The theory of characteristic cohomology classes of bundles especially vector bundles grew up in several contexts like the notion of cohomology of a space. It was natural to try and make calculations with bundles in terms of cohomology for two reasons. With homology and cohomology, there were combinatorial tools for computation. Then, cohomology and bundles each had contravariant properties under continuous mappings. The first definitions of characteristic classes were given by obstructions to existence to cross sections of a bundle or related fibre bundle. Examples of this can be found in the book by Steenrod (1951).
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References
Atiyah, M.F., Hirzebruch, F.: Vector bundles and homogeneous spaces. Proc. Symp. Pure Math. Amer. Math. Soc. 3:7–38 (1961)
Epstein, B.D.A., Steenrod, N.E.: Cohomology operations, Annals of Math. Studies 50, 1962
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Steenrod, N.E.: Topology of Fibre Bundles. Princeton Univ. Press, Princeton, 1951
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Husemöller, D., Joachim, M., Jurčo, B., Schottenloher, M. (2008). Basic Characteristic Classes. In: Basic Bundle Theory and K-Cohomology Invariants. Lecture Notes in Physics, vol 726. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-74956-1_11
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