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

Question

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

Thanks in advance!

S. L.

Answer 1

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: https://arxiv.org/abs/0810.2346 (Ryan Budney, Benjamin A. Burton: Embeddings of 3-manifolds in $S^4$ from the point of view of the 11-tetrahedron census)

Answer 2

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

Immersibility is a harder problem....