Theory of families of polytopes: fullerenes and Pogorelov polytopes

The paper is a review of the results of the eponymous cycle of author's works marked by the I. I. Shuvalov I degree prize 2018 for scientific research and recent results. We study families of three-dimensional simple polytopes defined by the condition of cyclic $k$-edge-connectivity, in particular, flag polytopes and Pogorelov polytopes, as well as related families of fullerenes and ideal right-angled hyperbolic polytopes. We describe methods for constructing families using operations of cutting off edges and a connected sum along faces, a construction of fullerenes using growth operations, a construction of cohomologically rigid families of three-dimensional and six-dimensional manifolds, and Thurston's geometrization of orientable three-dimensional manifolds corresponding to polytopes.