Abstract
We find a decomposition of simplicial complexes that implies and sharpens the characterization (due to Björner and Kalai) of thef-vector and Betti numbers of a simplicial complex. It generalizes a result of Stanley, who proved the acyclic case, and settles a conjecture of Stanley and Kalai.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
A. Björner,Face numbers of complexes and polytopes, inProc. Int. Congress of Math., Berkeley, 1986, pp. 1408–1418.
A. Björner and G. Kalai,An extended Euler-Poincaré theorem, Acta Math.161 (1988), 279–303.
A. Björner and G. Kalai,Extended Euler-Poincaré relations for cell complexes, inApplied Geometry and Discrete Mathematics: The Victor Klee Festschrift (P. Gritzmann and B. Sturmfels, eds.), DIMACS Series in Discrete Mathematics and Theoretical Computer Science4, American Mathematical Society, Providence, RI, 1991, pp. 81–89.
D. E. Daykin,A simple proof of the Kruskal-Katona theorem, J. Comb. Theory (Series A)17 (1974), 252–253.
P. Frankl,A new short proof for the Kruskal-Katona theorem, Discrete Math.48 (1984), 327–329.
A. M. Garsia,Combinatorial methods in the theory of Cohen-Macaulay rings, Adv. in Math.38 (1980), 229–266.
C. Greene and D. J. Kleitman,Strong versions of Sperner’s Theorem, J. Comb. Theory (Series A)20 (1976), 80–88.
C. Greene and D. J. Kleitman,Proof techniques in the theory of finite sets, inStudies in Combinatorics (G.-C. Rota, ed.), Mathematical Association of America, Washington, DC, 1978, pp. 22–79.
G. Kalai,Characterization of f-vectors of families of convex sets in Rd.Part I: Necessity of Eckhoff’s conditions, Israel J. Math.48 (1984), 175–195.
G. O. H. Katona,A theorem of finite sets, inTheory of Graphs (Proc. Colloq., Tihany, 1966, P. Erdös and G. Katona, eds.), Academic Press, New York and Akadémia Kiadó, Budapest, 1968, pp. 187–207.
J. B. Kruskal,The number of simplices in a complex, inMathematical Optimization Techniques (R. Bellman, ed.), University of California Press, Berkeley-Los Angeles, 1963, pp. 251–278.
J. Munkres,Elements of Algebraic Topology, Benjamin/Cummings, Menlo Park, CA, 1984.
H. J. Ryser,Combinatorial Mathematics, Mathematical Association of America, Washington, DC, 1963.
R. Stanley,The Upper Bound Conjecture and Cohen-Macaulay rings, Studies in Appl. Math.54 (1975), 135–142.
R. Stanley,Balanced Cohen-Macaulay complexes, Trans. Amer. Math. Soc.249 (1979), 139–157.
R. Stanley,A combinatorial decomposition of acyclic simplicial complexes, Discrete Math.118 (1993), to appear.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Duval, A.M. A combinatorial decomposition of simplicial complexes. Israel J. Math. 87, 77–87 (1994). https://doi.org/10.1007/BF02772984
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF02772984