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International Mathematics Research Notices (2008) Vol. 2008 : article ID rnm167, 39 pages, doi:10.1093/imrn/rnm167 published on February 8, 2008
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© The Author 2008. Published by Oxford University Press. All rights reserved. For permissions, please e-mail: journals.permissions@oxfordjournals.org

Shelling-Type Orderings of Regular CW-Complexes and Acyclic Matchings of the Salvetti Complex

Emanuele Delucchi

The Mathematical Sciences Research Institute, 17 Gauss Way, Berkeley, CA 94720-5070, USA

Correspondence: Correspondence to be sent to: delucchi{at}math.binghamton.edu

Motivated by the work of Salvetti and Settepanella [24, Combinatorial Morse theory and minimality of hyperplane arrangements, Remark 4.5], we give a purely combinatorial description of a class of discrete Morse functions having a minimal number of critical cells for the Salvetti complex of any linear arrangement. We start by studying certain total orderings of the cells of shellable regular CW-complexes, and use them to construct maximum acyclic matchings of the given complex. We apply this technique to the classical zonotope shellings. A new combinatorial stratification of the Salvetti complex allows us to paste such matchings and describe a class of maximum acyclic matchings of the whole complex. The construction can be done, so that the critical cells can be constructed from the chambers via the nbc sets. The results hold for abstract oriented matroids.



References

  1. Babson E., Hersh P. Discrete Morse functions from lexicographic orders. Transactions of the American Mathematical Society (2005) 357(2):509–34.[CrossRef][Web of Science]
  2. Baclawski K. Whitney numbers of geometric lattices. Advances in Mathematics (1975) 16:125–38.[CrossRef][Web of Science]
  3. Björner A. Posets, regular CW complexes and Bruhat order. European Journal of Combinatorics (1984) 5(1):7–16.
  4. Björner A. Shellable and Cohen–Macaulay partially ordered sets. Transactions of the American Mathematical Society (1980) 260(1):159–83.[CrossRef][Web of Science]
  5. Björner A. On the homology of geometric lattices. Algebra Universalis (1982) 14(1):107–28.[CrossRef][Web of Science]
  6. Björner A., Vergnas M. L., Sturmfels B., White N., Ziegler G. M. Oriented Matroids (1999) 2nd ed. Cambridge: Cambridge University Press. Encyclopedia of Mathematics and its Applications 46.
  7. Björner A., Wachs M. L. Shellable nonpure complexes and posets 1. Transactions of the American Mathematical Society (1996) 348(4):1299–1327.[CrossRef][Web of Science]
  8. Björner A., Wachs M. L. Shellable nonpure complexes and posets 2. Transactions of the American Mathematical Society (1997) 349(10):3945–75.[CrossRef][Web of Science]
  9. Bruggesser H., Mani P. Shellable decompositions of cells and spheres. Mathematica Scandinavica (1971) 29:197–205.[Web of Science]
  10. Brylawski T. The broken-circuit complex. Transactions of the American Mathematical Society (1977) 234(2):417–33.[CrossRef][Web of Science]
  11. Chari K. M. On discrete Morse functions and combinatorial decompositions. Discrete Mathematics (2000) 217(1–3):101–113.[CrossRef][Web of Science]
  12. Dimca A., Papadima S. Hypersurface complements, Milnor fibers and higher homotopy groups of arrangments. Annals of Mathematics (2003) 158(2):473–507.[Web of Science]
  13. Edelman H. P. A partial order on the regions of Rn dissected by hyperplanes. Transactions of the American Mathematical Society (1984) 283(2):617–31.[CrossRef][Web of Science]
  14. Edmonds J., Mandel A. Topology of oriented matroids. (1982) PhD thesis of A. Mandel, University of Waterloo.
  15. Forman R. Morse theory for cell complexes. Advances in Mathematics (1998) 134(1):90–145.[CrossRef][Web of Science]
  16. Jambu M., Leborgne D. Fonction de Möbius et arrangements d'hyperplans. Comptes Rendus Mathematique Academie des Sciences (1986) 303(7):311–14.
  17. Jewell K., Orlik P. Geometric relationship between cohomology of the complement of real and complexified arrangements. Topology and its Applications (2002) 118(1–2):113–29.[CrossRef][Web of Science]
  18. Kozlov N. D. Discrete Morse theory for free chain complexes. Comptes Rendus Mathematique Academie des Sciences (2005) 340(12):867–72.
  19. Kozlov N. D. Combinatorial algebraic topology. Algorithms and Computation in Mathematics (2007) 21. Berlin: Springer.
  20. Orlik P., Solomon L. Combinatorics and topology of complements of hyperplanes. Inventiones Mathematicae (1980) 56(2):167–89.[CrossRef][Web of Science]
  21. Orlik P., Terao H. Arrangements of hyperplanes. Grundlehren der Mathematischen Wissenschaften[Fundamental Principles of Mathematical Sciences] (1992) 300. Berlin: Springer.
  22. Randell R. Morse theory, Milnor fibers and minimality of hyperplane arrangements. Proceedings of the American Mathematical Society (2002) 130(9):2737–43.[CrossRef][Web of Science]
  23. Salvetti M. Topology of the complement of real hyperplanes in CN. Inventiones Mathematicae (1987) 88(3):603–18.[CrossRef][Web of Science]
  24. Salvetti M., Settepanella S. Combinatorial Morse theory and minimality of hyperplane arrangements. Geometry and Topology (2007) 11:1733–66.[CrossRef]
  25. Stanley P. R. Enumerative combinatorics (1997) 1. Cambridge: Cambridge University Press. Cambridge Studies in Advanced Mathematics 49.
  26. Yoshinaga M. Hyperplane arrangements and Lefschetz's hyperplane section theorem. Kodai Mathematical Journal (2007) 30(2):157–94.[CrossRef]
  27. Yuzvinsky S. Orlik–Solomon algebras in algebra and topology. Russian Mathematical Surveys (2001) 56(2):293–364.[CrossRef][Web of Science]

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This Article
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