Nilpotent Lie group homeomorphic to $\mathbb R^n$? Is it true that a connected, simply connected, nilpotent $n$-dimensional Lie group $G$ is homeomorphic to $\mathbb R^n$?
EDIT:
Maybe a possible argument is the following: Since $G$ is simply connected, $G$ cannot contain any non-trivial maximal compact subgroups. By a theorem associated to Iwasawa and Malcev, all maximal compact subgroups are conjugate and thus have the same dimension. By a theorem by Hochschild(?), $G/K$ is diffeomorphic to $\mathbb R^n$, where $K$ is a(ny) maximal compact subgroup. But for $G$ simply connected, connected, nilpotent, $K$ must be trivial, whence $G$ itself is diffeomorphic to $\mathbb R^n$.
 A: If a Lie group $G$ is nilpotent, its Lie algebra $\mathfrak g$ is nilpontent, and there is an ideal $\mathfrak h\subseteq\mathfrak g$ of codimension $1$. It follows that there is a normal subgroup $H\subseteq G$ whose Lie algebra is $\mathfrak h$, this subgroup is closed because $G$ is simply connected, and we have a short exact sequence of Lie groups $$1\to H\to G\to\mathbb R\to 0$$ This extension of groups is split, so that $G$ is in fact a semidirect product of $\mathbb R$ and $H$; to exhibit a splitting, let $\mathfrak a$ be a subspace of $\mathfrak g$ complementary to $\mathfrak h$: the $1$-dimensional closed Lie subgroup of $G$ tangent to it maps isomorphically to the $\mathbb R$ here, so its inverse splits the exact sequence.  Since $H$ is nilpotent and of smaller dimension than $G$, by induction we can assume that $H$ is diffeomorphic to $\mathbb R^n$ for some $n$ and then $G\cong\mathbb R\rtimes H$ is diffeomorphic to $\mathbb R^{n+1}$.
A: $S^1$ is abelian, hence nilpotent, but it is not homeomorphic to $\mathbb{R}$.
Edit: With the additional assumption that the Lie group $G$ is simply connected, yes.  The exponential map defines a local homeomorphism from $\mathbb{R}^n$ to $G$ (every point $x \in \mathbb{R}^n$ has a neighborhood $U$ such that $\exp$ restricted to $U$ is a homeomorphism onto its image).  
But if 2 connected and simply-connected spaces are locally homeomorphic, then they are homeomorphic. (There need to be some assumptions on the spaces to make this true but surely they are fulfilled for $\mathbb{R}^n$ and a smooth manifold $G$.)
Edit 2: This still isn't quite right.  See comments below.  For one thing, we haven't used the nilpotency assumption.  If $G$ is nilpotent, then $\exp$ is surjective, but we may need more to conclude local homeomorphism $\implies$ homeomorphism.
A: Let $G$ be a simply connected nilpotent Lie group with Lie algebra $\mathfrak g$. The nilpotency of $G$ implies that the Baker-Campbell-Hausdorff formula has only finitely many terms, so it converges everywhere on $\mathfrak g$. So it can be used to define a group law $\times$ on the Lie algebra $\mathfrak g$ such that $(\mathfrak g,\times)$ is a Lie group whose Lie algebra is isomorphic to $\mathfrak g$. Since two simply connected Lie groups with the same Lie algebra are isomorphic, $G$ is isomorphic to $(\mathfrak g,\times)$. In particular, $G$ is homeomorphic to $\mathfrak g$.
