Homology concordance and an infinite rank free subgroup

Abstract

Two knots are homology concordant if they are smoothly concordant in a homology cobordism. The group ${\hat{\mathcal{C}}}_{\mathbb{Z}}$ (resp. ${\mathcal{C}}_{\mathbb{Z}}$) was previously defined as the set of knots in homology spheres that bounds homology balls (resp. in $S^3$), modulo homology concordance. We prove ${\hat{\mathcal{C}}}_{\mathbb{Z}}/{\mathcal{C}}_{\mathbb{Z}}$ contains a ${\mathbb{Z}}^{\infty }$ subgroup. We construct our family of examples by applying the filtered mapping cone formula to $L$--space knots, and prove linear independence with the help of the connected knot complex.

arXiv comment27 pages, 11 figures. Minor revisions. This is the version to appear in the Journal of Topology