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I am a physics undergrad, and interested to learn Topology so far as it has use in Physics. Currently I am trying to study Topological solitons but bogged down by some topological concepts. I am not that interested for studying it for its own sake. Please could you mention the topics of Topology that are required in Physics? Could anyone recommend me a book that deals with these topics and also some applications to Physics. I have taken an introductory course in Real Analysis (Sherbert, Apostol, etc), and have no knowledge of complex analysis.

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If you want to learn topology wholesale, I would recommend Munkres' book, "Topology", which goes quite far in terms of introductory material.

However, in terms of what might be useful for physics I would recommend either:

  • Nakahara's "Geometry, Topology and Physics"
  • Naber's "Topology, Geometry and Gauge Fields: Foundations"

Personally, I haven't read much of Nakahara, but I've heard good things about it, although it may presuppose too many concepts. I've read selections of Naber and it seems fairly well written and understandable and starts from first principles, but again, it may not focus as much on the fundamentals, if that's what you're looking for.

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What about Topology for physicists by Schwarz? –  ramanujan_dirac Jun 12 '12 at 10:48
    
Sorry, I haven't heard anything of that book. However, judging by the absolutely ridiculous price on Amazon... Anyway, I was also flipping through Nash and Sen's book, and it seemed to treat topology in a very intuitive and clear manner, although at a mathematical price - Amazon reviewers claim that it isn't too mathematically rigorous/comprehensive. –  Nilay Kumar Jun 15 '12 at 8:40
    
I enjoyed reading Nash and Sen, It suited my taste, being less formal, and more intuitive. Nakahara is nice. Schwarz seemed good at first glance, but I havent read it. –  Prathyush Feb 14 '13 at 21:29
    
You also asked about topics in topology relevant for physics. Apart from the basic definitions and so on, one of the most applied concepts is Homotopy. It is beautiful in itself, and it formalizes the concept of winding numbers to higher dimension. In physics it is commonly used to enumerate the topological solitons present in your theory.There are others, but I found Homotopy to be very important and useful. –  Prathyush Feb 14 '13 at 21:39
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Don't hurry ramanujan, learn basic mathematical methods first (from Sadri-Hassani's "Mathematical Physics" for instance). Then the standard reference for you to learn grad-level mathematics would be Nakahara's "Geometry, Topology and Physics". If you think it's too much, you're right; this is a very serious advanced topic. But if you want to quickly pick some basic ideas, check out the 10th chapter of Ryder's "Quantum Field Theory". An advanced and physically oriented discussion would be found in Coleman's "Aspects of Symmetry".

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What about Topology for physicists by Schwarz? –  ramanujan_dirac Jun 12 '12 at 10:56
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