Cohomological rigidity of manifolds defined by 3-dimensional polytopes

V. M. Buchstaber   ; N. Yu. Erokhovets   ; M. Masuda   ; T. E. Panov   ; S. Park

Trudy Matematicheskogo Instituta imeni V.A. Steklova, Tome 72 (2017) no. 2, pp. 199-256

Résumé

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.

Détail

Keywords: quasitoric manifold, moment-angle manifold, hyperbolic manifold, small cover, simple polytope, right-angled polytope, cohomology ring, cohomological rigidity.

Bibliographie

[1] E. M. Andreev, “On convex polyhedra in Lobačevskiĭ spaces”, Math. USSR-Sb., 10:3 (1970), 413–440 | DOI | MR | Zbl

[2] O. Antolín-Camarena, G. R. Maloney, R. K. W. Roeder, “Computing arithmetic invariants for hyperbolic reflection groups”, Complex dynamics, A. K. Peters, Wellesley, MA, 2009, 597–631 ; 2007, 34 pp., arXiv: 0708.2109 | DOI | MR | Zbl

[3] A. Ayzenberg, Toric manifolds over $3$-polytopes, 2016, 11 pp., arXiv: 1607.03377

[4] I. V. Baskakov, “Massey triple products in the cohomology of moment-angle complexes”, Russian Math. Surveys, 58:5 (2003), 1039–1041 | DOI | DOI | MR | Zbl

[5] I. V. Baskakov, V. M. Bukhshtaber, T. E. Panov, “Cellular cochain algebras and torus actions”, Russian Math. Surveys, 59:3 (2004), 562–563 | DOI | DOI | MR | Zbl

[6] W. Browder, Surgery on simply-connected manifolds, Ergeb. Math. Grenzgeb., 65, Springer-Verlag, New York–Heidelberg, 1972, ix+132 pp. | MR | MR | Zbl | Zbl

[7] V. M. Buchstaber, “Lectures on toric topology”, Toric topology workshop (KAIST, 2008), Trends Math., 10, No 1, Information Center for Mathematical Sciences, KAIST, Daejeon, 2008, 1–64

[8] V. M. Buchstaber, N. Yu. Erokhovets, “Truncations of simple polytopes and applications”, Proc. Steklov Inst. Math., 289 (2015), 104–133 | DOI | DOI | MR | Zbl

[9] V. M. Buchstaber, N. Erokhovets, Construction of fullerenes, 2015, 20 pp., arXiv: 1510.02948

[10] V. M. Buchstaber, N. Yu. Erokhovets, Fullerenes, polytopes and toric topology, Lect. Notes Ser. Inst. Math. Sci. Natl. Univ. Singap., World Sci. Publ., River Edge, NJ (to appear); 2016, 117 pp., arXiv: 1609.02949

[11] V. M. Buchstaber, N. Yu. Erokhovets, “Finite sets of operations sufficient to construct any fullerene from $C_{20}$”, Structural Chemistry, 28:1 (2017), 225–234 ; (2016), 10 pp., arXiv: 1611.05298 | DOI

[12] V. M. Bukhshtaber, N. Yu. Erokhovets, “Konstruktsii semeistv trekhmernykh mnogogrannikov, kharakteristicheskie fragmenty fullerenov i mnogogranniki Pogorelova”, Izv. RAN. Ser. matem., 81:5 (2017)

[13] V. M. Buchstaber, T. E. Panov, “Torus actions, combinatorial topology, and homological algebra”, Russian Math. Surveys, 55:5 (2000), 825–921 | DOI | DOI | MR | Zbl

[14] V. M. Buchstaber, T. E. Panov, Toric topology, Math. Surveys Monogr., 204, Amer. Math. Soc., Providence, RI, 2015, xiv+518 pp. | DOI | MR | Zbl

