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.