Manchester, Alex2024-03-182024-03-182024Manchester, A. (2024). Action of the Mazur pattern up to topological concordance (arXiv:2212.13640v3). arXiv. https://doi.org/10.48550/arXiv.2212.13640https://hdl.handle.net/1911/115475In the '80s, Freedman showed that the Whitehead doubling operator acts trivally 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--Horn and Collins. In this paper, we give evidence that the Mazur pattern acts by the identity up to topological concordance. In particular, we show that two satellite operators $P_{K_0,\eta_0}$ and $P_{K_1,\eta_1}$ with $\eta_0$ and $\eta_1$ 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 $\rho$-invariants.engExcept where otherwise noted, this work is licensed under a Creative Commons Attribution (CC BY) license. Permission to reuse, publish, or reproduce the work beyond the terms of the license or beyond the bounds of fair use or other exemptions to copyright law must be obtained from the copyright holder.Journal articlesatellite knotsconcordanceMazur patternCochran--Orr--Teichner filtrationhttps://doi.org/10.48550/arXiv.2212.13640