Cohomological Rigidity of Families of Manifolds Associated with Ideal Right-Angled Hyperbolic 3-Polytopes

Nikolai Yu. Erokhovets

Trudy Matematicheskogo Instituta imeni V.A. Steklova, Toric Topology, Group Actions, Geometry, and Combinatorics. Part 2, Tome 318 (2022), pp. 99-138.

Résumé

In toric topology, every $n$-dimensional combinatorial simple convex polytope $P$ with $m$ facets is assigned an $(m+n)$-dimensional moment–angle manifold $\mathcal Z_P$ with an action of a compact torus $T^m$ such that $\mathcal Z_P/T^m$ is a convex polytope of combinatorial type $P$. A simple $n$-polytope $P$ is said to be $B$-rigid if any isomorphism of graded rings $H^*(\mathcal Z_P,\mathbb Z)= H^*(\mathcal Z_Q,\mathbb Z)$ for a simple $n$-polytope $Q$ implies that $P$ and $Q$ are combinatorially equivalent. An ideal almost Pogorelov polytope is a combinatorial $3$-polytope obtained by cutting off all the ideal vertices of an ideal right-angled polytope in the Lobachevsky (hyperbolic) space $\mathbb L^3$. These polytopes are exactly the polytopes obtained from arbitrary (not necessarily simple) convex $3$-polytopes by cutting off all the vertices followed by cutting off all the “old” edges. The boundary of the dual polytope is the barycentric subdivision of the boundary of the old polytope (and also of its dual polytope). We prove that any ideal almost Pogorelov polytope is $B$-rigid. A family of manifolds is said to be cohomologically rigid over a ring $R$ if two manifolds from the family are diffeomorphic whenever their graded cohomology rings over $R$ are isomorphic. As a result we obtain three cohomologically rigid families of manifolds over ideal almost Pogorelov polytopes: moment–angle manifolds, canonical six-dimensional quasitoric manifolds over $\mathbb Z$ or any field, and canonical three-dimensional small covers over $\mathbb Z_2$. The latter two classes of manifolds are known as pullbacks from the linear model.

Détail

Keywords: ideal right-angled polytope, $B$-rigidity, cohomological rigidity, almost Pogorelov polytope, pullback from the linear model.

Bibliographie

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

[2] E. M. Andreev, “On convex polyhedra of finite volume in Lobačevskiĭ space”, Math. USSR, Sb., 12:2 (1970), 255–259 | DOI | MR | Zbl

[3] Barnette D., “On generating planar graphs”, Discrete Math., 7:3–4 (1974), 199–208 | DOI | MR | Zbl

[4] Birkhoff G.D., “The reducibility of maps”, Amer. J. Math., 35:2 (1913), 115–128 | DOI | MR | Zbl

[5] Bobenko A.I., Springborn B.A., “Variational principles for circle patterns and Koebe's theorem”, Trans. Amer. Math. Soc., 356:2 (2004), 659–689 | DOI | MR | Zbl

[6] Bosio F., Two transformations of simple polytopes preserving moment–angle manifolds, E-print, 2017, arXiv: 1708.00399v1 [math.GT]

[7] Bosio F., Meersseman L., “Real quadrics in $\mathbf C^n$, complex manifolds and convex polytopes”, Acta math., 197:1 (2006), 53–127 | DOI | MR | Zbl

[8] Brinkmann G., Greenberg S., Greenhill C., McKay B.D., Thomas R., Wollan P., “Generation of simple quadrangulations of the sphere”, Discrete Math., 305:1–3 (2005), 33–54 | DOI | MR | Zbl

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

[10] Buchstaber V.M., Erokhovets N.Yu., “Fullerenes, polytopes and toric topology”, Combinatorial and toric homotopy: Introductory lectures, Lect. Notes Ser. Inst. Math. Sci. Natl. Univ. Singapore, 35, World Scientific, Hackensack, NJ, 2017, 67–178 ; arXiv: 1609.02949 [math.AT] | MR

[11] V. M. Buchstaber and N. Yu. Erokhovets, “Constructions of families of three-dimensional polytopes, characteristic patches of fullerenes, and Pogorelov polytopes”, Izv. Math., 81:5 (2017), 901–972 | DOI | MR | Zbl

[12] V. M. Buchstaber, N. Yu. Erokhovets, M. Masuda, T. E. Panov, and S. Park, “Cohomological rigidity of manifolds defined by 3-dimensional polytopes”, Russ. Math. Surv., 72:2 (2017), 199–256 | DOI | MR | Zbl

[13] Buchstaber V.M., Panov T.E., Toric topology, Math. Surv. Monogr., 204, Amer. Math. Soc., Providence, RI, 2015 | DOI | MR | Zbl

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

[15] Cho Y., Lee E., Masuda M., Park S., Unique toric structure on a Fano Bott manifold, E-print, 2020, arXiv: 2005.02740v1 [math.SG]

