Book on characteristic classes

Question

i am looking fo books on characteristic classes as main argument. I'd appreciate if you add a little comment on why you would study on it/them. Thank you.

Answer 1

The following is a celebrated classic. J. Milnor is a Fields medalist, famous for the power of his mathematical thinking and the clarity and precision of his style.

Milnor, John W.; Stasheff, James D., Characteristic classes, Annals of Mathematics Studies. No.76. Princeton, N.J.: Princeton University Press and University of Tokyo Press. VII, 331 p. (1974). ZBL0298.57008.

Answer 2

Also D. Husemoller - Fibre Bundles, 3rd edition, part III is a good reference.

Answer 3

For someone coming from the complex geometry perspective, I would suggest reading some combination of Chern (Complex Manifolds without Potential Theory), Wells (Differential Analysis on Complex Manifolds), and particularly Griffiths and Harris. Only the latter discusses the important interpretation of the $j$th Chern class of a vector bundle of rank $k$ in terms of the locus where $k-j+1$ generic smooth sections of the bundle become linearly dependent.