On f-Vectors and Relative Homology
Art M. Duval
DOI: 10.1023/A:1018687018440
Abstract
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.
Pages: 215–232
Keywords: $f$-vector; Betti sequence; relative homology; simplicial complex; decomposition
Full Text: PDF
References
1. A. Bj\ddot orner, “Face numbers of complexes and polytopes,” Proc. ICM-86, Berkeley, Vol. 2, pp. 1408-1418, American Mathematical Society, 1987.
2. A. Bj\ddot orner and G. Kalai, “An extended Euler-Poincaré theorem,” Acta Mathematica 161 (1988), 279-303.
3. A. Duval, “A combinatorial decomposition of simplicial complexes,” Israel Journal of Mathematics 87 (1994), 77-87.
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.
5. I. Herstein, Topics in Algebra, 2nd edition, Xerox College Publishing, Lexington, MA, 1975.
6. G. Katona, “A theorem of finite sets,” in Theory of Graphs (Proc. Colloq., Tihany, 1966), P. Erd\ddot os and G. Katona (Eds.), Academic Press, New York and Akadémia Kiadó, Budapest, 1968, pp. 187-207.
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.
8. J. Munkres, Elements of Algebraic Topology, Benjamin/Cummings, Menlo Park, CA, 1984.
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.
10. R. Stanley, “A combinatorial decomposition of acyclic simplicial complexes,” Discrete Mathematics 120 (1993), 175-182.
11. R. Stanley, Combinatorics and Commutative Algebra, 2nd edition, Birkh\ddot auser, Boston, 1996.
2. A. Bj\ddot orner and G. Kalai, “An extended Euler-Poincaré theorem,” Acta Mathematica 161 (1988), 279-303.
3. A. Duval, “A combinatorial decomposition of simplicial complexes,” Israel Journal of Mathematics 87 (1994), 77-87.
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.
5. I. Herstein, Topics in Algebra, 2nd edition, Xerox College Publishing, Lexington, MA, 1975.
6. G. Katona, “A theorem of finite sets,” in Theory of Graphs (Proc. Colloq., Tihany, 1966), P. Erd\ddot os and G. Katona (Eds.), Academic Press, New York and Akadémia Kiadó, Budapest, 1968, pp. 187-207.
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.
8. J. Munkres, Elements of Algebraic Topology, Benjamin/Cummings, Menlo Park, CA, 1984.
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.
10. R. Stanley, “A combinatorial decomposition of acyclic simplicial complexes,” Discrete Mathematics 120 (1993), 175-182.
11. R. Stanley, Combinatorics and Commutative Algebra, 2nd edition, Birkh\ddot auser, Boston, 1996.