Topological concordance of knots in homology spheres and the solvable filtration

TL;DR

This paper shows that the Cochran-Orr-Teichner filtration cannot distinguish certain knots in homology spheres from those in the 3-sphere, and proves bijectivity of winding number ±1 satellite operators on knot concordance modulo this filtration.

Contribution

It demonstrates that the solvable filtration does not detect differences in topological concordance classes of knots in homology spheres, and establishes the bijectivity of certain satellite operators.

Findings

The solvable filtration cannot distinguish knots in homology spheres from those in the 3-sphere.

Every knot in a homology sphere is equivalent to a knot in the 3-sphere modulo any term of the filtration.

Winding number ±1 satellite operators act bijectively on knot concordance modulo the solvable filtration.

Abstract

In 2016 Levine showed that there exists a knot in a homology 3-sphere which is not smoothly concordant to any knot in the 3-sphere where one allows concordances in any smooth homology cobordism. Whether the same is true if one allows topological concordances is not known. One might hope that such an example might be detected by the powerful filtration of knot concordance introduced by Cochran-Orr-Teichner. We prove that this is not the case, demonstrating that for any knot in any homology sphere there is a knot in the 3-sphere equivalent to the original knot modulo any term of this filtration. Our results apply equally well to link concordance. As an application we prove that every winding number +/-1 satellite operator acts bijectively on knot concordance, modulo any term of the solvable filtration.