Knots in homology spheres and the solvable filtration

The classical topological knot concordance group $\mathcal{C}$ consists knots in $S^3$ modulo those which bound flat disks in the 4-ball.  There is a natural generalization, $\widehat{\mathcal{C}}$ to the setting of knots in homology spheres.  Recently, Adam Levine showed that in the smooth (as opposed to topological) setting the map $\mathcal{C}\to \widehat{\mathcal{C}}$ is not a surjection, producing knots in homology spheres not smoothly concordant to any knot in $S^3$.  During this talk we will produce strong evidence that the opposite is true  topologically.  Namely we show that modulo any term of the solvable filtration of knot concordance introduced by Cochran-Orr-Teichner, the map $\mathcal{C}\to \widehat{\mathcal{C}}$ is an isomorphism.  Since every known invariant of topological concordance vanishes at some level of this filtration, our result implies that any possible difference between $\mathcal{C}$ and $\widehat{\mathcal{C}}$ will not be detected by any currently existent technology.