# Concordance, crossing changes, and knots in homology spheres

## TL;DR

This paper investigates when knots in homology spheres can be simplified to slice knots, establishing a connection between nullhomotopy in homology balls and concordance to slice knots, with implications for knot equivalence relations.

## Contribution

It characterizes nullhomotopic knots in homology spheres via concordance and homotopy, extending classical results from $S^3$ to homology spheres.

## Key findings

- A knot in a homology sphere is nullhomotopic in a homology ball iff it is concordant to a homotopic slice knot.
- The equivalence relation on knots in homology spheres is generated by concordance and homotopy in homology cobordisms.
- Provides a criterion linking nullhomotopy, concordance, and homotopy for knots in homology spheres.

## Abstract

Any knot in $S^3$ may be reduced to a slice knot by crossing changes. Indeed, this slice knot can be taken to be the unknot. In this paper we study the question of when the same holds for knots in homology spheres. We show that a knot in a homology sphere is nullhomotopic in a smooth homology ball if and only if that knot is smoothly concordant to a knot which is homotopic to a smoothly slice knot. As a consequence, we prove that the equivalence relation on knots in homology spheres given by cobounding immersed annuli in a homology cobordism is generated by concordance in homology cobordisms together with homotopy in a homology sphere.