Nerve complexes and moment-angle spaces of convex polytopes



We introduce spherical nerve complexes that are a far-reaching generalization of simplicial spheres, and consider the differential ring of simplicial complexes. We show that spherical nerve complexes form a subring of this ring, and define a homomorphism from the ring of polytopes to this subring that maps each polytope P to the nerve KP of the cover of the boundary ∂P by facets. We develop a theory of nerve complexes and apply it to the moment-angle spaces ZP of convex polytopes P. In the case of a polytope P with m facets, its moment-angle space ZP is defined by the canonical embedding in the cone ℝm. It is well-known that the space ZP is homeomorphic to the polyhedral product (D2, S1)∂P* if the polytope P is simple. We show that the homotopy equivalence \(\mathcal{Z}_P \simeq (D^2 ,S^1 )^{K_P }\) holds in the general case. On the basis of bigraded Betti numbers of simplicial complexes, we construct a new class of combinatorial invariants of convex polytopes. These invariants take values in the ring of polynomials in two variables and are multiplicative with respect to the direct product or join of polytopes. We describe the relation between these invariants and the well-known f-polynomials of polytopes. We also present examples of convex polytopes whose flag numbers (in particular, f-polynomials) coincide, while the new invariants are different.


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  1. 1.
    I. V. Baskakov, “Cohomology of K-Powers of Spaces and the Combinatorics of Simplicial Divisions,” Usp. Mat. Nauk 57(5), 147–148 (2002) [Russ. Math. Surv. 57, 989–990 (2002)].MathSciNetGoogle Scholar
  2. 2.
    I. V. Baskakov, V. M. Bukhshtaber, and T. E. Panov, “Cellular Cochain Algebras and Torus Actions,” Usp. Mat. Nauk 59(3), 159–160 (2004) [Russ. Math. Surv. 59, 562–563 (2004)].MathSciNetGoogle Scholar
  3. 3.
    V. M. Buchstaber, “Ring of Simple Polytopes and Differential Equations,” Tr. Mat. Inst. im. V.A. Steklova, Ross. Akad. Nauk 263, 18–43 (2008) [Proc. Steklov Inst. Math. 263, 13–37 (2008)].Google Scholar
  4. 4.
    V. M. Buchstaber and N. Yu. Erokhovets, “Polytopes, Fibonacci Numbers, Hopf Algebras, and Quasi-symmetric Functions,” Usp. Mat. Nauk 66(2), 67–162 (2011) [Russ. Math. Surv. 66, 271–367 (2011)].Google Scholar
  5. 5.
    V. M. Bukhshtaber and T. E. Panov, “Torus Actions, Combinatorial Topology, and Homological Algebra,” Usp. Mat. Nauk 55(5), 3–106 (2000) [Russ. Math. Surv. 55, 825–921 (2000)].MathSciNetGoogle Scholar
  6. 6.
    V. M. Bukhshtaber and T. E. Panov, Torus Actions in Topology and Combinatorics (MTsNMO, Moscow, 2004) [in Russian]; partial Engl. transl.: Torus Actions and Their Applications in Topology and Combinatorics (Am. Math. Soc., Providence, RI, 2002).MATHGoogle Scholar
  7. 7.
    A. A. Gaifullin, “The Construction of Combinatorial Manifolds with Prescribed Sets of Links of Vertices,” Izv. Ross. Akad. Nauk, Ser. Mat. 72(5), 3–62 (2008) [Izv. Math. 72, 845–899 (2008)].MathSciNetGoogle Scholar
  8. 8.
    D. Barnette, “Diagrams and Schlegel Diagrams,” in Combinatorial Structures and Their Applications: Proc. Int. Conf., Calgary, 1969 (Gordon and Breach, New York, 1970), pp. 1–4.Google Scholar
  9. 9.
