Knot concordance and higher-order Blanchfield duality

Cochran, Tim D¹ ; Harvey, Shelly¹ ; Leidy, Constance²

¹ Department of Mathematics, Rice University, Houston, Texas 77005-1892
² Wesleyan University, Wesleyan Station, Middletown, CT 06459

Geometry & topology, Tome 13 (2009) no. 3, pp. 1419-1482.

Résumé

In 1997, T Cochran, K Orr, and P Teichner [Ann. of Math. (2) 157 (2003) 433-519] defined a filtration of the classical knot concordance group $\mathcal{C}$,

The filtration is important because of its strong connection to the classification of topological $4$–manifolds. Here we introduce new techniques for studying $\mathcal{C}$ and use them to prove that, for each $n\in {\mathbb{N}}_0$, the group ${\mathcal{ℱ}}_n∕{\mathcal{ℱ}}_{n.5}$ has infinite rank. We establish the same result for the corresponding filtration of the smooth concordance group. We also resolve a long-standing question as to whether certain natural families of knots, first considered by Casson–Gordon and Gilmer, contain slice knots.

Détail

DOI : 10.2140/gt.2009.13.1419

Keywords: concordance, (n)-solvable, knot, slice knot, Blanchfield form, von Neumann signature

Bibliographie

[1] A Casson, M Freedman, Atomic surgery problems, from: "Four-manifold theory (Durham, N.H., 1982)" (editors C Gordon, R Kirby), Contemp. Math. 35, Amer. Math. Soc. (1984) 181

[2] A Casson, C M Gordon, On slice knots in dimension three, from: "Algebraic and geometric topology (Proc. Sympos. Pure Math., Stanford Univ., 1976), Part 2" (editor R J Milgram), Proc. Sympos. Pure Math. XXXII, Amer. Math. Soc. (1978) 39

[3] A 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

[4] J C Cha, The structure of the rational concordance group of knots, Mem. Amer. Math. Soc. 189 (2007)

[5] J C Cha, Topological minimal genus and $L^2$–signatures, Algebr. Geom. Topol. 8 (2008) 885

[6] J Cheeger, M Gromov, Bounds on the von Neumann dimension of $L^2$–cohomology and the Gauss–Bonnet theorem for open manifolds, J. Differential Geom. 21 (1985) 1

[7] T D Cochran, Noncommutative knot theory, Algebr. Geom. Topol. 4 (2004) 347

[8] T D Cochran, S Friedl, P Teichner, New constructions of slice links, to appear in Comment. Math. Helv.

[9] T D Cochran, S Harvey, C Leidy, Knot concordance and Blanchfield duality, Oberwolfach Reports 3 (2006)

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

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

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

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

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

[15] S Friedl, Eta invariants as sliceness obstructions and their relation to Casson–Gordon invariants, Algebr. Geom. Topol. 4 (2004) 893

[16] S Friedl, $L^2$–eta-invariants and their approximation by unitary eta-invariants, Math. Proc. Cambridge Philos. Soc. 138 (2005) 327

[17] S Friedl, P Teichner, New topologically slice knots, Geom. Topol. 9 (2005) 2129

[18] P M Gilmer, Configurations of surfaces in $4$–manifolds, Trans. Amer. Math. Soc. 264 (1981) 353

[19] P M Gilmer, Some Interesting Non-Ribbon Knots, Abstracts of papers presented to Amer. Math. Soc. 2 (1981) 448

[20] P M Gilmer, Slice knots in $S^{3}$, Quart. J. Math. Oxford Ser. $(2)$ 34 (1983) 305

[21] P M Gilmer, C Livingston, The Casson–Gordon invariant and link concordance, Topology 31 (1992) 475

[22] C M Gordon, Some aspects of classical knot theory, from: "Knot theory (Proc. Sem., Plans-sur-Bex, 1977)" (editor J C Hausmann), Lecture Notes in Math. 685, Springer (1978) 1

[23] S L Harvey, Higher-order polynomial invariants of $3$–manifolds giving lower bounds for the Thurston norm, Topology 44 (2005) 895

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

[25] B J Jiang, A simple proof that the concordance group of algebraically slice knots is infinitely generated, Proc. Amer. Math. Soc. 83 (1981) 189

[26] T Kim, Filtration of the classical knot concordance group and Casson–Gordon invariants, Math. Proc. Cambridge Philos. Soc. 137 (2004) 293

[27] C Lamm, Symmetric unions and ribbon knots, Osaka J. Math. 37 (2000) 537

[28] C Leidy, Higher-order linking forms for $3$–manifolds, preprint

[29] C Leidy, Higher-order linking forms for knots, Comment. Math. Helv. 81 (2006) 755

[30] C F Letsche, An obstruction to slicing knots using the eta invariant, Math. Proc. Cambridge Philos. Soc. 128 (2000) 301

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

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

[33] R A Litherland, Cobordism of satellite knots, from: "Four-manifold theory (Durham, N.H., 1982)" (editors C Gordon, R Kirby), Contemp. Math. 35, Amer. Math. Soc. (1984) 327

[34] C Livingston, Knots which are not concordant to their reverses, Quart. J. Math. Oxford Ser. $(2)$ 34 (1983) 323

[35] C Livingston, Links not concordant to boundary links, Proc. Amer. Math. Soc. 110 (1990) 1129

[36] C Livingston, Order 2 algebraically slice knots, from: "Proceedings of the Kirbyfest (Berkeley, CA, 1998)" (editors J Hass, M Scharlemann), Geom. Topol. Monogr. 2, Geom. Topol. Publ. (1999) 335

[37] C Livingston, Infinite order amphicheiral knots, Algebr. Geom. Topol. 1 (2001) 231

[38] C Livingston, A survey of classical knot concordance, from: "Handbook of knot theory" (editors W Menasco, M Thistlethwaite), Elsevier (2005) 319

[39] C Livingston, P Melvin, Abelian invariants of satellite knots, from: "Geometry and topology (College Park, Md., 1983/84)" (editors J Alexander, J Harer), Lecture Notes in Math. 1167, Springer (1985) 217

[40] W Lück, T Schick, Various $L^2$–signatures and a topological $L^2$–signature theorem, from: "High-dimensional manifold topology" (editors F T Farrell, W Lück), World Sci. Publ. (2003) 362

[41] D S Passman, The algebraic structure of group rings, Pure and Applied Math., Wiley-Interscience (1977)

[42] B Stenström, Rings of quotients. An introduction to methods of ring theory, Die Grund. der Math. Wissenschaften 217, Springer (1975)

[43] N W Stoltzfus, Unraveling the integral knot concordance group, Mem. Amer. Math. Soc. 12 (1977)

[44] C T C Wall, Surgery on compact manifolds, Math. Surveys and Monogr. 69, Amer. Math. Soc. (1999)