From Voronoi diagramms to the topology of geometric 3-dimensional manifolds

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 $P$ embedded into a 3-manifold $M$ is said to be a simple spine of $M$ if $M$ (punctured at a point if $M$ is a closed manifold) can be collapsed onto $P$. Spines are a useful tool in algorithmic topology.
Cut locus of a Riemannian manifold $M$ with respect to its point $x$ (that is, the closure of the set of points $y$ such that the shortest geodesic between $x$ and $y$ in $M$ is not unique) is a standard object to study in differential geometry.
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.