Substitutions of polytopes and of simplicial complexes, and multigraded betti numbers
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- by A. A. Ayzenberg
- Trans. Moscow Math. Soc. 2013, 175-202
- DOI: https://doi.org/10.1090/S0077-1554-2014-00224-7
- Published electronically: April 9, 2014
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Abstract:
For a simplicial complex $K$ on $m$ vertices and simplicial complexes $K_1,\ldots ,K_m$, we introduce a new simplicial complex $K(K_1,\ldots ,K_m)$, called a substitution complex. This construction is a generalization of the iterated simplicial wedge studied by A. Bari, M. Bendersky, F. R. Cohen, and S. Gitler. In a number of cases it allows us to describe the combinatorics of generalized joins of polytopes $P(P_1,\ldots ,P_m)$, as introduced by G. Agnarsson. 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 $K(K_1,\ldots ,K_m)$ 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 $K(K_1,\ldots ,K_m)$ are expressed in terms of those of the original complexes $K, K_1,\ldots ,K_m$. We also describe connections between the obtained results and the known results of other authors.References
- Geir Agnarsson, The flag polynomial of the Minkowski sum of simplices, Ann. Comb. 17 (2013), no. 3, 401–426. MR 3090169, DOI 10.1007/s00026-013-0189-2
- David J. Anick, Connections between Yoneda and Pontrjagin algebras, Algebraic topology, Aarhus 1982 (Aarhus, 1982) Lecture Notes in Math., vol. 1051, Springer, Berlin, 1984, pp. 331–350. MR 764587, DOI 10.1007/BFb0075575
- A. A. Aĭzenberg and V. M. Bukhshtaber, Nerve complexes and moment-angle spaces of convex polytopes, Tr. Mat. Inst. Steklova 275 (2011), no. Klassicheskaya i Sovremennaya Matematika v Pole Deyatel′nosti Borisa Nikolaevicha Delone, 22–54 (Russian, with Russian summary); English transl., Proc. Steklov Inst. Math. 275 (2011), no. 1, 15–46. MR 2962969, DOI 10.1134/S0081543811080025
- A. Bahri, M. Bendersky, F. R. Cohen, and S. Gitler, The polyhedral product functor: a method of decomposition for moment-angle complexes, arrangements and related spaces, Adv. Math. 225 (2010), no. 3, 1634–1668. MR 2673742, DOI 10.1016/j.aim.2010.03.026
- A. Bahri, M. Bendersky, F. R. Cohen, and S. Gitler, Operations on polyhedral products and a new topological construction of infinite families of toric manifolds. arXiv:1011.0094v4
- David Barnette, Diagrams and Schlegel diagrams, Combinatorial Structures and their Applications (Proc. Calgary Internat. Conf., Calgary, Alta., 1969) Gordon and Breach, New York, 1970, pp. 1–4. MR 0270266
- I. V. Baskakov, Cohomology of $K$-powers of spaces and the combinatorics of simplicial divisions, Uspekhi Mat. Nauk 57 (2002), no. 5(347), 147–148 (Russian); English transl., Russian Math. Surveys 57 (2002), no. 5, 989–990. MR 1992088, DOI 10.1070/RM2002v057n05ABEH000558
- V. M. Buchstaber and T. E. Panov, Torus actions in topology and combinatorics MCCME, Moscow, 2004. (Russian)
- V. M. Bukhshtaber and T. E. Panov, Actions of tori, combinatorial topology and homological algebra, Uspekhi Mat. Nauk 55 (2000), no. 5(335), 3–106 (Russian, with Russian summary); English transl., Russian Math. Surveys 55 (2000), no. 5, 825–921. MR 1799011, DOI 10.1070/rm2000v055n05ABEH000320
- V. M. Buchstaber and T. E. Panov, Toric topology, arXiv:1210.2368.
