$\mathbb{R}^n$ admits a [tessellation by permutohedra](http://en.wikipedia.org/wiki/Permutohedron#Tessellation_of_the_space). The corresponding identification of facets of a permutohedron therefore gives a well-defined space: call it $X\_n$. For example, $X\_2$, the hexagon with opposite sides identified, can be shown to be a 2-torus (see figure 2 in <http://arxiv.org/pdf/cond-mat/0703326v2>). Is $X\_n \cong (S^1)^n$? Why (or why not)?



The topology of periodic permutohedral boundary conditions
The quotient $X\_n$ is the same as the quotient of $\mathbb R^n$ by a subgroup of $\mathbb Z^n$ acting cocompactly through translations. Such a thing is always a torus.


