#### Volume 18, issue 2 (2018)

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$\Gamma\mkern-1.5mu$–structures and symmetric spaces

### Bernhard Hanke and Peter Quast

Algebraic & Geometric Topology 18 (2018) 877–895
##### 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.

##### Keywords
$\Gamma$–structures, Postnikov decompositions, rational cohomology, symmetric spaces
##### Mathematical Subject Classification 2010
Primary: 57T15
Secondary: 53C35, 55S45, 57T25