Abstract
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.
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References
Cha, J.C.: The structure of the rational concordance group of knots. Mem. Am. Math. Soc. 189(885), x+95 (2007)
Cochran, T.D.: Derivatives of links: Milnor’s concordance invariants and Massey’s products. Mem. Am. Math. Soc. 84(427), x+73 (1990)
Cochran, T.D., Orr, K.E., Teichner, P.: Knot concordance, Whitney towers and \(L^2\)-signatures. Ann. Math. (2) 157(2), 433–519 (2003)
Conant, J., Schneiderman, R., Teichner, P.: Jacobi identities in low-dimensional topology. Compos. Math. 143(3), 780–810 (2007)
Conant, J., Schneiderman, R., Teichner, P.: Higher-order intersections in low-dimensional topology. Proc. Natl. Acad. Sci. USA 108(20), 8131–8138 (2011)
Conant, J., Schneiderman, R., Teichner, P.: Tree homology and a conjecture of Levine. Geom. Topol. 16(1), 555–600 (2012)
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)
Conant, J., Schneiderman, R., Teichner, P.: Whitney tower concordance of classical links. Geom. Topol. 16(3), 1419–1479 (2012)
Conant, J., Schneiderman, R., Teichner, P.: Milnor invariants and twisted Whitney towers. J. Topol. 7(1), 187–224 (2014)
Conant, J., Teichner, P.: Grope cobordism and Feynman diagrams. Math. Ann. 328(1–2), 135–171 (2004)
Conant, J., Teichner, P.: Grope cobordism of classical knots. Topology 43(1), 119–156 (2004)
Dwyer, W.G.: Homology, Massey products and maps between groups. J. Pure Appl. Algebra 6(2), 177–190 (1975)
Freedman, M.H., Quinn, F.: Topology of 4-manifolds, Princeton Mathematical Series, vol. 39. Princeton University Press, Princeton (1990)
Freedman, M.H., Teichner, P.: \(4\)-manifold topology. II. Dwyer’s filtration and surgery kernels. Invent. Math. 122(3), 531–557 (1995)
Igusa, K., Orr, K.E.: Links, pictures and the homology of nilpotent groups. Topology 40(6), 1125–1166 (2001)
Krushkal, V.S.: Additivity properties of Milnor’s \({{\overline{\mu }}}\)-invariants. J. Knot Theory Ramif. 7(5), 625–637 (1998)
Krushkal, V.S., Teichner, P.: Alexander duality, gropes and link homotopy. Geom. Topol. 1, 51–69 (1997). (electronic)
Levine, J.P.: Homology cylinders: an enlargement of the mapping class group. Algebr. Geom. Topol. 1, 243–270 (2001). (electronic)
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)
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)
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)
Orr, K.E.: Homotopy invariants of links. Invent. Math. 95(2), 379–394 (1989)
Schneiderman, R.: Whitney towers and gropes in 4-manifolds. Trans. Am. Math. Soc 358(10), 4251–4278 (2006). (electronic)
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)
Stallings, J.: Homology and central series of groups. J. Algebra 2, 170–181 (1965)
Acknowledgements
The author thanks an anonymous referee for careful comments. This work was partially supported by NRF Grant 2013067043.
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Communicated by Thomas Schick.
