TOPOLOGICAL PROPERTIES OF GRAPHICAL ARRANGEMENTS

• Journal title : Honam Mathematical Journal
• Volume 36, Issue 2,  2014, pp.435-454
• Publisher : The Honam Mathematical Society
• DOI : 10.5831/HMJ.2014.36.2.435
Title & Authors
TOPOLOGICAL PROPERTIES OF GRAPHICAL ARRANGEMENTS
Nguyen, Thi A.; Kim, Sangwook;

Abstract
We show that for any graph G, the proper part of the intersection poset of the corresponding graphical arrangement $\small{\mathcal{A}_G}$ 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.
Keywords
graphical arrangement;EL-shallablity;
Language
English
Cited by
References
1.
Anders Bjorner, Shellable and Cohen-Macaulay partially ordered sets, Trans. Amer. Math. Soc., 260(1) (1980), 159-183.

2.
Anders Bjorner and Michelle L. Wachs, Shellable nonpure complexes and posets. I, Trans. Amer. Math. Soc., 348(4) (1996), 1299-1327.

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.

4.
Peter Orlik and Louis Solomon, Combinatorics and topology of complements of hyperplanes, Invent. Math., 56(2) (1980), 167-189.

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.

7.
Richard P. Stanley, Finite lattices and Jordan-Holder sets, Alg. Univ., 4 (1974), 361-371.

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.

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.