The structure of the rational concordance group of knots

TL;DR

This paper fully characterizes the algebraic structure of the rational concordance group of knots in rational homology spheres using new invariants linked to algebraic number theory, revealing complex torsion phenomena and higher-dimensional classifications.

Contribution

It introduces a complete set of invariants for the rational concordance group and relates them to Artin reciprocity, advancing the understanding of knot concordance in rational homology spheres.

Findings

Constructed infinitely many torsion elements.

Demonstrated the complex structure of the rational concordance group.

Developed new obstructions using von Neumann $L^2$-signatures.

Abstract

We study the group of rational concordance classes of codimension two knots in rational homology spheres. We give a full calculation of its algebraic theory by developing a complete set of new invariants. For computation, we relate these invariants with limiting behaviour of the Artin reciprocity over an infinite tower of number fields and analyze it using tools from algebraic number theory. In higher dimensions it classifies the rational concordance group of knots whose ambient space satisfies a certain cobordism theoretic condition. In particular, we construct infinitely many torsion elements. We show that the structure of the rational concordance group is much more complicated than the integral concordance group from a topological viewpoint. We also investigate the structure peculiar to knots in rational homology 3-spheres. To obtain further nontrivial obstructions in this dimension, we develop a technique of controlling a certain limit of the von Neumann $L^2$-signature invariants.