TITLE: Examples of manifolds that cannot be embedded in $\mathbb R^4$
QUESTION [9 upvotes]: 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.

REPLY [7 votes]: No compact nonorientable $(n-1)$-manifold embeds in $\mathbb R^n$:
this follows from the Alexander duality theorem.
Immersibility is a harder problem....