Fibered orbifolds and crystallographic groups

Ratcliffe, John G; Tschantz, Steven T

doi:10.2140/agt.2010.10.1627

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Fibered orbifolds and crystallographic groups

John G Ratcliffe and Steven T Tschantz

Algebraic & Geometric Topology 10 (2010) 1627–1664

DOI: 10.2140/agt.2010.10.1627

Abstract

In this paper, we prove that a normal subgroup $N$ of an $n$ –dimensional crystallographic group $Γ$ determines a geometric fibered orbifold structure on the flat orbifold $E^{n} ∕ Γ$ , and conversely every geometric fibered orbifold structure on $E^{n} ∕ Γ$ is determined by a normal subgroup $N$ of $Γ$ . In particular, we prove that $E^{n} ∕ Γ$ is a fiber bundle, with totally geodesic fibers, over a $β_{1}$ –dimensional torus, where $β_{1}$ is the first Betti number of $Γ$ .

Keywords

fibered orbifold, flat orbifold, crystallographic group, space group

Mathematical Subject Classification 2000

Primary: 20H15

Secondary: 55R65, 57M50, 57S30

References

Publication

Received: 7 January 2010

Revised: 3 May 2010

Accepted: 1 June 2010

Published: 21 July 2010

Authors

John G Ratcliffe
Department of Mathematics Vanderbilt University Nashville TN 37240

Steven T Tschantz
Department of Mathematics Vanderbilt University Nashville TN 37240

Volume 10, issue 3 (2010)