Onf-vectors and Betti numbers of multicomplexes


A multicomplexM is a collection of monomials closed under divisibility. For suchM we construct a cell complex ΔM whosei-dimensional cells are in bijection with thef i monomials ofM of degreei+1. The bijection is such that the inclusion relation of cells corresponds to divisibility of monomials. We then study relations between the numbersf i and the Betti numbers of ΔM. For squarefree monomials the construction specializes to the standard geometric realization of a simplicial complex.

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  1. [1]

    R. N. Andersen, M. N. Marjanović, andR. M. Schori: Symmetric products and higher-dimensional dunce hats,Topology Proc.,18 (1993), 7–17.

    Google Scholar 

  2. [2]

    A. Björner, andG. Kalai: An extended Euler-Poincaré theorem,Acta Math.,161 (1988), 279–303.

    Google Scholar 

  3. [3]

    A. Björner, andG. Kalai: Extended Euler-Poincaré relations for cell complexes, in:Applied Geometry and Discrete Mathematics (The V. Klee Festschrift) (eds. P. Gritzmann and B. Sturmfels) Amer. Math. Soc., Providence, R.I., 81–89, 1991.

    Google Scholar 

  4. [4]

    G. F. Clements, andB. Lindström: A generalization of a combinatorial theorem of Macaulay,J. Combinatorial Theory,7 (1969), 230–238.

    Google Scholar 

  5. [5]

    J. Eckhoff, andG. Wegner: Über einen Satz von Kruskal,Period. Math. Hungar.,6 (1975), 137–142.

    Google Scholar 

  6. [6]

    C. Greene, andD. Kleitman: Proof techniques in the theory of finite sets, in:Studies in Combinatorics, (ed.: G.-C. Rota) Math. Assoc. Amer., Washington, D.C., 22–79, 1978.

    Google Scholar 

  7. [7]

    G. Kalai: A characterization off-vectors of families of convex sets inR d, Part I: Necessity of Eckhoff's conditions,Israel J. Math.,48 (1984), 175–195.

    Google Scholar 

  8. [8]

    D. Quillen: Homotopy properties of the poset of nontrivialp-subgroups of a group,Advances in Math.,28 (1978), 101–128.

    Google Scholar 

  9. [9]

    R. P. Stanley: Hilbert functions of graded algebras,Advances in Math.,28 (1978), 57–83.

    Google Scholar 

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Additional information

This work was supported by the Mittag-Leffler Institute during the Combinatorial Year program 1991–92. The second author also acknowledges support from the Serbian Science Foundation, Grant No. 0401D.

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Björner, A., Vrećica, S. Onf-vectors and Betti numbers of multicomplexes. Combinatorica 17, 53–65 (1997). https://doi.org/10.1007/BF01196131

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Mathematics Subject Classification (1991)

  • 05 E 99
  • 55 M 99
  • 05 D 05
  • 55 N 99