Equivariant cohomology and the Maurer-Cartan equation

Details Equivariant cohomology and the Maurer-Cartan equation Equivariant cohomology and the Maurer-Cartan equation

2004-06-17

arxiv.org/abs/math/0406350v2

Duke Mathematical Journal 130 (2005), 479-521

Mathematics

Differential Geometry

33 pages. Final version, to appear in Duke Mathematical Journal

Scientific paper

Let G be a compact, connected Lie group, acting smoothly on a manifold M. Goresky-Kottwitz-MacPherson described a small Cartan model for the equivariant cohomology of M, quasi-isomorphic to the standard Cartan complex of equivariant differential forms. In this paper, we construct an explicit cochain map from the small Cartan model into the standard Cartan model, 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.

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