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Γ-structures and symmetric spaces

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Speaker: 
Bernhard Hanke
Affiliation: 
U Augsburg
Date: 
Thu, 2016-05-12 16:30 - 17:30
Location: 
MPIM Lecture Hall

Γ-structures are weak forms of multiplications on closed oriented manifolds. As 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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