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2011 | 9 | 4 | 715-730
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Moment-angle complexes from simplicial posets

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Treść / Zawartość
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Języki publikacji
EN
Abstrakty
EN
We extend the construction of moment-angle complexes to simplicial posets by associating a certain T m-space Z S to an arbitrary simplicial poset S on m vertices. Face rings ℤ[S] of simplicial posets generalise those of simplicial complexes, and give rise to new classes of Gorenstein and Cohen-Macaulay rings. Our primary motivation is to study the face rings ℤ[S] by topological methods. The space Z S has many important topological properties of the original moment-angle complex Z K associated to a simplicial complex K. In particular, we prove that the integral cohomology algebra of Z S is isomorphic to the Tor-algebra of the face ring ℤ[S]. This leads directly to a generalisation of Hochster’s theorem, expressing the algebraic Betti numbers of the ring ℤ[S] in terms of the homology of full subposets in S. Finally, we estimate the total amount of homology of Z S from below by proving the toral rank conjecture for the moment-angle complexes Z S.
Wydawca
Czasopismo
Rocznik
Tom
9
Numer
4
Strony
715-730
Opis fizyczny
Daty
wydano
2011-08-01
online
2011-05-26
Twórcy
Bibliografia
  • [1] 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., 2010, 225(3), 1634–1668 http://dx.doi.org/10.1016/j.aim.2010.03.026
  • [2] Buchstaber V.M., Panov T.E., Torus Actions and their Applications in Topology and Combinatorics, Univ. Lecture Ser., 24, American Mathematical Society, Providence, 2002
  • [3] Bukhshtaber V.M., Panov T.E., Torus actions and the combinatorics of polytopes, Proc. Steklov Inst. Math., 1999, 2(225), 87–120
  • [4] Bukhstaber V.M., Panov T.E., Combinatorics of simplicial cell complexes and torus actions, Proc. Steklov Inst. Math., 2004, 4(247), 33–49
  • [5] Cao X., Lü Z., Möbius transform, moment-angle complexes and Halperin-Carlsson conjecture, preprint available at http://arxiv.org/abs/0908.3174
  • [6] Davis M.W., Januszkiewicz T., Convex polytopes, Coxeter orbifolds and torus actions, Duke Math. J., 1991, 62(2), 417–451 http://dx.doi.org/10.1215/S0012-7094-91-06217-4
  • [7] Duval A.M., Free resolutions of simplicial posets, J. Algebra, 1997, 188(1), 363–399 http://dx.doi.org/10.1006/jabr.1996.6855
  • [8] Erokhovets N., Buchstaber invariant of simple polytopes, preprint available at http://arxiv.org/abs/0908.3407
  • [9] Fukukawa Y, Masuda M., Buchstaber invariants of skeleta of a simplex, preprint available at http://arxiv.org/abs/0908.3448
  • [10] Gitler S., Lopez de Medrano S., Intersections of quadrics, moment-angle manifolds and connected sums, preprint available at http://arxiv.org/abs/0901.2580
  • [11] Grbić J., Theriault S., The homotopy type of the complement of a coordinate subspace arrangement, Topology, 2007, 46(4), 357–396 http://dx.doi.org/10.1016/j.top.2007.02.006
  • [12] Maeda H., Masuda M., Panov T, Torus graphs and simplicial posets, Adv. Math., 2007, 212(2), 458–483 http://dx.doi.org/10.1016/j.aim.2006.10.011
  • [13] Masuda M., Panov T, On the cohomology of torus manifolds, Osaka J. Math., 2006, 43(3), 711–746
  • [14] Panov T.E., Cohomology of face rings, and torus actions, In: Surveys in Contemporary Mathematics, London Math. Soc. Lecture Note Ser., 347, Cambridge University Press, Cambridge, 2008, 165–201
  • [15] Panov T.E., Ray N., Categorical aspects of toric topology, In: Toric Topology, Contemp. Math., 460, American Mathematical Society, Providence, 2008, 293–322
  • [16] Panov T, Ray N., Vogt R., 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
  • [17] Stanley R.P, f-vectors and h-vectors of simplicial posets, J. Pure Appl. Algebra, 1991, 71(2–3), 319–331 http://dx.doi.org/10.1016/0022-4049(91)90155-U
  • [18] Ustinovsky Yu., Toral rank conjecture for moment-angle complexes, preprint available at http://arxiv.org/abs/0909.1053
Typ dokumentu
Bibliografia
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Identyfikator YADDA
bwmeta1.element.doi-10_2478_s11533-011-0041-z
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