Matthias Franz

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 $\mathbb{k}$. It is known that in this case the cohomology of the homogeneous space $G/K$ with coefficients in $\mathbb{k}$ and the torsion product of ${H}^{\ast}(BK)$ and $\mathbb{k}$ over ${H}^{\ast}(BG)$ are isomorphic as $\mathbb{k}$-modules. We show that this isomorphism is multiplicative and natural in the pair $(G,K)$ provided that 2 is invertible in $\mathbb{k}$. 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.