Characteristic class

The article on this topic in the Differential Geometry Wiki can be found at: characteristic class

Definition

Let $G$ be a topological group. A characteristic class of principal $G$-bundles is a natural transformation from the contravariant functor $b_G$ (which sends any topological space to the set of isomorphism classes of principal $G$-bundles on it) to the cohomology functor.

For a given topological space $X$, a characteristic class of principal $G$-bundles associates, to every principal $G$-bundle $P\to X$, an element $c(P)\in H^{*}(X)$, such that if $f:X\to Y$ is a continuous map, then $c(b_g^{*}(f)(P))=H^{*}(f)(c(P))$.

When we talk of characteristic classes of vector bundles, we are implicitly thinking of characteristic classes for the associated principal $GL(n)$-bundle.