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General Mathematics Seminar of the St. Petersburg Division of Steklov Institute of Mathematics, Russian Academy of Sciences
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From Voronoi diagramms to the topology of geometric 3-dimensional manifolds Sergei Anisov Utrecht University, the Netherlands |
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Abstract: The notion of a simple polyhedron (recall that neighborhoods of bertice of simple polyhedra are homeomorphic to the standard “6-wing batterfly” and all edges of it are triple lines) appeared in B. Casler's papers on three-dimensional topology. A simple polyhedron Cut locus of a Riemannian manifold Voronoi diagrams are a classical tool and a classical object to study in computational geometry. Simple polyhedra, “typical” cut loci in 3-manifolds, and “typical” Voronoi diagrasm have the same local structure. This simple observation enables us to apply ideas and methods of geometry and singularity theory to topological questions about spines of 3-manifolds. As a byproduct, one gets unexpected results from combinatorics. |