Massey products, toric topology and combinatorics of polytopes

V. M. Buchstaber ; I. Yu. Limonchenko

Izvestiya. Mathematics , Tome 83 (2019) no. 6, pp. 1081-1136.

Résumé

In this paper we introduce a direct family of simple polytopes $P^{0}\,{\subset}\, P^{1}\,{\subset}{\kern1pt}{\cdots}$ such that for any $2\,{\leq}\,k\,{\leq}\,n$ there are non-trivial strictly defined Massey products of order $k$ in the cohomology rings of their moment-angle manifolds $\mathcal Z_{P^n}$. We prove that the direct sequence of manifolds $*\subset S^{3}\hookrightarrow\dots\hookrightarrow\mathcal Z_{P^n}\hookrightarrow\mathcal Z_{P^{n+1}}\,{\hookrightarrow}\,{\cdots}$ has the following properties: every manifold $\mathcal Z_{P^n}$ is a retract of $\mathcal Z_{P^{n+1}}$, and one has inverse sequences in cohomology (over $n$ and $k$, where $k\to\infty$ as $n\to\infty$) of the Massey products constructed. As an application we get that there are non-trivial differentials $d_k$, for arbitrarily large $k$ as $n\to\infty$, in the Eilenberg–Moore spectral sequence connecting the rings $H^*(\Omega X)$ and $H^*(X)$ with coefficients in a field, where $X=\mathcal Z_{P^n}$.

Détail

Keywords: polyhedral product, moment-angle manifold, Massey product, Lusternik–Schnirelmann category, polytope family, flag polytope, generating series, nestohedron
Mots-clés : graph-associahedron.

Bibliographie

[1] H. Uehara, W. S. Massey, “The Jacobi identity for Whitehead products”, Algebraic geometry and topology, A symposium in honor of S. Lefschetz, Princeton Univ. Press, Princeton, N. J., 1957, 361–377 | MR | Zbl

[2] W. S. Massey, “Some higher order cohomology operations”, International symposium on algebraic topology, Universidad Nacional Autónoma de México and UNESCO, Mexico City, 1958, 145–154 | MR | Zbl

[3] J. P. May, “Matric Massey products”, J. Algebra, 12:4 (1969), 533–568 | DOI | MR | Zbl

[4] D. Kraines, “Massey higher products”, Trans. Amer. Math. Soc., 124:3 (1966), 431–449 | DOI | MR | Zbl

[5] E. J. O'Neill, “On Massey products”, Pacific J. Math., 76:1 (1978), 123–127 | DOI | MR | Zbl

[6] D. Sullivan, “Infinitesimal computations in topology”, Inst. Hautes Études Sci. Publ. Math., 47 (1977), 269–331 | DOI | MR | Zbl

[7] P. Deligne, Ph. Griffiths, J. Morgan, D. Sullivan, “Real homotopy theory of Kähler manifolds”, Invent. Math., 29:3 (1975), 245–274 | DOI | MR | Zbl

[8] I. K. Babenko, I. A. Taimanov, “Massey products in symplectic manifolds”, Sb. Math., 191:8 (2000), 1107–1146 | DOI | DOI | MR | Zbl

[9] A. Tralle, J. Oprea, Symplectic manifolds with no Kähler structure, Lecture Notes in Math., 1661, Springer-Verlag, Berlin, 1997, viii+207 pp. | DOI | MR | Zbl

[10] Yu. Rudyak, A. Tralle, “On Thom spaces, Massey products, and nonformal symplectic manifolds”, Internat. Math. Res. Notices, 2000:10, 495–513 ; Preprint MPI 1999-71, 1999 | DOI | MR | Zbl

[11] 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

[12] G. Denham, A. I. Suciu, “Moment-angle complexes, monomial ideals and Massey products”, Pure Appl. Math. Q., 3:1, Special issue: In honor of R. D. MacPherson. Part 3 (2007), 25–60 | DOI | MR | Zbl

