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?