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.

中文翻译：

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齐次空间的上同调环

让$G$是紧连通的李群，并且$ķ$一个封闭的连通子群。假设积分上同调中任何扭转元的阶数 $G$和 $ķ$在给定的主理想域中可逆 $\mathbb{ķ}$. 已知在这种情况下，齐次空间的上同调 $G/ķ$系数在 $\mathbb{ķ}$和的扭转积 $H^{*}(乙ķ)$和 $\mathbb{ķ}$超过 $H^{*}(乙G)$同构为$\mathbb{ķ}$-模块。我们证明了这种同构在对中是可乘的和自然的 $(G,ķ)$假设 2 是可逆的 $\mathbb{ķ}$. 证明以一种基本的方式使用同伦 Gerstenhaber 代数。特别是，我们证明了环面分类空间上的归一化奇异 cochains 是形式上的同伦 Gerstenhaber 代数。