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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
Hanke, Bernhard 1 ; Quast, Peter 1
@article{10_2140_agt_2018_18_877,
author = {Hanke, Bernhard and Quast, Peter},
title = {\ensuremath{\Gamma}{\textendash}structures and symmetric spaces},
journal = {Algebraic and Geometric Topology},
pages = {877--895},
publisher = {mathdoc},
volume = {18},
number = {2},
year = {2018},
doi = {10.2140/agt.2018.18.877},
url = {http://geodesic.mathdoc.fr/articles/10.2140/agt.2018.18.877/}
}
TY - JOUR AU - Hanke, Bernhard AU - Quast, Peter TI - Γ–structures and symmetric spaces JO - Algebraic and Geometric Topology PY - 2018 SP - 877 EP - 895 VL - 18 IS - 2 PB - mathdoc UR - http://geodesic.mathdoc.fr/articles/10.2140/agt.2018.18.877/ DO - 10.2140/agt.2018.18.877 ID - 10_2140_agt_2018_18_877 ER -
Hanke, Bernhard; Quast, Peter. Γ–structures and symmetric spaces. Algebraic and Geometric Topology, Tome 18 (2018) no. 2, pp. 877-895. doi: 10.2140/agt.2018.18.877
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