[15] V. M. Buchstaber, T. E. Panov, “On manifolds defined by $4$-colourings of simple $3$-polytopes”, Russian Math. Surveys, 71:6 (2016), 1137–1139 | DOI | DOI | MR

[16] V. M. Buchstaber, T. E. Panov, N. Ray, “Spaces of polytopes and cobordism of quasitoric manifolds”, Mosc. Math. J., 7:2 (2007), 219–242 | MR | Zbl

[17] R. Charney, M. W. Davis, “The $K(\pi,1)$-problem for hyperplane complements associated to infinite reflection groups”, J. Amer. Math. Soc., 8:3 (1995), 597–627 | DOI | MR | Zbl

[18] S. Choi, “Classification of Bott manifolds up to dimension $8$”, Proc. Edinb. Math. Soc. (2), 58:3 (2015), 653–659 | DOI | MR | Zbl

[19] S. Choi, M. Masuda, “Classification of $\mathbb{Q}$-trivial Bott manifolds”, J. Symplectic Geom., 10:3 (2012), 447–461 | DOI | MR | Zbl

[20] S. Choi, M. Masuda, S. Murai, “Invariance of Pontrjagin classes for Bott manifolds”, Algebr. Geom. Topol., 15:2 (2015), 965–986 | DOI | MR | Zbl

[21] S. Choi, M. Masuda, S.-I. Oum, “Classification of real Bott manifolds and acyclic digraphs”, Trans. Amer. Math. Soc., 369:4 (2017), 2987–3011 | DOI | MR | Zbl

[22] S. Choi, M. Masuda, D. Y. Suh, “Topological classification of generalized Bott towers”, Trans. Amer. Math. Soc., 362:2 (2010), 1097–1112 | DOI | MR | Zbl

[23] S. Choi, M. Masuda, D. Y. Suh, “Rigidity problems in toric topology: a survey”, Klassicheskaya i sovremennaya matematika v pole deyatelnosti Borisa Nikolaevicha Delone, Sbornik statei. K 120-letiyu so dnya rozhdeniya chlena-korrespondenta AN SSSR Borisa Nikolaevicha Delone, Tr. MIAN, 275, MAIK, M., 2011, 188–201 | MR | Zbl

[24] S. Choi, T. Panov, D. Y. Suh, “Toric cohomological rigidity of simple convex polytopes”, J. Lond. Math. Soc. (2), 82:2 (2010), 343–360 | DOI | MR | Zbl

[25] S. Choi, S. Park, “Projective bundles over toric surfaces”, Internat. J. Math., 27:4 (2016), 1650032, 30 pp. | DOI | MR | Zbl

[26] S. Choi, S. Park, D. Y. Suh, “Topological classification of quasitoric manifolds with second Betti number $2$”, Pacific J. Math., 256:1 (2012), 19–49 | DOI | MR | Zbl

[27] M. W. Davis, T. Januszkiewicz, “Convex polytopes, Coxeter orbifolds and torus actions”, Duke Math. J., 62:2 (1991), 417–451 | DOI | MR | Zbl

[28] M. Davis, T. Januszkiewicz, R. Scott, “Nonpositive curvature of blow-ups”, Selecta Math. (N. S.), 4:4 (1998), 491–547 | DOI | MR | Zbl

[29] C. Delaunay, “On hyperbolicity of toric real threefolds”, Int. Math. Res. Not., 2005:51 (2005), 3191–3201 | DOI | MR | Zbl

[30] G. Denham, A. I. Suciu, “Moment-angle complexes, monomial ideals and Massey products”, Pure Appl. Math. Q., 3:1 (2007), 25–60 | DOI | MR | Zbl

[31] M. Deza, M. Dyutur Sikirić, M. I. Shtogrin, “Fullerenes and disk-fullerenes”, Russian Math. Surveys, 68:4 (2013), 665–720 | DOI | DOI | MR | Zbl