[16] Choi S., Kim J.S., “Combinatorial rigidity of 3-dimensional simplicial polytopes”, Int. Math. Res. Not., 2011:8 (2011), 1935–1951 | MR | Zbl

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

[18] Choi S., Park K., “Example of $C$-rigid polytopes which are not $B$-rigid”, Math. Slovaca, 69:2 (2019), 437–448 | DOI | MR | Zbl

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

[20] Davis M.W., Okun B., “Vanishing theorems and conjectures for the 2-homology of right-angled Coxeter groups”, Geom. Topol., 5 (2001), 7–74 | DOI | MR | Zbl

[21] Došlić T., “On lower bounds of number of perfect matchings in fullerene graphs”, J. Math. Chem., 24:4 (1998), 359–364 | DOI | MR

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

[23] N. Yu. Erokhovets, “Three-dimensional right-angled polytopes of finite volume in the Lobachevsky space: Combinatorics and constructions”, Proc. Steklov Inst. Math., 305 (2019), 78–134 | DOI | MR | Zbl

[24] Erokhovets N., $B$-rigidity of the property to be an almost Pogorelov polytope, E-print, 2020, arXiv: 2004.04873v5 [math.AT] | MR

[25] Erokhovets N., $B$-rigidity of ideal almost Pogorelov polytopes, E-print, 2020, arXiv: 2005.07665v3 [math.AT]

[26] Erokhovets N.Yu., “$B$-rigidity of the property to be an almost Pogorelov polytope”, Topology, geometry, and dynamics: V.A. Rokhlin–memorial, Contemp. Math., 772, Amer. Math. Soc., Providence, RI, 2021, 107–122 | DOI | MR | Zbl

[27] Erokhovets N.Yu., “Kanonicheskaya geometrizatsiya orientiruemykh trekhmernykh mnogoobrazii, opredelyaemykh vektornymi raskraskami trekhmernykh mnogogrannikov”, Mat. sb., 213:6 (2022), 21–70 ; arXiv: 2011.11628v2 [math.GT] | MR

[28] Fan F., Ma J., Wang X., $B$-rigidity of flag 2-spheres without 4-belt, E-print, 2015, arXiv: 1511.03624 [math.AT]

[29] Fan F., Ma J., Wang X., Some rigidity problems in toric topology. I, E-print, 2020, arXiv: 2004.03362v2 [math.AT]

[30] Fan F., Wang X., On the cohomology of moment–angle complexes associated to Gorenstein* complexes, E-print, 2015, arXiv: 1508.00159v1 [math.AT]

[31] I. V. Izmestiev, “Three-dimensional manifolds defined by coloring a simple polytope”, Math. Notes, 69:3 (2001), 340–346 | DOI | MR | Zbl

[32] M. Joswig, “The group of projectivities and colouring of the facets of a simple polytope”, Russ. Math. Surv., 56:3 (2001), 584–585 | DOI | MR | Zbl

[33] A. D. Mednykh, “Automorphism groups of three-dimensional hyperbolic manifolds”, Sov. Math., Dokl., 32 (1985), 633–636 | MR | Zbl

[34] Mednykh A.D., Vesnin A.Yu., “On three-dimensional hyperbolic manifolds of Löbell type”, Complex analysis and applications (Proc. Conf., Varna, 1985), ed. by L. Iliev et al., Publ. House Bulg. Acad. Sci., Sofia, 1986, 440–446 | MR

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

[36] Rivin I., “Euclidean structures on simplicial surfaces and hyperbolic volume”, Ann. Math. Ser. 2, 139:3 (1994), 553–580 | DOI | MR | Zbl

[37] Rivin I., “A characterization of ideal polyhedra in hyperbolic 3-space”, Ann. Math. Ser. 2, 143:1 (1996), 51–70 | DOI | MR | Zbl

[38] Schramm O., “How to cage an egg”, Invent. math., 107:3 (1992), 543–560 | DOI | MR | Zbl

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

[40] Thurston W.P., The geometry and topology of three-manifolds (electron. vers. 1.1), MSRI, Berkeley, CA, 2002 http://www.msri.org/publications/books/gt3m/

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

[42] A. Yu. Vesnin, “Right-angled polyhedra and hyperbolic 3-manifolds”, Russ. Math. Surv., 72:2 (2017), 335–374 | DOI | MR | Zbl

[43] Vesnin A.Yu., Egorov A.A., “Ideal right-angled polyhedra in Lobachevsky space”, Chebyshev. sb., 21:2 (2020), 65–83 | MR | Zbl

[44] E. B. Vinberg and O. V. Shvartsman, “Discrete groups of motions of spaces of constant curvature”, Geometry II: Spaces of Constant Curvature, Encycl. Math. Sci., 29, Springer, Berlin, 1993, 139–248 ; Tsigler G.M., Teoriya mnogogrannikov, MTsNMO, M., 2014 | MR

[45] G. M. Ziegler, Lectures on Polytopes, Grad. Texts Math., 152, Springer, Berlin, 1995 | DOI | MR | Zbl