Oliver Singh

If $\mathrm{\Sigma}$ and ${\mathrm{\Sigma}}^{\prime}$ 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 ${d}_{\mathrm{sing}}(\mathrm{\Sigma},{\mathrm{\Sigma}}^{\prime})$ and the stabilisation distance ${d}_{st}(\mathrm{\Sigma},{\mathrm{\Sigma}}^{\prime})$. Using techniques similar to those used by Gabai in his proof of the 4‐dimensional light bulb theorem, we prove that ${d}_{st}(\mathrm{\Sigma},{\mathrm{\Sigma}}^{\prime})\u2a7d{d}_{\mathrm{sing}}(\mathrm{\Sigma},{\mathrm{\Sigma}}^{\prime})+1$.