Corollary 2

Let H be a closed subgroup of G.

a) The canonical injection $\Omega (G/H)^G\rightarrow \Omega (G/H)$ is a homotopism.

b) The complex $\Omega (G/H)^G$ can be identified with the subcomplex of Alt($\mathfrak{g}$) consisting of the elements $\omega$ of Alt($\mathfrak{g}$) that are invariant under the adjoint representation of H and are such that $i(\xi )\omega = 0$ for all $\xi \in L(H)$. If, in addition, H is connected, this subcomplex consists of the $\omega \in$ Alt($\mathfrak{g}$) such that $\theta (\xi )\omega = 0$ and $i(\xi )\omega = 0$ for all $\xi \in L(H)$.