A Small Model for the Cohomology of Some Principal Bundles

Let G be a compact, connected and simply connected Lie group, and {\Omega}G the space of the loops in G based at the identity. This note shows a way to compute the cohomology of the total space of a principal {\Omega}G-bundle over a manifold M, from the cohomology of G, the differential forms on M and the characteristic classes of the bundle. The equivariant situation is also treated.