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I'm a student of Physics and I have interest on the theory of Fiber Bundles because of the applications they have in Physics (gauge theory for example). What are good books to learn the theory of fiber bundles and connections that are rigorous but at the same time gives what we need to apply in Physics?
The obvious reference for understanding fiber bundles is Mikio Nakahara's book entitled Geometry, Topology and Physics (http://www.amazon.com/Geometry-Topology-Physics-Edition-Graduate/dp/0750306068).
Furthermore, a book that I find really interesting is the book Topology, Geometry and Gauge Fields: Foundations, by Gregory Naber and note that it focuses more on understanding gauge theories through differential geometry (http://www.amazon.com/Topology-Geometry-Gauge-fields-Foundations/dp/1461426820/ref=sr_1_1?s=books&ie=UTF8&qid=1413385257&sr=1-1&keywords=topology+geometry+and+gauge+fields). It has another volume on interactions.
Finally a good reference are thew following lecture notes by Lionel Brits but they serve more as a reminder rather than to learn from zero (http://www.thetangentbundle.net/papers/gauge.pdf).
Here's a permanent link in the wayback machine for the lecture notes by Brits, as I've noticed that he's been taking down files from the tangent bundle recently (the precious wikis are now completely gone).
Did you try to ask him why the useful and precious content of the wikis is getting removed?
@Dilaton I tried once, but the email bounced - I think even his email isn't working any longer.
I think I good book for that may be C. J. Isham's Modern Differential Geometry for Physicists. I haven't gotten to the chapter of fiber bundles, but what I've read seems to be quite rigorous. And as it is written for physicists, I think it could please your needs.
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