$\Gamma$–structures are weak forms of multiplications on closed oriented manifolds. As was shown by Hopf the rational cohomology algebras of manifolds admitting $\Gamma$–structures are free over odd-degree generators. We prove that this condition is also sufficient for the existence of $\Gamma$–structures on manifolds which are nilpotent in the sense of homotopy theory. This includes homogeneous spaces with connected isotropy groups.
Passing to a more geometric perspective we show that on compact oriented Riemannian symmetric spaces with connected isotropy groups and free rational cohomology algebras the canonical products given by geodesic symmetries define $\Gamma$–structures. This extends work of Albers, Frauenfelder and Solomon on $\Gamma$–structures on Lagrangian Grassmannians.
"$\Gamma$–structures and symmetric spaces." Algebr. Geom. Topol. 18 (2) 877 - 895, 2018. https://doi.org/10.2140/agt.2018.18.877