Spectral Sequences on Combinatorial Simplicial Complexes
Dmitry N. Kozlov
Institute for Advanced Study School of Mathematics Olden Lane Princeton NJ USA
DOI: 10.1023/A:1011209803008
Abstract
The goal of this paper is twofold. First, we give an elementary introduction to the usage of spectral sequences in the combinatorial setting. Second we list a number of applications.
In the first group of applications the simplicial complex is the nerve of a poset; we consider general posets and lattices, as well as partition-type posets. Our last application is of a different nature: the S n \mathcal{S}_n -quotient of the complex of directed forests is a simplicial complex whose cell structure is defined combinatorially.
Pages: 27–48
Keywords: spectral sequences; posets; graphs; homology groups; shellability
Full Text: PDF
References
1. K. Baclawski, “Whitney numbers of geometric lattices,” Advances in Math. 16 (1975), 125-138.
2. K. Baclawski, “Galois connections and the Leray spectral sequence,” Advances in Math. 25 (1977), 191-215.
3. K. Baclawski, “Cohen-Macaulay ordered sets,” J. Algebra 63 (1980), 226-258.
4. A. Bj\ddot orner, “Subspace arrangements,” in First European Congress of Mathematics, Paris 1992, A. Joseph et al. (Eds.), Progress in Math., Vol. 119, Birkh\ddot auser, Basel, 1994, pp. 321-370.
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6. A. Bj\ddot orner and J.W. Walker, “A homotopy complementation formula for partially ordered sets,” European J. Combin. 4 (1983), 11-19.
7. G.E. Bredon, Introduction to Compact Transformation Groups, Academic Press, San Diego, 1972.
8. E.M. Feichtner and D.N. Kozlov, “On subspace arrangements of type D,” in Proceedings of FPSAC'96, Discrete Math., Vol. 210, No. 1-3, 2000, pp. 27-54.
9. P. Hanlon, “The generalized Dowling lattices,” Trans. Amer. Math. Soc. 325 (1991), 1-37.
10. M.I. Kargapolov and Ju.L. Merzljakov, Fundamentals of the Theory of Groups, Graduate Texts in Mathematics, Vol. 62, Springer-Verlag, Berlin, 1979 (English translation of Osnovy teorii grupp, Nauka, Moscow, 1977).
11. D.N. Kozlov, “General lexicographic shellability and orbit arrangements,” Ann. Comb. 1 (1) (1997), 67-90.
12. D.N. Kozlov, “Complexes of directed trees,” J. Comb. Theory A 88(1) (1999), 112-122.
13. W.S. Massey, “Exact couples in algebraic topology I, II,” Ann. of Math. 56 (1952), 363-396.
14. J. McCleary, User's Guide to Spectral Sequences, Publish or Perish, Wilmington, 1985.
15. D. Quillen, Higher Algebraic K-Theory I, Lecture Notes in Mathematics, Vol. 341, Springer-Verlag, Berlin, 1973, 85-148.
16. D. Quillen, “Homotopy properties of the poset of nontrivial p-subgroups of a group,” Adv. in Math. 28 (1978), 101-128.
17. E. Spanier, Algebraic Topology, McGraw-Hill, New York, 1966.
18. R.P. Stanley, Enumerative Combinatorics, Vol. I, Wadsworth, Belmont, CA, 1986.
19. R.P. Stanley, private communication, 1997.
20. S. Sundaram, “On the topology of two partition posets with forbidden blocksizes,” J. Pure Appl. Algebra 155(2-3) (2001), 271-304.
21. V. Welker, Partition lattices, group actions on subspace arrangements and combinatorics of discriminants, Habilitationsshrift, Essen, 1996.
2. K. Baclawski, “Galois connections and the Leray spectral sequence,” Advances in Math. 25 (1977), 191-215.
3. K. Baclawski, “Cohen-Macaulay ordered sets,” J. Algebra 63 (1980), 226-258.
4. A. Bj\ddot orner, “Subspace arrangements,” in First European Congress of Mathematics, Paris 1992, A. Joseph et al. (Eds.), Progress in Math., Vol. 119, Birkh\ddot auser, Basel, 1994, pp. 321-370.
5. A. Bj\ddot orner, “Topological methods,” in Handbook of Combinatorics, R. Graham, M. Gr\ddot otschel, and L. Lovász (Eds.), North-Holland, Amsterdam, 1995, pp. 1819-1872.
6. A. Bj\ddot orner and J.W. Walker, “A homotopy complementation formula for partially ordered sets,” European J. Combin. 4 (1983), 11-19.
7. G.E. Bredon, Introduction to Compact Transformation Groups, Academic Press, San Diego, 1972.
8. E.M. Feichtner and D.N. Kozlov, “On subspace arrangements of type D,” in Proceedings of FPSAC'96, Discrete Math., Vol. 210, No. 1-3, 2000, pp. 27-54.
9. P. Hanlon, “The generalized Dowling lattices,” Trans. Amer. Math. Soc. 325 (1991), 1-37.
10. M.I. Kargapolov and Ju.L. Merzljakov, Fundamentals of the Theory of Groups, Graduate Texts in Mathematics, Vol. 62, Springer-Verlag, Berlin, 1979 (English translation of Osnovy teorii grupp, Nauka, Moscow, 1977).
11. D.N. Kozlov, “General lexicographic shellability and orbit arrangements,” Ann. Comb. 1 (1) (1997), 67-90.
12. D.N. Kozlov, “Complexes of directed trees,” J. Comb. Theory A 88(1) (1999), 112-122.
13. W.S. Massey, “Exact couples in algebraic topology I, II,” Ann. of Math. 56 (1952), 363-396.
14. J. McCleary, User's Guide to Spectral Sequences, Publish or Perish, Wilmington, 1985.
15. D. Quillen, Higher Algebraic K-Theory I, Lecture Notes in Mathematics, Vol. 341, Springer-Verlag, Berlin, 1973, 85-148.
16. D. Quillen, “Homotopy properties of the poset of nontrivial p-subgroups of a group,” Adv. in Math. 28 (1978), 101-128.
17. E. Spanier, Algebraic Topology, McGraw-Hill, New York, 1966.
18. R.P. Stanley, Enumerative Combinatorics, Vol. I, Wadsworth, Belmont, CA, 1986.
19. R.P. Stanley, private communication, 1997.
20. S. Sundaram, “On the topology of two partition posets with forbidden blocksizes,” J. Pure Appl. Algebra 155(2-3) (2001), 271-304.
21. V. Welker, Partition lattices, group actions on subspace arrangements and combinatorics of discriminants, Habilitationsshrift, Essen, 1996.
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