Whitney tower concordance and knots in homology spheres

Abstract

In a groundbreaking work A. Levine proved the surprising result that there exist knots in homology spheres which are not smoothly concordant to any knot in $S^3$, even if one allows for concordances in homology cobordisms. Since then subsequent works due to Hom-Levine-Lidman and Zhou have strengthened this result showing that there are many knots in homology spheres which are not smoothly concordant to knots in $S^3$. In this paper we present evidence that the opposite is true topologically. We study the Whitney tower filtration of concordance due to Cochran-Orr-Teichner and prove that modulo any term in this filtration every knot (or link) in a homology sphere is equivalent to a knot (or link) in $S^3$. As an application we recover the main result of [Davis2019], namely that the solvable filtration similarly fails to distinguish links in homology spheres from links in $S^3$.