  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 $k$. It is known that
in this case the cohomology of the homogeneous space $G/K$ with coefficients in
$k$ and the torsion product of $H^{*}(BK)$ and $k$ over $H^{*}(BG)$ are
isomorphic as $k$-modules. We show that this isomorphism is multiplicative and
natural in the pair $(G,K)$ provided that 2 is invertible in $k$. The proof
uses homotopy Gerstenhaber algebras in an essential way. In particular, we show
that the normalized singular cochains on the classifying space of a torus are
formal as a homotopy Gerstenhaber algebra.
