
TL;DR
This paper proves a theorem about the adjacency of exceptional vertices in triangulated spheres with degree conditions, using coloring monodromy and connections to branched covers, Belyi surfaces, and cone-metrics.
Contribution
It introduces a coloring monodromy approach to analyze vertex degree conditions and extends the theory to branched covers and geometric structures.
Findings
Exceptional vertices are not adjacent when all but two have degrees divisible by k.
Coloring monodromy helps prove the case for k=2 and relates to branched covers.
Connections are established with Belyi surfaces and cone-metrics of constant curvature.
Abstract
If all but two vertices of a triangulated sphere have degrees divisible by , then the exceptional vertices are not adjacent. This theorem is proved for with the help of the coloring monodromy. For colorings by the vertices of platonic solids have to be used. With a coloring monodromy one can associate a branched cover. This generalizes to a space of germs between two triangulated surfaces. We also discuss relations with Belyi surfaces and with cone-metrics of constant curvature.
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Taxonomy
TopicsGeometric Analysis and Curvature Flows · Homotopy and Cohomology in Algebraic Topology · Algebraic Geometry and Number Theory
