Cohomological rigidity of manifolds defined by 3-dimensional polytopes

Abstract

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 of 3-dimensional combinatorial simple polytopes 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 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 . The first family consists of 3-dimensional small covers of polytopes in , or equivalently, hyperbolic 3-manifolds of Löbell type. The second family consists of 6-dimensional quasitoric manifolds over polytopes in . Our main result is that both families are cohomologically rigid, that is, two manifolds and from either family are diffeomorphic if and only if their cohomology rings are isomorphic. It is also proved that if and are diffeomorphic, then their corresponding polytopes and 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. Bibliography: 69 titles.