[32] T. Došlić, “Cyclical edge-connectivity of fullerene graphs and $(k,6)$-cages”, J. Math. Chem., 33:2 (2003), 103–112 | DOI | MR | Zbl

[33] F. Fan, J. Ma, X. Wang, $B$-Rigidity of flag $2$-spheres without $4$-belt, 2015, 11 pp., arXiv: 1511.03624

[34] F. Fan, X. Wang, On the cohomology of moment-angle complexes associated to Gorenstein* complexes, 2016 (v1 – 2015), 49 pp., arXiv: 1508.00159

[35] M. Gromov, “Hyperbolic groups”, Essays in group theory, Math. Sci. Res. Inst. Publ., 8, Springer, New York, 1987, 75–263 | DOI | MR | Zbl

[36] B. Grünbaum, “Some analogues of Eberhard's theorem on convex polytopes”, Israel J. Math., 6:4 (1968), 398–411 | DOI | MR | Zbl

[37] S. Hasui, “On the classification of quasitoric manifolds over dual cyclic polytopes”, Algebr. Geom. Topol., 15:3 (2015), 1387–1437 | DOI | MR | Zbl

[38] T. Inoue, “Organizing volumes of right-angled hyperbolic polyhedra”, Algebr. Geom. Topol., 8:3 (2008), 1523–1565 | DOI | MR | Zbl

[39] H. Ishida, Y. Fukukawa, M. Masuda, “Topological toric manifolds”, Mosc. Math. J., 13:1 (2013), 57–98 | MR | Zbl

[40] P. E. Jupp, “Classification of certain $6$-manifolds”, Proc. Cambridge Philos. Soc., 73:2 (1973), 293–300 | DOI | MR | Zbl

[41] Y. Kamishima, M. Masuda, “Cohomological rigidity of real Bott manifolds”, Algebr. Geom. Topol., 9:4 (2009), 2479–2502 | DOI | MR | Zbl

[42] I. Yu. Limonchenko, “Massey products in cohomology of moment-angle manifolds for 2-truncated cubes”, Russian Math. Surveys, 71:2 (2016), 376–378 | DOI | DOI | MR | Zbl

[43] E. A. Lord, A. L. Mackay, S. Ranganathan, New geometries for new materials, Cambridge Univ. Press, Cambridge, 2006, x+238 pp. | Zbl

[44] G. A. Margulis, “Arithmetic properties of discrete groups”, Russian Math. Surveys, 29:1 (1974), 107–156 | DOI | MR | Zbl

[45] M. Masuda, “Cohomological non-rigidity of generalized real Bott manifolds of height 2”, Differentsialnye uravneniya i topologiya. I, Sbornik statei. K 100-letiyu so dnya rozhdeniya akademika Lva Semenovicha Pontryagina, Tr. MIAN, 268, MAIK, M., 2010, 252–257 | MR | Zbl

[46] M. Masuda, T. E. Panov, “Semifree circle actions, Bott towers and quasitoric manifolds”, Sb. Math., 199:8 (2008), 1201–1223 | DOI | DOI | MR | Zbl

[47] M. Masuda, D. Y. Suh, “Classification problems of toric manifolds via topology”, Toric topology, Contemp. Math., 460, Amer. Math. Soc., Providence, RI, 2008, 273–286 | DOI | MR | Zbl

[48] A. E. Mironov, “New examples of Hamilton-minimal and minimal Lagrangian submanifolds in $\mathbb{C}^n$ and $\mathbb{C}\mathrm P^n$”, Sb. Math., 195:1 (2004), 85–96 | DOI | DOI | MR | Zbl

[49] A. E. Mironov, T. E. Panov, “Intersections of quadrics, moment-angle manifolds, and Hamiltonian-minimal Lagrangian embeddings”, Funct. Anal. Appl., 47:1 (2013), 38–49 | DOI | DOI | MR | Zbl

