The cohomology rings of homogeneous spaces

Abstract

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.