Inventiones mathematicae

, 179:435 | Cite as

Collapsing irreducible 3-manifolds with nontrivial fundamental group

  • L. Bessières
  • G. Besson
  • M. Boileau
  • S. Maillot
  • J. Porti
Article

Abstract

Let M be a closed, orientable, irreducible, non-simply connected 3-manifold. We prove that if M admits a sequence of Riemannian metrics which volume-collapses and whose sectional curvature is locally controlled, then M is a graph manifold. This is the last step in Perelman’s proof of Thurston’s Geometrisation Conjecture.

References

  1. 1.
    Bessières, L., Besson, G., Boileau, M., Maillot, S., Porti, J.: Geometrisation of 3-manifolds (2009, submitted) Google Scholar
  2. 2.
    Boileau, M., Leeb, B., Porti, J.: Geometrization of 3-dimensional orbifolds. Ann. Math. (2) 162(1), 195–290 (2005) MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Boileau, M., Maillot, S., Porti, J.: Three-Dimensional Orbifolds and Their Geometric Structures. Panoramas et Synthèses (Panoramas and Syntheses), vol. 15. Société Mathématique de France, Paris (2003) MATHGoogle Scholar
  4. 4.
    Boileau, M., Porti, J.: Geometrization of 3-orbifolds of cyclic type. Astérisque 272, 208 (2001). Appendix A by Michael Heusener and Porti MathSciNetGoogle Scholar
  5. 5.
    Cao, J., Ge, J.: A proof of perelman’s collapsing theorem for 3-manifolds, 28 October 2009. arXiv:math.DG/0908.3229
  6. 6.
    Cao, H.-D., Zhu, X.-P.: A complete proof of the Poincaré and geometrization conjectures—application of the Hamilton-Perelman theory of the Ricci flow. Asian J. Math. 10(2), 165–492 (2006) MATHMathSciNetGoogle Scholar
  7. 7.
    Cheeger, J., Ebin, D.G.: Comparison Theorems in Riemannian Geometry. North-Holland Mathematical Library, vol. 9. North-Holland, Amsterdam (1975) MATHGoogle Scholar
  8. 8.
    Cheeger, J., Gromoll, D.: On the structure of complete manifolds of nonnegative curvature. Ann. Math. (2) 96, 413–443 (1972) CrossRefMathSciNetGoogle Scholar
  9. 9.
    Cheeger, J., Gromov, M., Taylor, M.: Finite propagation speed, kernel estimates for functions of the Laplace operator, and the geometry of complete Riemannian manifolds. J. Differ. Geom. 17(1), 15–53 (1982) MATHMathSciNetGoogle Scholar
  10. 10.
    Fukaya, K.: Hausdorff convergence of Riemannian manifolds and its applications. In: Recent Topics in Differential and Analytic Geometry. Adv. Stud. Pure Math., vol. 18, pp. 143–238. Academic Press, Boston (1990) Google Scholar
  11. 11.
    Gómez-Larrañaga, J.C., González-Acuña, F.: Lusternik-Schnirelmann category of 3-manifolds. Topology 31(4), 791–800 (1992) MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Gromov, M.: Structures Métriques Pour les Variétés Riemanniennes. Textes Mathématiques (Mathematical Texts), vol. 1. CEDIC, Paris (1981). Edited by J. Lafontaine and P. Pansu MATHGoogle Scholar
  13. 13.
    Gromov, M.: Volume and bounded cohomology. Inst. Hautes Études Sci. Publ. Math. 56, 5–99 (1983) MathSciNetGoogle Scholar
  14. 14.
    Hamilton, R.S.: Three-manifolds with positive Ricci curvature. J. Differ. Geom. 17(2), 255–306 (1982) MATHMathSciNetGoogle Scholar
  15. 15.
    Hamilton, R.S.: Four-manifolds with positive curvature operator. J. Differ. Geom. 24(2), 153–179 (1986) MATHMathSciNetGoogle Scholar
  16. 16.
    Hamilton, R.S.: A compactness property for solutions of the Ricci flow. Am. J. Math. 117(3), 545–572 (1995) MATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Hempel, J.: 3-Manifolds. Ann. of Math. Studies, vol. 86. Princeton University Press, Princeton (1976) MATHGoogle Scholar
  18. 18.
    Ivanov, N.V.: Foundations of the theory of bounded cohomology. Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 143, 69–109 (1985). Studies in topology, V MATHMathSciNetGoogle Scholar
  19. 19.
