Equivariant cohomology and the Maurer-Cartan equation

Abstract

Let $G$ be a compact, connected Lie group acting smoothly on a manifold $M$. In their 1998 article [7], Goresky, Kottwitz, and MacPherson described a small Cartan model for the equivariant cohomology of $M$, quasi-isomorphic to the standard (large) Cartan complex of equivariant differential forms. In this article, we construct an explicit cochain map from the small Cartan model into the large Cartan model, intertwining the $(S{\mathfrak{g}}^{*}){}_{\mathrm{inv}}$-module structures and inducing an isomorphism in cohomology. The construction involves the solution of a remarkable inhomogeneous Maurer-Cartan equation. This solution has further applications to the theory of transgression in the Weil algebra and to the Chevalley-Koszul theory of the cohomology of principal bundles