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43
Shellable nonpure complexes and posets. I
 TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY
, 1996
"... The concept of shellability of complexes is generalized by deleting the requirement of purity (i.e., that all maximal faces have the same dimension). The usefulness of this level of generality was suggested by certain examples coming from the theory of subspace arrangements. We develop several of ..."
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Cited by 130 (8 self)
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The concept of shellability of complexes is generalized by deleting the requirement of purity (i.e., that all maximal faces have the same dimension). The usefulness of this level of generality was suggested by certain examples coming from the theory of subspace arrangements. We develop several of the basic properties of the concept of nonpure shellability. Doubly indexed fvectors and hvectors are introduced, and the latter are shown to be nonnegative in the shellable case. Shellable complexes have the homotopy type of a wedge of spheres of various dimensions, and their StanleyReisner rings admit a combinatorially induced direct sum decomposition. The technique of lexicographic shellability for posets is similarly extended from pure posets (all maximal chains of the same length) to the general case. Several examples of nonpure lexicographically shellable posets are given, such as the kequal partition lattice (the intersection lattice of the kequal subspace arrangement) and the Tamari lattices of binary trees. This leads to simplified computation of Betti numbers for the kequal arrangement. It also determines the homotopy type of intervals in a Tamari lattice and in the lattice of number partitions ordered by dominance, thus strengthening some known Möbius function formulas. The extension to regular CW complexes is briefly discussed and shown to be related to the concept of lexicographic shellability.
Hyperplane arrangement cohomology and monomials in the exterior algebra
, 2000
"... Abstract. We show that if X is the complement of a complex hyperplane arrangement, then the homology of X has linear free resolution as a module over the exterior algebra on the first cohomology of X. We study invariants of X that can be deduced from this resolution. A key ingredient is a result of ..."
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Cited by 22 (4 self)
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Abstract. We show that if X is the complement of a complex hyperplane arrangement, then the homology of X has linear free resolution as a module over the exterior algebra on the first cohomology of X. We study invariants of X that can be deduced from this resolution. A key ingredient is a result of Aramova, Avramov, and Herzog (2000) on resolutions of monomial ideals in the exterior algebra. We give a new conceptual proof of this result. Let X be the complement of a complex hyperplane arrangement A. Inthispaper we study the singular homology H∗(X) as a module over the exterior algebra E on the first singular cohomology V: = H1 (X) always with coefficients in a fixed field K. Our first main result (Section 1) asserts that H∗(X) is generated in a single degree and has a linear free resolution; this amounts to an infinite sequence of statements about the multiplication in the OrlikSolomon algebra H ∗ (X). We also analyze other topological examples from the point of view of resolutions over the exterior algebra. In Section 2 we study an invariant of an Emodule N called the singular variety,
Two decompositions in topological combinatorics with applications to matroid complexes
 Trans. Amer. Math. Soc
, 1997
"... Abstract. This paper introduces two new decomposition techniques which are related to the classical notion of shellability of simplicial complexes, and uses the existence of these decompositions to deduce certain numerical properties for an associated enumerative invariant. First, we introduce the n ..."
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Cited by 21 (1 self)
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Abstract. This paper introduces two new decomposition techniques which are related to the classical notion of shellability of simplicial complexes, and uses the existence of these decompositions to deduce certain numerical properties for an associated enumerative invariant. First, we introduce the notion of Mshellability, which is a generalization to pure posets of the property of shellability of simplicial complexes, and derive inequalities that the ranknumbers of Mshellable posets must satisfy. We also introduce a decomposition property for simplicial complexes called a convex eardecomposition, and, using results of Kalai and Stanley on hvectors of simplicial polytopes, we show that hvectors of pure rankd simplicial complexes that have this property satisfy h0 ≤ h1 ≤ ·· · ≤ h [d/2] and hi ≤ hd−i for 0 ≤ i ≤ [d/2]. We then show that the abstract simplicial complex formed by the collection of independent sets of a matroid (or matroid complex) admits a special type of convex eardecomposition called a PS eardecomposition. This enables us to construct an associated Mshellable poset, whose set of ranknumbers is the hvector of the matroid complex. This results in a combinatorial proof of a conjecture of Hibi [17] that the hvector of a matroid complex satisfies the above two sets of inequalities. 1.
Decompositions of Simplicial Balls and Spheres With Knots Consisting of Few Edges
, 1999
"... Constructibility is a condition on pure simplicial complexes that is weaker than shellability. In this paper we show that nonconstructible triangulations of the ddimensional sphere exist for every d 3. This answers a question of Danaraj & Klee [8]; it also strengthens a result of Lickorish [1 ..."
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Cited by 17 (5 self)
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Constructibility is a condition on pure simplicial complexes that is weaker than shellability. In this paper we show that nonconstructible triangulations of the ddimensional sphere exist for every d 3. This answers a question of Danaraj & Klee [8]; it also strengthens a result of Lickorish [13] about nonshellable spheres. Furthermore, we provide a hierarchy of combinatorial decomposition properties that follow from the existence of a nontrivial knot with "few edges" in a 3sphere or 3ball, and a similar hierarchy for 3balls with a knotted spanning arc that consists of "few edges."
