TITLE: Is the universal covering of an open subset of $\mathbb{R}^n$ diffeomorphic to an open subset of  $\mathbb{R}^n$ ?
QUESTION [30 upvotes]: Is the universal covering of a connected open subset $U$ of ℝn diffeomorphic to an open subset of ℝn  (standard differentiable structure)?
If not true in general, is there any condition on $U$ which guarantees a positive answer?

REPLY [25 votes]: The answer is no, and there is a counter-example in dimension $4$. 
A theorem of Whitney and Massey states that the total space of a disc-bundle over a non-orientable surface $\Sigma$ embeds in $S^4$ if and only if the normal Euler class of the disc bundle is one of the integers:
$$\{2\chi -4, 2\chi, \cdots, 4-2\chi\}$$
where $\chi$ is the Euler characteristic of the surface. 
So for example, if $\Sigma = \mathbb RP^2$, $\chi = 1$.  So normal Euler classes $-2$ and $+2$ appear for embeddings $\mathbb RP^2 \to S^4$.   These come from the standard embeddings of $\mathbb RP^2$ in $S^4$. 
The universal cover of this total space is the pull-back of that bundle along the covering map $S^2 \to \mathbb RP^2$.  But the total space of this bundle is orientable so it can't embed in $S^4$ as it's Euler class is not zero -- the pull-back bundle must be isomorpic to the tangent bundle of $S^2$.  And this does not embed in $S^4$.