Action of the Mazur pattern up to topological concordance

Manchester, Alex¹

¹Department of Mathematics, Rice University, Houston, TX, United States

Algebraic and Geometric Topology, Tome 25 (2025) no. 4, pp. 2115-2142

Résumé

In the 1980s, Freedman showed that the Whitehead doubling operator acts trivially up to topological concordance. On the other hand, Akbulut showed that the Whitehead doubling operator acts nontrivially up to smooth concordance. The Mazur pattern is a natural candidate for a satellite operator which acts by the identity up to topological concordance but not up to smooth concordance. Recently there has been a resurgence of study of the action of the Mazur pattern up to concordance in the smooth and topological categories. Examples showing that the Mazur pattern does not act by the identity up to smooth concordance have been given by Cochran, Franklin, Hedden and Horn and by Collins. We give evidence that the Mazur pattern acts by the identity up to topological concordance.

In particular, we show that two satellite operators PK0,η0 and PK1,η1 with η0 and η1 freely homotopic have the same action on the topological concordance group modulo the subgroup of (1)-solvable knots, which gives evidence that they act in the same way up to topological concordance. In particular, the Mazur pattern and the identity operator are related in this way, and so this is evidence for the topological side of the analogy to the Whitehead doubling operator. We give additional evidence that they have the same action on the full topological concordance group by showing that, up to topological concordance, they cannot be distinguished by Casson–Gordon invariants or metabelian ρ-invariants.

Détail

DOI : 10.2140/agt.2025.25.2115

Keywords: Mazur pattern, satellite knots, topological concordance, Cochran–Orr–Teichner filtration

Bibliographie

[1] 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

[2] J Cheeger, M Gromov, Bounds on the von Neumann dimension of L2-cohomology and the Gauss–Bonnet theorem for open manifolds, J. Differential Geom. 21 (1985) 1

[3] T D Cochran, C W Davis, A Ray, Injectivity of satellite operators in knot concordance, J. Topol. 7 (2014) 948 | DOI

[4] T D Cochran, B D Franklin, M Hedden, P D Horn, Knot concordance and homology cobordism, Proc. Amer. Math. Soc. 141 (2013) 2193 | DOI

[5] T Cochran, S Friedl, P Teichner, New constructions of slice links, Comment. Math. Helv. 84 (2009) 617 | DOI

[6] T Cochran, S Harvey, The geometry of the knot concordance space, Algebr. Geom. Topol. 18 (2018) 2509 | DOI

[7] T D Cochran, S Harvey, P Horn, Filtering smooth concordance classes of topologically slice knots, Geom. Topol. 17 (2013) 2103 | DOI

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

[9] T D Cochran, S Harvey, C Leidy, Derivatives of knots and second-order signatures, Algebr. Geom. Topol. 10 (2010) 739 | DOI

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

[11] T D Cochran, S Harvey, M Powell, Grope metrics on the knot concordance set, J. Topol. 10 (2017) 669 | DOI

[12] T D Cochran, K E Orr, P Teichner, Knot concordance, Whitney towers and L2-signatures, Ann. of Math. 157 (2003) 433 | DOI

[13] M M Cohen, A course in simple-homotopy theory, Vol. 10, Springer (1973) | DOI

[14] S Collins, Homology cobordism, smooth concordance, and the figure eight knot, preprint (2022)

[15] A Conway, Homotopy ribbon discs with a fixed group, Algebr. Geom. Topol. 24 (2024) 4575 | DOI

[16] C W Davis, A Ray, Satellite operators as group actions on knot concordance, Algebr. Geom. Topol. 16 (2016) 945 | DOI

[17] M H Freedman, A new technique for the link slice problem, Invent. Math. 80 (1985) 453 | DOI

[18] M H Freedman, F Quinn, Topology of 4-manifolds, 39, Princeton Univ. Press (1990)

[19] M H Freedman, P Teichner, 4-manifold topology, II : Dwyer’s filtration and surgery kernels, Invent. Math. 122 (1995) 531 | DOI

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

[21] R E Gompf, K Miyazaki, Some well-disguised ribbon knots, Topology Appl. 64 (1995) 117 | DOI

[22] R E Gompf, A I Stipsicz, 4-manifolds and Kirby calculus, 20, Amer. Math. Soc. (1999) | DOI

[23] M Hedden, J Pinzón-Caicedo, Satellites of infinite rank in the smooth concordance group, Invent. Math. 225 (2021) 131 | DOI

[24] A S Levine, Nonsurjective satellite operators and piecewise-linear concordance, Forum Math. Sigma 4 (2016) | DOI

[25] T Lidman, A N Miller, J Pinzón-Caicedo, Linking number obstructions to satellite homomorphisms, Quantum Topol. (2024) | DOI

[26] R A Litherland, Cobordism of satellite knots, from: "Four-manifold theory" (editors C Gordon, R Kirby), Contemp. Math. 35, Amer. Math. Soc. (1984) 327 | DOI

[27] R A Litherland, Symmetries of twist-spun knots, from: "Knot theory and manifolds" (editor D Rolfsen), Lecture Notes in Math. 1144, Springer (1985) 97 | DOI

[28] C Livingston, P Melvin, Abelian invariants of satellite knots, from: "Geometry and topology" (editors J Alexander, J Harer), Lecture Notes in Math. 1167, Springer (1985) 217 | DOI

[29] C Livingston, S Naik, Introduction to knot concordance, book project (2011)

[30] A N Miller, Homomorphism obstructions for satellite maps, Trans. Amer. Math. Soc. Ser. B 10 (2023) 220 | DOI

[31] A N Miller, L Piccirillo, Knot traces and concordance, J. Topol. 11 (2018) 201 | DOI

[32] J Milnor, Whitehead torsion, Bull. Amer. Math. Soc. 72 (1966) 358 | DOI

[33] A Ray, Satellite operators with distinct iterates in smooth concordance, Proc. Amer. Math. Soc. 143 (2015) 5005 | DOI

[34] H Seifert, On the homology invariants of knots, Quart. J. Math. Oxford Ser. 1 (1950) 23 | DOI

[35] K Yasui, Corks, exotic 4-manifolds and knot concordance, preprint (2015)