Theory of families of polytopes: fullerenes and Pogorelov polytopes
This is a review on older and recent results (in part by the author) on certain families of three-dimensional polytopes. They can be defined via the notion of \(k\)-belt. This is a cyclic sequence of \(k\) faces of the given polytope in which only successive faces are adjacent and no three faces have a common vertex. The author defines five classes of simple \(3\)-polytopes by existence, non-existence, or triviality (defined by surrounding a face) of \(k\)-belts for various \(k\). Among them are flag polytopes and Pogorelov polytopes. These polytope classes play an important role in hyperbolic geometry and algebraic topology, but also in chemistry, namely the fullerenes. The latter have only \(5\)-gonal and \(6\)-gonal faces, and an \(n\)-disk fullerene, which plays a role here, has in addition one \(n\)-gonal face with \(n\not=5,6\). The author reviews properties and characterizations of these classes of polytopes, their role in hyperbolic geometry, constructions to obtain such polytopes from simpler ones, and the construction of Fullerenes using growth operations. The last two sections provide a comprehensive survey of the use of the considered polytopes, in particular Pogorelov polytopes, in the construction of cohomologically rigid families of manifolds and in Thurston's geometrization of three-dimensional manifolds.
scientific article; zbMATH DE number 7392401
scientific article