Algebraic & Geometric Topology

$\Gamma$–structures and symmetric spaces

Bernhard Hanke and Peter Quast

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$\Gamma$–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.

Article information

Algebr. Geom. Topol., Volume 18, Number 2 (2018), 877-895.

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

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 57T15: Homology and cohomology of homogeneous spaces of Lie groups
Secondary: 53C35: Symmetric spaces [See also 32M15, 57T15] 55S45: Postnikov systems, $k$-invariants 57T25: Homology and cohomology of H-spaces

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


Hanke, Bernhard; Quast, Peter. $\Gamma$–structures and symmetric spaces. Algebr. Geom. Topol. 18 (2018), no. 2, 877--895. doi:10.2140/agt.2018.18.877.

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  • J F Adams, On the non-existence of elements of Hopf invariant one, Ann. of Math. 72 (1960) 20–104
  • P Albers, U Frauenfelder, J P Solomon, A $\Gamma$–structure on Lagrangian Grassmannians, Comment. Math. Helv. 89 (2014) 929–936
  • S Araki, On the Brouwer degrees of some maps of compact symmetric spaces, Topology 3 (1965) 281–290
  • A Borel, Sur la cohomologie des espaces fibrés principaux et des espaces homogènes de groupes de Lie compacts, Ann. of Math. 57 (1953) 115–207
  • F E Burstall, J H Rawnsley, Twistor theory for Riemannian symmetric spaces, Lecture Notes in Mathematics 1424, Springer (1990)
  • H Cartan, Détermination des algèbres $H_*(\pi,n;\mathbb{Z}_p)$ et $H^*(\pi,n;\mathbb{Z}_p)$, $p$ premier impair, from “Algèbres d'Eilenberg–Mac Lane et homotopie”, Séminaire Henri Cartan (1954/55) tome 7, Secrétariat Mathématique, Paris (1956) Exposé 9
  • H Cartan, Détermination des algèbres $H_*(\pi,n;\mathbb{Z}_2)$ et $H^*(\pi,n;\mathbb{Z}_2)$; groupes stables modulo $p$, from “Algèbres d'Eilenberg–Mac Lane et homotopie”, Séminaire Henri Cartan (1954/55) tome 7, Secrétariat Mathématique, Paris (1956) Exposé 10
  • J Dieudonné, A history of algebraic and differential topology: 1900–1960, Birkhäuser, Boston (1989)
  • J-H Eschenburg, A-L Mare, P Quast, Pluriharmonic maps into outer symmetric spaces and a subdivision of Weyl chambers, Bull. Lond. Math. Soc. 42 (2010) 1121–1133
  • O Goertsches, The equivariant cohomology of isotropy actions on symmetric spaces, Doc. Math. 17 (2012) 79–94
  • W Greub, S Halperin, R Vanstone, Connections, curvature, and cohomology, III: Cohomology of principal bundles and homogeneous spaces, Pure and Applied Mathematics 47, Academic Press, New York (1976)
  • A Hatcher, Algebraic topology, Cambridge Univ. Press (2002)
  • S Helgason, Differential geometry, Lie groups, and symmetric spaces, Pure and Applied Mathematics 80, Academic Press, New York (1978)
  • H Hopf, Sur la topologie des groupes clos de Lie et de leurs généralisations, C. R. Acad. Sci. Paris 208 (1939) 1266–1267
  • H Hopf, Über die Topologie der Gruppen-Mannigfaltigkeiten und ihre Verallgemeinerungen, Ann. of Math. 42 (1941) 22–52
  • A W Knapp, Lie groups beyond an introduction, Progress in Mathematics 140, Birkhäuser, Boston (1996)
  • D Kotschick, S Terzić, On formality of generalized symmetric spaces, Math. Proc. Cambridge Philos. Soc. 134 (2003) 491–505
  • O Loos, Symmetric spaces, I: General theory, W. A. Benjamin, New York (1969)
  • O Loos, Symmetric spaces, II: Compact spaces and classification, W. A. Benjamin, New York (1969)
  • J P May, K Ponto, More concise algebraic topology: localization, completion, and model categories, Univ. of Chicago Press (2012)
  • J McCleary, A user's guide to spectral sequences, 2nd edition, Cambridge Studies in Advanced Mathematics 58, Cambridge Univ. Press (2001)
  • M Mimura, H Toda, Topology of Lie groups, I and II, Translations of Mathematical Monographs 91, Amer. Math. Soc., Providence, RI (1991)
  • S Murakami, Sur la classification des algèbres de Lie réelles et simples, Osaka J. Math. 2 (1965) 291–307
  • T Sakai, Riemannian geometry, Translations of Mathematical Monographs 149, Amer. Math. Soc., Providence, RI (1996)
  • J-P Serre, Cohomologie modulo $2$ des complexes d'Eilenberg–Mac Lane, Comment. Math. Helv. 27 (1953) 198–232
  • M Spivak, A comprehensive introduction to differential geometry, V, 3rd edition, Publish or Perish, Boston (1999)
  • D Sullivan, Infinitesimal computations in topology, Inst. Hautes Études Sci. Publ. Math. 47 (1977) 269–331
  • M Takeuchi, On Pontrjagin classes of compact symmetric spaces, J. Fac. Sci. Univ. Tokyo Sect. I 9 (1962) 313–328
  • J A Wolf, Spaces of constant curvature, 5th edition, Publish or Perish, Houston, TX (1984)