Equivariant cohomology and the Maurer-Cartan equation

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) Cat-tan complex of equivariant differential forms. In this article, We construct all explicit cochain map from the small Cartan model into the large Cartan model, intertwining the (Sg*)(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.