Journal of Algebraic Combinatorics

, Volume 9, Issue 3, pp 215–232 | Cite as

On f-Vectors and Relative Homology

  • Art M. Duval


We find strong necessary conditions on the f-vectors, Betti sequences, and relative Betti sequence of a pair of simplicial complexes. We also present an example showing that these conditions are not sufficient. If only the difference between two Betti sequences is specified, and not the individual Betti sequences, then the characterization is complete, and the characterization of all pairs of simplicial complexes matches the characterization of pairs of near-cones. Our necessary conditions rely upon a combinatorial decomposition of pairs of simplicial complexes that reflects the homology and relative homology of the complexes.

f-vector Betti sequence relative homology simplicial complex decomposition 


  1. 1.
    A. Björner, “Face numbers of complexes and polytopes,” Proc.ICM-86, Berkeley, Vol.2, pp.1408–1418, American Mathematical Society, 1987.Google Scholar
  2. 2.
    A. Björner and G. Kalai, “An extended Euler-Poincaré theorem,” Acta Mathematica 161 (1988), 279–303.Google Scholar
  3. 3.
    A. Duval, “Acombinatorial decomposition of simplicial complexes,” Israel Journal of Mathematics 87 (1994), 77–87.Google Scholar
  4. 4.
    C. Greene and D. Kleitman, “Proof techniques in the theory of finite sets,” in Studies in Combinatorics, G.-C. Rota (Ed.), Mathematical Association of America, Washington, DC, 1978, pp.22–79.Google Scholar
  5. 5.
    I. Herstein, Topics in Algebra, 2nd edition, Xerox College Publishing, Lexington, MA, 1975.Google Scholar
  6. 6.
    G. Katona, “A theorem of finite sets,” in Theory of Graphs (Proc.Colloq., Tihany, 1966), P. Erdös and G. Katona (Eds.), Academic Press, New York and Akadémia Kiadó, Budapest, x1968x, pp.187–207.Google Scholar
  7. 7.
    J. Kruskal, “The number of simplices in a complex,” in Mathematical Optimization Techniques, R. Bellman (Ed.), University of California Press, Berkeley-Los Angeles, 1963, pp.251–278.Google Scholar
  8. 8.
    J. Munkres, Elements of Algebraic Topology, Benjamin/Cummings, Menlo Park, CA, 1984.Google Scholar
  9. 9.
    R. Stanley, “Generalized h-vectors, intersection cohomology of toric varieties, and related results,” in Commutative Algebra and Combinatorics, M. Nagata and H. Matsumura (Eds.), Advanced Studies in Pure Mathematics 11, Kinokuniya, Tokyo, and North-Holland, Amsterdam/New York, 1987, pp.187–213.Google Scholar
  10. 10.
    R. Stanley, “A combinatorial decomposition of acyclic simplicial complexes,” Discrete Mathematics 120 (1993), 175–182.Google Scholar
  11. 11.
    R. Stanley, Combinatorics and Commutative Algebra, 2nd edition, Birkhäuser, Boston, 1996.Google Scholar

Copyright information

© Kluwer Academic Publishers 1999

Authors and Affiliations

  • Art M. Duval
    • 1
  1. 1.Department of Mathematical SciencesUniversity of Texas at El PasoEl Paso

Personalised recommendations