My question is basically in the title:
Does for every $d$-dimensional connected manifold $M$ exist a surjective continuous map $\phi:\Bbb R^d\to M$?
I first thought that in the case of Riemannian manifolds the exponential map shall do the trick, but it might not be surjective, right? So if such a $\phi$ does not always exist, what are the weakest restrictions I need to make it exist, e.g. should $M$ be compact, differentiable, ...?