This question is a follow-up to  [my previous question](https://mathoverflow.net/questions/90819/branched-coverings-of-the-4-sphere-branched-along-a-knotted-surface) . The statement of the question is the title. 


Note that the $4$-dimensional real projective space is non-orientable and a characteristic class argument gives that it does not embed in $7$-space. Right now, I am more interested in orientable $4$-manifolds.



Which closed orientable $4$-dimensional manifolds cannot be embedded in $6$-space?
This is true if and only if $X^4$ is spin and its signature vanishes. This is on p. 345 in Gompf/Stipsicz (4-manifolds and Kirby calculus), who cite Ruberman: Imbedding four-manifold and slicing links, 1982.


**EDIT** Of course I mean that $X^4$ CAN be embedded in 6-dimensional space iff the conditions are met.


