Whitney towers, gropes and Casson–Gordon style invariants of links

Kim, Min Hoon¹

¹Department of Mathematics, Pohang University of Science and Technology, Gyungbuk 790-784, South Korea

Algebraic and Geometric Topology, Tome 15 (2015) no. 3, pp. 1813-1845

Résumé

In this paper, we prove a conjecture of Friedl and Powell that their Casson–Gordon type invariant of 2–component links with linking number one is actually an obstruction to being height-3.5 Whitney tower/grope concordant to the Hopf link. The proof employs the notion of solvable cobordism of 3–manifolds with boundary, which was introduced by Cha. We also prove that the Blanchfield form and the Alexander polynomial of links in S3 give obstructions to height-3 Whitney tower/grope concordance. This generalizes the results of Hillman and Kawauchi.

Détail

DOI : 10.2140/agt.2015.15.1813

Classification : 57M25, 57M27, 57N70
Keywords: link concordance, Whitney tower concordance, grope concordance, Casson–Gordon invariant

Bibliographie

[1] R C Blanchfield, Intersection theory of manifolds with operators with applications to knot theory, Ann. of Math. 65 (1957) 340

[2] J R Burke, Infection by string links and new structure in the knot concordance group, Algebr. Geom. Topol. 14 (2014) 1577

[3] A J Casson, C M Gordon, On slice knots in dimension three, from: "Algebraic and geometric topology" (editor R J Milgram), Proc. Sympos. Pure Math. 32, Amer. Math. Soc. (1978) 39

[4] A J Casson, C M Gordon, Cobordism of classical knots, from: "À la recherche de la topologie perdue" (editors L Guillou, A Marin), Progr. Math. 62, Birkhäuser (1986) 181

[5] J C Cha, Structure of the string link concordance group and Hirzebruch-type invariants, Indiana Univ. Math. J. 58 (2009) 891

[6] J C Cha, Link concordance, homology cobordism and Hirzebruch-type defects from iterated $p$–covers, J. Eur. Math. Soc. (JEMS) 12 (2010) 555

[7] J C Cha, Amenable $L^2$–theoretic methods and knot concordance, Int. Math. Res. Not. 2014 (2014) 4768

[8] J C Cha, Symmetric Whitney tower cobordism for bordered $3$–manifolds and links, Trans. Amer. Math. Soc. 366 (2014) 3241

[9] J C Cha, M Powell, Nonconcordant links with homology cobordant zero-framed surgery manifolds, Pacific J. Math. 272 (2014) 1

[10] D Cimasoni, A geometric construction of the Conway potential function, Comment. Math. Helv. 79 (2004) 124

[11] T D Cochran, S Harvey, C Leidy, Link concordance and generalized doubling operators, Algebr. Geom. Topol. 8 (2008) 1593

[12] T D Cochran, S Harvey, C Leidy, Knot concordance and higher-order Blanchfield duality, Geom. Topol. 13 (2009) 1419

[13] T D Cochran, S Harvey, C Leidy, Primary decomposition and the fractal nature of knot concordance, Math. Ann. 351 (2011) 443

[14] T D Cochran, T Kim, Higher-order Alexander invariants and filtrations of the knot concordance group, Trans. Amer. Math. Soc. 360 (2008) 1407

[15] T D Cochran, K E Orr, P Teichner, Knot concordance, Whitney towers and $L^2$–signatures, Ann. of Math. 157 (2003) 433

[16] T D Cochran, K E Orr, P Teichner, Structure in the classical knot concordance group, Comment. Math. Helv. 79 (2004) 105

[17] T D Cochran, P Teichner, Knot concordance and von Neumann $\rho$–invariants, Duke Math. J. 137 (2007) 337

[18] J Conant, R Schneiderman, P Teichner, Higher-order intersections in low-dimensional topology, Proc. Natl. Acad. Sci. USA 108 (2011) 8131

[19] D Cooper, The universal abelian cover of a link, from: "Low-dimensional topology" (editors R Brown, T L Thickstun), London Math. Soc. Lecture Note Ser. 48, Cambridge Univ. Press (1982) 51

[20] R H Crowell, R H Fox, Introduction to knot theory, Graduate Texts in Math. 57, Springer (1977)

[21] C W Davis, Von Neumann rho-invariants and torsion in the topological knot concordance group, Algebr. Geom. Topol. 12 (2012) 753

[22] B D Franklin, The effect of infecting curves on knot concordance, Int. Math. Res. Not. 2013 (2013) 184

[23] M H Freedman, F Quinn, Topology of $4$–manifolds, Princeton Math. Series 39, Princeton Univ. Press (1990)

[24] S Friedl, M Powell, An injectivity theorem for Casson–Gordon type representations relating to the concordance of knots and links, Bull. Korean Math. Soc. 49 (2012) 395

[25] S Friedl, M Powell, Links not concordant to the Hopf link, Math. Proc. Cambridge Philos. Soc. 156 (2014) 425

[26] S L Harvey, Homology cobordism invariants and the Cochran–Orr–Teichner filtration of the link concordance group, Geom. Topol. 12 (2008) 387

[27] A Hatcher, Algebraic topology, Cambridge Univ. Press (2002)

[28] J Hillman, Algebraic invariants of links, Series on knots and everything 52, World Scientific Publishing Co. Pte. Ltd. (2012)

[29] P D Horn, The nontriviality of the grope filtrations of the knot and link concordance groups, Comment. Math. Helv. 85 (2010) 751

[30] A Kawauchi, On the Alexander polynomials of cobordant links, Osaka J. Math. 15 (1978) 151

[31] C Kearton, Cobordism of knots and Blanchfield duality, J. London Math. Soc. 10 (1975) 406

[32] J P Levine, Knot cobordism groups in codimension two, Comment. Math. Helv. 44 (1969) 229

[33] J P Levine, The module of a $2$–component link, Comment. Math. Helv. 57 (1982) 377

[34] J P Levine, Link invariants via the eta-invariant, Comment. Math. Helv. 69 (1994) 82

[35] R Schneiderman, Whitney towers and gropes in $4$–manifolds, Trans. Amer. Math. Soc. 358 (2006) 4251