# Stabilization distance between surfaces

## TL;DR

This paper introduces the stabilization distance between surfaces in 4-manifolds, constructs examples with arbitrary distances, and uses homology techniques to distinguish different slice discs and their properties.

## Contribution

It defines the stabilization distance, constructs examples with arbitrary stabilization distances, and applies homology methods to distinguish slice discs in 4-manifolds.

## Key findings

- Constructed pairs of 2-knots with stabilization distance m
- Exhibited knots with slice discs having generalized stabilization distance m
- Used homology to distinguish slice discs with identical cyclic cover invariants

## Abstract

Define the 1-handle stabilization distance between two surfaces properly embedded in a fixed 4-dimensional manifold to be the minimal number of 1-handle stabilizations necessary for the surfaces to become ambiently isotopic. For every nonnegative integer $m$ we find a pair of 2-knots in the 4-sphere whose stabilization distance equals $m$. Next, using a generalized stabilization distance that counts connected sum with arbitrary 2-knots as distance zero, for every nonnegative integer $m$ we exhibit a knot $J_m$ in the 3-sphere with two slice discs in the 4-ball whose generalized stabilization distance equals $m$. We show this using homology of cyclic covers. Finally, we use metabelian twisted homology to show that for each $m$ there exists a knot and pair of slice discs with generalized stabilization distance at least $m$, with the additional property that abelian invariants associated to cyclic covering spaces coincide. This detects different choices of slicing discs corresponding to a fixed metabolising link on a Seifert surface.