Satellites of Infinite Rank in the Smooth Concordance Group

TL;DR

This paper investigates satellite operations in the smooth concordance group, proposing a conjecture about their rank, and introduces a gauge theory criterion to identify when these operations generate infinite rank subgroups.

Contribution

It introduces a gauge theory-based criterion for satellite operators to generate infinite rank subgroups in the smooth concordance group, especially for winding number zero satellites.

Findings

The criterion applies to many unknotted patterns with zero topological concordance.

Satellite operations are conjectured to be either constant or of infinite rank.

The paper raises questions about the interaction between satellite operators and concordance.

Abstract

We conjecture that satellite operations are either constant or have infinite rank in the concordance group. We reduce this to the difficult case of winding number zero satellites, and use $SO(3)$ gauge theory to provide a general criterion sufficient for the image of a satellite operation to generate an infinite rank subgroup of the smooth concordance group $\mathcal{C}$. Our criterion applies widely; notably to many unknotted patterns for which the corresponding operators on the topological concordance group are zero. We raise some questions and conjectures regarding satellite operators and their interaction with concordance.