Cohomology of Homogeneous Spaces H₁/H₂

Let $G$ be a compact connected Lie group with Lie algebra $\mathfrak{g}$ , $H_1$ a normal subgroup of $G$ with Lie algebra ${\mathfrak{h}}_1$ , and $H_2？H_1$ a normal subgroup with Lie algebra ${\mathfrak{h}}_2$ . This paper establishes an isomorphism between the de Rham cohomology of the quotient manifold $H_1/H_2$ and the Lie algebra cohomology of the quotient Lie algebra ${\mathfrak{h}}_1/{\mathfrak{h}}_2$ . By constructing explicit isomorphisms of cochain complexes, we prove $H_{dR}^{？}\left(H_1/H_2\right)？H^{？}\left({\mathfrak{h}}_1/{\mathfrak{h}}_2,？\right)？{}^{}$