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__Some background about me:__

I am not a physicist but I'd like to pursue a PhD in theoretical physics (after a year or two) and work on topological quantum computing. I am familiar with quantum mechanics and solid state physics (at the level of Ashcroft), concepts like fractional quantum hall effects, topological insulators etc that I learned in my advanced condensed matter course. I have no issues as such with the usual mathematical methods that are used by physicists and engineers; and proof methods,set theory etc. I'm studying rigorous linear algebra (Hoffman/Kunze) and quantum computation (Nielson and Chuang) now.

1. Should I go through pure topology books by Munkres or Mendelson, or look at books like Nakahara or Frankel that talk about physical applications?

2. What kind of background is required to study topology? Can I dive into topology right after acquiring the necessary background in formal mathematics or should I do analysis and/or algebra first?

3. What kind of computing/theoretical computer science knowledge is required?

4. Do theoretical physicists and people who work in this field actually learn pure math (or take classes on abstract algebra, topology etc) or do they use the math as and when needed?

I apologize if the questions are silly or stupid. Any input would be appreciated.