# Examples of manifolds that cannot be embedded in $\mathbb R^4$

Could someone give me an example of a (smooth) $n$-manifold $(n=2, 3)$ which cannot be embedded (or immersed) in $\mathbb R^4$?

S. L.

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All 2-manifolds embed in $\mathbb R^4$.

For $3$-manifolds, $\mathbb RP^3$ doesn't embed in $\mathbb R^4$. An interesting aspect for 3-manifolds in $\mathbb R^4$ is the subject fractures into tame topological embeddings and smooth/PL embeddings. For example, Poincare Dodecahedral space has a tame topological embedding in $\mathbb R^4$ but not a smooth or PL-embedding.

If you want more examples, check out this pre-print: http://front.math.ucdavis.edu/0810.2346

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Cheers! And some random text... – Sam Nov 29 '10 at 8:56

No compact nonorientable $(n-1)$-manifold embeds in $\mathbb R^n$: this follows from the Alexander duality theorem.

Immersibility is a harder problem....

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From the homotopy theory perspective immersibility amounts to finding tangent bundle immersions. There's no similar reduction for embedding problems. – Ryan Budney Jun 10 '11 at 23:04