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Equivariant cohomology (Cartan model)

A cohomology theory for manifolds with a Lie group action, computed by the Cartan complex of equivariant differential forms.

Equivariant cohomology (Cartan model)

Let $G$ be a Lie group acting smoothly on a manifold $M$ . Write $\mathfrak{g}$ for the Lie algebra of $G$ .

Equivariant cohomology can be defined topologically as the cohomology of the Borel construction $EG\times_G M$ , where $EG\to BG$ is the universal principal bundle . When $G$ is compact, it admits a de Rham model: the Cartan model.

Definition (Cartan complex and equivariant cohomology)

Let $\Omega^*(M)$ denote the graded algebra of differential forms (see differential forms ), and let $S(\mathfrak{g}^*)$ be the symmetric algebra on $\mathfrak{g}^*$ , graded so that elements of $\mathfrak{g}^*$ have degree 2.

The Cartan complex of the $G$ -manifold $M$ is

\Omega_G^*(M) := \bigl(S(\mathfrak{g}^*)\otimes \Omega^*(M)\bigr)^G,

the $G$ -invariant elements, equipped with the Cartan differential $d_G$ defined by the rule

(d_G\alpha)(\xi) = d(\alpha(\xi)) - \iota_{\xi_M}\alpha(\xi),

where $d$ is the exterior derivative , $\xi_M$ is the fundamental vector field on $M$ generated by $\xi\in\mathfrak{g}$ , and $\alpha$ is viewed as a polynomial map $\mathfrak{g}\to \Omega^*(M)$ .

The equivariant cohomology of $M$ (Cartan model) is

H_G^*(M) := H^*(\Omega_G^*(M), d_G).

Examples

Point. If $M=\mathrm{pt}$ , then $\Omega^*(M)=\mathbb{R}$ and $H_G^*(\mathrm{pt}) \cong H^*(BG)$ .
Trivial action. If $G$ acts trivially on $M$ , then $H_G^*(M)$ is naturally isomorphic to $H^*(M)\otimes H^*(BG)$ (over a field of characteristic 0).
Free action. If $G$ acts freely and properly so that $M\to M/G$ is a principal bundle, then $H_G^*(M)\cong H^*(M/G)$ .