ORBIFOLDS OF CUBIC CRYSTALLOGRAPHIC GROUPS William D. Dunbar, Simon's Rock College, 84 Alford Road, Great Barrington, MA 01230
Associated with each crystallographic group G, there is the space X of equivalence classes of positions in Euclidean space, where two points in Euclidean space are considered equivalent if some symmetry of the group takes one to the other. This space X inherits a topological structure, and in fact a metric structure, from Euclidean space. X is an orbifold, i.e. a topological space in which every point has a neighborhood which is a quotient space of a finite group action on an open ball in Euclidean space (this generalizes the definition of a surface as a space in which every point has a neighborhood homeomorphic to a disk -- the group in this case consists of the identity element alone). X will have singular points corresponding to (equivalence classes of) points in Euclidean space which are fixed by some symmetry of G.
When G is one of the 36 groups of cubic type, the orbifold X will be homeomorphic to a hypersphere in 12 cases and to projective space in 1 case, when G consists only of direct motions, X will be homeomorphic to a ball in 11 cases, to the suspension of the projective plane in 10 cases and to the cone on the projective plane in 2 cases, when G contains reflections and/or inversions. The singular sets of these orbifolds will be discussed, as well as the way they reveal subgroup relationships between the crystallographic groups, and some of the implications for the study of equivariant Morse functions.