Primary decomposition of knot concordance and von Neumann rho-invariants

Kim, Min Hoon; Kim, Se-Goo; Kim, Taehee

doi:10.1090/proc/15282

Primary decomposition of knot concordance and von Neumann rho-invariants
HTML articles powered by AMS MathViewer

by Min Hoon Kim, Se-Goo Kim and Taehee Kim PDF

Proc. Amer. Math. Soc. 149 (2021), 439-447 Request permission

Abstract:

We address the primary decomposition of the knot concordance group in terms of the solvable filtration and higher order von Neumann $\rho$-invariants by Cochran, Orr, and Teichner. We show that for a non-negative integer $n$, if the connected sum of two $n$-solvable knots with coprime Alexander polynomials is slice, then each of the knots has vanishing von Neumann $\rho$-invariants of order $n$. This gives positive evidence for the conjecture that nonslice knots with coprime Alexander polynomials are not concordant. As an application, we show that if $K$ is one of Cochran–Orr–Teichner’s knots which are the first examples of nonslice knots with vanishing Casson–Gordon invariants, then $K$ is not concordant to any knot with Alexander polynomial coprime to that of $K$.

References

Jeff Cheeger and Mikhael Gromov, Bounds on the von Neumann dimension of $L^2$-cohomology and the Gauss-Bonnet theorem for open manifolds, J. Differential Geom. 21 (1985), no. 1, 1–34. MR 806699
Jae Choon Cha, Amenable $L^2$-theoretic methods and knot concordance, Int. Math. Res. Not. IMRN 17 (2014), 4768–4803. MR 3257550, DOI 10.1093/imrn/rnt092
Jae Choon Cha, Primary decomposition in the concordance group of topologically slice knots, arXiv:1910.14629, 2019.
Jae Choon Cha and Kent E. Orr, $L^2$-signatures, homology localization, and amenable groups, Comm. Pure Appl. Math. 65 (2012), no. 6, 790–832. MR 2903800, DOI 10.1002/cpa.21393
Tim D. Cochran, Shelly Harvey, and Constance Leidy, Primary decomposition and the fractal nature of knot concordance, Math. Ann. 351 (2011), no. 2, 443–508. MR 2836668, DOI 10.1007/s00208-010-0604-5
Tim D. Cochran, Shelly Harvey, and Constance Leidy, 2-torsion in the $n$-solvable filtration of the knot concordance group, Proc. Lond. Math. Soc. (3) 102 (2011), no. 2, 257–290. MR 2769115, DOI 10.1112/plms/pdq020
Tim D. Cochran, Kent E. Orr, and Peter Teichner, Knot concordance, Whitney towers and $L^2$-signatures, Ann. of Math. (2) 157 (2003), no. 2, 433–519. MR 1973052, DOI 10.4007/annals.2003.157.433
Tim D. Cochran, Kent E. Orr, and Peter Teichner, Structure in the classical knot concordance group, Comment. Math. Helv. 79 (2004), no. 1, 105–123. MR 2031301, DOI 10.1007/s00014-001-0793-6
Tim D. Cochran and Peter Teichner, Knot concordance and von Neumann $\rho$-invariants, Duke Math. J. 137 (2007), no. 2, 337–379. MR 2309149, DOI 10.1215/S0012-7094-07-13723-2
Michael H. Freedman and Frank Quinn, Topology of 4-manifolds, Princeton Mathematical Series, vol. 39, Princeton University Press, Princeton, NJ, 1990. MR 1201584
Michael H. Freedman, A surgery sequence in dimension four; the relations with knot concordance, Invent. Math. 68 (1982), no. 2, 195–226. MR 666159, DOI 10.1007/BF01394055
Hye Jin Jang, Two-torsion in the grope and solvable filtrations of knots, Internat. J. Math. 28 (2017), no. 4, 1750023, 34. MR 3639041, DOI 10.1142/S0129167X17500239
Se-Goo Kim, Polynomial splittings of Casson-Gordon invariants, Math. Proc. Cambridge Philos. Soc. 138 (2005), no. 1, 59–78. MR 2127228, DOI 10.1017/S0305004104008023
Se-Goo Kim and Taehee Kim, Polynomial splittings of metabelian von Neumann rho-invariants of knots, Proc. Amer. Math. Soc. 136 (2008), no. 11, 4079–4087. MR 2425750, DOI 10.1090/S0002-9939-08-09372-6
Se-Goo Kim and Taehee Kim, Splittings of von Neumann rho-invariants of knots, J. Lond. Math. Soc. (2) 89 (2014), no. 3, 797–816. MR 3217650, DOI 10.1112/jlms/jdu008
J. Levine, Invariants of knot cobordism, Invent. Math. 8 (1969), 98–110; addendum, ibid. 8 (1969), 355. MR 253348, DOI 10.1007/BF01404613
J. Levine, Knot cobordism groups in codimension two, Comment. Math. Helv. 44 (1969), 229–244. MR 246314, DOI 10.1007/BF02564525