Mathematische Zeitschrift

, Volume 288, Issue 3–4, pp 1081–1101 | Cite as

Classification and characterization of rationally elliptic manifolds in low dimensions

  • Martin Herrmann


We give a characterization of closed, simply connected, rationally elliptic 6-manifolds in terms of their rational cohomology rings and a partial classification of their real cohomology rings. We classify rational, real and complex homotopy types of closed, simply connected, rationally elliptic 7-manifolds. We give partial results in dimensions 8 and 9.


Rationally elliptic spaces Rationally elliptic manifolds Minimal models Cohomology ring 

Mathematics Subject Classification

Primary 55P62 Secondary 57R19 


  1. 1.
    Barden, D.: Simply connected five-manifolds. Ann. Math. 2(82), 365–385 (1965)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Barge, J.: Structures différentiables sur les types d’homotopie rationnelle simplement connexes. Ann. Sci. École Norm. Sup. (4) 9(4), 469–501 (1976)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Brieskorn, E., Knörrer, H.: Plane algebraic curves. In: Modern Birkhäuser Classics. Birkhäuser/Springer Basel AG, Basel (1986). Translated from the German original by John Stillwell [2012] reprint of the 1986 editionGoogle Scholar
  4. 4.
    Borel, A.: Sur la cohomologie des espaces fibrés principaux et des espaces homogènes de groupes de Lie compacts. Ann. Math. 2(57), 115–207 (1953)CrossRefzbMATHGoogle Scholar
  5. 5.
    DeVito, J.: The classification of simply connected biquotients of dimension at most 7 and 3 new examples of almost positively curved manifolds. PhD thesis, University of Pennsylvania (2011)Google Scholar
  6. 6.
    DeVito, J.: The classification of compact simply connected biquotients in dimension 6 and 7. Math. Ann. 368(3–4), 1493–1541 (2017)Google Scholar
  7. 7.
    Friedlander, J.B., Halperin, S.: An arithmetic characterization of the rational homotopy groups of certain spaces. Invent. Math. 53(2), 117–133 (1979)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Félix, Y., Halperin, S., Thomas, J.-C.: Rational homotopy theory, Graduate Texts in Mathematics, vol. 205. Springer, New York (2001)Google Scholar
  9. 9.
    Félix, Y., Oprea, J., Tanré, D.: Algebraic models in geometry, Oxford Graduate Texts in Mathematics, vol. 17 . Oxford University Press, Oxford (2008)Google Scholar
  10. 10.
    Galaz-Garcia, F., Kerin, M., Radeschi, M., Wiemeler, M.: Torus orbifolds, rational ellipticity and non-negative curvature (2014). arXiv:1404.3903v1 [math.DG]
  11. 11.
    Grove, K., Halperin, S.: Contributions of rational homotopy theory to global problems in geometry. Inst. Hautes Études Sci. Publ. Math. 56, 171–177 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Grove, K., Halperin, S.: Dupin hypersurfaces, group actions and the double mapping cylinder. J. Differ. Geom. 26(3), 429–459 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Halperin, S.: Finiteness in the minimal models of Sullivan. Trans. Am. Math. Soc. 230, 173–199 (1977)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Herrmann, M.: Rationale Elliptizität, Krümmung und Kohomologie. Dissertation, Karlsruhe Institute of Technology (2014).
  15. 15.
    Hoelscher, C.A.: Classification of cohomogeneity one manifolds in low dimensions. Pac. J. Math. 246(1), 129–185 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Hoelscher, C.A.: On the homology of low-dimensional cohomogeneity one manifolds. Transform. Groups 15(1), 115–133 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Klaus, S.: Einfach-zusammenhängende Kompakte Homogene Räume bis zur Dimension Neun. Diplomarbeit, Johannes Gutenberg Universität Mainz, Juni (1988)Google Scholar
  18. 18.
    Kraft, H.: Geometrische Methoden in der Invariantentheorie. Aspects of Mathematics, D1. Friedr. Vieweg & Sohn, Braunschweig (1984)Google Scholar
  19. 19.
    Matsumura, H.: Commutative ring theory, Cambridge Studies in Advanced Mathematics, vol. 8. Cambridge University Press, Cambridge (1986). Translated from the Japanese by M. ReidGoogle Scholar
  20. 20.
    McKay, B.: Lagrangian submanifolds in affine symplectic geometry. Differ. Geom. Appl. 24(6), 670–689 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Miller, T.J.: On the formality of \((k-1)\)-connected compact manifolds of dimension less than or equal to \(4k-2\). Ill. J. Math. 23(2), 253–258 (1979)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Pavlov, A.V.: Estimates for the betti numbers of rationally elliptic spaces. Sib. Math. J. 43(6), 1080–1085 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Paternain, G.P., Petean, J.: Minimal entropy and collapsing with curvature bounded from below. Invent. Math. 151(2), 415–450 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Stanley, R.P.: Hilbert functions of graded algebras. Adv. Math. 28(1), 57–83 (1978)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Sullivan, D.: Infinitesimal computations in topology. Inst. Hautes Études Sci. Publ. Math. 47, 269–331 (1977)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Totaro, B.: Curvature, diameter, and quotient manifolds. Math. Res. Lett. 10(2–3), 191–203 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Wall, C.T.C.: Classification problems in differential topology. V. On certain \(6\)-manifolds. Invent. Math. 1, 355–374 (1966). corrigendum ibid. 2, 306 (1966)Google Scholar

Copyright information

© Springer-Verlag GmbH Deutschland 2017

Authors and Affiliations

  1. 1.Fakultät für MathematikKarlsruher Institut für TechnologieKarlsruheGermany

Personalised recommendations