$\Gamma$-structures and symmetric spaces

TL;DR

This paper characterizes when certain geometric manifolds admit weak multiplication structures called $\Gamma$-structures, linking algebraic cohomology conditions with geometric symmetric space structures.

Contribution

It proves that free rational cohomology algebras imply the existence of $\Gamma$-structures on nilpotent manifolds, including symmetric spaces, extending previous work.

Findings

Manifolds with free rational cohomology admit $\Gamma$-structures.

Geodesic symmetries induce $\Gamma$-structures on symmetric spaces.

Extension of $\Gamma$-structure existence to a broader class of manifolds.

Abstract

$\Gamma$-structures are weak forms of multiplications on closed oriented manifolds. As 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.