Branched coverings and combinatorial holonomy
Project leaders: M. Joswig, G. M. Ziegler
The theory of simplicial and polyhedral surfaces has combinatorial notions of paths and geodesics, quite distinct from the case of smooth surfsces. Thus it requires a separate theory of "how to move information along paths on the surface", which also has useful and interesting extensions beyond the two-dimensional case of surfaces. In this project we also study higher-dimensional generalizations including, in particular, the case of complex surfaces.
The transport of information "along the surface" and thus "in tangent direction" is in some sense transversal/orthogonal to the moduli that any (unparametrized, but embedded/immersed) surface would admit, that is, to motions "in normal directions".
In particular, inconsistencies in motions along closed paths along a manifold may be captured in holonomy groups and may partially be resolved by a construction of covering surfaces/manifolds. Moreover, in the combinatorial (triangulated/polyhedral) case branched coverings arise for which the branching locus is suggested/determined by combinatorial data. Surprisingly rich classes of manifolds may be obtained that way, and connections to rather deep combinatorial (coloring) problems arise naturally. Already the surface case is rich in this respect: The theory of branched coverings of the 2-sphere has an inherently combinatorial flavor. Related topics include graph embeddings into surfaces and group actions on graphs. For example, we are interessted in Koebe-type theorems for branched coverings, as pursued as part of Project B1.
We center our project around the interplay of combinatorics and topology with a focus on graph colorings and related subjects. This field is both classical and very active. The current state of research suggests that an infusion of ideas from differential geometry must lead to considerable progress.
A branched covering Tē -> Sē
This project has expired and is continued by the follow-up project
Non-positive curvature and cubical surfaces.