Equivariant Cohomology and the Cartan Model

Abstract

If a compact Lie group G acts on a manifold M , the space M/G of orbits of the action is usually a singular space. Nonetheless, it is often possible to develop a ’differential geometry’ of the orbit space in terms of appropriately defined equivariant objects on M . In this article, we will be mostly concerned with ’differential forms onM/G’. A first idea would be to work with the complex of ’basic’ forms on M , but for many purposes this complex turns out to be too small. A much more useful complex of equivariant differential forms on M was introduced by H. Cartan in 1950, in [2, Section 6]. In retrospect, Cartan’s approach presented a differential form model for the equivariant cohomology of M , as defined by A. Borel [8] some ten years later. Borel’s construction replaces the quotient M/G by a better behaved (but usually infinite-dimensional) homotopy quotient MG, and Cartan’s complex should be viewed as a model for forms on MG. One of the features of equivariant cohomology are the localization formulas for the integrals of equivariant cocycles. The first instance of such an integration formula was the ’exact stationary phase formula’, discovered by Duistermaat-Heckman [12] in 1980. This formula was quickly recognized, by Berline-Vergne [5] and Atiyah-Bott [3], as a localization principle in equivariant cohomology. Today, equivariant localization is a basic tool in mathematical physics, with numerous applications. In this article, we will begin with Borel’s topological definition of equivariant cohomology. We then proceed to describe H. Cartan’s more algebraic approach, and conclude with a discussion of localization principles. As additional references for the material covered here, we particularly recommend the books by BerlineGetzler-Vergne [4] and Guillemin-Sternberg [17].