Γ-structures and symmetric spaces

Γ-structures are weak forms of multiplications on closed oriented manifolds. As shown by Hopf the rational cohomology algebras of manifolds admitting Γ-structures are free over odd degree generators. We prove that this condition is also sufficient for the existence of Γ-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 Γ-structures. This extends work of Albers, Frauenfelder and Solomon on Γ-structures on Lagrangian Grassmannians.