# Distances between surfaces in 4-manifolds

## TL;DR

This paper introduces and compares two integer-valued distances between homotopic embedded surfaces in 4-manifolds, establishing an inequality relating stabilisation and singularity distances using techniques inspired by Gabai's 4D light-bulb theorem.

## Contribution

It defines two new notions of distance between surfaces and proves an inequality relating them, advancing understanding of surface relations in 4-manifolds.

## Key findings

- Proves that stabilisation distance is at most singularity distance plus one.
- Establishes a relationship between two types of surface distances in 4-manifolds.
- Uses techniques similar to Gabai's proof of the 4D light-bulb theorem.

## Abstract

If $\Sigma$ and $\Sigma'$ are homotopic embedded surfaces in a $4$-manifold then they may be related by a regular homotopy (at the expense of introducing double points) or by a sequence of stabilisations and destabilisations (at the expense of adding genus). This naturally gives rise to two integer-valued notions of distance between the embeddings: the singularity distance $d_{\text{sing}}(\Sigma,\Sigma')$ and the stabilisation distance $d_{\text{st}}(\Sigma,\Sigma')$. Using techniques similar to those used by Gabai in his proof of the 4-dimensional light-bulb theorem, we prove that $d_{\text{st}}(\Sigma,\Sigma')\leq d_{\text{sing}}(\Sigma,\Sigma')+1$.