ИСТИНА

A (mathematical) fullerene is a simple convex 3-polytope with all facets 5- and 6-gons. A k-belt of a simple 3-polytope is a cyclic sequence of facets with empty common intersection such that facets intersect if and only if they follow each other. It is known that fullerenes has no 3- and 4-belts. Results by A.V.~Pogorelov and E.M.~Andreev imply that this condition is the criterion for a simple 3-polytope to be realized in Lobachevky (hyperbolic) space with right angles. Such polytopes we call Pogorelov polytopes. By results of F.Fan, J.Ma and X.Wang two Pogorelov polytopes P and Q are combinatorially equivalent, if the graded cohomology rings H^*(Z_P) and H^*(Z_Q) of moment-angle manifolds are isomorphic. We will discuss recent result by V.M.Buchstaber, N.Yu.Erokhovets, M.Masuda, T.E.Panov and S.Park, which states that for Pogorelov polytopes characteristic pairs (P_1,\Lambda_1) and (P_2,\Lambda_2) are equivalent, if the graded cohomology rings H^*(M(P_1,\Lambda_1)) and H^*(M(P_2,\Lambda_2)) of quasitoric manifolds are isomorphic. The example of a Pogorelov polytope is given by a k-barrel -- a polytope with surface glued from two patches, each consisting of a k-gon surrounded by a k-belt of 5-gons. We will discuss the characterization of the class of Pogorelov polytopes as polytopes that are either barrels or can be combinatorially constructed from a 5- or a 6-barrel using operations of truncation of two adjacent edges of a k-gon, k>= 6, and connected sum with the dodecahedron along a 5-gon. In the case of fullerenes there is a special 1-parametric family consisting of the dodecahedron and polytopes obtained from it by connected sum with the dodecahedron along a 5-gon surrounded by 5-gons. We prove that all other fullerenes can be combinatorially constructed from the 6-barrel by truncations of two adjacent edges of a 6- or a 7-gon such that on intermediate steps we have either a fullerene or a simple polytope with 5-, 6- and one 7-gonal facet adjacent to some 5-gon. These results are obtained in joint work with V.M.~Buchstaber.