Geometry & Topology

Knot concordance and higher-order Blanchfield duality

Tim D Cochran, Shelly Harvey, and Constance Leidy

Full-text: Open access

Abstract

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 C,

n 1 0 . 5 0 C .

The filtration is important because of its strong connection to the classification of topological 4–manifolds. Here we introduce new techniques for studying C and use them to prove that, for each n0, the group nn.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.

Article information

Source
Geom. Topol., Volume 13, Number 3 (2009), 1419-1482.

Dates
Received: 10 September 2008
Accepted: 1 December 2008
First available in Project Euclid: 20 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.gt/1513800250

Digital Object Identifier
doi:10.2140/gt.2009.13.1419

Mathematical Reviews number (MathSciNet)
MR2496049

Zentralblatt MATH identifier
1175.57004

Subjects
Primary: 57M25: Knots and links in $S^3$ {For higher dimensions, see 57Q45}
Secondary: 57M10: Covering spaces

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

Citation

Cochran, Tim D; Harvey, Shelly; Leidy, Constance. Knot concordance and higher-order Blanchfield duality. Geom. Topol. 13 (2009), no. 3, 1419--1482. doi:10.2140/gt.2009.13.1419. https://projecteuclid.org/euclid.gt/1513800250


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References

  • A Casson, M Freedman, Atomic surgery problems, from: “Four-manifold theory (Durham, N.H., 1982)”, (C Gordon, R Kirby, editors), Contemp. Math. 35, Amer. Math. Soc. (1984) 181–199
  • 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”, (R J Milgram, editor), Proc. Sympos. Pure Math. XXXII, Amer. Math. Soc. (1978) 39–53
  • A Casson, C M Gordon, Cobordism of classical knots, from: “À la recherche de la topologie perdue”, (L Guillou, A Marin, editors), Progr. Math. 62, Birkhäuser, Boston (1986) 181–199 With an appendix by P M Gilmer
  • J C Cha, The structure of the rational concordance group of knots, Mem. Amer. Math. Soc. 189 (2007) x+95
  • J C Cha, Topological minimal genus and $L\sp 2$–signatures, Algebr. Geom. Topol. 8 (2008) 885–909
  • J Cheeger, M Gromov, Bounds on the von Neumann dimension of $L\sp 2$–cohomology and the Gauss–Bonnet theorem for open manifolds, J. Differential Geom. 21 (1985) 1–34
  • T D Cochran, Noncommutative knot theory, Algebr. Geom. Topol. 4 (2004) 347–398
  • T D Cochran, S Friedl, P Teichner, New constructions of slice links, to appear in Comment. Math. Helv.
  • T D Cochran, S Harvey, C Leidy, Knot concordance and Blanchfield duality, Oberwolfach Reports 3 (2006)
  • T D Cochran, S Harvey, C Leidy, Link concordance and generalized doubling operators, Algebr. Geom. Topol. 8 (2008) 1593–1646
  • T D Cochran, T Kim, Higher-order Alexander invariants and filtrations of the knot concordance group, Trans. Amer. Math. Soc. 360 (2008) 1407–1441
  • T D Cochran, K E Orr, P Teichner, Knot concordance, Whitney towers and $L\sp 2$–signatures, Ann. of Math. $(2)$ 157 (2003) 433–519
  • T D Cochran, K E Orr, P Teichner, Structure in the classical knot concordance group, Comment. Math. Helv. 79 (2004) 105–123
  • T D Cochran, P Teichner, Knot concordance and von Neumann $\rho$–invariants, Duke Math. J. 137 (2007) 337–379
  • S Friedl, Eta invariants as sliceness obstructions and their relation to Casson–Gordon invariants, Algebr. Geom. Topol. 4 (2004) 893–934
  • S Friedl, $L\sp 2$–eta-invariants and their approximation by unitary eta-invariants, Math. Proc. Cambridge Philos. Soc. 138 (2005) 327–338
  • S Friedl, P Teichner, New topologically slice knots, Geom. Topol. 9 (2005) 2129–2158