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

[14] J. Herzog, T. Hibi, Monomial ideals, Grad. Texts in Math., 260, Springer-Verlag London, Ltd., London, 2011, xvi+305 pp. | DOI | MR | Zbl

[15] T. Panov, N. Ray, R. Vogt, “Colimits, Stanley–Reisner algebras, and loop spaces”, Categorical decomposition techniques in algebraic topology (Isle of Skye, 2001), Progr. Math., 215, Birkhäuser, Basel, 2004, 261–291 | MR | Zbl

[16] T. E. Panov, N. Ray, “Categorical aspects of toric topology”, Toric topology, Contemp. Math., 460, Amer. Math. Soc., Providence, RI, 2008, 293–322 ; arXiv: 0707.0300 | DOI | MR | Zbl

[17] I. Limonchenko, “Topology of moment-angle manifolds arising from flag nestohedra”, Chin. Ann. Math. Ser. B, 38:6 (2017), 1287–1302 ; arXiv: 1510.07778 | DOI | MR | Zbl

[18] I. Yu. Limonchenko, “O vysshikh proizvedeniyakh Massi i ratsionalnoi formalnosti moment-ugol-mnogoobrazii nad multivstavkami”, Tr. MIAN, 305, MAIK «Nauka/Interperiodika», M., 2019 (to appear) ; arXiv: 1711.00461v2

[19] E. G. Zhuravleva, “Massey products in the cohomology of the moment-angle manifolds corresponding to Pogorelov polytopes”, Math. Notes, 105:4 (2019), 519–527 ; arXiv: 1707.07365v2 | DOI | DOI | MR | Zbl

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

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

[22] V. M. Buchstaber, N. Yu. Erokhovets, M. Masuda, T. E. Panov, S. Park, “Cohomological rigidity of manifolds defined by 3-dimensional polytopes”, Russian Math. Surveys, 72:2 (2017), 199–256 ; arXiv: 1610.07575 | DOI | DOI | MR | Zbl

[23] 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

[24] J.-P. Serre, Algèbre locale. Multiplicités, Cours au Collège de France, 1957–1958, rédigé par P. Gabriel, Collège de France, Paris, 1960, 157 pp. | MR | Zbl

[25] E. S. Golod, “On the homology of some local rings”, Soviet Math. Dokl., 3 (1962), 745–749 | MR | Zbl

[26] T. H. Gulliksen, G. Levin, Homology of local rings, Queen's Papers in Pure and Appl. Math., 20, Queen's Univ., Kingston, Ont., 1969, x+192 pp. | MR | Zbl

[27] J. Grbić, S. Theriault, “The homotopy type of the complement of a coordinate subspace arrangement”, Topology, 46:4 (2007), 357–396 | DOI | MR | Zbl

[28] J. Grbić, S. Theriault, “The homotopy type of the polyhedral product for shifted complexes”, Adv. Math., 245 (2013), 690–715 | DOI | MR | Zbl

[29] J. Grbić, S. Theriault, “Homotopy theory in toric topology”, Russian Math. Surveys, 71:2 (2016), 185–251 | DOI | DOI | MR | Zbl

[30] K. Iriye, D. Kishimoto, “Decompositions of polyhedral products for shifted complexes”, Adv. Math., 245 (2013), 716–736 | DOI | MR | Zbl

[31] K. Iriye, D. Kishimoto, “Fat-wedge filtration and decomposition of polyhedral products”, Kyoto J. Math., 59:1 (2019), 1–51 ; arXiv: 1412.4866v4 | DOI | MR | Zbl

[32] J. Grbić, T. Panov, S. Theriault, Jie Wu, “Homotopy types of moment-angle complexes for flag complexes”, Trans. Amer. Math. Soc., 368:9 (2016), 6663–6682 ; arXiv: 1211.0873 | DOI | MR | Zbl

