Knot concordance and homology sphere groups

TL;DR

This paper investigates the structure of homomorphisms from the knot concordance group to the rational homology sphere group, revealing new insights into their images, kernels, and implications for knot concordance and sliceness.

Contribution

It provides new proofs and results on the structure of homomorphisms related to homology spheres and knot concordance, including the trivial intersection with lens spaces and the infinite rank of the kernel.

Findings

The image of the inclusion homomorphism intersects trivially with lens space subgroup.

The cokernel of the inclusion homomorphism is infinitely generated.

The kernel of the homomorphism from the knot concordance group contains a d7 summand.

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.