Inventiones mathematicae

, Volume 205, Issue 2, pp 413–457 | Cite as

Casson towers and slice links

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Abstract

We prove that a Casson tower of height 4 contains a flat embedded disc bounded by the attaching circle, and we prove disc embedding results for height 2 and 3 Casson towers which are embedded into a 4-manifold, with some additional fundamental group assumptions. In the proofs we create a capped grope from a Casson tower and use a refined height raising argument to establish the existence of a symmetric grope which has two layers of caps, data which is sufficient for a topological disc to exist, with the desired boundary. As applications, we present new slice knots and links by giving direct applications of the disc embedding theorem to produce slice discs, without first constructing a complementary 4-manifold. In particular we construct a family of slice knots which are potential counterexamples to the homotopy ribbon slice conjecture.

Mathematics Subject Classification

57N13 57N70 57M25 

References

  1. 1.
    Abe, T., Tange, M.: A construction of slice knots via annulus twists (2013). arXiv:1305.7492
  2. 2.
    Bižaca, Ž.: A reimbedding algorithm for Casson handles. Trans. Am. Math. Soc. 345(2), 435–510 (1994)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Bonahon, F.: Ribbon fibred knots, cobordism of surface diffeomorphisms and pseudo-Anosov diffeomorphisms. Math. Proc. Camb. Philos. Soc. 94(2), 235–251 (1983)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Casson, A.J.: Three lectures on new-infinite constructions in \(4\)-dimensional manifolds, À la recherche de la topologie perdue, Progr. Math., vol. 62, Birkhäuser, Boston, pp. 201–244 (1986) (with an appendix by L. Siebenmann)Google Scholar
  5. 5.
    Casson, A.J., Freedman, M.H.: Atomic surgery problems. In: Durham, N.H. (ed.) Four-Manifold Theory, Contemp. Math., vol. 35. Amer. Math. Soc., Providence, pp. 181–199 (1982)Google Scholar
  6. 6.
    Cochran, T.D., Friedl, S., Teichner, P.: New constructions of slice links. Comment. Math. Helv. 84(3), 617–638 (2009)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Casson, A.J., Gordon, C.McA.: A loop theorem for duality spaces and fibred ribbon knots. Invent. Math. 74(1), 119–137 (1983)Google Scholar
  8. 8.
    Casson, A.J., Gordon, C.McA.: Cobordism of classical knots, À la recherche de la topologie perdue. Birkhäuser, Boston, pp. 181–199 (1986) (with an appendix by P. M. Gilmer)Google Scholar
  9. 9.
    Cochran, T.D., Horn, P.D.: Structure in the bipolar filtration of topologically slice knots. Algebr. Geom. Topol. 15(1), 415–428 (2015)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Cochran, T.D., Harvey, S., Horn, P.: Filtering smooth concordance classes of topologically slice knots. Geom. Topol. 17(4), 2103–2162 (2013)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Cochran, T.D., Orr, K.E., Teichner, P.: Knot concordance, Whitney towers and \(L^2\)-signatures. Ann. Math. (2) 157(2), 433–519 (2003)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Cochran, T.D., Orr, K.E., Teichner, P.: Structure in the classical knot concordance group. Comment. Math. Helv. 79(1), 105–123 (2004)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Cappell, S.E., Shaneson, J.L.: The codimension two placement problem and homology equivalent manifolds. Ann. Math. (2) 99, 277–348 (1974)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Cochran, T.D., Teichner, P.: Knot concordance and von Neumann \(\rho \)-invariants. Duke Math. J. 137(2), 337–379 (2007)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Davis, J.F.: A two component link with Alexander polynomial one is concordant to the Hopf link. Math. Proc. Cambr. Philos. Soc. 140(2), 265–268 (2006)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Davis, J.F., Naik, S.: Alexander polynomials of equivariant slice and ribbon knots in \(S^3\). Trans. Am. Math. Soc. 358(7), 2949–2964 (2006, electronic)Google Scholar
  17. 17.
    Edwards, R.D.: The solution of the \(4\)-dimensional annulus conjecture (after Frank Quinn), Four-manifold theory (Durham, N.H., 1982), Contemp. Math., vol. 35, pp. 211–264. Amer. Math. Soc., Providence (1984)Google Scholar
  18. 18.
    Freedman, M.H., and Freedman, T.: Bing topology and Casson handles (2013). http://people.mpim-bonn.mpg.de/sbehrens/files/Freedman2013.pdf
  19. 19.
    Freedman, M.H., Lin, X.-S.: On the \((A, B)\)-slice problem. Topology 28(1), 91–110 (1989)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Freedman, M.H., Quinn, F.: Topology of 4-manifolds, Princeton Mathematical Series, vol. 39. Princeton University Press, Princeton (1990)Google Scholar
  21. 21.
