Cohomological rigidity of manifolds defined by right-angled 3-dimensional polytopes

TL;DR

This paper proves cohomological rigidity for certain 3- and 6-dimensional manifolds derived from specific 3-polytopes, linking topological, combinatorial, and geometric properties using toric topology techniques.

Contribution

It establishes cohomological rigidity for families of manifolds associated with polytopes in class P, connecting cohomology rings with manifold diffeomorphism types.

Findings

Cohomology ring isomorphism implies diffeomorphism for the manifolds.

Diffeomorphic manifolds have combinatorially equivalent polytopes.

Results connect topology, geometry, and combinatorics of 3-polytopes.

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, 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.