Topological concordance of knots in homology spheres and 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.