup$ – tomasz Dec 20 '13 at 11:45
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    $\begingroup$ Please don't bold random sections of your question. $\endgroup$ – Potato Dec 20 '13 at 18:48
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Yes. In fact, I recall the Stanford University Mathematics Camp doing what was basically a brief, partially-rigorous course in algebraic topology for high school students as their Program II, some years back. I imagine that your ideal program of study would have a substantial overlap with that course.

For example, the classification of closed surfaces is a very worthwhile theorem that can be proved using completely elementary methods, at least if one takes the Jordan Curve Theorem as a black box. I think that there are many such examples of important concepts in topology which could be learned before the so-called "foundations" of the subject.

In principle, one could learn quite a bit about homotopy and homology groups, line bundles, or whatever else strikes your fancy, without ever needing to dive into analytical foundations. It might even be preferable—for some—to learn the subject this way first, building geometric intuition, then returning later to fill in the gaps in rigor. Knowing in broad strokes how the hairy ball theorem works, for example, can motivate (co)homology and the Euler characteristic, which can be confusing ideas if first learned in a highly technical manner.

In general, don't let anything stop you from learning the material you want to learn, when you want to learn it, if your primary motivation is enjoyment. You do want solid foundations, but straying from the beaten path from time to time can be a very efficient way of developing unique personal strengths.

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