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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.