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