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Prerequisites for topology, differential geometry and Morse theory

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You could study from the differential topology textbook by Milnor himself (pretty short, about 80 pages) or by Pollack and Guillemin. You can also check Milnor’s lectures on the subject on YouTube.
Linear algebra and a good course in multivariable calculus is pretty much all you need before you start these but knowing some analysis and point-set topology should be good. You can check Rudin for these but there are better textbooks out there.
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You should learn some differential geometry, Loring Tu's book is a good introduction.
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For point set topology, Munkres is quite good and very clear. I don't know Morse theory, but I think a standard undergrad topology class covers the first 3 chapters or so worth of material.
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It might be a bit much to do in 1-2 years. I studied geometry primarily through UG and my MA, even then I only touched on Morse theory as an extra topic for one of my Masters classes. If you can get the prereqs for Lee's book 'Smooth Manifolds' and then read up to chapter 18, you can start on Morse theory.
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While normally I would certainly advise someone to learn Morse theory before discrete Morse theory, it is certainly not necessary. In your case, it might be better to jump straight into discrete Morse theory. The only thing I would recommend is to first find a topology or geometry professor at your school and have an hour long conversation with them where they explain what the idea of Morse theory is at a very high level (meaning without wandering into technical parts of the theory).

One thing you will have to accept is that any of these textbooks will have terms that are not familiar to you. You will have to constantly be looking at alternative resources while you learn this stuff.
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If you're willing to put a bit more time into it, you could try John Lee's *Intro to {Topological, Smooth, Riemannian} Manifolds* trilogy

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