  • P M Gilmer, Configurations of surfaces in $4$–manifolds, Trans. Amer. Math. Soc. 264 (1981) 353–380
  • P M Gilmer, Some Interesting Non-Ribbon Knots, Abstracts of papers presented to Amer. Math. Soc. 2 (1981) 448
  • P M Gilmer, Slice knots in $S\sp{3}$, Quart. J. Math. Oxford Ser. $(2)$ 34 (1983) 305–322
  • P M Gilmer, C Livingston, The Casson–Gordon invariant and link concordance, Topology 31 (1992) 475–492
  • C M Gordon, Some aspects of classical knot theory, from: “Knot theory (Proc. Sem., Plans-sur-Bex, 1977)”, (J-C Hausmann, editor), Lecture Notes in Math. 685, Springer, Berlin (1978) 1–60
  • S L Harvey, Higher-order polynomial invariants of $3$–manifolds giving lower bounds for the Thurston norm, Topology 44 (2005) 895–945
  • S L Harvey, Homology cobordism invariants and the Cochran–Orr–Teichner filtration of the link concordance group, Geom. Topol. 12 (2008) 387–430
  • B J Jiang, A simple proof that the concordance group of algebraically slice knots is infinitely generated, Proc. Amer. Math. Soc. 83 (1981) 189–192
  • T Kim, Filtration of the classical knot concordance group and Casson–Gordon invariants, Math. Proc. Cambridge Philos. Soc. 137 (2004) 293–306
  • C Lamm, Symmetric unions and ribbon knots, Osaka J. Math. 37 (2000) 537–550
  • C Leidy, Higher-order linking forms for $3$–manifolds, preprint
  • C Leidy, Higher-order linking forms for knots, Comment. Math. Helv. 81 (2006) 755–781
  • C F Letsche, An obstruction to slicing knots using the eta invariant, Math. Proc. Cambridge Philos. Soc. 128 (2000) 301–319
  • J P Levine, Knot cobordism groups in codimension two, Comment. Math. Helv. 44 (1969) 229–244
  • J P Levine, Link invariants via the eta invariant, Comment. Math. Helv. 69 (1994) 82–119
  • R A Litherland, Cobordism of satellite knots, from: “Four-manifold theory (Durham, N.H., 1982)”, (C Gordon, R Kirby, editors), Contemp. Math. 35, Amer. Math. Soc. (1984) 327–362
  • C Livingston, Knots which are not concordant to their reverses, Quart. J. Math. Oxford Ser. $(2)$ 34 (1983) 323–328
  • C Livingston, Links not concordant to boundary links, Proc. Amer. Math. Soc. 110 (1990) 1129–1131
  • C Livingston, Order 2 algebraically slice knots, from: “Proceedings of the Kirbyfest (Berkeley, CA, 1998)”, (J Hass, M Scharlemann, editors), Geom. Topol. Monogr. 2, Geom. Topol. Publ., Coventry (1999) 335–342
  • C Livingston, Infinite order amphicheiral knots, Algebr. Geom. Topol. 1 (2001) 231–241
  • C Livingston, A survey of classical knot concordance, from: “Handbook of knot theory”, (W Menasco, M Thistlethwaite, editors), Elsevier, Amsterdam (2005) 319–347
  • C Livingston, P Melvin, Abelian invariants of satellite knots, from: “Geometry and topology (College Park, Md., 1983/84)”, (J Alexander, J Harer, editors), Lecture Notes in Math. 1167, Springer, Berlin (1985) 217–227
  • W Lück, T Schick, Various $L\sp 2$–signatures and a topological $L\sp 2$–signature theorem, from: “High-dimensional manifold topology”, (F T Farrell, W Lück, editors), World Sci. Publ., River Edge, NJ (2003) 362–399
  • D S Passman, The algebraic structure of group rings, Pure and Applied Math., Wiley-Interscience, New York (1977)
  • B Stenstr öm, Rings of quotients. An introduction to methods of ring theory, Die Grund. der Math. Wissenschaften 217, Springer, New York (1975)
  • N W Stoltzfus, Unraveling the integral knot concordance group, Mem. Amer. Math. Soc. 12 (1977) iv+91
  • C T C Wall, Surgery on compact manifolds, second edition, Math. Surveys and Monogr. 69, Amer. Math. Soc. (1999) Edited and with a foreword by A A Ranicki