Topology needed for differential geometry

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, metrics and 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 the topics in differential geometry that are needed in physics in sufficient detail (without too much emphasis on mathematical rigour)? I have heard of the following textbook authors: Nakhara, Fecko, Spivak. Would you recommend these?

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You shouldn’t need much. Almost all you need to know about topology (especially of the point-set variety) should have been covered in a course in advanced calculus. That is to say, you really need to know about “stuff” in $\mathbb{R}^n$. (The one main exception is when you study instantons and some existence results are topological in nature; for that you will need to know a little bit about fundamental groups and homotopy.) The reason is that differential topology and differential geometry study objects which locally look like Euclidean spaces. This dramatically rules out lots of the more esoteric examples that point-set topologists and functional analysts like to consider. So most introductory books in differential geometry will quickly sketch some of the basic topological facts you will need to get going.

In terms of topology needed for differential geometry, one of the texts I highly recommend would be

  • J.M. Lee’s Introduction to Smooth Manifolds

It is quite mathematical and quite advanced, and covers large chunks of what you will call differential geometry also. One can complement that with his Riemannian Manifolds to get some Riemannian geometry also.

But since you are asking from the point of view of a Physics Undergrad, perhaps better for you would be to start with either (or both of)

  • Nakahara’s Geometry, Topology, and Physics
  • Choquet-Bruhat’s Analysis, Manifolds, and Physics: Vol. 1 and Vol. 2

and follow-up with

  • Greg Naber’s two book series: Topology, Geometry, and Gauge Fields: Foundations and Topology, Geometry, and Gauge Fields: Interactions