A Homological Invariant of a Bieberbach Group with Dihedral Extension

Abstract

A Bieberbach group is a crystallographic group which is an extension of a free abelian group of finite rank by a finite extension. Meanwhile, research on homological invariants has been on interest of many authors since it is related to the study of the properties of the crystal using mathematical approach. One of the homological invariants is the exterior square. In this paper, the exterior square of a Bieberbach group of dimension four with dihedral extension is computed theoretically.