On Higher Massey Products and Rational Formality for Moment--Angle Manifolds over Multiwedges

Ivan Yu. Limonchenko

Informatics and Automation, Algebraic topology, combinatorics, and mathematical physics, Tome 305 (2019), pp. 174-196.

Résumé

We prove that certain conditions on multigraded Betti numbers of a simplicial complex $K$ imply the existence of a higher Massey product in the cohomology of a moment–angle complex $\mathcal Z_K$, and this product contains a unique element (a strictly defined product). Using the simplicial multiwedge construction, we find a family $\mathcal F$ of polyhedral products being smooth closed manifolds such that for any $l,r\geq 2$ there exists an $l$-connected manifold $M\in \mathcal F$ with a nontrivial strictly defined $r$-fold Massey product in $H^*(M)$. As an application to homological algebra, we determine a wide class of triangulated spheres $K$ such that a nontrivial higher Massey product of any order may exist in the Koszul homology of their Stanley–Reisner rings. As an application to rational homotopy theory, we establish a combinatorial criterion for a simple graph $\Gamma $ to provide a (rationally) formal generalized moment–angle manifold $\mathcal Z_P^J=(\underline {D}^{2j_i},\underline {S}^{2j_i-1})^{\partial P^*}$, $J=(j_1,\dots ,j_m)$, over a graph-associahedron $P=P_{\Gamma }$, and compute all the diffeomorphism types of formal moment–angle manifolds over graph-associahedra.

Détail

Keywords: polyhedral product, moment–angle manifold, Stanley–Reisner ring, Massey product
Mots-clés : simplicial multiwedge, graph-associahedron.

Bibliographie

[1] A. A. Ayzenberg, “Substitutions of polytopes and of simplicial complexes, and multigraded Betti numbers”, Trans. Moscow Math. Soc., 2013 (2013), 175–202 | Zbl

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

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

[4] Bahri A., Bendersky M., Cohen F.R., Gitler S., “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 [math.AT] | DOI | Zbl

[5] Bahri A., Bendersky M., Cohen F.R., Gitler S., “On free loop spaces of a toric space”, Bol. Soc. Mat. Mex. Ser. 3, 23:1 (2017), 257–265 ; arXiv: 1410.6458v2 [math.AT] | DOI | Zbl

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

[7] Beben P., Grbić J., $LS$-category of moment–angle manifolds, Massey products, and a generalization of the Golod property, E-print, 2016, arXiv: 1601.06297v2 [math.AT]

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

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

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

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

[12] Buchstaber V.M., Panov T.E., Torus actions and their applications in topology and combinatorics, Univ. Lect. Ser., 24, Amer. Math. Soc., Providence, RI, 2002 | DOI | Zbl

[13] V. M. Buchstaber and T. E. Panov, Torus Actions in Topology and Combinatorics, MTsNMO, Moscow, 2004

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

[15] V. M. Buchstaber and V. D. Volodin, “Sharp upper and lower bounds for nestohedra”, Izv. Math., 75:6 (2011), 1107–1133 | DOI | DOI | Zbl

[16] Cai L., “On products in a real moment–angle manifold”, J. Math. Soc. Jpn., 69:2 (2017), 503–528 ; arXiv: 1410.5543 [math.AT] | DOI | Zbl

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

[18] Choi S., Park H., “Wedge operations and torus symmetries”, Tohoku Math. J. Ser. 2, 68:1 (2016), 91–138 | DOI | Zbl

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

[20] Deligne P., Griffiths P., Morgan J., Sullivan D., “Real homotopy theory of Kähler manifolds”, Invent. math., 29 (1975), 245–274 | DOI | Zbl

[21] Delzant T., “Hamiltoniens périodiques et images convexes de l'application moment”, Bull. Soc. math. France, 116:3 (1988), 315–339 | DOI | Zbl

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

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

[24] Félix Y., Halperin S., Thomas J.-C., Rational homotopy theory, Grad. Texts Math., 205, Springer, New York, 2001 | DOI | Zbl

[25] Félix Y., Oprea J., Tanré D., Algebraic models in geometry, Oxford Grad. Texts Math., 17, Oxford Univ. Press, Oxford, 2008 | Zbl

[26] Félix Y., Tanré D., “Rational homotopy of the polyhedral product functor”, Proc. Amer. Math. Soc., 137:3 (2009), 891–898 | DOI | Zbl

[27] Frankhuizen R., “$A_\infty $-resolutions and the Golod property for monomial rings”, Algebr. Geom. Topol., 18:6 (2018), 3403–3424 ; arXiv: 1612.02737v2 [math.AT] | DOI | Zbl

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

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

[30] Grbić J., Panov T., Theriault S., Wu J., “The homotopy types of moment–angle complexes for flag complexes”, Trans. Amer. Math. Soc., 368:9 (2016), 6663–6682 ; arXiv: 1211.0873 [math.AT] | DOI | Zbl

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

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

[33] J. Grbić and S. Theriault, “Homotopy theory in toric topology”, Russ. Math. Surv., 71:2 (2016), 185–251 | DOI | DOI | Zbl

[34] Hochster M., “Cohen–Macaulay rings, combinatorics, and simplicial complexes”, Ring theory II, Proc. 2nd Oklahoma Conf. (1975), Lect. Notes Pure Appl. Math., 26, M. Dekker, New York, 1977, 171–223

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

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

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

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

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

[40] Massey W.S., “Some higher order cohomology operations”, Symposium internacional de Topología algebráica, Univ. Nac. Autón. México, UNESCO, Mexico City, 1958, 145–154

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

[42] McGavran D., “Adjacent connected sums and torus actions”, Trans. Amer. Math. Soc., 251 (1979), 235–254 | DOI | Zbl

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

[44] Notbohm D., Ray N., “On Davis–Januszkiewicz homotopy types. I: Formality and rationalisation”, Algebr. Geom. Topol., 5 (2005), 31–51 | DOI | Zbl

[45] Panov T., Ray N., “Categorical aspects of toric topology”, Toric topology, Contemp. Math., 460, ed. by M. Harada et al., Amer. Math. Soc., Providence, RI, 2008, 293–322 | DOI | Zbl

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

[47] Papadima S., Suciu A.I., “Algebraic invariants for right-angled Artin groups”, Math. Ann., 334:3 (2006), 533–555 | DOI | Zbl

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

[49] Sullivan D., “Infinitesimal computations in topology”, Publ. math. Inst. hautes étud. sci., 47 (1977), 269–331 | DOI | Zbl

[50] Uehara H., Massey W.S., “The Jacobi identity for Whitehead products”, Algebraic geometry and topology: A symposium in honor of S. Lefschetz, Princeton Math. Ser., 12, Princeton Univ. Press, Princeton, NJ, 1957, 361–377 | Zbl

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