Table of Contents
ISRN Geometry
Volume 2014 (2014), Article ID 694106, 9 pages
http://dx.doi.org/10.1155/2014/694106
Review Article

A Survey on Seifert Fiber Space Theorem

Laboratoire d'Analyse Topologie et Probabilités, UMR CNRS 7353, Aix-Marseille Université, 39 rue F.Joliot-Curie, 13453 Marseille Cedex 13, France

Received 18 September 2013; Accepted 29 October 2013; Published 5 March 2014

Academic Editors: G. Martin and J. Porti

Copyright © 2014 Jean-Philippe Préaux. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. H. Seifert and W. Threlfall, Lehrbuch der Topologie, 89 Teubner, 1934. View at MathSciNet
  2. E. H. Spanier, Algebraic Topology, McGraw-Hill, New York, NY, USA, 1966. View at MathSciNet
  3. J. Hempel, 3-Manifolds, Princeton University Press, Princeton, NJ, USA, 1976. View at MathSciNet
  4. H. Seifert, “Topologie dreidimensionaler gefaserter Räume,” Acta Mathematica, vol. 60, no. 1, pp. 147–288, 1933, An english translation by W. Heil in H. Seifert and W. Threlfall, A Textbook of Topology, Pure and Applied Mathematics no. 89, Academic Press, 1980. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  5. H. Seifert, “Topology of 3-dimensional fibered spaces,” Acta Mathematica, vol. 60, no. 1, pp. 147–288, 1933. View at Publisher · View at Google Scholar
  6. D. B. A. Epstein, “Periodic flows on three-manifolds,” Annals of Mathematics, vol. 95, pp. 66–82, 1972. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  7. P. Scott, “The geometries of 3-manifolds,” The Bulletin of the London Mathematical Society, vol. 15, no. 5, pp. 401–487, 1983. View at Publisher · View at Google Scholar · View at MathSciNet
  8. J. L. Tollefson, “The compact 3-manifolds covered by S2×1,” Proceedings of the American Mathematical Society, vol. 45, pp. 461–462, 1974. View at Google Scholar · View at MathSciNet
  9. P. Orlik, Seifert Manifolds, vol. 291 of Lecture Notes in Mathematics, Springer, Berlin, Germany, 1972. View at MathSciNet
  10. F. Waldhausen, “Gruppen mit Zentrum und 3-dimensionale Mannigfaltigkeiten,” Topology, vol. 6, pp. 505–517, 1967. View at Google Scholar · View at MathSciNet
  11. C. M. Gordon and W. Heil, “Cyclic normal subgroups of fundamental groups of 3-manifolds,” Topology, vol. 14, no. 4, pp. 305–309, 1975. View at Google Scholar · View at MathSciNet
  12. W. Jaco and P. B. Shalen, Seifert Fibered Spaces in 3-Manifolds, vol. 21 of Memoirs of the AMS, no. 220, 1979. View at MathSciNet
  13. P. Scott, “There are no fake Seifert fibre spaces with infinite π1,” Annals of Mathematics, vol. 117, no. 1, pp. 35–70, 1983. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  14. G. Mess, “Centers of 3-manifold groups, and groups which are coarse quasi-isometric to planes,” 1988, Unpublished. View at Google Scholar
  15. P. Tukia, “Homeomorphic conjugates of Fuchsian groups,” Journal für die Reine und Angewandte Mathematik, vol. 391, pp. 1–54, 1988. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  16. A. Casson and D. Jungreis, “Convergence groups and Seifert fibered 3-manifolds,” Inventiones Mathematicae, vol. 118, no. 3, pp. 441–456, 1994. View at Publisher · View at Google Scholar · View at MathSciNet
  17. D. Gabai, “Convergence groups are Fuchsian groups,” Annals of Mathematics, vol. 136, no. 3, pp. 447–510, 1992. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  18. S. Maillot, “Quasi-isometries of groups, graphs and surfaces,” Commentarii Mathematici Helvetici, vol. 76, no. 1, pp. 29–60, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  19. S. Maillot, “Open 3-manifolds whose fundamental groups have infinite center, and a torus theorem for 3-orbifolds,” Transactions of the American Mathematical Society, vol. 355, no. 11, pp. 4595–4638, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  20. B. H. Bowditch, “Planar groups and the Seifert conjecture,” Journal für die Reine und Angewandte Mathematik, vol. 576, pp. 11–62, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  21. W. Whitten, “Recognizing nonorientable Seifert bundles,” Journal of Knot Theory and Its Ramifications, vol. 1, no. 4, pp. 471–475, 1992. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  22. W. Heil and W. Whitten, “The Seifert fiber space conjecture and torus theorem for nonorientable 3-manifolds,” Canadian Mathematical Bulletin, vol. 37, no. 4, pp. 482–489, 1994. View at Publisher · View at Google Scholar · View at MathSciNet
  23. P. de la Harpe and J. P. Préaux, “Groupes fondamentaux des variétés de dimension 3 et algèbres d'opérateurs,” Annales de la Faculté des Sciences de Toulouse, vol. 16, no. 3, pp. 561–589, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  24. R. Kirby, “Problems in low dimensional topology,” 1995, http://math.berkeley.edu/%7Ekirby/.
  25. K. Murasugi, “Remarks on torus knots,” Proceedings of the Japan Academy, vol. 37, p. 222, 1961. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  26. L. Neuwirth, “A note on torus knots and links determined by their groups,” Duke Mathematical Journal, vol. 28, pp. 545–551, 1961. View at Publisher · View at Google Scholar · View at MathSciNet
  27. G. Burde and H. Zieschang, “Eine Kennzeichnung der Torusknoten,” Mathematische Annalen, vol. 167, pp. 169–176, 1966. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  28. F. Waldhausen, “On the determination of some bounded 3-manifolds by their fundamental groups alone,” in Proceedings of the International Symposium on Topology and Its Applications, pp. 331–332, Herceg Novi, Montenegro, 1968.
  29. C. D. Feustel, “On the torus theorem and its applications,” Transactions of the American Mathematical Society, vol. 217, pp. 1–43, 1976. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  30. C. D. Feustel, “On the torus theorem for closed 3-manifolds,” Transactions of the American Mathematical Society, vol. 217, pp. 45–57, 1976. View at Google Scholar · View at MathSciNet
  31. P. Scott, “A new proof of the annulus and torus theorems,” The American Journal of Mathematics, vol. 102, no. 2, pp. 241–277, 1980. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  32. L. Bessières, G. Besson, M. Boileau, S. Maillot, and J. Porti, “Geometrisation of 3-manifolds,” EMS Tracts in Mathematics, vol. 13, 2010. View at Publisher · View at Google Scholar