Characteristic classes in Borel cohomology (Q580770)

Consider a closed normal subgroup \(\Pi\) of a topological group \(\Gamma\) with quotient group G. If Y is a \(\Gamma\)-space such that \(\Pi\) acts freely on it then the projection \(Y\to Y/\Pi\) is a G-map and it is called a principal \((G,\Pi)\)-bundle. There exists a universal (G,\(\Pi)\)-bundle E(G,\(\Pi)\to B(G,\Pi).\) The authors shows that \(H^*_ G(B(G,\Pi)) = H^*(B\Gamma)\) where \(H^*_ G(-) = H^*(EG\times_{G}-)\) is the Borel cohomology. In the special case when \(\Gamma =G\times \Pi\) he observes that characteristic classes of a (G,\(\Pi)\)-bundle over a G-space X with values in the Borel cohomology are defined by the \(H^*(BG)\)-module structure on \(H^*_ G(X)\) and the ordinary characteristic classes of the \(\Pi\)-bundle \(EG\times_ GY\to EG\times_ GX\).