Selecta Mathematica

, Volume 18, Issue 4, pp 839–854 | Cite as

Dehn surgeries and negative-definite four-manifolds



Given a knot K in the three-sphere, we address the question: Which Dehn surgeries on K bound negative-definite four-manifolds? We show that the answer depends on a number m(K), which is a smooth concordance invariant. We study the properties of this invariant and compute it for torus knots.


Dehn surgery Smooth negative-definite four-manifold Torus knot Concordance 

Mathematics Subject Classification (2000)

57M27 57Q60 


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  1. 1.
    Donaldson S.K.: An application of gauge theory to four-dimensional topology. J. Differ. Geom. 18, 279–315 (1983)MathSciNetMATHGoogle Scholar
  2. 2.
    Gompf, R.E., Stipsicz, A.I.: 4-manifolds and Kirby calculus, Graduate Studies in Math, vol. 20, Am. Math. Soc. (1999)Google Scholar
  3. 3.
    Greene, J.E.: L-space surgeries, genus bounds, and the cabling conjecture. arXiv:1009.1130 (2010)Google Scholar
  4. 4.
    Hirzebruch, F., Neumann W.D., Koh, S.S.: Differentiable manifolds and quadratic forms. Lecture Notes in Pure and Applied Math, vol. 4, Marcel Dekker (1971)Google Scholar
  5. 5.
    Lecuona A., Lisca P.: Stein fillable Seifert fibered 3-manifolds. Algebr. Geom. Topol. 11, 625–642 (2011)MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Lisca P., Stipsicz A.I.: Ozsváth-Szabó invariants and tight contact three-manifolds, I. Geom. Topol. 8, 925–945 (2004)MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Moser L.: Elementary surgery along a torus knot. Pac. J. Math. 38, 737–745 (1971)MATHGoogle Scholar
  8. 8.
    Neumann W.: A calculus for plumbing applied to the topology of complex surface singularities and degenerating complex curves. Trans. Am. Math. Soc. 268, 299–344 (1981)MATHCrossRefGoogle Scholar
  9. 9.
    Owens, B., Strle, S.: A characterisation of the \({Z^n \oplus Z(\delta)}\) lattice and definite nonunimodular intersection forms. arXiv:0802.1495 (2008)Google Scholar
  10. 10.
    Ozsváth P., Szabó Z.: Holomorphic disks and genus bounds. Geom. Topol. 8, 311–334 (2004)MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    Popescu-Pampu, P.: The geometry of continued fractions and the topology of surface singularities. Singularities in geometry and topology 2004, pp. 119–195, Adv. Stud. Pure Math. vol. 46, Math. Soc. Japan (2007)Google Scholar

Copyright information

© Springer Basel AG 2012

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsUniversity of GlasgowGlasgowUK
  2. 2.Faculty of Mathematics and PhysicsUniversity of LjubljanaLjubljanaSlovenia

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