Cohomological rigidity of manifolds defined by 3-dimensional polytopes

Abstract

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.

Massey Products in the Cohomology of the Moment-Angle Manifolds Corresponding to Pogorelov Polytopes

“…Now we consider the problem of existence of nontrivial Massey products in H * (Z P ) for Pogorelov polytopes P . As noted in[6, Proposition 4.8], triple Massey products of 3-dimensional cohomology classes (3.1) are trivial for simple polytopes P without 4-belts, in particular, for Pogorelov polytopes. In this paper we prove the following: For any Pogorelov polytope P , there exists a nontrivial triple Massey product α…”

“…This class consists of combinatorial 3-dimensional simple polytopes which do not have 3-belts and 4-belts of facets. It is known that the Pogorelov class consists precisely of combinatorial 3-polytopes which admit a right-angled realization in Lobachevsky space L 3 , and such a realization is unique up to isometry (see [4], [5], [6]). There is a family of hyperbolic 3-manifolds associated with Pogorelov polytopes, known as hyperbolic manifolds of Löbell type (see [7]).…”

Example of C-rigid polytopes which are not B-rigid

“…Since the "C-rigidity" requires the existence of quasitoric manifold over a given polytope, there is no canonical way to define the C-rigidity of such polytopes. Thus there has been confusion; some literatures such as [7] define that such polytope is not C-rigid, but some literatures such as [3] define that such polytopes are C-rigid as their conventions. However, it does not matter because the Crigidity of P should be considered only when P supports a quasitoric manifold.…”

“…In [5], the first named author found the counterexample of the reverse implication of (1); there is a 3-dimensional B-rigid simple polytope with 11 facets which is not A-rigid. However, the reverse implication of (2) has been open (see the remark in Section 3 of [3]).…”

Mirror symmetries of hyperbolic tetrahedral manifolds

“…The isometry group of the Al-Jubouri and the Seifert-Weber manifolds was found in the papers [4] and [15] respectively. Further generalization of tetrahedral manifolds known as Löbell type manifolds was done in series of papers ( [20], [21], [3], [5], [6], [16], [2]). The arithmetic properties of the tetrahedral manifolds investigated in [14].…”