I would like to know if given an $n$-dimensional Lie algebra $\mathfrak{g}$, there is a way to know whether the simply-connected Lie group integrating it is topologically $\mathbb{R}^n$.
At the moment, I am looking at certain Lie algebras of which I don't know the integration, so I am generally interested in knowing how much of the topology of the simply connected Lie group is "saved" by the Lie algebra.