Which closed orientable \$4\$-dimensional manifolds cannot be embedded in \$6\$-space? - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T13:37:01Z http://mathoverflow.net/feeds/question/90872 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/90872/which-closed-orientable-4-dimensional-manifolds-cannot-be-embedded-in-6-space Which closed orientable \$4\$-dimensional manifolds cannot be embedded in \$6\$-space? Scott Carter 2012-03-11T03:06:36Z 2012-03-11T04:11:31Z <p>This question is a follow-up to <a href="http://mathoverflow.net/questions/90819/branched-coverings-of-the-4-sphere-branched-along-a-knotted-surface" rel="nofollow"> my previous question </a>. The statement of the question is the title. </p> <p>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.</p> http://mathoverflow.net/questions/90872/which-closed-orientable-4-dimensional-manifolds-cannot-be-embedded-in-6-space/90877#90877 Answer by Igor Rivin for Which closed orientable \$4\$-dimensional manifolds cannot be embedded in \$6\$-space? Igor Rivin 2012-03-11T03:54:24Z 2012-03-11T04:11:31Z <p>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.</p> <p><strong>EDIT</strong> Of course I mean that \$X^4\$ CAN be embedded in 6-dimensional space iff the conditions are met.</p> http://mathoverflow.net/questions/90872/which-closed-orientable-4-dimensional-manifolds-cannot-be-embedded-in-6-space/90878#90878 Answer by Mark Grant for Which closed orientable \$4\$-dimensional manifolds cannot be embedded in \$6\$-space? Mark Grant 2012-03-11T03:56:14Z 2012-03-11T03:56:14Z <p>\$\mathbb{C}P^2\$ does not embed in \$\mathbb{R}^6\$. See</p> <p>Feder, S.; Segal, D. M. <em>Immersions and embeddings of projective spaces,</em> Proc. Amer. Math. Soc. 35 (1972), 590–592. </p>