Mathematische Annalen

, Volume 358, Issue 1–2, pp 351–359 | Cite as

Contact structures on \(M \times S^2\)

  • Jonathan Bowden
  • Diarmuid Crowley
  • András I. Stipsicz
Article

Abstract

We show that if a manifold \(M\) admits a contact structure, then so does \(M \times S^2\). Our proof relies on surgery theory, a theorem of Eliashberg on contact surgery and a theorem of Bourgeois showing that if \(M\) admits a contact structure then so does \(M \times T^2\).

References

  1. 1.
    Bourgeois, F.: Odd dimensional tori are contact manifolds. Int. Math. Res. Notice 1571–1574 (2002)Google Scholar
  2. 2.
    Bowden, J., Crowley, D., Stipsicz, A.: The topology of Stein fillable manifolds in high dimensions I. (arXiv:1306.2746)Google Scholar
  3. 3.
    Casals, R., Pancholi, D., Presas, F.: Almost contact 5-manifolds are contact (arXiv:1203.2166)Google Scholar
  4. 4.
    Cieliebak, K., Eliashberg, Y.: From Stein to Weinstein and Back: Symplectic Geometry of Affine Complex Manifold. American Mathematical Society Colloquium Publications, vol 59. American Mathematical Society, Providence (2012)Google Scholar
  5. 5.
    Eliashberg, Y.: Topological characterization of Stein manifolds of dimension \({\>}2\). Int. J. Math. 1, 29–46 (1990)CrossRefMATHMathSciNetGoogle Scholar
  6. 6.
    Etnyre, J.: Contact structures on 5-manifolds (arXiv:1210.5208)Google Scholar
  7. 7.
    Geiges, H.: Contact topology in dimension greater than three. In: European Congress of Mathematics, vol. II (Barcelona, 2000), pp. 535–545 (Progr. Math. Birkhäuser, Basel 202, 2001)Google Scholar
  8. 8.
    Giroux, E.: Géométrie de contact: de la dimension trois vers les dimensions supérieures. In: Proceedings of the International Congress of Mathematicians, vol. II, pp. 405–414. Higher edn. Press, Beijing (2002)Google Scholar
  9. 9.
    Hajduk, B., Walczak, R.: Constructions of contact forms on products and piecewise fibred manifolds (arXiv:1204.1692)Google Scholar
  10. 10.
    Kreck, M.: Surgery and duality. Ann. Math. 149, 707–754 (1999)CrossRefMATHMathSciNetGoogle Scholar
  11. 11.
    Lück, W.: A basic introduction to surgery theory. In: ICTP Lecture Notes Series 9, Band 1. School on high-dimensional manifold theory Trieste 2001. ICTP, Trieste (2002). http://www.him.uni-bonn.de/lueck/publications.php
  12. 12.
    Lutz, R.: Sur la géométrie des structures contact invariantes. Ann. Inst. Fourier 29, 283–306 (1979)CrossRefMATHMathSciNetGoogle Scholar
  13. 13.
    Martinet, J.: Formes de contact sur les variétés de dimension 3. In: Proceedings of Liverpool Singularities Symposium II, Lecture Notes in Mathematics vol 209, pp. 142–163. Springer, Berlin (1971)Google Scholar
  14. 14.
    Wall, C.T.C.: Geometrical connectivity I. J. Lond. Math. Soc. 3, 597–604 (1971)CrossRefMATHGoogle Scholar
  15. 15.
    Weinstein, A.: Contact surgery and symplectic handlebodies. Hokkaido Math. J. 20, 241–251 (1991)CrossRefMATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Jonathan Bowden
    • 1
  • Diarmuid Crowley
    • 2
  • András I. Stipsicz
    • 3
  1. 1.Mathematisches InstitutUniversität AugsburgAugsburgGermany
  2. 2.Max Planck Institut für MathematikBonnGermany
  3. 3.Rényi Institute of MathematicsBudapestHungary

Personalised recommendations