I am a physics undergrad, and need to study differential geometry ASAP to supplement my studies on solitons and instantons. How much topology do I need to know. I know some basic concepts reading from the Internet on topological spaces, connectedness, compactness, metric, quotient Hausdorff spaces. Do I need to go deeper? Also, could you suggest me some chapters from topology textbooks to brush up this knowledge. Could you please also suggest a good differential geometry books that covers diff. geom. needed in physics in sufficient detail, but not too mathematical? I heard some names such as Nakahara, Fecko, Spivak. How are these?
You need to know the rudiments of the application of algebraic topology to the classification of bundles on manifolds. If you're self teaching using the internet, it would be useful to look up "characteristic classes", and work backwards from there, filling in the gaps that you need.
Nakahara is a good introduction to this material, as is Eguchi, Gilkey and Hanson
I'm an undergrad myself studying string theory and I think every physicist should have "Nakahara M. Geometry, Topology and Physics".
In fact I became a bit of a math junky after my first real math classes and bought a ton of books (including some mentioned above by other commenters). They were all a waste of money (not completely) but Nakahara's book has pretty much all the math i've ever needed in a much easier format. For instance I find Hatcher's book nice but daunting because of how dense/huge the sections are on certain topics.
Nakahara's book is short and succinct but with the best notation (consistent at least with QFT/string books I read) and if you need any extra details you can probably just use wikipedia.
Pretty much every time I dive into this book I gain a deeper insight into something I didn't even realize I had a "fuzzy" understanding of.
Ok that's enough of the commercial but I seriously can't recommend this book enough!
As for algebraic topology you start with Armstrong's Basic topology or the last portion of Munkre'sTopology then move to Hatcher's AT [http://www.math.cornell.edu/~hatcher/AT/ATpage.html].if want to learn differential geometry online see Zaitsev D. Differential Geometry: Lecture Notes (FREE DOWNLOAD) and Hicks N.J. Notes on Differential Geometry(FREE DOWNLOAD). some others are Spivak's Calculus on Manifolds: A Modern Approach to Classical Theorems of Advanced Calculus,Fecko's Differential Geometry and Lie Groups for Physicists,Isham C.J. Modern Differential Geometry for Physicists, Nakahara M. Geometry, Topology and Physics, Nash C. and Sen S. Topology and Geometry for Physicists and the free online S.Waner's Introduction to Differential Geometry and General Relativity. Hope this will be useful.
My favourite book is Charles Nash and Siddhartha Sen Topology and geometry for Physicists. It has been clearly, concisely written and gives an Intuitive picture over a more axiomatic and rigorous one.
For differential geometry take a look at Gauge field, Knots and Gravity by John Baez. Its very well written starts from the very basics and works its way upwards.