Action of the Mazur pattern up to topological concordance

TL;DR

This paper investigates whether the Mazur pattern acts as the identity on the topological concordance group, providing evidence that it does, contrasting with its nontrivial action in the smooth category.

Contribution

The paper demonstrates that the Mazur pattern acts by the identity up to topological concordance and shows it cannot be distinguished from the identity using certain invariants.

Findings

Mazur pattern acts as the identity up to topological concordance

Equivalent actions of satellite operators with homotopic axes on the topological concordance group

Mazur pattern and identity operator indistinguishable by Casson-Gordon signatures and rho-invariants in the topological setting

Abstract

In 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 but not 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 signatures or metabelian $\rho$-invariants.