Theorem 2

The canonical injection $\iota :\Omega (X)^G\rightarrow \Omega (X)$ is a homotopism of complexes (Algèbre, Chap. X, p. 33, déf. 5); the map $\omega \rightarrow \omega^{\sharp}=\int_G\tau_g^*\omega dg$ is a homotopism, inverse to it up to homotopy. In particular, the map $H(\iota ) : H(\Omega (X)^G)\rightarrow H(\Omega (X))$ is bijective.

By Cor. 3 of no. 4, the map $\omega \rightarrow \omega^{\sharp}$ is a morphism of complexes from $\Omega (X)$ to $\Omega (X)^G$ that induces the identity on the subcomplex $\Omega (X)^G$; thus, to prove the theorem it suffices to construct a homomorphism $s:\Omega (X)\rightarrow \Omega$(X), graded of degree $-1$, such that

$\omega^{\sharp}-\omega = (d\circ s+s\circ d)(\omega )$ for all $\omega \in \Omega (X)$. (17)

By Lemma 1 of Integration, Chapter IX, §2, no. 4 and Remark 1 of §2, no. 2, there exists a positive measure $d\xi$ on $\mathfrak{g}$ of compact support whose image under the exponential map is equal to $dg$. For $\omega \in \Omega$(X), put

we have to show that formula (17) is satisfied. As in the proof of Cor. 1 (no. 4), we verify the formula

We now deduce from formulas (15) and (16) the equalities

hence Th. 2.

We apply the theorem in the case X = G, for the action of G by left translations. Recall (Chap. III, §3, no. 13, Prop. 50) that associating to a differential form on G its value at the identity element gives an isomorphism from $\Omega (G)^G$ to the graded algebra Alt($\mathfrak{g}$) of alternating multilinear forms on $\mathfrak{g}$. Identify $\Omega (G)^G$ with Alt($\mathfrak{g}$) by means of this isomorphism. The operator $d$ is then given by the formula (Chap. III, §3, no. 14, Prop. 51)

for $\omega$ in Alt$^p(\mathfrak{g})$ and $a_1, . . . , a_{p+1}$ in $\mathfrak{g}$.

For $\xi \in \mathfrak{g}$, let $L_{\xi}$ be the corresponding left-invariant vector field (defined by means of the action of G on itself by right translations, cf. Chap. III, §3, no. 6). The operators $\theta (L_{\xi}), i(L_{\xi})$ commute with the action of G on $\Omega (G)$ defined by left translation, and hence induce operators $\theta (\xi ), i(\xi )$ on $\Omega (G)^G$; with the preceding identifications, these are expressed by the formulas (Differentiable and Analytic Manifolds, Results, 8.3.2 and 8.4.2)

for $\omega$ in Alt$^p(\mathfrak{g})$ and $a_1, . . . , a_p$ in $\mathfrak{g}$.

The subcomplex $^G\Omega (G)^G$ of biinvariant forms (Chap. III, §3, no. 13) can be identified with the subcomplex Alt($\mathfrak{g}$)$^G$ of alternating multilinear forms on $\mathfrak{g}$ invariant under the adjoint representation (that is, such that $\theta (\xi )\omega = 0$ for all $\xi \in \mathfrak{g}$). Thus, we have a commutative diagram of complexes

(18)

Alt($\mathfrak{g}$)$^G\longrightarrow$ Alt($\mathfrak{g}$)

where the horizontal arrows are the canonical injections, and the vertical arrows are the isomorphisms induced by the map $\omega \rightarrow \omega (e)$.