Fundamental Group of a Euclidean 3-Orbifold. Carroll K. Johnson, Chemical & Analytical Sciences Division, Oak Ridge National Laboratory*, Oak Ridge, Tennessee 37831-6197. ckj@ornl.gov

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.