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This article is cited in 15 scientific papers (total in 16 papers)
Cohomological rigidity of manifolds defined by 3-dimensional polytopes
V. M. Buchstaberabc, N. Yu. Erokhovetsb, M. Masudad, T. E. Panovbec, S. Parkd a Steklov Mathematical Institute of Russian Academy of Sciences, Moscow
b Moscow State University
c Institute for Information Transmission Problems of the Russian Academy of Sciences, Moscow
d Osaka City University, Osaka, Japan
e Institute for Theoretical and Experimental Physics, Moscow
Abstract:
A family of closed manifolds is said to be cohomologically rigid if a cohomology ring isomorphism implies a diffeomorphism for any two manifolds in the family. Cohomological rigidity is established here for large families of 3-dimensional and 6-dimensional manifolds defined by 3-dimensional polytopes. The class $\mathscr{P}$ of 3-dimensional combinatorial simple polytopes $P$ different from tetrahedra and without facets forming 3- and 4-belts is studied. This class includes mathematical fullerenes, that is, simple 3-polytopes with only 5-gonal and 6-gonal facets. By a theorem of Pogorelov, any polytope in $\mathscr{P}$ admits in Lobachevsky 3-space a right-angled realisation which is unique up to isometry. Our families of smooth manifolds are associated with polytopes in the class $\mathscr{P}$. The first family consists of 3-dimensional small covers of polytopes in $\mathscr{P}$, or equivalently, hyperbolic 3-manifolds of Löbell type. The second family consists of 6-dimensional quasitoric manifolds over polytopes in $\mathscr{P}$. Our main result is that both families are cohomologically rigid, that is, two manifolds $M$ and $M'$ from either family are diffeomorphic if and only if their cohomology rings are isomorphic. It is also proved that if $M$ and $M'$ are diffeomorphic, then their corresponding polytopes $P$ and $P'$ are combinatorially equivalent. These results are intertwined with classical subjects in geometry and topology such as the combinatorics of 3-polytopes, the Four Colour Theorem, aspherical manifolds, a diffeomorphism classification of 6-manifolds, and invariance of Pontryagin classes. The proofs use techniques of toric topology.
Bibliography: 69 titles.
Keywords:
quasitoric manifold, moment-angle manifold, hyperbolic manifold, small cover, simple polytope, right-angled polytope, cohomology ring, cohomological rigidity.
DOI:
https://doi.org/10.4213/rm9759
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English version:
Russian Mathematical Surveys, 2017, 72:2, 199–256
Bibliographic databases:
MSC: Primary 57R91, 57M50; Secondary 05C15, 14M25, 52A55, 52B10 Received: 20.12.2016
Citation:
V. M. Buchstaber, N. Yu. Erokhovets, M. Masuda, T. E. Panov, S. Park, “Cohomological rigidity of manifolds defined by 3-dimensional polytopes”, Uspekhi Mat. Nauk, 72:2(434) (2017), 3–66; Russian Math. Surveys, 72:2 (2017), 199–256
Citation in format AMSBIB
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http://mi.mathnet.ru/eng/umn9759https://doi.org/10.4213/rm9759 http://mi.mathnet.ru/eng/umn/v72/i2/p3
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V. M. Buchstaber, N. Yu. Erokhovets, “Constructions of families of three-dimensional polytopes, characteristic patches of fullerenes, and Pogorelov polytopes”, Izv. Math., 81:5 (2017), 901–972
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V. M. Buchstaber, N. Yu. Erokhovets, “Fullerenes, polytopes and toric topology”, Combinatorial and toric homotopy, Lect. Notes Ser. Inst. Math. Sci. Natl. Univ. Singap., 35, World Sci. Publ., Hackensack, NJ, 2018, 67–178
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N. Erokhovets, “Construction of fullerenes and Pogorelov polytopes with 5-, 6- and one 7-gonal face”, Symmetry, 10:3 (2018), 67, 28 pp.
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E. G. Zhuravleva, “Massey Products in the Cohomology of the Moment-Angle Manifolds Corresponding to Pogorelov Polytopes”, Math. Notes, 105:4 (2019), 519–527
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V. M. Buchstaber, I. Yu. Limonchenko, “Massey products, toric topology and combinatorics of polytopes”, Izv. Math., 83:6 (2019), 1081–1136
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S. Allen, J. La Luz, “Local face rings and diffeomorphisms of quasitoric manifolds”, Homology Homotopy Appl., 21:1 (2019), 303–322
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