I am an undergraduate and just finished my uni's last undergraduate analysis course. Our text was Munkres' Analysis on Manifolds, which I read cover-to-cover and enjoyed. Now, I am looking for a related 'next read.' In particular, I want to read something that goes into more detail on any subset of the following set: $\{$
- De Rham cohomology (or in general, connections to algebraic topology$-$I have also just finished a first graduate course in algebraic topology.)
- The theory of fiber bundles (I'm not actually certain what these are, but I think they're supposedly a good 'next step.')
- (Co)bordism
- A categorical development to the same content (preferably appropriate for someone who does not know much category theory)
$\}$.
Any recommendations would be appreciated. Let me know if you think there are other topics worth tackling first, or your thoughts on the topics I have listed.
Bott and Tu is a beautifully-written book that hits the points you mention. (If it has a flaw, it's that it focuses on the smooth category, but that's not a problem if you're interested in De Rham cohomology.) Milnor and Stasheff is the canonical reference for the general theory of vector bundles, though Husemoller might be a better choice if you're interested in more than just classifying spaces and characteristic classes. I don't remember how much those two books get into cobordism, but I'd recommend May's "Concise Course in Algebraic Topology" for it. (Note, though, that even though it's a very well-written book, it's designed for people who already have some familiarity with the subject, and the idea behind the book is to say, "Well, here's what was secretly going on the whole time." Think of it as analogous to revisiting an intro-level calculus class using the machinery of modern measure theory instead of Riemann integration.)