We now use singular set diagrams, underlying topological
spaces, extended Wyckoff tables, lattice complexes, and
Heegaard surfaces  to characterize Euclidean 3-orbifolds
(E3Os) from crystallographic space groups. We plan to add
fundamental groups (FGs) to the E3O atlas  by combining
recent studies involving groupoid, sheaf, topos, and
orbifold theory  with crystallographic group presentation
results . FGs bridge geometric and algebraic topology to
provide new computational opportunities. We also will
explore the feasibility of reformulating crystallographic
topology  using categories  as was done for orbifolds
In general, screw axis, glide plane, and Bravais lattice elements of the space group determine the underlying topological space and orbifolding process (wrapping up of an asymmetric unit to form the E3O), with mirrors, if present, forming an E3O boundary. Inversion center, rotation axis, and mirror elements (i.e., special Wyckoff sites) become spherical 2-orbifolds, which form the singular set of the E3O. Space groups such as P212121, Cc, and Pca21 form Euclidean 3-manifolds with full FGs and no singular sets, while Pmmm, Fm3m, etc. form simply connected E3Os with elaborate singular sets and trivial FGs. All other E3Os lie between these two extremes.
 C. K. Johnson,
"Heegaard splitting of Euclidean 3-orbifolds", Trends in Math. Phys. Conf.,
 I. Moerdijk and D. A. Pronk, "Orbifolds, sheaves and groupoids" K-Theory 12, 2-21 (1997); D. A. Pronk "A presentation of the fundamental group of a triangulated orbifold" preprint (1998),
 E. Molnar, "Minimal presentation of crystallographic groups by fundamental polyhedra," Comp. Math. Appl. 16, 507 (1988).
*Operated by Lockheed Martin Energy Research Corp. for the U.S. Dept. of Energy under Contract No. DE-AC05-96OR22464.