Homology concordance and an infinite rank free subgroup

TL;DR

This paper demonstrates that the quotient of certain knot concordance groups contains an infinitely generated free abelian subgroup, using advanced techniques involving L-space knots and the filtered mapping cone formula.

Contribution

It establishes the existence of an infinite rank free subgroup in the homology concordance group quotient, a novel result in knot theory.

Findings

The group quotient contains a  subgroup.

Construction of examples via the filtered mapping cone formula.

Linear independence shown using the connected knot complex.

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.