Iterated Homology of Simplicial Complexes
Art M. Duval
and Lauren L. Rose2
2dagger
DOI: 10.1023/A:1011216008191
Abstract
We develop an iterated homology theory for simplicial complexes. Thistheory is a variation on one due to Kalai. For 
a simplicial complex of dimension d - 1, and each r = 0, ..., d, we define rth iterated homology groups of . When r = 0, this corresponds to ordinary homology. If is a cone over 
, then when r = 1, we get the homology of . If a simplicial complex is (nonpure) shellable, then its iterated Betti numbers give the restriction numbers, h k,j , of the shelling. Iterated Betti numbers are preserved by algebraic shifting, and may be interpreted combinatorially in terms of the algebraically shifted complex in several ways. In addition, the depth of a simplicial complex can be characterized in terms of its iterated Betti numbers.
a simplicial complex of dimension d - 1, and each r = 0, ..., d, we define rth iterated homology groups of . When r = 0, this corresponds to ordinary homology. If is a cone over , then when r = 1, we get the homology of . If a simplicial complex is (nonpure) shellable, then its iterated Betti numbers give the restriction numbers, h k,j , of the shelling. Iterated Betti numbers are preserved by algebraic shifting, and may be interpreted combinatorially in terms of the algebraically shifted complex in several ways. In addition, the depth of a simplicial complex can be characterized in terms of its iterated Betti numbers.Pages: 279–294
Keywords: shellability; algebraic shifting; depth; Betti numbers; simplicial complex
Full Text: PDF
References
1. A. Bj\ddot orner, “Face numbers, Betti numbers and depth,” in preparation.
2. A. Bj\ddot orner, “Nonpure shellability, f -vectors, subspace arrangements and complexity,” in Formal Power Series and Algebraic Combinatorics, DIMACS Workshop, May 1994, L.J. Billera, C. Greene, R. Simion, and R. Stanley (Eds.), Amer. Math. Soc., Providence, RI, 1996, pp. 25-53. DIMACS Series in Discrete Math. and Theor. Computer Science.
3. A. Bj\ddot orner and G. Kalai, “On f -vectors and homology,” in Combinatorial Mathematics: Proceedings of the Third International Conference G. Bloom, R. Graham, and J. Malkevitch (Eds.); Ann. N. Y. Acad. Sci. 555 (1989), 63-80.
4. A. Bj\ddot orner and G. Kalai, “An extended Euler-Poincaré theorem,” Acta Math. 161 (1988), 279-303.
5. A. Bj\ddot orner and M. Wachs, “Shellable nonpure complexes and posets I,” Trans. Amer. Math. Soc. 348 (1996), 1299-1327.
6. A. Bj\ddot orner and M. Wachs, “Shellable nonpure complexes and posets II,” Trans. Amer. Math. Soc. 349 (1997), 3945-3975.
7. M. Chari, “On Discrete Morse functions and combinatorial decompositions,” extended abstract, Eleventh International Conference on Formal Power Series and Algebraic Combinatorics, Barcelona, 1999.
8. A. Duval, “A combinatorial decomposition of simplicial complexes,” Israel J. Math. 87 (1994), 77-87.
9. T. Hibi, “Quotient algebras of Stanley-Reisner rings and local cohomology,” J. Alg. 140 (1991), 336-343.
10. 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.
11. G. Kalai, “Symmetric matroids,” J. Combin. Theory (Series B) 50 (1990), 54-64.
12. G. Kalai, Algebraic shifting, unpublished manuscript (July 1993 version), http://www.ma.huji.ac. il/\sim kalai/as93.ps.
13. J. Munkres, “Topological results in combinatorics,” Michigan Math. J. 31 (1984), 113-127.
14. G. Reisner, “Cohen-Macaulay quotients of polynomial rings,” thesis, University of Minnesota, 1974; Adv. Math. 21 (1976), 30-49.
15. D. Smith, “On the Cohen-Macaulay property in commutative algebra and simplicial topology,” Pac. J. Math. 141 (1990), 165-196.
16. R. Stanley, Combinatorics and Commutative Algebra, 2nd edition, Birkh\ddot auser, Boston, 1996.
17. R. Stanley, “A combinatorial decomposition of acyclic simplicial complexes,” Disc. Math. 120 (1993), 175-182.
2. A. Bj\ddot orner, “Nonpure shellability, f -vectors, subspace arrangements and complexity,” in Formal Power Series and Algebraic Combinatorics, DIMACS Workshop, May 1994, L.J. Billera, C. Greene, R. Simion, and R. Stanley (Eds.), Amer. Math. Soc., Providence, RI, 1996, pp. 25-53. DIMACS Series in Discrete Math. and Theor. Computer Science.
3. A. Bj\ddot orner and G. Kalai, “On f -vectors and homology,” in Combinatorial Mathematics: Proceedings of the Third International Conference G. Bloom, R. Graham, and J. Malkevitch (Eds.); Ann. N. Y. Acad. Sci. 555 (1989), 63-80.
4. A. Bj\ddot orner and G. Kalai, “An extended Euler-Poincaré theorem,” Acta Math. 161 (1988), 279-303.
5. A. Bj\ddot orner and M. Wachs, “Shellable nonpure complexes and posets I,” Trans. Amer. Math. Soc. 348 (1996), 1299-1327.
6. A. Bj\ddot orner and M. Wachs, “Shellable nonpure complexes and posets II,” Trans. Amer. Math. Soc. 349 (1997), 3945-3975.
7. M. Chari, “On Discrete Morse functions and combinatorial decompositions,” extended abstract, Eleventh International Conference on Formal Power Series and Algebraic Combinatorics, Barcelona, 1999.
8. A. Duval, “A combinatorial decomposition of simplicial complexes,” Israel J. Math. 87 (1994), 77-87.
9. T. Hibi, “Quotient algebras of Stanley-Reisner rings and local cohomology,” J. Alg. 140 (1991), 336-343.
10. 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.
11. G. Kalai, “Symmetric matroids,” J. Combin. Theory (Series B) 50 (1990), 54-64.
12. G. Kalai, Algebraic shifting, unpublished manuscript (July 1993 version), http://www.ma.huji.ac. il/\sim kalai/as93.ps.
13. J. Munkres, “Topological results in combinatorics,” Michigan Math. J. 31 (1984), 113-127.
14. G. Reisner, “Cohen-Macaulay quotients of polynomial rings,” thesis, University of Minnesota, 1974; Adv. Math. 21 (1976), 30-49.
15. D. Smith, “On the Cohen-Macaulay property in commutative algebra and simplicial topology,” Pac. J. Math. 141 (1990), 165-196.
16. R. Stanley, Combinatorics and Commutative Algebra, 2nd edition, Birkh\ddot auser, Boston, 1996.
17. R. Stanley, “A combinatorial decomposition of acyclic simplicial complexes,” Disc. Math. 120 (1993), 175-182.