Geometry & Topology

Filtering smooth concordance classes of topologically slice knots

Tim D Cochran, Shelly Harvey, and Peter Horn

Full-text: Open access

Abstract

We propose and analyze a structure with which to organize the difference between a knot in S3 bounding a topologically embedded 2–disk in B4 and it bounding a smoothly embedded disk. The n–solvable filtration of the topological knot concordance group, due to Cochran–Orr–Teichner, may be complete in the sense that any knot in the intersection of its terms may well be topologically slice. However, the natural extension of this filtration to what is called the n–solvable filtration of the smooth knot concordance group, is unsatisfactory because any topologically slice knot lies in every term of the filtration. To ameliorate this we investigate a new filtration, {n}, that is simultaneously a refinement of the n–solvable filtration and a generalization of notions of positivity studied by Gompf and Cochran. We show that each nn+1 has infinite rank. But our primary interest is in the induced filtration, {Tn}, on the subgroup, T, of knots that are topologically slice. We prove that TT0 is large, detected by gauge-theoretic invariants and the τ, s, ϵ–invariants, while the nontriviality of T0T1 can be detected by certain d–invariants. All of these concordance obstructions vanish for knots in T1. Nonetheless, going beyond this, our main result is that T1T2 has positive rank. Moreover under a “weak homotopy-ribbon” condition, we show that each TnTn+1 has positive rank. These results suggest that, even among topologically slice knots, the fundamental group is responsible for a wide range of complexity.

Article information

Source
Geom. Topol., Volume 17, Number 4 (2013), 2103-2162.

Dates
Received: 1 June 2012
Accepted: 1 April 2013
First available in Project Euclid: 20 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.gt/1513732648

Digital Object Identifier
doi:10.2140/gt.2013.17.2103

Mathematical Reviews number (MathSciNet)
MR3109864

Zentralblatt MATH identifier
1282.57006

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

Keywords
concordance slice knot 4–manifold

Citation

Cochran, Tim D; Harvey, Shelly; Horn, Peter. Filtering smooth concordance classes of topologically slice knots. Geom. Topol. 17 (2013), no. 4, 2103--2162. doi:10.2140/gt.2013.17.2103. https://projecteuclid.org/euclid.gt/1513732648


