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 $H^{\ast }(BK)$ and $k$ over $H^{\ast }(BG)$ 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.

52 pages; new Sections 2.2 (Notation) and 13 (Examples), appendix expanded, minor changes

References in corpus (1)

• Homotopy Gerstenhaber algebras are strongly homotopy commutative

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