David Auckly, Rustam Sadykov

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.