Distances between surfaces in 4‐manifolds

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 $d_{\text{st}}(\Sigma,\Sigma')\leq d_{\text{sing}}(\Sigma,\Sigma')+1$.