## Algebraic & Geometric Topology

### $\Gamma$–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.

#### Article information

Source
Algebr. Geom. Topol., Volume 18, Number 2 (2018), 877-895.

Dates
Revised: 5 September 2017
Accepted: 8 November 2017
First available in Project Euclid: 22 March 2018

https://projecteuclid.org/euclid.agt/1521684024

Digital Object Identifier
doi:10.2140/agt.2018.18.877

Mathematical Reviews number (MathSciNet)
MR3773742

Zentralblatt MATH identifier
06859608

#### Citation

Hanke, Bernhard; Quast, Peter. $\Gamma$–structures and symmetric spaces. Algebr. Geom. Topol. 18 (2018), no. 2, 877--895. doi:10.2140/agt.2018.18.877. https://projecteuclid.org/euclid.agt/1521684024

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