# Characteristic class

From Diffgeom

*The article on this topic in the Topology Wiki can be found at:* characteristic class

## Definition

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

For a given topological space , a characteristic class of principal -bundles associates, to every principal -bundle , an element , such that if is a continuous map, then .