Фуллерены И Диск-Фуллерены

Abstract. 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 P , 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 Löbell 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 P and P 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.

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Фуллерены И Диск-Фуллерены

Cited by 7 publications

References 51 publications

Zigzags of Fullerenes and $$c$$ -Disk-Fullerenes

Zigzags of Fullerenes and $$c$$ -Disk-Fullerenes

Constructions of families of three-dimensional polytopes, characteristic patches of fullerenes, and Pogorelov polytopes

Cohomological rigidity of manifolds defined by 3-dimensional polytopes

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