The structure of the rational concordance group of knots

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.