## Which closed orientable $4$-dimensional manifolds cannot be embedded in $6$-space?

This question is a follow-up to my previous question . 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.

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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.

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$\mathbb{C}P^2$ does not embed in $\mathbb{R}^6$. See

Feder, S.; Segal, D. M. Immersions and embeddings of projective spaces, Proc. Amer. Math. Soc. 35 (1972), 590–592.

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