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show/hide this revision's text 2 an important lapse in terminology corrected

Here is one example. When $G$ is a compact Lie group and $H$ is a Lie subgroup, the real cohomology of the homogeneous space $G/H$ is the same as the relative cohomology $H^*(g,h,\mathbf{R})$ where $g$ and $h$ are the Lie algebras of $G$, respectively $H$. This can be proven by averaging, just as in the case when $H$ is trivial. In general the differential in the relative cochain complex is not acycliczero, but when $G/H$ is symmetric, it is, for a simple reason: the symmetric involution acts as $(-1)^d$ on the degree $d$ part of the complex; since this action should commute with the differential, the differential must be 0; for more details see Felix, Halperin, Thomas, Rational homotopy theory, p. 162.

So symmetric spaces are formal. However, in general compact homogeneous spaces need not be formal; e.g. $SU(n)/Sp(n)$ is not formal for $n\geq 5$, see Greub, Halperin, Vanstone, Curvature, connections and cohomology.
show/hide this revision's text 1

Here is one example. When $G$ is a compact Lie group and $H$ is a Lie subgroup, the real cohomology of the homogeneous space $G/H$ is the same as the relative cohomology $H^*(g,h,\mathbf{R})$ where $g$ and $h$ are the Lie algebras of $G$, respectively $H$. This can be proven by averaging, just as in the case when $H$ is trivial. In general the relative cochain complex is not acyclic, but when $G/H$ is symmetric, it is, for a simple reason: the symmetric involution acts as $(-1)^d$ on the degree $d$ part of the complex; since this action should commute with the differential, the differential must be 0; for more details see Felix, Halperin, Thomas, Rational homotopy theory, p. 162.

So symmetric spaces are formal. However, in general compact homogeneous spaces need not be formal; e.g. $SU(n)/Sp(n)$ is not formal for $n\geq 5$, see Greub, Halperin, Vanstone, Curvature, connections and cohomology.