You answer and Michael comment, let me think that I shoud add that in fact I know that $M$ and $N$ are h-cobordant. I even know that the cobordism $W \subset M \times [0,1]$ with the border $\partial W \simeq (M \times {0}) \cup N$.

In your example, there is a surjection from $L_1$ to $L_2$, because all points in $L_2$ are on exactly one of the fiber of $L_1 \times {t}$ of $W$. And this surjection seems differentiable. Right ?

thanks for this answer. and for lower dimension?

Thanks for you help Ryan, you might be right too! I am not expert in that field...

And topological is may be a bit to weak. My manifold are DIFF. I edit the post.

Can you elaborate, the only counter example I see from whitehead is with V and W non compact (and in fact non simply connected at infinity).

Is this a counter example ? You inversed U and W ? U is the variety that appears on both side in my question. Moreover, in my specific case, U is fixed to be the real line although the general question is interesting.