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The cohomology rings of homogeneous spaces
Journal of Topology  (IF1.582),  Pub Date : 2021-11-22, DOI: 10.1112/topo.12213
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 $k$. It is known that in this case the cohomology of the homogeneous space $G / K$ with coefficients in $k$ and the torsion product of $H ∗ ( B K )$ and $k$ over $H ∗ ( B G )$ 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.