We now use singular set diagrams, underlying topological
spaces, extended Wyckoff tables, lattice complexes, and
Heegaard surfaces [1] to characterize Euclidean 3-orbifolds
(E3Os) from crystallographic space groups. We plan to add
fundamental groups (FGs) to the E3O atlas [2] by combining
recent studies involving groupoid, sheaf, topos, and
orbifold theory [3] with crystallographic group presentation
results [4]. FGs bridge geometric and algebraic topology to
provide new computational opportunities. We also will
explore the feasibility of reformulating crystallographic
topology [2] using categories [5] as was done for orbifolds
in [3].
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.
[1] C. K. Johnson,
"Heegaard splitting of Euclidean 3-orbifolds", Trends in Math. Phys. Conf.,
(1998)
[2]
http://www.ornl.gov/ortep/topology.html.
[3] 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),
http://www.math.uu.nl/people/moerdijk/index.html.
[4] E. Molnar, "Minimal presentation of crystallographic
groups by fundamental polyhedra," Comp. Math. Appl. 16, 507
(1988).
[5]
http://plato.stanford.edu/entries/category-theory/.
*Operated by Lockheed Martin Energy Research Corp. for the
U.S. Dept. of Energy under Contract No. DE-AC05-96OR22464.