$\Gamma$–structures and symmetric spaces

Abstract

$\Gamma$–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.