Cohomological rigidity of manifolds defined by right-angled 3-dimensional polytopes
A family of closed manifolds is called cohomologically rigid if a cohomology ring isomorphism implies a diffeomorphism for any two manifolds in the family. We establish cohomological rigidity for large families of 3-dimensional and 6-dimensional manifolds defined by 3-dimensional polytopes. We consider the class P of 3-dimensional combinatorial simple polytopes, different from a tetrahedron, whose facets do not form 3- and 4-belts. This class includes mathematical fullerenes, i.e. simple 3-polytopes with only 5-gonal and 6-gonal facets. By a theorem of Pogorelov, any polytope from P admits a right-angled realisation in Lobachevsky 3-space, which is unique up to isometry. Our families of smooth manifolds are associated with polytopes from the class P. The first family consists of 3-dimensional small covers of polytopes from P, or hyperbolic 3-manifolds of Loebell type. The second family consists of 6-dimensional quasitoric manifolds over polytopes from P. Our main result is that both families are cohomologically rigid, i.e. two manifolds M and M' from either of the families are diffeomorphic if and only if their cohomology rings are isomorphic. We also prove that if M and M' are diffeomorphic, then their corresponding polytopes are combinatorially equivalent. These results are intertwined with the classical subjects of geometry and topology, such as combinatorics of 3-polytopes, the Four Colour Theorem, aspherical manifolds, diffeomorphism classification of 6-manifolds and invariance of Pontryagin classes. The proofs use techniques of toric topology.