By a *nilmanifold* I mean a quotient $M =\Gamma \backslash G$ of a connected, simply-connected nilpotent real Lie group $G$ by the left action of a maximal lattice , i.e. a discrete cocompact subgroup.


Suppose we have two maximal lattices $\Gamma\_1$ and $\Gamma\_2$ of $G$. What can be said about the nilmanifolds $\Gamma\_1 \backslash G$ and $\Gamma\_2 \backslash G$? Are they homeomorphic? Or perphaps one is a covering space of the other?



Two nilmanifolds of the same Lie group
The manifold $\Gamma\_1/G$ is homeomorphic to $\Gamma\_2/G$ implies that $\Gamma\_1$ is isomorphic to $\Gamma\_2$ since $G$ is $1$-connected and $\Gamma\_1,\Gamma\_2$ are discrete, thus $\pi\_1(\Gamma\_1/G)=\Gamma\_1$ and two homeomorphic manifolds have isomorphic fundamental groups. This implies that $\Gamma\_1$ is isomorphic to $\Gamma\_2$. In fact the converse is also true: $\Gamma\_1$ is isomorphic to $\Gamma\_2$, implies that $\Gamma\_1/G$ is homeomorphic to $\Gamma\_2/G$. This is a consequence of:


Raghunathan. Discrete subgroups of Lie groups Corollary 2. p.34


