Конструкции Семейств Трехмерных Многогранников, Характеристические Фрагменты Фуллеренов И Многогранники Погорелова

Бухштабер, Виктор Матвеевич; Buchstaber, Victor Matveevich; Ероховец, Николай Юрьевич; Erokhovets, Nikolai Yur'evich

doi:10.4213/im8665

Cited by 9 publications

References 47 publications

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Massey products, toric topology and combinatorics of polytopes

Buchstaber¹,

Limonchenko²

2019

Izv. Math.

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In this paper we introduce a direct family of simple polytopes such that for any there are non-trivial strictly defined Massey products of order in the cohomology rings of their moment-angle manifolds . We prove that the direct sequence of manifolds has the following properties: every manifold is a retract of , and one has inverse sequences in cohomology (over and , where as ) of the Massey products constructed. As an application we get that there are non-trivial differentials , for arbitrarily large as , in the Eilenberg–Moore spectral sequence connecting the rings and with coefficients in a field, where .

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Massey products, toric topology and combinatorics of polytopes

Buchstaber¹,

Limonchenko²

2019

Izv. Math.

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Cohomological rigidity of manifolds defined by 3-dimensional polytopes

Buchstaber¹,

Erokhovets²,

Masuda³

et al. 2017

Russ. Math. Surv.

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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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Canonical geometrization of orientable $3$-manifolds defined by vector colourings of $3$-polytopes

Erokhovets

2022

Sb. Math.

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The geometrization conjecture of Thurston (finally proved by Perelman) says that any oriented 3-manifold can canonically be partitioned into pieces, which have a geometric structure modelled on one of the eight geometries:Nil and Sol. In a seminal paper (1991) Davis and Januszkiewicz introduced a wide class of n-dimensional manifolds, small covers over simple n-polytopes. We give a complete answer to the following problem: build an explicit canonical decomposition of any orientable 3-manifold defined by a vector colouring of a simple 3-polytope, in particular, of a small cover. The proof is based on an analysis of results in this direction obtained previously by different authors.Bibliography: 44 titles.

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Конструкции Семейств Трехмерных Многогранников, Характеристические Фрагменты Фуллеренов И Многогранники Погорелова

Cited by 9 publications

References 47 publications

Massey products, toric topology and combinatorics of polytopes

Massey products, toric topology and combinatorics of polytopes

Cohomological rigidity of manifolds defined by 3-dimensional polytopes

Canonical geometrization of orientable $3$-manifolds defined by vector colourings of $3$-polytopes

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