We find conditions under which a non‐orientable closed surface smoothly embedded into an orientable $4$‐manifold $X$ can be represented by a connected sum of an embedded closed surface in $X$ and an unknotted projective plane in a $4$‐sphere. This allows us to extend the Gabai $4$‐dimensional light bulb theorem and the Auckly–Kim–Melvin–Ruberman–Schwartz ‘one is enough’ theorem to the case of non‐orientable surfaces.

中文翻译：

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在嵌入4个歧管的不可定向表面上

我们找到了一种条件，在这种情况下，不可定向的封闭表面可以平滑地嵌入可定向的表面 $4$歧管 $X$ 可以用内嵌的封闭曲面的连接和表示 $X$ 和一架未打结的射影飞机 $4$-领域。这使我们能够扩大加拜$4$三维灯泡定理和Auckly–Kim–Melvin–Ruberman–Schwartz“一个就足够”定理，对于无法定向的曲面来说。