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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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up vote 16 down vote accepted

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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$CP^2$ does not even immerse in R^6. Proof: If such an immersion exists, then the normal Euler class has the property, that its square is the normal Pontrjagin class, and that is -3 times the signature (when evaluated on the fundamental homology class). But in $H^2(CP^2)$ there is no such a class $x$, for which $x^2$ evaluated on the fundamental class is -3. QED.

Moreover the following theorem is true (It is essentially due to Hughs) Theorem: In the 4-dimensional oriented cobordism group $\Omega_4 \approx Z$ precisely the even elements contain a manifold that admits an immersion into $R^6.$

About embeddings: The conditions (the manifold must be spin and have zero signature) are clearly necessary: An embedded manifold in a Euclidean space has zero normal Euler class. Hence in the present case both $p_1$ and $w_2$ are zero. The opposite is non-trivial, it is the containt of Ruberman's paper mentioned above.

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Thanks Andras (forgive the lack of accent). That fact is quite helpful. I have been trying for some time to use chart movies to construct an immersed braiding of CP with branch set a standardly embedded ${\mathbb R}P^2$. Your remark explains why this is doomed. Now I think I can desingularize the construction in 7-space. – Scott Carter Jul 17 '13 at 1:35

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