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2018 $\Gamma$–structures and symmetric spaces
Bernhard Hanke, Peter Quast
Algebr. Geom. Topol. 18(2): 877-895 (2018). DOI: 10.2140/agt.2018.18.877

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.

Citation

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Bernhard Hanke. Peter Quast. "$\Gamma$–structures and symmetric spaces." Algebr. Geom. Topol. 18 (2) 877 - 895, 2018. https://doi.org/10.2140/agt.2018.18.877

Information

Received: 5 November 2016; Revised: 5 September 2017; Accepted: 8 November 2017; Published: 2018
First available in Project Euclid: 22 March 2018

zbMATH: 06859608
MathSciNet: MR3773742
Digital Object Identifier: 10.2140/agt.2018.18.877

Subjects:
Primary: 57T15
Secondary: 53C35, 55S45, 57T25

Rights: Copyright © 2018 Mathematical Sciences Publishers

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Vol.18 • No. 2 • 2018
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