Characteristic class

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The article on this topic in the Topology 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)).