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Distances between surfaces in 4‐manifolds
Journal of Topology  (IF1.582),  Pub Date : 2020-05-02, DOI: 10.1112/topo.12148
Oliver Singh

If $Σ$ and $Σ ′$ 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 $d sing ( Σ , Σ ′ )$ and the stabilisation distance $d st ( Σ , Σ ′ )$. Using techniques similar to those used by Gabai in his proof of the 4‐dimensional light bulb theorem, we prove that $d st ( Σ , Σ ′ ) ⩽ d sing ( Σ , Σ ′ ) + 1$.