Action of the Mazur pattern up to topological concordance

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dc.contributor.authorManchester, Alexen_US
dc.date.accessioned2024-03-18T15:16:21Z
dc.date.available2024-03-18T15:16:21Z
dc.date.issued2024en_US
dc.description.abstractIn 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.en_US
dc.identifier.citationManchester, A. (2024). Action of the Mazur pattern up to topological concordance (arXiv:2212.13640v3). arXiv. https://doi.org/10.48550/arXiv.2212.13640en_US
dc.identifier.doihttps://doi.org/10.48550/arXiv.2212.13640en_US
dc.identifier.urihttps://hdl.handle.net/1911/115475
dc.language.isoengen_US
dc.publisherarXiven_US
dc.rightsExcept 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.en_US
dc.rights.urihttps://creativecommons.org/licenses/by/4.0/en_US
dc.subject.keywordsatellite knotsen_US
dc.subject.keywordconcordanceen_US
dc.subject.keywordMazur patternen_US
dc.subject.keywordCochran--Orr--Teichner filtrationen_US
dc.titleAction of the Mazur pattern up to topological concordanceen_US
dc.typeJournal articleen_US
dc.type.dcmiTexten_US
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