Knot concordance and homology sphere groups

Abstract

We study two homomorphisms to the rational homology sphere group. If $\psi$ denotes the inclusion homomorphism from the integral homology sphere group, then using work of Lisca we show that the image of $\psi$ intersects trivially with the subgroup of the rational homology sphere group generated by lens spaces. As corollaries this gives a new proof that the cokernel of $\psi$ is infinitely generated, and implies that a connected sum $K$ of 2-bridge knots is concordant to a knot with determinant 1 if and only if $K$ is smoothly slice. Furthermore, if $\beta$ denotes the homomorphism from the knot concordance group defined by taking double branched covers of knots, we prove that the kernel of $\beta$ contains a $\mathbb{Z}^{\infty}$ summand by analyzing the Tristram-Levine signatures of a family of knots whose double branched covers all bound rational homology balls.