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.
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References
A. Björner, “Face numbers of complexes and polytopes,” Proc.ICM-86, Berkeley, Vol.2, pp.1408–1418, American Mathematical Society, 1987.
A. Björner and G. Kalai, “An extended Euler-Poincaré theorem,” Acta Mathematica 161 (1988), 279–303.
A. Duval, “Acombinatorial decomposition of simplicial complexes,” Israel Journal of Mathematics 87 (1994), 77–87.
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.
I. Herstein, Topics in Algebra, 2nd edition, Xerox College Publishing, Lexington, MA, 1975.
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.
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.
J. Munkres, Elements of Algebraic Topology, Benjamin/Cummings, Menlo Park, CA, 1984.
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.
R. Stanley, “A combinatorial decomposition of acyclic simplicial complexes,” Discrete Mathematics 120 (1993), 175–182.
R. Stanley, Combinatorics and Commutative Algebra, 2nd edition, Birkhäuser, Boston, 1996.
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Duval, A.M. On f-Vectors and Relative Homology. Journal of Algebraic Combinatorics 9, 215–232 (1999). https://doi.org/10.1023/A:1018687018440
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DOI: https://doi.org/10.1023/A:1018687018440