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 Uspekhi Mat. Nauk, 2017, Volume 72, Issue 2(434), Pages 3–66 (Mi umn9759)

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.

 Funding Agency Grant Number Russian Foundation for Basic Research 17-01-0067116-51-55017-ÃÔÅÍ Contest «Young Russian Mathematics» Japan Society for the Promotion of Science 16K05152 The research of the first, second, and fourth authors was supported by the Russian Foundation for Basic Research (grant nos. 17-01-00671 and 16-51-55017-ÃÔÅÍ). The second author was supported by the Young Russian Mathematics Award. The third author was supported by the JSPS Grant-in-Aid for Scientific Research (C) (grant no. 16K05152).

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

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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This publication is cited in the following articles:
1. A. Yu. Vesnin, “Right-angled polyhedra and hyperbolic 3-manifolds”, Russian Math. Surveys, 72:2 (2017), 335–374
2. 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
3. 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
4. N. Erokhovets, “Construction of fullerenes and Pogorelov polytopes with 5-, 6- and one 7-gonal face”, Symmetry, 10:3 (2018), 67, 28 pp.
5. V. Gómez-Gutiérrez, S. López de Medrano, “Topology of the intersections of ellipsoids in $\mathbf{R}^n$”, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM, 112:3 (2018), 879–891
6. A. Yu. Vesnin, S. V. Matveev, E. A. Fominykh, “New aspects of complexity theory for 3-manifolds”, Russian Math. Surveys, 73:4 (2018), 615–660
7. D. A. Derevnin, A. D. Mednykh, “Mirror symmetries of hyperbolic tetrahedral manifolds”, Sib. elektron. matem. izv., 15 (2018), 1850–1856
8. Panov T., Veryovkin Ya., “On the Commutator Subgroup of a Right-Angled Artin Group”, J. Algebra, 521 (2019), 284–298
9. E. G. Zhuravleva, “Proizvedeniya Massi v kogomologiyakh moment-ugol mnogoobrazii, sootvetstvuyuschikh mnogogrannikam klassa Pogorelova”, Matem. zametki, 105:4 (2019), 526–536
10. Choi S., Park K., “Example of C-Rigid Polytopes Which Are Not B-Rigid”, Math. Slovaca, 69:2 (2019), 437–448
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