Matroid Shellability, β -Systems, and Affine Hyperplane Arrangements
Günter M. Ziegler
DOI: 10.1023/A:1022492019120
Abstract
The broken-circuit complex is fundamental to the shellability and homology of matroids, geometric lattices, and linear hyperplane arrangements. This paper introduces and studies the [ `( BC)] ( M) \overline {BC} (M) , and the basic cycles are explicitly constructed. Similarly, an EL-shelling for the geometric semilattice associated with M is produced,_and it is shown that the [ `( BC)] ( M) \overline {BC} (M) The intersection poset of any (real or complex) afflnehyperplane arrangement 
is a geometric semilattice. Thus the construction yields a set of basic cycles, indexed by 
nbc( M), for the union 
of such an arrangement.
is a geometric semilattice. Thus the construction yields a set of basic cycles, indexed by nbc( M), for the union of such an arrangement.Pages: 283–300
Keywords: matroid; $beta$-invariant; broken-circuit complex; shellability; affine hyperplane arrangement
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References
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2. A. Bjorner, "Shellable and Cohen-Macaulay partially ordered sets," Trans. Amer. Math. Soc. 260 (1980), 159-183.
3. A. Bjorner, "The homology and shellability of matroids and geometric lattices," in Matroid Applications, N. White, ed., Cambridge University Press, Cambridge, 1992, pp. 226-283.
4. A. Bjorner, "Topological methods," in Handbook of Combinatorics, R. Graham, M. Grotschel, and L. Lovasz, eds., North-Holland, Amsterdam, to appear.
5. A. Bjorner, M. Las Vergnas, B. Sturmfels, N. White, and G.M. Ziegler, Oriented Matroids, Encyclopedia of Mathematics, Cambridge University Press, Cambridge, 1992.
6. A. Bjorner, L. Lovasz, and A. Yao, "Linear decision trees: volume estimates and topological bounds," Rep. No. 5 (1991-1992), Institut Mittag-Leffler, Djursholm, 1991.
7. A. Bjorner and M.L. Wachs: "On lexicographically shellable posets," Trans. Amer. Math. Soc. 277 (1983), 323-341.
8. A. Bjorner and G.M. Ziegler, "Broken circuit complexes: factorizations and generalizations," J. Combin. Theory Ser. B 51 (1991), 96-126.
9. A. Bjorner and G.M. Ziegler, "Combinatorial stratification of complex arrangements," J. Amer. Math. Soc. S (1992), 105-149.
10. T. Brylawski, "A combinatorial model for series-parallel networks," Trans. Amer. Math. Soc. 154 (1971), 1-22.
11. T. Brylawski, "The broken-circuit complex," Trans. Amer. Math. Soc. 234 (1977), 417-433.
12. H. Crapo, "A higher invariant for matroids," J. Combin. Theory 2 (1967), 406-417.
13. B.H. Dayton and C.A. Weibel, "K-theory of hyperplanes," Trans. Amer. Math. Soc. 257 (1980), 119-141.
14. M. Goresky and R. MacPherson, Stratified Morse Theory, Series 3, Vol. 14 of Ergebnisse der Mathematik und ihrer Grenzgebiete, Springer, New York, 1988.
15. M. Laurent and M. Deza, "Bouquets of geometric lattices: some algebraic and topological aspects," Discrete Math. 75 (1989), 279-313. ZIEGLER
16. J.-P. Roudneff, "Cells with many facets in arrangements of hyperplanes," Discrete Math. 98 (1991), 185-191.
17. M.L. Wachs and J.W. Walker, "On geometric semilattices," Order 2 (1986), 367-385.
18. T. Zaslavsky, "Facing up to arrangements: face-count formulas for partitions of space by hyperplanes," Mem. Amer. Math. Soc. 1(154) (1975).
19. G.M. Ziegler, "Combinatorial models for subspace arrangements," Habilitationsschrift, Technical University of Berlin, Berlin, 1992.
20. G.M. Ziegler and R.T. Zivaljevic, "Homotopy types of subspace arrangements via diagrams of spaces," Rep. No. 10 (1991-1992), Institut Mittag-Leffler, Djursholm, 1991, Math. Annalen, to appear.