Honam Mathematical Journal (호남수학학술지)

The Honam Mathematical Society (호남수학회)
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TOPOLOGICAL PROPERTIES OF GRAPHICAL ARRANGEMENTS

  • Nguyen, Thi A. (Department of Mathematics, Chonnam National University) ;
  • Kim, Sangwook (Department of Mathematics, Chonnam National University)
  • Received : 2014.04.03
  • Accepted : 2014.05.28
  • Published : 2014.06.25
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Abstract

We show that for any graph G, the proper part of the intersection poset of the corresponding graphical arrangement $\mathcal{A}_G$ has the homotopy type of a wedge of spheres. Furthermore, we also indicate the number of spheres in the wedge, based on the number of spanning forests of G and other graphs that are obtained from G.

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References

  1. Anders Bjorner, Shellable and Cohen-Macaulay partially ordered sets, Trans. Amer. Math. Soc., 260(1) (1980), 159-183. https://doi.org/10.1090/S0002-9947-1980-0570784-2
  2. Anders Bjorner and Michelle L. Wachs, Shellable nonpure complexes and posets. I, Trans. Amer. Math. Soc., 348(4) (1996), 1299-1327. https://doi.org/10.1090/S0002-9947-96-01534-6
  3. Curtis Greene and Thomas Zaslavsky, On the interpretation of Whitney numbers through arrangements of hyperplanes, zonotopes, non-Radon partitions, and orientations of graphs, Trans. Amer. Math. Soc., 280(1) (1983), 97-126. https://doi.org/10.1090/S0002-9947-1983-0712251-1
  4. Peter Orlik and Louis Solomon, Combinatorics and topology of complements of hyperplanes, Invent. Math., 56(2) (1980), 167-189. https://doi.org/10.1007/BF01392549
  5. Peter Orlik and Hiroaki Terao, Arrangements of hyperplanes, volume 300 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Springer-Verlag, Berlin, 1992.
  6. Richard P. Stanley, Supersolvable lattices, Algebra Universalis, 2 (1972), 197-217. https://doi.org/10.1007/BF02945028
  7. Richard P. Stanley, Finite lattices and Jordan-Holder sets, Alg. Univ., 4 (1974), 361-371. https://doi.org/10.1007/BF02485748
  8. Richard P. Stanley, Enumerative Combinatorics, vol 2, Cambridge Studies in Advanced Mathematics, 62, Cambridge University Press, 1999.
  9. Richard P. Stanley, An introduction to hyperplane arrangements, In Geometric combinatorics, volume 13 of IAS/Park City Math. Ser., pages 389-496. Amer. Math. Soc., Providence, RI, 2007.
  10. Thomas Zaslavsky, Facing up to arrangements: face-count formulas for partitions of space by hyperplanes, Mem. Amer. Math. Soc., 1(issue 1, 154):vii+102, 1975.
  11. Michelle L. Wachs, A basis for the homology of the d-divisible partition lattice, Adv. Math., 117(2) (1996), 294-318. https://doi.org/10.1006/aima.1996.0014
  12. Michelle L. Wachs, Poset topology: tools and applications, In Geometric combinatorics, volume 13 of IAS/Park City Math. Ser., pages 497-615. Amer. Math. Soc., Providence, RI, 2007.