Let \(G\) be a compact connected Lie group and \(K\) a closed connected subgroup. Assume that the order of any torsion element in the integral cohomology of \(G\) and \(K\) is invertible in a given principal ideal domain \(R\). It has been known for a long time that the cohomology of the homogeneous space \(G/K\) with coefficients in \(R\) and the torsion product of \(H^*(BK)\) and \(R\) over \(H^*(BG)\) are isomorphic as \(R\)-modules in this case. I will explain that this isomorphism is in fact multiplicative and natural in the pair \((G,K)\) provided that 2 is invertible in \(R\). The proof uses homotopy Gerstenhaber algebras in an essential way.