[50] V. V. Nikulin, “On the classification of arithmetic groups generated by reflections in Lobachevsky spaces”, Math. USSR-Izv., 18:1 (1982), 99–123 | DOI | MR | Zbl

[51] S. P. Novikov, Homotopically equivalent smooth manifolds. I, 97 pp., \par http://www.mi.ras.ru/~snovikov/10.pdf | MR | Zbl

[52] S. P. Novikov, “Topology”, Topology I, Encyclopaedia Math. Sci., 12, Springer-Verlag, Berlin, 1996, 1–310 | MR | Zbl

[53] T. E. Panov, “Cohomology of face rings, and torus actions”, Surveys in contemporary mathematics, London Math. Soc. Lecture Note Ser., 347, Cambridge Univ. Press, Cambridge, 2008, 165–201 ; 2007 (v1 – 2005), 28 pp., arXiv: math/0506526 | MR | Zbl

[54] T. E. Panov, “Geometric structures on moment-angle manifolds”, Russian Math. Surveys, 68:3 (2013), 503–568 | DOI | DOI | MR | Zbl

[55] T. E. Panov, Ya. A. Veryovkin, “Polyhedral products and commutator subgroups of right-angled Artin and Coxeter groups”, Sb. Math., 207:11 (2016), 1582–1600 | DOI | DOI | MR

[56] A. V. Pogorelov, “A regular partition of Lobachevskian space”, Math. Notes, 1:1 (1967), 3–5 | DOI | MR | Zbl

[57] L. Potyagailo, E. Vinberg, “On right-angled reflection groups in hyperbolic spaces”, Comment. Math. Helv., 80:1 (2005), 63–73 | DOI | MR | Zbl

[58] Y. Suyama, “Examples of smooth compact toric varieties that are not quasitoric manifolds”, Algebr. Geom. Topol., 14:5 (2014), 3097–3106 | DOI | MR | Zbl

[59] Yu. Suyama, “Simplicial $2$-spheres obtained from non-singular complete fans”, Dalnevost. matem. zhurn., 15:2 (2015), 277–288 ; (2014), 9 pp., arXiv: 1409.3713 | Zbl

[60] W. P. Thurston, “Shapes of polyhedra and triangulations of the sphere”, The Epstein birthday schrift, Geom. Topol. Monogr., 1, Geom. Topol. Publ., Coventry, 1998, 511–549 | DOI | MR | Zbl

[61] A. Yu. Vesnin, “Three-dimensional hyperbolic manifolds of Löbell type”, Sib. Math. J., 28:5 (1987), 731–734 | DOI | MR | Zbl

[62] A. Yu. Vesnin, “Three-dimensional hyperbolic manifolds with general fundamental polyhedron”, Math. Notes, 49:6 (1991), 575–577 | DOI | MR | Zbl

[63] A. Yu. Vesnin, “Pryamougolnye mnogogranniki i trekhmernye giperbolicheskie mnogoobraziya”, UMN, 72:2 (2017), 67–110

[64] È. B. Vinberg, “Discrete groups generated by reflections in Lobačevskiĭ spaces”, Math. USSR-Sb., 1:3 (1967), 429–444 | DOI | MR | Zbl

[65] È. B. Vinberg, “The non-existence of crystallographic groups of reflections in Lobachevskiĭ spaces of large dimension”, Trans. Moscow Math. Soc., 1985 (1985), 75–112 | MR | Zbl

[66] C. T. C. Wall, “Classification problems in differential topology. V. On certain $6$-manifolds”, Invent. Math., 1:4 (1966), 355–374 | DOI | MR | Zbl

[67] A. V. Zhubr, “Closed simply connected six-dimensional manifolds: proofs of classification theorems”, St. Petersburg Math. J., 12:4 (2001), 605–680 | MR | Zbl

[68] G. M. Ziegler, Lectures on polytopes, Grad. Texts in Math., 152, Springer-Verlag, New York, 1995, x+370 pp. | DOI | MR | Zbl