Poset fiber theorems
 TRANS. AMER. MATH. SOC
, 2004
"... Suppose that f: P → Q is a poset map whose fibers f −1 (Q≤q) are sufficiently well connected. Our main result is a formula expressing the homotopy type of P in terms of Q and the fibers. Several fiber theorems from the literature (due to Babson, Baclawski and Quillen) are obtained as consequences o ..."
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Cited by 12 (2 self)
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Suppose that f: P → Q is a poset map whose fibers f −1 (Q≤q) are sufficiently well connected. Our main result is a formula expressing the homotopy type of P in terms of Q and the fibers. Several fiber theorems from the literature (due to Babson, Baclawski and Quillen) are obtained as consequences or special cases. Homology, CohenMacaulay, and equivariant versions are given, and some applications are discussed.
Small examples of nonconstructible simplicial balls and spheres
 SIAM J. Discrete Math
, 2004
"... We construct nonconstructible simplicial dspheres with d + 10 vertices and nonconstructible, nonrealizable simplicial dballs with d + 9 vertices for d≥3. 1 ..."
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Cited by 11 (5 self)
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We construct nonconstructible simplicial dspheres with d + 10 vertices and nonconstructible, nonrealizable simplicial dballs with d + 9 vertices for d≥3. 1
Polygon dissections and some generalizations of cluster complexes
, 2005
"... Abstract. Let W be a Weyl group corresponding to the root system An−1 or Bn. We define a simplicial complex ∆m W in terms of polygon dissections for such a group and any positive integer m. For m = 1, ∆m W is isomorphic to the cluster complex corresponding to W, defined in [8]. We enumerate the face ..."
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Cited by 11 (3 self)
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Abstract. Let W be a Weyl group corresponding to the root system An−1 or Bn. We define a simplicial complex ∆m W in terms of polygon dissections for such a group and any positive integer m. For m = 1, ∆m W is isomorphic to the cluster complex corresponding to W, defined in [8]. We enumerate the faces of ∆m W and show that the entries of its hvector are given by the generalized Narayana numbers Nm W (i), defined in [3]. We also prove that for any m≥1 the complex ∆m W is shellable and hence CohenMacaulay. 1. Introduction and
Nonconstructible complexes and the bridge index
, 1999
"... We show that if a 3dimensional polytopal complex has a knot in its 1skeleton, where the bridge index of the knot is larger than the number of edges of the knot, then the complex is not constructible, and hence, not shellable. As an application we settle a conjecture of Hetyei concerning the shella ..."
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Cited by 10 (2 self)
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We show that if a 3dimensional polytopal complex has a knot in its 1skeleton, where the bridge index of the knot is larger than the number of edges of the knot, then the complex is not constructible, and hence, not shellable. As an application we settle a conjecture of Hetyei concerning the shellability of cubical barycentric subdivisions of 3spheres. We also obtain similar bounds concluding that a 3sphere or 3ball is nonshellable or not vertex decomposable. These two last bounds are sharp.
Signable Posets and Partitionable Simplicial Complexes
, 1996
"... The notion of a partitionable simplicial complex is extended to that of a signable partially ordered set. It is shown in a unified way that face lattices of shellable polytopal complexes, polyhedral cone fans, and oriented matroid polytopes, are all signable. Each of these classes, which are belie ..."
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Cited by 9 (5 self)
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The notion of a partitionable simplicial complex is extended to that of a signable partially ordered set. It is shown in a unified way that face lattices of shellable polytopal complexes, polyhedral cone fans, and oriented matroid polytopes, are all signable. Each of these classes, which are believed to be mutually incomparable, strictly contains the class of convex polytopes. A general sufficient condition, termed total signability, for a simplicial complex to satisfy McMullen's Upper Bound Theorem on the numbers of faces, is provided. The simplicial members of each of the three classes above are concluded to be partitionable and to satisfy the upper bound theorem. The computational complexity of face enumeration and of deciding partitionability is discussed. It is shown that under a suitable presentation, the face numbers of a signable simplicial complex can be efficiently computed. In particular, the face numbers of simplicial fans can be computed in polynomial time, extending the analogous statement for convex polytopes.
Vertex decomposable graphs and obstructions to shellability
, 2009
"... Inspired by several recent papers on the edge ideal of a graph G, we study the equivalent notion of the independence complex of G. Using the tool of vertex decomposability from geometric combinatorics, we show that 5chordal graphs with no chordless 4cycles are shellable and sequentially CohenMac ..."
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Cited by 9 (1 self)
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Inspired by several recent papers on the edge ideal of a graph G, we study the equivalent notion of the independence complex of G. Using the tool of vertex decomposability from geometric combinatorics, we show that 5chordal graphs with no chordless 4cycles are shellable and sequentially CohenMacaulay. We use this result to characterize the obstructions to shellability in flag complexes, extending work of Billera, Myers, and Wachs. We also show how vertex decomposability may be used to show that certain graph constructions preserve shellability.