# The cohomology rings of homogeneous spaces

## TL;DR

This paper proves that under certain conditions, the cohomology of homogeneous spaces G/K has a natural multiplicative structure, using homotopy Gerstenhaber algebras, especially when 2 is invertible in the coefficient ring.

## Contribution

It establishes the multiplicative and natural isomorphism of cohomology rings of homogeneous spaces with a novel use of homotopy Gerstenhaber algebras.

## Key findings

- Cohomology of G/K is isomorphic to the torsion product of H*(BK) and k over H*(BG).
- Normalized singular cochains on classifying spaces of tori are formal as homotopy Gerstenhaber algebras.
- The isomorphism is multiplicative and natural when 2 is invertible in the coefficient ring.

## 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 $H^{*}(BK)$ and $k$ over $H^{*}(BG)$ 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.