Crystallographic Morse Functions on Orbifolds

Morse functions and orbifolds are separate concepts used in global analysis and geometric topology, respectively. We find that a combination of the two is useful for classification purposes in the unofficial crystallographic orbifold atlas we are compiling.

Using peak-pass-pale-pit terminology for the four types of non-degenerate critical points (zero gradient points) in 3-Euclidean space, a "Crystallographic Morse Function" (CMF) is an infinite pattern of 3-D isotropic normal probability density functions (pdf) centered on special space group orbit (Wyckoff) positions. The peak positions are at the pdf centers and the overlapping pdf tails determine the pass, pale, pit, and gradient-flow separatrices arrangement. The arrangement is then "orbi-folded" to form a 3-Euclidean orbifold with Morse function superimposed.

The model utilizes and extends the crystallographic lattice-complex concept
(defined in Chapter 14 of the "International Tables for Crystallography,
Vol. A (1995)" as "the set of all point configurations that may be
generated within one type of Wyckoff set (orbit)"). For example, the basic
CMF orbifold in space group *Im*3*m* is the body centered cubic arrangement
that contains eight lattice complexes, *i.e.*, four 0-D critical points,
three 1-D topologically unique gradient-flow separatrices joining
consecutive critical points, and one shortest path between the peak and
pit. Each of these elements is contained within a single 2-elliptic
orbifold domain within the 3-Euclidean orbifold. A node-link directed
graph, representing this configuration and the associated Wyckoff
multiplicities, characterizes both the CMF and the space-group orbifold.
The Morse function provides a unique format for drawing and comparing
orbifold graphs because of the Morse mapping onto a 1-D Euclidean space. A
"duplex" matroid representation also exists with the elliptic 2-orbifold
singular-set arrangement in one direction and Morse CW-complex poset
arrangement in the other.

We conjecture that in conjunction with our previously described "color orbifold covers" specification of group / normal subgroup pairs of orbifolds based on color groups, the crystallographic 3-Euclidean orbifolds should be classified and documented using the embedding characteristics for some basis set of CMFs. For crystal chemistry research, this would provide a very useful and geometrically intuitive classification. We are currently working with the cubic orbifolds.

Critiques on this approach and pointers to related literature and research combining orbifolds and Morse theory would be appreciated.

General background is given in two abstracts from the 1995 American Crystallographic Association annual meeting in Montreal: Crystal Structure Topology Illustrations: Critical Polyhedra in Euclidean and Orbifold Spaces and Critical Polyhedra on Orbifolds: Topology in Crystal Structure Prediction. An easy introduction to orbifolds is given at http://www.geom.umn.edu/~strauss/sym.2/sym.2.4.html. The principal Morse Theory reference is M. Goresky and R. MacPherson, (1988), "Stratified Morse Theory". The only paper we have found combining orbifolds and Morse theory is E. Lerman and S. Tolman, "Hamiltonian torus actions on symplectic orbifolds and toric varieties ", at http://www.math.uiuc.edu/~lerman/research.html. Matroids are discussed in various papers at http://www.math.TU-Berlin.DE/~ziegler/. An example embedding of the simplest possible CMF (cubic primitive) in a series of space groups using tiling rather than Morse theory is given in the paper by Emil Molnar on "Symmetry breaking of the cube tiling ..." at http://www.zblmath.fiz-karlsruhe.de/e-journals/BAG/vol.35/no.2/.

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*Page last revised: April 18, 1996*