Cohomological rigidity of manifolds defined by 3-dimensional polytopes

A family of closed manifolds is said to be cohomologically rigid if a cohomology ring isomorphism implies a diffeomorphism for any two manifolds in the family. Cohomological rigidity is established here for large families of 3-dimensional and 6-dimensional manifolds defined by 3-dimensional polytopes. The class P of 3-dimensional combinatorial simple polytopes P different from tetrahedra and without facets forming 3and 4-belts is studied. This class includes mathematical fullerenes, that is, simple 3-polytopes with only 5-gonal and 6-gonal facets. By a theorem of Pogorelov, any polytope in P admits in Lobachevsky 3-space a right-angled realisation which is unique up to isometry. Our families of smooth manifolds are associated with polytopes in the class P. The first family consists of 3-dimensional small covers of polytopes in P, or equivalently, hyperbolic 3-manifolds of L "obell type. The second family consists of 6-dimensional quasitoric manifolds over polytopes in P. Our main result is that both families are cohomologically rigid, that is, two manifolds M and M' from either family are diffeomorphic if and only if their cohomology rings are isomorphic. It is also proved that if M and M' are diffeomorphic, then their corresponding polytopes P and P' are combinatorially equivalent. These results are intertwined with classical subjects in geometry and topology such as the combinatorics of 3-polytopes, the Four Colour Theorem, aspherical manifolds, a diffeomorphism classification of 6-manifolds, and invariance of Pontryagin classes. The proofs use techniques of toric topology.