Knot concordance in homology spheres

The knot concordance group $\mathcal{C}$ consists of knots in $S^3$ modulo knots that bound smooth disks in $B^4$. We consider $\widehat{\mathcal{C}}_{\mathbb{Z}}$, the group of knots in homology spheres that bound homology balls modulo knots that bound smooth disks in a homology ball. Matsumoto asked if the natural map from $\mathcal{C}$ to $\widehat{\mathcal{C}}_{\mathbb{Z}}$ is an isomorphism. Adam Levine answered this question in the negative by showing the map is not surjective. We show that the image of $\mathcal{C}$ in $\widehat{\mathcal{C}}_{\mathbb{Z}}$ is of infinite index; more specifically, it contains a subgroup isomorphic to the integers. This is joint work with Adam Levine and Tye Lidman.