Mathematische Zeitschrift

, Volume 274, Issue 3–4, pp 1299–1325 | Cite as

Non-formal homogeneous spaces

  • Manuel AmannEmail author


Several large classes of homogeneous spaces are known to be formal—in the sense of rational homotopy theory. However, it seems that far fewer examples of non-formal homogeneous spaces are known. In this article we provide several construction principles and characterisations for non-formal homogeneous spaces, which will yield a lot of examples. This will enable us to prove that, from dimension 72 on, such a space can be found in each dimension.


Homogeneous space Non-formal manifold 

Mathematics Subject Classification (2000)

Primary 57N65 Secondary 57T15 57T20 



The author is very grateful to Anand Dessai for various fruitful discussions and to Jim Stasheff for commenting on a previous version of this article. The author would also like to thank the referee for several helpful suggestions.


  1. 1.
    Allday, C., Puppe, V.: Cohomological methods in transformation groups. Cambridge Studies in Advanced Mathematics, vol. 32. Cambridge University Press, Cambridge (1993)Google Scholar
  2. 2.
    Amann, M.: Positive Quaternion Kähler manifolds. PhD thesis, WWU Münster (2009).
  3. 3.
    Amann, M.: Computational complexity of topological invariants (2011). arXiv:1112.0812v1Google Scholar
  4. 4.
    Amann, M., Kapovitch, V.: On fibrations with formal elliptic fibers. Adv. Math. 231(3–4), 2048–2068 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Belegradek, I., Kapovitch, V.: Obstructions to nonnegative curvature and rational homotopy theory. J. Am. Math. Soc. 16(2), 259–284 (2003). electronicGoogle Scholar
  6. 6.
    Besse, A.L.: Einstein manifolds. In: Classics in Mathematics. Springer, Berlin (2008, Reprint of the 1987 edition)Google Scholar
  7. 7.
    Deligne, P., Griffiths, P., Morgan, J., Sullivan, D.: Real homotopy theory of Kähler manifolds. Invent. Math. 29(3), 245–274 (1975)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Félix, Y., Halperin, S., Thomas, J.-C.: Elliptic spaces. Bull. Am. Math. Soc. (N.S.) 25(1), 69–73 (1991)zbMATHCrossRefGoogle Scholar
  9. 9.
    Félix, Y., Halperin, S., Thomas, J.-C.: Rational homotopy theory. In: Graduate Texts in Mathematics, vol. 205. Springer, New York (2001)Google Scholar
  10. 10.
    Félix, Y., Oprea, J., Tanré, D.: Algebraic models in geometry. In: Oxford Graduate Texts in Mathematics, vol. 17. Oxford University Press, Oxford (2008)Google Scholar
  11. 11.
    Fernández, M., Muñoz, V.: Non-formal compact manifolds with small betti numbers (2006). arXiv:math/0504396v2Google Scholar
  12. 12.
    Fernández, M., Muñoz, V.: On non-formal simply connected manifolds. Topol. Appl. 135(1–3), 111–117 (2004)zbMATHCrossRefGoogle Scholar
  13. 13.
    Fernández, M., Muñoz, V.: Formality of Donaldson submanifolds. Math. Z. 250(1), 149–175 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Garvín, A., Lechuga, L.: The computation of the Betti numbers of an elliptic space is a NP-hard problem. Topol. Appl. 131(3), 235–238 (2003)zbMATHCrossRefGoogle Scholar
  15. 15.
    Greub, W., Halperin, S., Vanstone, R.: Connections, curvature, and cohomology. vol. III: Cohomology of principal bundles and homogeneous spaces. In: Pure and Applied Mathematics, vol. 47-III. Academic Press [Harcourt Brace Jovanovich Publishers], New York (1976)Google Scholar
  16. 16.
    Grove, K., Verdiani, L., Ziller, W.: A positively curved manifold homeomorphic to \(T_1{\mathbb{S}}^4\) (2009). arXiv:0809.2304v3Google Scholar
  17. 17.
    Kapovitch, V.: A note on rational homotopy of biquotients (2012, preprint)Google Scholar
  18. 18.
    Kotschick, D.: On products of harmonic forms. Duke Math. J. 107(3), 521–531 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    Kotschick, D., Terzić, S.: On formality of generalized symmetric spaces. Math. Proc. Camb. Philos. Soc. 134(3), 491–505 (2003)zbMATHCrossRefGoogle Scholar
  20. 20.
    Kotschick, D., Terzić, S.: Chern numbers and the geometry of partial flag manifolds. Comment. Math. Helv. 84(3), 587–616 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    Kotschick, D., Terzić, S.: Geometric formality of homogeneous spaces and of biquotients. Pac. J. Math. 249(1), 157–176 (2011)zbMATHCrossRefGoogle Scholar
  22. 22.
    Lupton, G.: Variations on a conjecture of Halperin. In: Homotopy and geometry (Warsaw, 1997). Banach Center Publ., vol. 45, pp. 115–135. Polish Acad. Sci., Warsaw (1998)Google Scholar
  23. 23.
    Mimura, M., Toda, H.: Topology of Lie groups. I, II. In: Translations of Mathematical Monographs, vol. 91. American Mathematical Society, Providence (1991). Translated from the 1978 Japanese edition by the authorsGoogle Scholar
  24. 24.
    Neisendorfer, J., Miller, T.: Formal and coformal spaces. Ill. J. Math. 22(4), 565–580 (1978)MathSciNetzbMATHGoogle Scholar
  25. 25.
    Onishchik, A.L.: Topology of Transitive Transformation Groups. Johann Ambrosius Barth Verlag GmbH, Leipzig (1994)zbMATHGoogle Scholar
  26. 26.
    Petersen, P., Wilhelm, F.: An exotic sphere with positive sectional curvature (2008). arXiv:0805.0812v3Google Scholar
  27. 27.
    Stȩpień, Z.: On formality of a class of compact homogeneous spaces. Geom. Dedicata 93, 37–45 (2002)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Tralle, A.: On compact homogeneous spaces with nonvanishing Massey products. In: Differential Geometry and Its Applications (Opava, 1992). Math. Publ., , vol. 1, pp. 47–50. Silesian Univ. Opava (1993)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  1. 1.Department of Mathematics, David Rittenhouse LaboratoryUniversity of PennsylvaniaPhiladelphiaUSA

Personalised recommendations