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

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.

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$\Gamma$–structures, Postnikov decompositions, rational cohomology, symmetric spaces
Mathematical Subject Classification 2010
Primary: 57T15
Secondary: 53C35, 55S45, 57T25
Received: 5 November 2016
Revised: 5 September 2017
Accepted: 8 November 2017
Published: 12 March 2018
Bernhard Hanke
Institut für Mathematik
Universität Augsburg
Peter Quast
Institut für Mathematik
Universität Augsburg