Volume 10, issue 3 (2010)

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Fibered orbifolds and crystallographic groups

John G Ratcliffe and Steven T Tschantz

Algebraic & Geometric Topology 10 (2010) 1627–1664
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 ${E}^{n}∕\phantom{\rule{0.3em}{0ex}}\Gamma$, and conversely every geometric fibered orbifold structure on ${E}^{n}∕\phantom{\rule{0.3em}{0ex}}\Gamma$ is determined by a normal subgroup $N$ of $\Gamma$. In particular, we prove that ${E}^{n}∕\phantom{\rule{0.3em}{0ex}}\Gamma$ is a fiber bundle, with totally geodesic fibers, over a ${\beta }_{1}$–dimensional torus, where ${\beta }_{1}$ is the first Betti number of $\Gamma$.

Keywords
fibered orbifold, flat orbifold, crystallographic group, space group
Mathematical Subject Classification 2000
Primary: 20H15
Secondary: 55R65, 57M50, 57S30