    M. M. Bayer and L. J. Billera, “Generalized Dehn-Sommerville Relations for Polytopes, Spheres and Eulerian Partially Ordered Sets,” Invent. Math. 79(1), 143–157 (1985).MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    L. J. Billera, P. Filliman, and B. Sturmfels, “Constructions and Complexity of Secondary Polytopes,” Adv. Math. 83(2), 155–179 (1990).MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    F. Bosio and L. Meersseman, “Real Quadrics in ℂn, Complex Manifolds and Convex Polytopes,” Acta Math. 197(1), 53–127 (2006).MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    W. Bruns and J. Gubeladze, “Combinatorial Invariance of Stanley-Reisner Rings,” Georgian Math. J. 3(4), 315–318 (1996).MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    V. M. Buchstaber, T. E. Panov, and N. Ray, “Spaces of Polytopes and Cobordism of Quasitoric Manifolds,” Moscow Math. J. 7(2), 219–242 (2007).MathSciNetMATHGoogle Scholar
  14. 14.
    M. W. Davis and T. Januszkiewicz, “Convex Polytopes, Coxeter Orbifolds and Torus Actions,” Duke Math. J. 62(2), 417–451 (1991).MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    M. Franz, “The Integral Cohomology of Toric Manifolds,” Tr. Mat. Inst. im. V.A. Steklova, Ross. Akad. Nauk 252, 61–70 (2006) [Proc. Steklov Inst. Math. 252, 53–62 (2006)].MathSciNetGoogle Scholar
  16. 16.
    P. S. Hirschhorn, Model Categories and Their Localizations (Am. Math. Soc., Providence, RI, 2003), Math. Surv. Monogr. 99.MATHGoogle Scholar
  17. 17.
    M. Hochster, “Cohen-Macaulay Rings, Combinatorics, and Simplicial Complexes,” in Ring Theory II: Proc. 2nd Oklahoma Conf., 1975 (M. Dekker, New York, 1977), Lect. Notes Pure Appl. Math. 26, pp. 171–223.Google Scholar
  18. 18.
    S. Mac Lane, Categories for the Working Mathematician, 2nd ed. (Springer, New York, 1998), Grad. Texts Math. 5.MATHGoogle Scholar
  19. 19.
    T. E. Panov, “Cohomology of Face Rings, and Torus Actions,” in Surveys in Contemporary Mathematics (Cambridge Univ. Press, Cambridge, 2008), LMS Lect. Note Ser. 347, pp. 165–201.Google Scholar
  20. 20.
    T. E. Panov and N. Ray, “Categorical Aspects of Toric Topology,” in Toric Topology, Ed. by M. Harada, Y. Karshon, M. Masuda, and T. Panov (Am. Math. Soc., Providence, RI, 2008), Contemp. Math. 460, pp. 293–322.CrossRefGoogle Scholar
  21. 21.
    G.-C. Rota, “On the Foundations of Combinatorial Theory. I: Theory of Möbius Functions,” Z. Wahrscheinlichkeitstheor. Verwandte Geb. 2(4), 340–368 (1964).MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    R. P. Stanley, Enumerative Combinatorics (Wadsworth and Brooks/Cole, Monterey, CA, 1986), Vol. 1.MATHGoogle Scholar
  23. 23.
    R. P. Stanley, Combinatorics and Commutative Algebra (Birkhäuser, Boston, MA, 1996), Prog. Math. 41.MATHGoogle Scholar
  24. 24.
    V. Welker, G. M. Ziegler, and R. T. Živaljević, “Homotopy Colimits-Comparison Lemmas for Combinatorial Applications,” J. Reine Angew. Math. 509, 117–149 (1999).MathSciNetMATHGoogle Scholar
  25. 25.
    G. M. Ziegler, Lectures on Polytopes (Springer, New York, 2007).Google Scholar

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© Pleiades Publishing, Ltd. 2011

Authors and Affiliations

  1. 1.Faculty of Mechanics and MathematicsMoscow State UniversityMoscowRussia
  2. 2.Steklov Mathematical InstituteRussian Academy of SciencesMoscowRussia

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