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 $d_{st}(\mathrm{\Sigma },{\mathrm{\Sigma }}^{\prime })⩽d_{\mathrm{sing}}(\mathrm{\Sigma },{\mathrm{\Sigma }}^{\prime })+1$.

中文翻译：

---

曲面之间的距离为4个流形

如果 $\mathrm{\Sigma }$ 和 ${\mathrm{\Sigma }}^{\prime }$是4歧管中的同位嵌入表面，那么它们可能通过规则的同伦（以引入双点为代价）或一系列稳定和去稳定（以添加属为代价）相关联。这自然产生了两个嵌入之间距离的整数值概念：奇异距离$d_{唱}（\mathrm{\Sigma }，{\mathrm{\Sigma }}^{\prime }）$ 和稳定距离 $d_{圣}（\mathrm{\Sigma }，{\mathrm{\Sigma }}^{\prime }）$。使用类似于Gabai在证明4维灯泡定理时所使用的技术，我们证明了$d_{圣}（\mathrm{\Sigma }，{\mathrm{\Sigma }}^{\prime }）⩽d_{唱}（\mathrm{\Sigma }，{\mathrm{\Sigma }}^{\prime }）+\mathrm{1个}$。