Γ–structures and symmetric spaces

Hanke, Bernhard; Quast, Peter

doi:10.2140/agt.2018.18.877

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

Bernhard Hanke and Peter Quast

Algebraic & Geometric Topology 18 (2018) 877–895

DOI: 10.2140/agt.2018.18.877

Abstract

$Γ$ –structures are weak forms of multiplications on closed oriented manifolds. As was shown by Hopf the rational cohomology algebras of manifolds admitting $Γ$ –structures are free over odd-degree generators. We prove that this condition is also sufficient for the existence of $Γ$ –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 $Γ$ –structures. This extends work of Albers, Frauenfelder and Solomon on $Γ$ –structures on Lagrangian Grassmannians.

Keywords

$\Gamma$–structures, Postnikov decompositions, rational cohomology, symmetric spaces

Mathematical Subject Classification 2010

Primary: 57T15

Secondary: 53C35, 55S45, 57T25

References

Publication

Received: 5 November 2016

Revised: 5 September 2017

Accepted: 8 November 2017

Published: 12 March 2018

Authors

Bernhard Hanke
Institut für Mathematik Universität Augsburg Augsburg Germany

Peter Quast
Institut für Mathematik Universität Augsburg Augsburg Germany

Volume 18, issue 2 (2018)