[33] A. Berglund, M. Jöllenbeck, “On the Golod property of Stanley–Reisner rings”, J. Algebra, 315:1 (2007), 249–273 | DOI | MR | Zbl

[34] L. Katthän, “A non-Golod ring with a trivial product on its Koszul homology”, J. Algebra, 479 (2017), 244–262 | DOI | MR | Zbl

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

[36] S. López de Medrano, “Topology of the intersection of quadrics in $\mathbb{R}^n$”, Algebraic topology (Arcata, CA, 1986), Lecture Notes in Math., 1370, Springer, Berlin, 1989, 280–292 | DOI | MR | Zbl

[37] S. Gitler, S. López de Medrano, “Intersections of quadrics, moment-angle manifolds and connected sums”, Geom. Topol., 17:3 (2013), 1497–1534 | DOI | MR | Zbl

[38] I. Yu. Limonchenko, “Stanley–Reisner rings of generalized truncation polytopes and their moment–angle manifolds”, Proc. Steklov Inst. Math., 286 (2014), 188–197 ; arXiv: 1401.2124 | DOI | DOI | MR | Zbl

[39] I. Yu. Limonchenko, “Semeistva minimalno negolodovskikh kompleksov i poliedralnye proizvedeniya”, Dalnevost. matem. zhurn., 15:2 (2015), 222–237 ; arXiv: 1509.04302 | MR | Zbl

[40] P. Beben, J. Grbić, “LS-category of moment-angle manifolds, Massey products, and a generalization of the Golod property”, Trans. Amer. Math. Soc., 2016, 1–21 (Submitted); arXiv: 1601.06297v2

[41] V. Buchstaber, I. Limonchenko, Embeddings of moment-angle manifolds and sequences of Massey products, arXiv: 1808.08851

[42] V. Buchstaber, I. Limonchenko, Direct families of polytopes with nontrivial Massey products, arXiv: 1811.02221v1

[43] V. M. Buchstaber, “Ring of simple polytopes and differential equations”, Proc. Steklov Inst. Math., 263 (2008), 13–37 | DOI | MR | Zbl

[44] V. M. Buchstaber, V. D. Volodin, “Sharp upper and lower bounds for nestohedra”, Izv. Math., 75:6 (2011), 1107–1133 ; Upper and lower bound theorems for graph-associahedra, arXiv: 1005.1631v2 | DOI | DOI | MR | Zbl

[45] A. Bahri, M. Bendersky, F. R. Cohen, S. Gitler, “Operations on polyhedral products and a new topological construction of infinite families of toric manifolds”, Homology Homotopy Appl., 17:2 (2015), 137–160 ; arXiv: 1011.0094 | DOI | MR | Zbl

[46] W. Bruns, J. Gubeladze, “Combinatorial invariance of Stanley–Reisner rings”, Georgian Math. J., 3:4 (1996), 315–318 | DOI | MR | Zbl

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

[48] A. Bahri, M. Bendersky, F. R. Cohen, S. Gitler, “The polyhedral product functor: a method of decomposition for moment-angle complexes, arrangements and related spaces”, Adv. Math., 225:3 (2010), 1634–1668 | DOI | MR | Zbl

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

[50] V. M. Bukhshtaber, T. E. Panov, Toricheskie deistviya v topologii i kombinatorike, MTsNMO, M., 2004, 272 pp. | MR | Zbl

[51] 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 ; arXiv: math/0506526 | MR | Zbl

[52] V. M. Buchstaber, N. Ray, “Tangential structures on toric manifolds, and connected sums of polytopes”, Internat. Math. Res. Notices, 2001:4 (2001), 193–219 | DOI | MR | Zbl

[53] F. A. Bogomolov, “The actions of a circumference on smooth odd-dimensional spheres”, Math. Notes, 5:4 (1969), 241–244 | DOI | MR | Zbl

