Substitutions of polytopes and of simplicial complexes, and multigraded Betti numbers

A. A. Aizenberg

Trudy Moskovskogo matematičeskogo obŝestva, Trudy Moskovskogo Matematicheskogo Obshchestva, Tome 74 (2013) no. 2, pp. 211-245.

Résumé

For a simplicial complex $K$ on $m$ vertices and simplicial complexes $K_1,\dots,K_m$, we introduce a new simplicial complex $K(K_1,\dots,K_m)$, called a substitution complex. This construction is a generalization of the iterated simplicial wedge studied by A. Bari et al. [Geom. Topol. 17, No. 3, 1497–1534 (2013; Zbl 1276.14087)]. In a number of cases it allows us to describe the combinatorics of generalized joins of polytopes $P(P_1,\dots,P_m)$, as introduced by G. Agnarsson [Ann. Comb. 17, No. 3, 401–426 (2013; Zbl 1272.05005)]. The substitution gives rise to an operad structure on the set of finite simplicial complexes in which a simplicial complex on $m$ vertices is considered as an $m$-ary operation. We prove the following main results: (1) the complex $K(K_1,\dots,K_m)$ is a simplicial sphere if and only if $K$ is a simplicial sphere and the $K_i$ are the boundaries of simplices, (2) the class of spherical nerve-complexes is closed under substitution, (3) multigraded betti numbers of $K(K_1,\dots,K_m)$ are expressed in terms of those of the original complexes $K,K_1,\dots,K_m$. We also describe connections between the obtained results and the known results of other authors.

Bibliographie

[1] Agnarsson Geir, The flag polynomial of the Minkowski sum of simplices, arXiv: 1006.5928 | MR

[2] Anick D., “Connections between Yoneda and Pontrjagin algebras”, Algebraic topology (Aarhus, 1982), Lecture Notes in Math., 1051, Springer, Berlin, 1984, 331–350 | DOI | MR

[3] Aizenberg A. A., Bukhshtaber V. M., “Nerv-kompleksy i moment-ugol-prostranstva vypuklykh mnogogrannikov”, Tr. MIAN, 275, 2011, 22–54 | MR

[4] 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. in Math., 225:3 (2010), 1634–1668 | DOI | MR | Zbl

[5] Bahri A., Bendersky M., Cohen F. R., Gitler S., Operations on polyhedral products and a new topological construction of infinite families of toric manifolds, arXiv: 1011.0094v4

[6] Barnette D., “Diagrams and Schlegel diagrams”, Combinatorial Structures and their Applications, Proc. Calgary Internat. Conf. (Calgary, Alta), Gordon an Breach, New York, 1970, 1–4 | MR

[7] Baskakov I. V., “Kogomologii $K$-stepenei prostranstv i kombinatorika simplitsialnykh razbienii”, UMN, 57:5(347) (2002), 147–148 | DOI | MR | Zbl

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

[9] Bukhshtaber V. M., Panov T. E., “Deistviya torov, kombinatornaya topologiya i gomologicheskaya algebra”, UMN, 55:5(335) (2000), 3–106 | DOI | MR

[10] Buchstaber V. M., Panov T. E., Toric topology, arXiv: 1210.2368 | MR

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

[12] Erokhovets N. Yu., “Moment-ugol mnogoobraziya prostykh $n$-mernykh mnogogrannikov s $n+3$ gipergranyami”, UMN, 66:5(401) (2011), 187–188 | DOI | MR | Zbl

[13] Erokhovets N. Yu., Maksimalnye deistviya torov na moment-ugol mnogoobraziyakh, Dissertatsiya na soiskanie uchenoi stepeni kandidata fiziko-matematicheskikh nauk, MGU im. M. V. Lomonosova, mekh.-mat. fakultet, 2011

[14] Grünbaum B., Convex polytopes, Graduate texts in mathematics, 221, second edition, Springer-Verlag, New York, 2003 | DOI | MR

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

[16] Mani P., “Spheres with few vertices”, J. Comb. Theory. Ser. A, 13 (1972), 346–352 | DOI | MR | Zbl

[17] Miller E., Sturmfels B., Combinatorial commutative algebra, Graduate texts in mathematics, 227, Springer-Verlag, New York, 2005 | MR

[18] Murai S., “Spheres arising from multicomplexes”, J. Comb. Theory. Ser. A, 118:8 (2011), 2167–2184 | DOI | MR | Zbl

[19] Panov T., Ray N., “Categorical aspects of toric topology”, Toric Topology, Contemp. Math., 460, AMS, Providence, RI, 2008, 293–322 | DOI | MR | Zbl

[20] Provan J. S., Billera L. J., “Decompositions of simplicial complexes related to diameters of convex polyhedra”, Math. Oper. Res., 5 (1980), 576–594 | DOI | MR | Zbl

[21] Stanley R., Combinatorics and Commutative Algebra, Progress in Mathematics, 41, Birkhäuser, Boston, MA, 1996 | MR | Zbl

[22] Ustinovskii Yu. M., “Operatsiya udvoeniya mnogogrannikov i deistviya tora”, UMN, 64:5(389) (2009), 181–182 | DOI | MR | Zbl

[23] Ustinovsky Yury, Toral rank conjecture for moment-angle complexes, arXiv: 0909.1053v2

[24] Welker V., Ziegler M. G., Živaljević T. R., “Homotopy colimits — comparison lemmas for combinatorial applications”, J. reine und angew. Math., 509, April (1999), 117–149 | DOI | MR | Zbl

[25] Ziegler M. Günter, Lectures on Polytopes, Springer-Verlag, New York, 2007