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 $\mathscr{P}$ of 3-dimensional combinatorial simple polytopes $P$ different from tetrahedra and without facets forming 3- and 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 $\mathscr{P}$ 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 $\mathscr{P}$. The first family consists of 3-dimensional small covers of polytopes in $\mathscr{P}$, or equivalently, hyperbolic 3-manifolds of Löbell type. The second family consists of 6-dimensional quasitoric manifolds over polytopes in $\mathscr{P}$. 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.
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