Fibered orbifolds and crystallographic groups

Abstract

In this paper, we prove that a normal subgroup $N$ of an $n$–dimensional crystallographic group $\Gamma$ determines a geometric fibered orbifold structure on the flat orbifold $E^n∕\Gamma$, and conversely every geometric fibered orbifold structure on $E^n∕\Gamma$ is determined by a normal subgroup $N$ of $\Gamma$. In particular, we prove that $E^n∕\Gamma$ is a fiber bundle, with totally geodesic fibers, over a ${\beta }_1$–dimensional torus, where ${\beta }_1$ is the first Betti number of $\Gamma$.