- Victor M. Buchstaber, Taras E. Panov, and Nigel Ray, Spaces of polytopes and cobordism of quasitoric manifolds, Mosc. Math. J. 7 (2007), no. 2, 219–242, 350 (English, with English and Russian summaries). MR 2337880, DOI 10.17323/1609-4514-2007-7-2-219-242
- N. Yu. Erokhovets, Moment-angle manifolds of simple $n$-dimensional polytopes with $n+3$ facets, Uspekhi Mat. Nauk 66 (2011), no. 5(401), 187–188 (Russian); English transl., Russian Math. Surveys 66 (2011), no. 5, 1006–1008. MR 2919276, DOI 10.1070/RM2011v066n05ABEH004767
- N. Yu. Erokhovets, Maximal torus actions on moment-angle manifolds, Ph.D. thesis, MSU, Department of Mechanics and Mathematics, 2011.
- Branko Grünbaum, Convex polytopes, 2nd ed., Graduate Texts in Mathematics, vol. 221, Springer-Verlag, New York, 2003. Prepared and with a preface by Volker Kaibel, Victor Klee and Günter M. Ziegler. MR 1976856, DOI 10.1007/978-1-4613-0019-9
- Melvin Hochster, Cohen-Macaulay rings, combinatorics, and simplicial complexes, Ring theory, II (Proc. Second Conf., Univ. Oklahoma, Norman, Okla., 1975) Lecture Notes in Pure and Appl. Math., Vol. 26, Dekker, New York, 1977, pp. 171–223. MR 0441987
- P. Mani, Spheres with few vertices, J. Combinatorial Theory Ser. A 13 (1972), 346–352. MR 317175, DOI 10.1016/0097-3165(72)90068-4
- Ezra Miller and Bernd Sturmfels, Combinatorial commutative algebra, Graduate Texts in Mathematics, vol. 227, Springer-Verlag, New York, 2005. MR 2110098
- Satoshi Murai, Spheres arising from multicomplexes, J. Combin. Theory Ser. A 118 (2011), no. 8, 2167–2184. MR 2834171, DOI 10.1016/j.jcta.2011.04.015
- Taras E. Panov and Nigel Ray, Categorical aspects of toric topology, Toric topology, Contemp. Math., vol. 460, Amer. Math. Soc., Providence, RI, 2008, pp. 293–322. MR 2428364, DOI 10.1090/conm/460/09026
- J. Scott Provan and Louis J. Billera, Decompositions of simplicial complexes related to diameters of convex polyhedra, Math. Oper. Res. 5 (1980), no. 4, 576–594. MR 593648, DOI 10.1287/moor.5.4.576
- Richard P. Stanley, Combinatorics and commutative algebra, 2nd ed., Progress in Mathematics, vol. 41, Birkhäuser Boston, Inc., Boston, MA, 1996. MR 1453579
- Yu. M. Ustinovskiĭ, Doubling operation for polytopes and torus actions, Uspekhi Mat. Nauk 64 (2009), no. 5(389), 181–182 (Russian); English transl., Russian Math. Surveys 64 (2009), no. 5, 952–954. MR 2589970, DOI 10.1070/RM2009v064n05ABEH004647
- Yu. Ustinovsky, Toral rank conjecture for moment-angle complexes, arXiv:0909.1053v2
- Volkmar Welker, Günter M. Ziegler, and Rade T. Živaljević, Homotopy colimits—comparison lemmas for combinatorial applications, J. Reine Angew. Math. 509 (1999), 117–149. MR 1679169, DOI 10.1515/crll.1999.035
- Günter M. Ziegler, Lectures on polytopes, Graduate Texts in Mathematics, vol. 152, Springer-Verlag, New York, 1995. MR 1311028, DOI 10.1007/978-1-4613-8431-1
Bibliographic Information
- A. A. Ayzenberg
- Affiliation: Moscow, MSU, Department of Mechanics and Mathematics, Chair of Geometric Methods of Mathematical Physics
- Email: ayzenberga@gmail.com
- Published electronically: April 9, 2014
- Additional Notes: Supported by the RFFI Grant 12-01-92104–YaFa and the RF Government Grant 2010-220-01-077.
- © Copyright 2014 American Mathematical Society
- Journal: Trans. Moscow Math. Soc. 2013, 175-202
- MSC (2010): Primary 05E45; Secondary 52B11, 52B05, 55U10, 13F55
- DOI: https://doi.org/10.1090/S0077-1554-2014-00224-7
- MathSciNet review: 3235795