Knot concordance in homology cobordisms

TL;DR

This paper investigates the structure of knot concordance in homology cobordisms, revealing that the non-locally-flat concordance group is infinitely generated with elements of infinite order, using Heegaard Floer homology tools.

Contribution

It demonstrates that the cokernel of the natural map from the smooth knot concordance group to the homology cobordism group is infinitely generated and contains elements of infinite order.

Findings

The cokernel is infinitely generated.

Contains elements of infinite order.

Uses Heegaard Floer homology techniques.

Abstract

Let $\widehat{\mathcal{C}}_{\mathbb{Z}}$ denote the group of knots in homology spheres that bound homology balls, modulo smooth concordance in homology cobordisms. Answering a question of Matsumoto, the second author previously showed that the natural map from the smooth knot concordance group $\mathcal{C}$ to $\widehat{\mathcal{C}}_{\mathbb{Z}}$ is not surjective. Using tools from Heegaard Floer homology, we show that the cokernel of this map, which can be understood as the non-locally-flat piecewise-linear concordance group, is infinitely generated and contains elements of infinite order.