Whitney tower concordance and knots in homology spheres

A Levine (Forum Math.Sigma 4 (2016) art.id.e34) proved the surprising result that there exist knots in homology spheres which are not smoothly concordant to any knot in S 3 , where one allows for concordances in homology cobordisms.Since then subsequent works due to Hom, Levine and Lidman (Duke Math.J. 171 (2022) 3089-3131) and Zhou (J.Topol.14 (2021) 1369-1395) 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 (Ann. of Math.157 (2003) 433-519) 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 a result of the author (J.Topol.13 (2020) 343-355), namely that the solvable filtration similarly fails to distinguish links in homology spheres from links in S 3 .