    Freedman, M.H.: A surgery sequence in dimension four; the relations with knot concordance. Invent. Math. 68(2), 195–226 (1982)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Freedman, M.H.: The topology of four-dimensional manifolds. J. Differ. Geom. 17(3), 357–453 (1982)MathSciNetMATHGoogle Scholar
  23. 23.
    Freedman, M.H.: A new technique for the link slice problem. Invent. Math. 80(3), 453–465 (1985)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Freedman, M.H.: White head\(_{3}\) is a “slice” link. Invent. Math. 94(1), 175–182 (1988)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Freedman, M.H.: Link compositions and the topological slice problem. Topology 32(1), 145–156 (1993)MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Friedl, S.: Eta invariants as sliceness obstructions and their relation to Casson–Gordon invariants. Algebr. Geom. Topol. 4, 893–934 (2004, electronic)Google Scholar
  27. 27.
    Freedman, M.H., Teichner, P.: \(4\)-manifold topology. I. Subexponential groups. Invent. Math. 122(3), 509–529 (1995)MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    Michael, H., Teichner, P.: \(4\)-manifold topology. II. Dwyer’s filtration and surgery kernels. Invent. Math. 122(3), 531–557 (1995)MathSciNetMATHGoogle Scholar
  29. 29.
    Friedl, S., Peter, T.: New topologically slice knots. Geom. Topol. 9, 2129–2158 (2005, electronic)Google Scholar
  30. 30.
    Gompf, R.E., Singh, S.: On Freedman’s reimbedding theorems, Four-manifold theory (Durham, N.H., 1982), Contemp. Math., vol. 35, pp. 277–309. Amer. Math. Soc., Providence (1984)Google Scholar
  31. 31.
    Gompf, R.E., Stipsicz, A.I.: \(4\)-Manifolds and Kirby Calculus, Graduate Studies in Mathematics, vol. 20. American Mathematical Society, Providence (1999)MATHGoogle Scholar
  32. 32.
    Gompf, R.E., Scharlemann, M., Thompson, A.: Fibered knots and potential counterexamples to the property 2R and slice-ribbon conjectures. Geom. Topol. 14(4), 2305–2347 (2010)MathSciNetCrossRefMATHGoogle Scholar
  33. 33.
    Garoufalidis, S., Teichner, P.: On knots with trivial Alexander polynomial. J. Differ. Geom. 67(1), 167–193 (2004)MathSciNetMATHGoogle Scholar
  34. 34.
    Hedden, M., Livingston, C., Ruberman, D.: Topologically slice knots with nontrivial Alexander polynomial. Adv. Math. 231(2), 913–939 (2012)MathSciNetCrossRefMATHGoogle Scholar
  35. 35.
    Kervaire, M.A.: Les nœuds de dimensions supérieures. Bull. Soc. Math. Fr. 93, 225–271 (1965)MathSciNetMATHGoogle Scholar
  36. 36.
    Kirby, R.C.: Problems in low dimensional topology . In: Proceedings of Georgia Topology Conference, Part 2, pp. 35–473. Press (1995)Google Scholar
  37. 37.
    Krushkal, V.S., Quinn, F.: Subexponential groups in 4-manifold topology. Geom. Topol. 4, 407–430 (2000, electronic)Google Scholar
  38. 38.
    Krushkal, V.S.: On the relative slice problem and four-dimensional topological surgery. Math. Ann. 315(3), 363–396 (1999)MathSciNetCrossRefMATHGoogle Scholar
  39. 39.
    Krushkal, V.S.: A counterexample to the strong version of Freedman’s conjecture. Ann. Math. (2) 168(2), 675–693 (2008)MathSciNetCrossRefMATHGoogle Scholar
  40. 40.
    Krushkal, V.: “Slicing” the Hopf link. Geom. Topol. 19(3), 1657–1683 (2015)MathSciNetCrossRefMATHGoogle Scholar
  41. 41.
    Kirby, R.C., and Taylor, L.R.: A survey of 4-manifolds through the eyes of surgery, Surveys on surgery theory, vol. 2, Ann. of Math. Stud., vol. 149. Princeton Univ. Press, Princeton, pp. 387–421 (2001)Google Scholar
  42. 42.
    Levine, J.P.: Knot cobordism groups in codimension two. Comment. Math. Helv. 44, 229–244 (1969)MathSciNetCrossRefMATHGoogle Scholar
  43. 43.
    Levine, A.S.: Slicing mixed Bing–Whitehead doubles. J. Topol. 5(3), 713–726 (2012)MathSciNetCrossRefMATHGoogle Scholar
  44. 44.
    Levine, J.P., Orr, K.E.: A survey of applications of surgery to knot and link theory, Surveys on surgery theory, vol. 1, Ann. of Math. Stud., vol. 145, pp. 345–364. Princeton Univ. Press, Princeton (2000)Google Scholar
  45. 45.
    Milnor, J.W.: Link groups. Ann. Math. (2) 59, 177–195 (1954)MathSciNetCrossRefMATHGoogle Scholar
  46. 46.
    Miyazaki, K.: Nonsimple, ribbon fibered knots. Trans. Am. Math. Soc. 341(1), 1–44 (1994)MathSciNetCrossRefMATHGoogle Scholar
  47. 47.
    Otto, C.: The \((n)\)-solvable filtration of link concordance and Milnor’s invariants. Algebr. Geom. Topol. 14(5), 2627–2654 (2014)MathSciNetCrossRefMATHGoogle Scholar
  48. 48.
    Quinn, F.: Ends of maps. III. Dimensions \(4\) and \(5\). J. Differ. Geom. 17(3), 503–521 (1982)MathSciNetMATHGoogle Scholar
  49. 49.
    Ray, A.: Casson towers and filtrations of the knot concordance group. Algebr. Geom. Topol. (2013, preprint). arXiv:1309.7532

Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.Department of MathematicsPOSTECHPohangRepublic of Korea
  2. 2.School of MathematicsKorea Institute for Advanced StudySeoulRepublic of Korea
  3. 3.Départment de MathématiquesUQAMMontrealCanada

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