Mathematische Annalen

, Volume 370, Issue 3–4, pp 963–992 | Cite as

Rational Whitney tower filtration of links

  • Jae Choon Cha


We present complete classifications of links in the 3-sphere modulo framed and twisted Whitney towers in a rational homology 4-ball. This provides a geometric characterization of the vanishing of the Milnor invariants of links in terms of Whitney towers. Our result also says that the higher order Arf invariants, which are conjectured to be nontrivial, measure the potential difference between the Whitney tower theory in rational homology 4-balls and that in the 4-ball extensively developed by Conant, Schneiderman and Teichner.

Mathematics Subject Classification

57N13 57N70 57M25 



The author thanks an anonymous referee for careful comments. This work was partially supported by NRF Grant 2013067043.


  1. 1.
    Cha, J.C.: The structure of the rational concordance group of knots. Mem. Am. Math. Soc. 189(885), x+95 (2007)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Cochran, T.D.: Derivatives of links: Milnor’s concordance invariants and Massey’s products. Mem. Am. Math. Soc. 84(427), x+73 (1990)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Cochran, T.D., Orr, K.E., Teichner, P.: Knot concordance, Whitney towers and \(L^2\)-signatures. Ann. Math. (2) 157(2), 433–519 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Conant, J., Schneiderman, R., Teichner, P.: Jacobi identities in low-dimensional topology. Compos. Math. 143(3), 780–810 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Conant, J., Schneiderman, R., Teichner, P.: Higher-order intersections in low-dimensional topology. Proc. Natl. Acad. Sci. USA 108(20), 8131–8138 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Conant, J., Schneiderman, R., Teichner, P.: Tree homology and a conjecture of Levine. Geom. Topol. 16(1), 555–600 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Conant, J., Schneiderman, R., Teichner, P.: Universal quadratic forms and Whitney tower intersection invariants. In: Proceedings of the Freedman Fest, Geom. Topol. Monogr., vol. 18, Geom. Topol. Publ., Coventry, pp. 35–60 (2012)Google Scholar
  8. 8.
    Conant, J., Schneiderman, R., Teichner, P.: Whitney tower concordance of classical links. Geom. Topol. 16(3), 1419–1479 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Conant, J., Schneiderman, R., Teichner, P.: Milnor invariants and twisted Whitney towers. J. Topol. 7(1), 187–224 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Conant, J., Teichner, P.: Grope cobordism and Feynman diagrams. Math. Ann. 328(1–2), 135–171 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Conant, J., Teichner, P.: Grope cobordism of classical knots. Topology 43(1), 119–156 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Dwyer, W.G.: Homology, Massey products and maps between groups. J. Pure Appl. Algebra 6(2), 177–190 (1975)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Freedman, M.H., Quinn, F.: Topology of 4-manifolds, Princeton Mathematical Series, vol. 39. Princeton University Press, Princeton (1990)Google Scholar
  14. 14.
    Freedman, M.H., Teichner, P.: \(4\)-manifold topology. II. Dwyer’s filtration and surgery kernels. Invent. Math. 122(3), 531–557 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Igusa, K., Orr, K.E.: Links, pictures and the homology of nilpotent groups. Topology 40(6), 1125–1166 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Krushkal, V.S.: Additivity properties of Milnor’s \({{\overline{\mu }}}\)-invariants. J. Knot Theory Ramif. 7(5), 625–637 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Krushkal, V.S., Teichner, P.: Alexander duality, gropes and link homotopy. Geom. Topol. 1, 51–69 (1997). (electronic)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Levine, J.P.: Homology cylinders: an enlargement of the mapping class group. Algebr. Geom. Topol. 1, 243–270 (2001). (electronic)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Levine, J.P.: Addendum and correction to: homology cylinders: an enlargement of the mapping class group. Algebr. Geom. Topol. 1 (2001), 243–270; MR1823501 (2002m:57020), Algebr. Geom. Topol. 2, 1197–1204 (electronic) (2002)Google Scholar
  20. 20.
    Milnor, J.W.: Isotopy of links. Algebraic Geometry and Topology, A symposium in honor of S. Lefschetz, Princeton University Press, Princeton, pp. 280–306 (1957)Google Scholar
  21. 21.
    Magnus, W., Karrass, A., Solitar, D.: Combinatorial Group Theory: Presentations of Groups in Terms of Generators and Relations. Interscience Publishers (Wiley), New York-London-Sydney (1966)zbMATHGoogle Scholar
  22. 22.
    Orr, K.E.: Homotopy invariants of links. Invent. Math. 95(2), 379–394 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Schneiderman, R.: Whitney towers and gropes in 4-manifolds. Trans. Am. Math. Soc 358(10), 4251–4278 (2006). (electronic)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Schneiderman, R., Teichner, P.: Whitney towers and the Kontsevich integral. In: Proceedings of the Casson Fest, Geom. Topol. Monogr., vol. 7, Geom. Topol. Publ., Coventry, pp. 101–134 (electronic) (2004)Google Scholar
  25. 25.
    Stallings, J.: Homology and central series of groups. J. Algebra 2, 170–181 (1965)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  1. 1.Department of MathematicsPOSTECHPohangRepublic of Korea
  2. 2.School of MathematicsKorea Institute for Advanced StudySeoulRepublic of Korea

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