Knot Concordance and Homology Sphere Groups. (29th May 2017) - Aceto, Paolo; Larson, Kyle
Abstract: We study two homomorphisms to the rational homology sphere group $Θ^3_\mathbb{Q}$ . If $ψ$ denotes the inclusion homomorphism from the integral homology sphere group $Θ^3_\mathbb{Z}$, then using work of Lisca we show that the image of $ψ$ intersects trivially with the subgroup of $Θ^3_\mathbb{Q}$ generated by lens spaces. As corollaries this gives a new proof that the cokernel of $ψ$ is infinitely generated, and implies that a connected sum $K$ of two-bridge knots is concordant to a knot with determinant 1 if and only if $K$ is smoothly slice. Furthermore, if $β$ denotes the homomorphism from the knot concordance group $\mathcal{C}$ defined by taking double branched covers of knots, we prove that the kernel of $β$ contains a $\mathbb{Z}^{∞}$ summand by analyzing the Tristram–Levine signatures of a family of knots whose double branched covers all bound rational homology balls.