# Embedding covers of manifolds

I am considering $k$-fold covers of smooth manifolds (with smooth covering maps).

Let $f:M^m\to N^m$ be a smooth finite covering map.

--

The following implication is not true: $M$ can be embedded into $\mathbb{R}^n$ $\Rightarrow$ $N$ can be embedded into $\mathbb{R}^n$.

The easiest counterexample is $M=\mathbb{S}^{n-1}$ and $N=RP^{n-1}$.

--

But what about the converse of the implication:

$N$ can be embedded into $\mathbb{R}^n$ $\Rightarrow$ $M$ can be embedded into $\mathbb{R}^n$.

Intuitively, this makes sense since if $N$ can be embedded, we should be able to embed $M$ which is obtained by "cutting, unfolding and gluing several copies of $N$" and is therefore "less complicated".

Or is there an obvious counterexample I'm missing?

NB: I'm of course aware of the Whitney Embedding Theorem. The point is, however, to derive something better than $n=2m$ for $M$ based on the fact that I know something about the embeddability of $N$.

 It appears that adding the hypothesis of "closed" doesn't save us -- Section Three of (http://arxiv.org/abs/0810.2346) gives a few examples of closed three-manifolds (of Nil and Solv geometry) that embed in $S^4$ and which have (infinitely many) finite covers that do not embed in $S^4$.