Chern–Weil homomorphism

Summary

The Chern–Weil homomorphism turns invariant polynomials on the Lie algebra of a group $G$ into de Rham cohomology classes. These classes live on the base of a principal $G$-bundle. One chooses a connection, plugs its curvature form into the polynomial, and obtains a closed form whose cohomology class does not depend on the connection. The classes obtained are the real characteristic classes of the bundle, such as Chern and Pontryagin classes. It links local differential geometry, through curvature, with global topology.