Characteristic classes are cohomology classes in the base space of a vector bundle, defined through obstruction theory, which are (perhaps partial) obstructions to the existence of everywhere linearly independent vector fields on the vector bundle. The most common examples of characteristic classes are the Chern, Pontryagin, and Stiefel-Whitney classes.
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Weisstein, Eric W. "Characteristic Class." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/CharacteristicClass.html