Fibered orbifolds and crystallographic groups, II

TL;DR

This paper studies fibered structures of flat orbifolds derived from crystallographic groups, providing a generalized Calabi construction, criteria for extension splitting, and classifying fibrations, especially in three dimensions.

Contribution

It introduces a generalized Calabi construction for fibered orbifolds, characterizes the structure group, and classifies all fibrations of 3-dimensional flat orbifolds.

Findings

The structure group is finite iff the orbifold has an orthogonally dual fibered structure.

Isometry group of E^n/G is a compact Lie group with a torus component.

Complete classification of fibrations of 3-orbifolds over 1-orbifolds.

Abstract

Let G be an n-dimensional crystallographic group (n-space group). If G is a Z-reducible, then the flat n-orbifold E^n/G has a nontrivial fibered orbifold structure. We prove that this structure can be described by a generalized Calabi construction, that is, E^n/G is represented as the quotient of the Cartesian product of two flat orbifolds under the diagonal action of a structure group of isometries. We determine the structure group and prove that it is finite if and only if the fibered orbifold structure has an orthogonally dual fibered orbifold structure. A geometric fibration of E^n/G corresponds to a space group extension 1 -> N -> G -> G/N -> 1. We give a criterion for the splitting of a space group extension in terms of the structure group action that is strong enough to detect the splitting of all the space group extensions corresponding to the standard Seifert fibrations of a compact, connected, flat 3-orbifold. If G is an arbitrary n-space group, we prove that the group Isom(E^n/G) of isometries of E^n/G is a compact Lie group whose component of the identity is a torus of dimension equal to the first Betti number of G. This implies that Isom(E^n/G) is finite if and only if G/[G,G] is finite. We describe how to classify all the geometric fibrations of compact, connected, flat n-orbifolds, over a 1-orbifold, up to affine equivalence. We apply our classification theory to the scientifically important case n = 3, and classify all the geometric fibrations of compact, connected, flat 3-orbifolds, over a 1-orbifold, up to affine equivalence.