    Ivanov, N.V.: Foundations of the theory of bounded cohomology. Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 143, 177–178 (1985). Studies in topology, V Google Scholar
  20. 20.
    Jaco, W.: Lectures on Three-Manifold Topology. CBMS Regional Conference Series in Mathematics, vol. 43. American Mathematical Society, Providence (1980) MATHGoogle Scholar
  21. 21.
    Kapovitch, V.: Perelman’s stability theorem. In: Surveys in Differential Geometry. Vol. XI. Surv. Differ. Geom., vol. 11, pp. 103–136. Int. Press, Somerville (2007) Google Scholar
  22. 22.
    Kleiner, B., Lott, J.: Locally collapsed 3-manifolds (in preparation) Google Scholar
  23. 23.
    Kleiner, B., Lott, J.: Notes on Perelman’s papers. Geom. Topol. 12(5), 2587–2855 (2008) MATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    Kuessner, T.: Multicomplexes, bounded cohomology and additivity of simplicial volume (2001). math.AT/0109057
  25. 25.
    McMullen, C.: Iteration on Teichmüller space. Invent. Math. 99(2), 425–454 (1990) MATHCrossRefMathSciNetGoogle Scholar
  26. 26.
    Morgan, J., Tian, G.: Ricci Flow and the Poincaré Conjecture. Clay Mathematics Monographs, vol. 3. American Mathematical Society, Providence (2007) MATHGoogle Scholar
  27. 27.
    Morgan, J., Tian, G.: Completion of the proof of the geometrization conjecture (2008). arXiv:0809.4040
  28. 28.
    Otal, J.-P.: Thurston’s hyperbolization of Haken manifolds. In: Surveys in Differential Geometry, vol. III, Cambridge, MA, 1996, pp. 77–194. Interface Press, Boston (1998) Google Scholar
  29. 29.
    Otal, J.-P.: The Hyperbolization Theorem for Fibered 3-manifolds. SMF/AMS Texts and Monographs, vol. 7. American Mathematical Society, Providence (2001). Translated from the 1996 French original by Leslie D. Kay MATHGoogle Scholar
  30. 30.
    Perelman, G.: Alexandrov spaces with curvatures bounded from below II. Preprint (1991). http://www.math.psu.edu/petrunin/papers/papers.html
  31. 31.
    Perelman, G.: The entropy formula for the Ricci flow and its geometric applications (2002). math.DG/0211159
  32. 32.
    Perelman, G.: Finite extinction time for the solutions to the Ricci flow on certain three-manifolds (2003). math.DG/0307245
  33. 33.
    Perelman, G.: Ricci flow with surgery on three-manifolds (2003). math.DG/0303109
  34. 34.
    Peters, S.: Convergence of Riemannian manifolds. Compos. Math. 62(1), 3–16 (1987) MATHGoogle Scholar
  35. 35.
    Petersen, P.: Riemannian Geometry. Graduate Texts in Mathematics, vol. 171. Springer, New York (1998) MATHGoogle Scholar
  36. 36.
    Shioya, T., Yamaguchi, T.: Volume collapsed three-manifolds with a lower curvature bound. Math. Ann. 333(1), 131–155 (2005) MATHCrossRefMathSciNetGoogle Scholar
  37. 37.
    Soma, T.: The Gromov invariant of links. Invent. Math. 64(3), 445–454 (1981) MATHCrossRefMathSciNetGoogle Scholar
  38. 38.
    Thurston, W.P.: Hyperbolic structures on 3-manifolds. I. Deformation of acylindrical manifolds. Ann. Math. (2) 124(2), 203–246 (1986) CrossRefMathSciNetGoogle Scholar
  39. 39.
    Yamaguchi, T.: A convergence theorem in the geometry of Alexandrov spaces. In: Actes de la Table Ronde de Géométrie Différentielle, Luminy, 1992. Sémin. Congr., vol. 1, pp. 601–642. Soc. Math. France, Paris (1996) Google Scholar

Copyright information

© Springer-Verlag 2009

Authors and Affiliations

  • L. Bessières
    • 1
  • G. Besson
    • 1
  • M. Boileau
    • 2
  • S. Maillot
    • 3
  • J. Porti
    • 4
  1. 1.Institut Fourier, Université Joseph Fourier (Grenoble I)UMR 5582 CNRS-UJFSaint-Martin-d’HèresFrance
  2. 2.Institut Mathématique de Toulouse, UMR 5219 CNRS-UPSUniversité Paul SabatierToulouse Cedex 9France
  3. 3.Institut de Mathématiques et de Modélisation de Montpellier, Université de MontpellierUMR 5149 CNRSMontpellier Cedex 5France
  4. 4.Departament de MatemàtiquesUniversitat Autònoma de BarcelonaBellaterraSpain

Personalised recommendations