English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Tools annual

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools monthly

Global: Zendy Plus monthly

Presents the access of premium content as premium feature

Premium Content

Global: Zendy Plus annual

Global: Zendy Free Plan

A family of closed manifolds is called cohomologically rigid if a cohomologyring isomorphism implies a diffeomorphism for any two manifolds in the family.We establish cohomological rigidity for large families of 3-dimensional and6-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. Thisclass includes mathematical fullerenes, i.e. simple 3-polytopes with only5-gonal and 6-gonal facets. By a theorem of Pogorelov, any polytope from Padmits a right-angled realisation in Lobachevsky 3-space, which is unique up toisometry.  Our families of smooth manifolds are associated with polytopes from the classP. The first family consists of 3-dimensional small covers of polytopes from P,or hyperbolic 3-manifolds of Loebell type. The second family consists of6-dimensional quasitoric manifolds over polytopes from P. Our main result isthat both families are cohomologically rigid, i.e. two manifolds M and M' fromeither of the families are diffeomorphic if and only if their cohomology ringsare isomorphic. We also prove that if M and M' are diffeomorphic, then theircorresponding polytopes are combinatorially equivalent. These results areintertwined with the classical subjects of geometry and topology, such ascombinatorics of 3-polytopes, the Four Colour Theorem, aspherical manifolds,diffeomorphism classification of 6-manifolds and invariance of Pontryaginclasses. The proofs use techniques of toric topology.

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