Science China Mathematics

, Volume 60, Issue 4, pp 569–580 | Cite as

Teichmüller space of negatively curved metrics on complex hyperbolic manifolds is not contractible

Articles Invited Articles


We prove that for all n = 4k − 2 and k ≥ 2 there exists a closed smooth complex hyperbolic manifold M with real dimension n having non-trivial π 1(T <0(M)). T <0(M) denotes the Teichmüller space of all negatively curved Riemannian metrics on M, which is the topological quotient of the space of all negatively curved metrics modulo the space of self-diffeomorphisms of M that are homotopic to the identity


space of Riemannian metrics negative curvature complex hyperbolic space 


58D27 58D17 53C20 57R19 53C55 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Ardanza S. Exotic smooth structures on non-locally symmetric negatively curved manifolds. PhD Thesis. Binghamton: Binghamton University, 2000Google Scholar
  2. 2.
    Browder W. On the action of Tn(?p). In: Differential and Combinatorial Topology: A Symposium in Honor of Marston Morse. Princeton: Princeton University Press, 1965, 23–26Google Scholar
  3. 3.
    Conner P E, Raymond F. Deforming homotopy equivalences to homeomorphisms in aspherical manifolds. Bull Amer Math Soc, 1977, 83: 36–85MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Farrell F T. Topological rigidity and geometric applications. In: Geometry, Analysis and Topology of Discrete Groups. Advanced Lectures in Mathematics, vol. 6. Somerville: International Press, 2008, 163–195Google Scholar
  5. 5.
    Farrell F T, Jones L E. Negatively curved manifolds with exotic smooth structures. J Amer Math Soc, 1989, 2: 899–908MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Farrell F T, Jones L E. Topological rigidity for compact non-positively curved manifolds. Proc Sympos Pure Math, 1993, 54: 229–274MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Farrell F T, Jones L E. Complex hyperbolic manifolds and exotic smooth structures. Invent Math, 1993, 117: 57–74MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Farrell F T, Ontaneda P. The Teichmüller space of pinched negatively curved metrics on a hyperbolic manifold is not contractible. Ann of Math, 2009, 170: 45–65MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Farrell F T, Ontaneda P. Teichmüller spaces and negatively curved fiber bundles. Geom Funct Anal, 2010, 20: 1397–1430MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Gromov M, Thurston W. Pinching constants for hyperbolic manifolds. Invent Math, 1987, 89: 1–12MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Hirzebruch F. Topological Methods in Algebraic Geometry, 3rd ed. Berlin: Springer-Verlag, 1966CrossRefMATHGoogle Scholar
  12. 12.
    Kervaire M, Milnor J W. Groups of homotopy spheres: I. Ann of Math, 1962, 112: 321–360Google Scholar
  13. 13.
    Mostow G. Quasi-conformal mappings in n-space and the rigidity of the hyperbolic space forms. Publ Math Inst Hautes Études Sci, 1968, 34: 53–104MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Ontaneda P. Hyperbolic manifolds with negatively curved exotic triangulations in dimension six. J Differential Geom, 1994, 40: 7–22MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Sorcar G. Teichmüller space of negatively curved metrics on Gromov-Thurston manifolds is not contractible. J Topol Anal, 2014, 6: 541–555MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Sullivan D. Hyperbolic geometry and homeomorphisms. In: Geometric Topology: Proceedings of the Georgia Topology Conference. New York: Academic Press, 1979, 543–555CrossRefGoogle Scholar

Copyright information

© Science China Press and Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  1. 1.Yau Mathematical Sciences Center and Department of Mathematical SciencesTsinghua UniversityBeijingChina
  2. 2.Department of MathematicsOhio State UniversityColumbusUSA

Personalised recommendations