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

[55] M. Hochster, “Cohen–Macaulay rings, combinatorics, and simplicial complexes”, Ring theory II (Univ. Oklahoma, Norman, OK, 1975), Lecture Notes in Pure and Appl. Math., 26, Dekker, New York, 1977, 171–223 | MR | Zbl

[56] 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 ; arXiv: 1603.06902 | DOI | DOI | MR | Zbl

[57] A. Postnikov, “Permutohedra, associahedra, and beyond”, Int. Math. Res. Not. IMRN, 2009:6 (2009), 1026–1106 ; arXiv: math.CO/0507163 | DOI | MR | Zbl

[58] A. Postnikov, V. Reiner, L. Williams, “Faces of generalized permutohedra”, Doc. Math., 13 (2008), 207–273 ; arXiv: math/0609184v2 | MR | Zbl

[59] 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 | DOI | DOI | MR | Zbl

[60] V. M. Buchstaber, E. V. Koritskaya, “Quasilinear Burgers–Hopf equation and Stasheff polytopes”, Funct. Anal. Appl., 41:3 (2007), 196–207 | DOI | DOI | MR | Zbl

[61] V. M. Buchstaber, V. D. Volodin, “Combinatorial $2$-truncated cubes and applications”, Associahedra, Tamari lattices, and related structures, Tamari memorial Festschrift, Prog. Math. Phys., 299, Birkhäuser/Springer, Basel, 2012, 161–186 | DOI | MR | Zbl

[62] E. M. Feichtner, B. Sturmfels, “Matroid polytopes, nested sets and Bergman fans”, Port. Math. (N.S.), 62:4 (2005), 437–468 ; arXiv: math/0411260 | MR | Zbl

[63] M. Carr, S. L. Devadoss, “Coxeter complexes and graph-associahedra”, Topology Appl., 153:12 (2006), 2155–2168 | DOI | MR | Zbl

[64] R. Bott, C. Taubes, “On the self-linking of knots”, Topology and physics, J. Math. Phys., 35:10 (1994), 5247–5287 | DOI | MR | Zbl

[65] J. D. Stasheff, “Homotopy associativity of $H$-spaces. I”, Trans. Amer. Math. Soc., 108:2 (1963), 275–292 | DOI | MR | Zbl

[66] V. M. Buchstaber, A. N. Kholodov, “Boas–Buck structures on sequences of polynomials”, Funct. Anal. Appl., 23:4 (1989), 266–276 | DOI | MR | Zbl

[67] V. M. Buchstaber, N. Yu. Erokhovets, “Polytopes, Fibonacci numbers, Hopf algebras, and quasi-symmetric functions”, Russian Math. Surveys, 66:2 (2011), 271–367 | DOI | DOI | MR | Zbl

[68] J. McCleary, A user's guide to spectral sequences, Cambridge Stud. Adv. Math., 58, 2nd ed., Cambridge Univ. Press, Cambridge, 2001, xvi+561 pp. | MR | Zbl

[69] V. M. Bukhshtaber, Kogomologicheskie operatsii v obobschennykh teoriyakh kogomologii, Diss. ... kand. fiz.-matem. nauk, Mekh.-matem. f-t MGU, M., 1969

[70] J. Milnor, “Construction of universal bundles. I”, Ann. of Math. (2), 63:2 (1956), 272–284 ; II:3, 430–436 | DOI | MR | Zbl | DOI | MR | Zbl

[71] A. Dold, R. Lashof, “Principal quasifibrations and fibre homotopy equivalence of bundles”, Illinois J. Math., 3:2 (1959), 285–305 | DOI | MR | Zbl

[72] A. Dold, Halbexakte Homotopiefunktoren, Lecture Notes in Math., 12, Springer-Verlag, Berlin–New York, 1966, 156 pp. | DOI | MR | Zbl | Zbl

[73] S. Eilenberg, J. C. Moore, “Homology and fibrations. I. Coalgebras, cotensor product and its derived functors”, Comment. Math. Helv., 40 (1966), 199–236 | DOI | MR | Zbl