Export citation

References

  • S Akbulut, R Kirby, Branched covers of surfaces in $4$–manifolds, Math. Ann. 252 (1979/80) 111–131
  • A J Casson, C M Gordon, On slice knots in dimension three, from: “Algebraic and geometric topology, Part 2”, (R J Milgram, editor), Proc. Sympos. Pure Math. 32, Amer. Math. Soc. (1978) 39–53
  • A J 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, MA (1986) 181–199
  • J C Cha, Topological minimal genus and $L\sp 2$–signatures, Algebr. Geom. Topol. 8 (2008) 885–909
  • J C Cha, Amenable $L\sp{2}$–theoretic methods and knot concordance, Inter. Math. Res. Notices (2013)
  • T D Cochran, R E Gompf, Applications of Donaldson's theorems to classical knot concordance, homology $3$–spheres and property $P$, Topology 27 (1988) 495–512
  • T Cochran, S Harvey, C Leidy, Link concordance and generalized doubling operators, Algebr. Geom. Topol. 8 (2008) 1593–1646
  • T D Cochran, S Harvey, C Leidy, Knot concordance and higher-order Blanchfield duality, Geom. Topol. 13 (2009) 1419–1482
  • T D Cochran, S Harvey, C Leidy, 2–torsion in the $n$–solvable filtration of the knot concordance group, Proc. Lond. Math. Soc. 102 (2011) 257–290
  • T D Cochran, S Harvey, C Leidy, Primary decomposition and the fractal nature of knot concordance, Math. Ann. 351 (2011) 443–508
  • T D Cochran, W B R Lickorish, Unknotting information from $4$–manifolds, Trans. Amer. Math. Soc. 297 (1986) 125–142
  • T D Cochran, K E Orr, P Teichner, Knot concordance, Whitney towers and $L\sp 2$–signatures, Ann. of Math. 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
  • C W Davis, Linear independence of knots arising from iterated infection without the use of Tristram–Levine signature, Int. Math. Res. Not. 2013 (2013)
  • S K Donaldson, An application of gauge theory to four-dimensional topology, J. Differential Geom. 18 (1983) 279–315
  • N D Elkies, A characterization of the ${\bf Z}\sp n$ lattice, Math. Res. Lett. 2 (1995) 321–326
  • H Endo, Linear independence of topologically slice knots in the smooth cobordism group, Topology Appl. 63 (1995) 257–262
  • R Fintushel, R J Stern, Pseudofree orbifolds, Ann. of Math. 122 (1985) 335–364
  • R Fintushel, R J Stern, Instanton homology of Seifert fibred homology three spheres, Proc. London Math. Soc. 61 (1990) 109–137
  • M H Freedman, F Quinn, Topology of $4$–manifolds, Princeton Mathematical Series 39, Princeton Univ. Press (1990)
  • M Furuta, Homology cobordism group of homology $3$–spheres, Invent. Math. 100 (1990) 339–355
  • P M Gilmer, Configurations of surfaces in $4$–manifolds, Trans. Amer. Math. Soc. 264 (1981) 353–380
  • P M Gilmer, C Livingston, On surgery curves for genus one slice knots, to appear in Pacific Journal Math.
  • P Gilmer, C Livingston, The Casson–Gordon invariant and link concordance, Topology 31 (1992) 475–492
  • P Gilmer, C Livingston, Discriminants of Casson–Gordon invariants, Math. Proc. Cambridge Philos. Soc. 112 (1992) 127–139
  • R E Gompf, Smooth concordance of topologically slice knots, Topology 25 (1986) 353–373
  • R E Gompf, A I Stipsicz, $4$–manifolds and Kirby calculus, Graduate Studies in Mathematics 20, Amer. Math. Soc. (1999)
  • C M Gordon, Some aspects of classical knot theory, from: “Knot theory”, (J-C Hausmann, editor), Lecture Notes in Math. 685, Springer, Berlin (1978) 1–60
  • J Greene, S Jabuka, The slice-ribbon conjecture for $3$–stranded pretzel knots, Amer. J. Math. 133 (2011) 555–580
  • J E Grigsby, D Ruberman, S Strle, Knot concordance and Heegaard Floer homology invariants in branched covers, Geom. Topol. 12 (2008) 2249–2275
  • M Hedden, Knot Floer homology of Whitehead doubles, Geom. Topol. 11 (2007) 2277–2338
  • M Hedden, S-G Kim, C Livingston, Topologically slice knots of smooth concordance order two
  • M Hedden, P Kirk, Instantons, concordance, and Whitehead doubling, J. Differential Geom. 91 (2012) 281–319
  • M Hedden, C Livingston, D Ruberman, Topologically slice knots with nontrivial Alexander polynomial, Adv. Math. 231 (2012) 913–939
  • J Hom, The knot Floer complex and the smooth concordance group, to appear in Comm. Math. Helv.
  • J Hoste, A formula for Casson's invariant, Trans. Amer. Math. Soc. 297 (1986) 547–562
  • S Jabuka, Concordance invariants from higher order covers, Topology Appl. 159 (2012) 2694–2710
  • S Jabuka, S Naik, Order in the concordance group and Heegaard Floer homology, Geom. Topol. 11 (2007) 979–994
  • P B Kronheimer, T S Mrowka, Gauge theory and Rasmussen's invariant, Journal of Topology (2013)
  • R A Litherland, Cobordism of satellite knots, from: “Four-manifold theory”, (C Gordon, R Kirby, editors), Contemp. Math. 35, Amer. Math. Soc. (1984) 327–362
  • C Manolescu, B Owens, A concordance invariant from the Floer homology of double branched covers, Int. Math. Res. Not. 2007 (2007) 21
  • B Owens, S Strle, Rational homology spheres and the four-ball genus of knots, Adv. Math. 200 (2006) 196–216
  • B Owens, S Strle, A characterization of the $\Bbb Z\sp n\oplus\Bbb Z(\delta)$ lattice and definite nonunimodular intersection forms, Amer. J. Math. 134 (2012) 891–913
  • P Ozsváth, Z Szabó, Absolutely graded Floer homologies and intersection forms for four-manifolds with boundary, Adv. Math. 173 (2003) 179–261
  • P Ozsváth, Z Szabó, Knot Floer homology and the four-ball genus, Geom. Topol. 7 (2003) 615–639
  • P Ozsváth, Z Szabó, Holomorphic disks and knot invariants, Adv. Math. 186 (2004) 58–116
  • P Ozsváth, Z Szabó, On knot Floer homology and lens space surgeries, Topology 44 (2005) 1281–1300
  • P S Ozsváth, Z Szabó, Knot Floer homology and integer surgeries, Algebr. Geom. Topol. 8 (2008) 101–153
  • J Rasmussen, Khovanov homology and the slice genus, Invent. Math. 182 (2010) 419–447
  • V A Rohlin, Two-dimensional submanifolds of four-dimensional manifolds, Funkcional. Anal. i Priložen. 5 (1971) 48–60 In Russian; translated in Functional Anal. Appl. 5 (1971) 39–48
  • D Ruberman, S Strle, Concordance properties of parallel links
  • A Scorpan, The wild world of $4$–manifolds, Amer. Math. Soc. (2005)
  • O J Viro, Branched coverings of manifolds with boundary, and invariants of links, I, Izv. Akad. Nauk SSSR Ser. Mat. 37 (1973) 1241–1258 In Russian; translated in Math. USSR-Izv. 7 (1973) 1239–1256