The cohomology of homogeneous spaces in historical context

Abstract

The real singular cohomology ring of a homogeneous space $G/K$, interpreted as the real Borel equivariant cohomology $H^*_K(G)$, was historically the first computation of equivariant cohomology of any nontrivial connected group action. After early approaches using the Cartan model for equivariant cohomology with real coefficients and the Serre spectral sequence, post-1962 work computing the groups and rings $H^*(G/K)$ and $H^*_H(G/K)$ with more general coefficient rings motivated the development of minimal models in rational homotopy theory, the Eilenberg-Moore spectral sequence, and A-infinity algebras. In this essay, we survey the history of these ideas and the associated results.