Equivariant cohomology (Cartan model)

Let $G$ be a Lie group acting smoothly on a smooth manifold $M$ . Let $\mathfrak g$ be the Lie algebra of $G$ , and let $\mathfrak g^*$ be its real linear dual.

Let $\Omega^*(M)$ denote the graded algebra of real differential forms on $M$ . Let $S(\mathfrak g^*)$ denote the symmetric algebra of $\mathfrak g^*$ , viewed as polynomial functions on $\mathfrak g$ . Assign degree two to each linear polynomial.

The Cartan complex is

\Omega_G^*(M)=\bigl(S(\mathfrak g^*)\otimes_{\mathbb R}\Omega^*(M)\bigr)^G.

Here $\otimes_{\mathbb R}$ is the tensor product, and the superscript $G$ means the elements invariant under the induced group action. Total degree is twice polynomial degree plus form degree.

View an element $\alpha$ as a polynomial map from $\mathfrak g$ to $\Omega^*(M)$ . For $\xi\in\mathfrak g$ , let $\xi_M$ be the fundamental vector field generated by the action. The Cartan differential is

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

where $d$ is the exterior derivative and $\iota_{\xi_M}$ is contraction with $\xi_M$ . It has degree one and satisfies $d_G^2=0$ on the invariant elements. The Cartan equivariant cohomology is the cohomology of this complex:

H^*_{G,\mathrm{Cartan}}(M)=H^*(\Omega_G^*(M),d_G).

Group action and sign convention

For $g\in G$ , let $a_g:M\to M$ be the map $a_g(x)=g\cdot x$ . Let $\operatorname{Ad}_g$ denote the adjoint action on $\mathfrak g$ , and let $(a_{g^{-1}})^*$ denote pullback of forms. The invariance condition in the Cartan complex means precisely that

\alpha(\operatorname{Ad}_g\xi)=(a_{g^{-1}})^*\alpha(\xi) \qquad(g\in G,\ \xi\in\mathfrak g).

The sign convention for the fundamental vector field is

\xi_M(x)=\left.\frac{d}{dt}\right|_{0}\exp(t\xi)\cdot x,

where $\exp$ is the exponential map.

Comparison with topological equivariant cohomology

The Borel definition is $H_G^*(M;R)=H^*(EG\times_GM;R)$ , using the universal principal bundle and any coefficient ring $R$ . For compact $G$ , the real Cartan complex above computes $H_G^*(M;\mathbb R)$ . Complexifying it computes complex coefficients; real differential forms do not directly compute arbitrary characteristic-zero coefficients, such as $\mathbb Q$ .

For a noncompact group, the Cartan complex is still defined, but this comparison can fail. For example, $G=(\mathbb R,+)$ acting on a point has Cartan cohomology $\mathbb R[u]$ , with $\deg u=2$ , whereas its Borel cohomology is just $\mathbb R$ because $G$ is contractible.

Examples for compact groups

Point: $H^*_{G,\mathrm{Cartan}}(\mathrm{pt})=S(\mathfrak g^*)^G\cong H^*(BG;\mathbb R)$ .
Trivial action: $H^*_{G,\mathrm{Cartan}}(M)\cong H^*(M;\mathbb R)\otimes H^*(BG;\mathbb R)$ .
Free action: $H^*_{G,\mathrm{Cartan}}(M)\cong H^*(M/G;\mathbb R)$ .

References

Eckhard Meinrenken, “Equivariant cohomology and the Cartan model,” Encyclopedia of Mathematical Physics (2006), §§5–6, especially formula (19), Theorem 6.1 and Remark 6.2. Author's text.