Characteristic class

Summary

A characteristic class assigns to each bundle of a given type over a space $X$ a cohomology class of $X$, compatibly with pullbacks. It measures how far a bundle is from being trivial: the trivial bundle has only the trivial classes. The main families are the Stiefel–Whitney classes, the Chern classes, the Pontryagin classes and the Euler class. Through the classifying space, characteristic classes of principal $G$-bundles correspond to cohomology classes of $BG$.