Texts on Principal Bundles, Characteristic Classes, Intro to 4-manifolds / Gauge Theory

Question

I am looking for a textbook that might serve as an introduction to principal bundles, curvature forms and characteristic classes, and perhaps towards 4-manifolds and gauge theory.

Currently, the only books I know of in this regard are:

• "From Calculus to Cohomology" (Madsen, Tornehave)
• "Geometry of Differential Forms" (Morita)
• "Differential Forms in Algebraic Topology" (Bott, Tu)

I have been reading both "Calculus to Cohomology" and "Geometry of Differential Forms," but am occasionally frustrated by the lack of thoroughness. Both are at the perfect level for me, and cover almost exactly what I'm looking for, but I really prefer textbooks which are as thorough as possible, ideally to the extent of, say, John Lee's books (which I adore). Meanwhile, Bott and Tu is a little advanced for me right now.

Of course, I don't mean to be picky, but I also can't believe that the three I've listed are the most thorough accounts of the subject.

Answer 1

You might find the following useful: G.L. Naber, Topology, Geometry, and Gauge Fields: Foundations, 2nd.. It has a specific aim and purpose though: it's oriented towards those who want to learn the math foundations for gauge theory within a rigorous setting. Maybe pure math students might like a more broader approach.

Answer 2

If you want to focus on four manifolds, don't forget the classic of Donaldson and Kronheimer, Geometry of four manifolds. It may be a tad on the advanced side, but does contain some information specifc to 4 dimensions not available in the other books you listed.

Answer 3

C.H. Taubes, Differential Geometry: Bundles, Connections, Metrics and Curvature (Oxford Graduate Texts in Mathematics) might also be helpful. ... It hasn't been released yet, but given the author's fame and stature I think it might be a good pick. ...