[74] J. F. Adams, “On the non-existence of elements of Hopf invariant one”, Ann. of Math. (2), 72:1 (1960), 20–104 | DOI | MR | Zbl

[75] O. Cornea, G. Lupton, J. Oprea, D. Tanré, Lusternik–Schnirelmann category, Math. Surveys Monogr., 103, Amer. Math. Soc., Providence, RI, 2003, xviii+330 pp. | DOI | MR | Zbl

[76] M. Ginsburg, “On the Lusternik–Schnirelmann category”, Ann. of Math. (2), 77:3 (1963), 538–551 | DOI | MR | Zbl

[77] T. Ganea, “Lusternik–Schnirelmann category and strong category”, Illinois J. Math., 11:3 (1967), 417–427 | DOI | MR | Zbl

[78] H. Toda, Composition methods in homotopy groups of spheres, Ann. of Math. Stud., 49, Princeton Univ. Press, Princeton, N.J., 1962, v+193 pp. | DOI | MR | MR | Zbl | Zbl

[79] J. M. Cohen, “The decomposition of stable homotopy”, Ann. of Math. (2), 87:2 (1968), 305–320 | DOI | MR | Zbl

[80] J. P. May, “The cohomology of principal bundles, homogeneous spaces, and two-stage Postnikov systems”, Bull. Amer. Math. Soc., 74:2 (1968), 334–339 | DOI | MR | Zbl

[81] V. M. Bukhshtaber, “Groups of polynomial transformations of a line, non-formal symplectic manifolds, and the Landweber–Novikov algebra”, Russian Math. Surveys, 54:4 (1999), 837–838 | DOI | DOI | MR | Zbl

[82] V. M. Buhštaber, “The Chern–Dold character in cobordisms. I”, Math. USSR-Sb., 12:4 (1970), 573–594 | DOI | MR | Zbl

[83] L. V. Goncharova, “The cohomologies of Lie algebras of formal vector fields on the line”, Funct. Anal. Appl., 7:2 (1973), 91–97 | DOI | MR | Zbl

[84] L. V. Goncharova, “The cohomologies of Lie algebras of formal vector fields on the straight line”, Funct. Anal. Appl., 7:3 (1973), 194–203 | DOI | MR | Zbl

[85] V. M. Bukhshtaber, A. V. Shokurov, “The Landweber–Novikov algebra and formal vector fields on the line”, Funct. Anal. Appl., 12:3 (1978), 159–168 | DOI | MR | Zbl

[86] I. V. Artel'nykh, “Massey products and the Bukhshtaber spectral sequence”, Russian Math. Surveys, 55:3 (2000), 559–561 | DOI | DOI | MR | Zbl

[87] D. V. Millionschikov, “Algebra of formal vector fields on the line and Buchstaber's conjecture”, Funct. Anal. Appl., 43:4 (2009), 264–278 | DOI | DOI | MR | Zbl

[88] A. S. Švarc, “The genus of a fiber space. I”, Amer. Math. Soc. Transl. Ser. 2, 48, Amer. Math. Soc., Providence, R.I., 1965, 151–155 | DOI | MR | Zbl

[89] S. Halperin, J.-M. Lemaire, “Notions of category in differential algebra”, Algebraic topology — rational homotopy (Louvain-la-Neuve, 1986), Lecture Notes in Math., 1318, Springer, Berlin, 1988, 138–154 | DOI | MR | Zbl

[90] B. Jessup, “New estimates for the collapse of the Milnor–Moore spectral sequence over a field”, Proc. Amer. Math. Soc., 114:4 (1992), 1115–1117 | DOI | MR | Zbl

[91] K. Iriye, T. Yano, “A Golod complex with non-suspension moment-angle complex”, Topology Appl., 225 (2017), 145–163 ; arXiv: 1601.03472v4 | DOI | MR | Zbl