Matroid Shellability, β-Systems,and Affine Hyperplane Arrangements

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 -system of a matroid, nbc(M), whose cardinality is Crapo's -invariant. In studying the shellability and homology of base-pointed matroids, geometric semilattices, and afflne hyperplane arrangements, it is found that the -system acts as the afflne counterpart to the broken-circuit complex. In particular, it is shown that the -system indexes the homology facets for the lexicographic shelling of the reduced broken-circuit complex , 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 -system labels its decreasing chains.Basic cycles can be carried over from 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.     

Quotient Complexes and Lexicographic Shellability

Let _n,k,k and _n,k,h , h < k, denote the intersection lattices of the k-equal subspace arrangement of type _n and the k,h-equal subspace arrangement of type _n respectively. Denote by the group of signed permutations. We show that ( _n,k,k )/ is collapsible. For ( _n,k,h )/ , h < k, we show the following. If n 0 (mod k), then it is homotopy equivalent to a sphere of dimension . If n h (mod k), then it is homotopy equivalent to a sphere of dimension . Otherwise, it is contractible. Immediate consequences for the multiplicity of the trivial characters in the representations of on the homology groups of ( _n,k,k ) and ( _n,k,h ) are stated.The collapsibility of ( _n,k,k )/ is established using a discrete Morse function. The same method is used to show that ( _n,k,h )/ , h < k, is homotopy equivalent to a certain subcomplex. The homotopy type of this subcomplex is calculated by showing that it is shellable. To do this, we are led to introduce a lexicographic shelling condition for balanced cell complexes of boolean type. This extends to the non-pure case work of P. Hersh (Preprint, 2001) and specializes to the CL-shellability of A. Björner and M. Wachs (Trans. Amer. Math. Soc. 4 (1996), 1299–1327) when the cell complex is an order complex of a poset.     

Spectral Sequences on Combinatorial Simplicial Complexes

The goal of this paper is twofold. First, we give an elementary introduction to the usage of spectral sequences in the combinatorial setting. Second we list a number of applications.In the first group of applications the simplicial complex is the nerve of a poset; we consider general posets and lattices, as well as partition-type posets. Our last application is of a different nature: the -quotient of the complex of directed forests is a simplicial complex whose cell structure is defined combinatorially.     

Simplicial shellable spheres via combinatorial blowups

The construction of the Bier sphere for a simplicial complex is due to Bier (1992). Björner, Paffenholz, Sjöstrand and Ziegler (2005) generalize this construction to obtain a Bier poset from any bounded poset and any proper ideal . They show shellability of for the case , the boolean lattice, and thereby obtain `many shellable spheres' in the sense of Kalai (1988).

We put the Bier construction into the general framework of the theory of nested set complexes of Feichtner and Kozlov (2004). We obtain `more shellable spheres' by proving the general statement that combinatorial blowups, hence stellar subdivisions, preserve shellability.

     

On the poset of conjugacy classes of subgroups of π-power index

We investigate tile poset S^π(G)/G of conjugacy clases of subgroups of π-power index in a finite group G. In particular, we are concerned with combinatorial and topological properties of the order complex of S^π(G)/G. We show that the order complex of S^π(G)/G iS homotopic to a join of orbit spaces of order complexes of posets, which bear structural information on the cheif factors of the group. Moreover, for π-solvable groups and in case π = {p} we reveal a shellable subposer of S^π(G)/G of the same homotopy type. This complements the study of the poset S^π(G) of subgroups of π-power index performed in [20]. For the analysis of the order complexes we develop some new lemmata on the topology of order complexes of posets and in the theory of shellability.     

Zonotopal Subdivisions of Cyclic Zonotopes

The cyclic zonotope (n, d) is the zonotope in ^d generated by any n distinct vectors of the form (1, t, t ²,..., t ^d–1). It is proved that the refinement poset of all proper zonotopal subdivisions of (n, d) which are induced by the canonical projection : (n, d) (n, d), in the sense of Billera and Sturmfels, is homotopy equivalent to a sphere and that any zonotopal subdivision of (n, d) is shellable. The first statement gives an affirmative answer to the generalized Baues problem in a new special case and refines a theorem of Sturmfels and Ziegler on the extension space of an alternating oriented matroid. An important ingredient in the proofs is the fact that all zonotopal subdivisions of (n, d) are stackable in a suitable direction. It is shown that, in general, a zonotopal subdivision is stackable in a given direction if and only if a certain associated oriented matroid program is Euclidean, in the sense of Edmonds and Mandel.     

Shellability of noncrossing partition lattices

We give a case-free proof that the lattice of noncrossing partitions associated to any finite real reflection group is EL-shellable. Shellability of these lattices was open for the groups of type and those of exceptional type and rank at least three.

     

Deformation and Cohen-Macaulayness of the multicone over the flag variety

A general theory of LS algebras over a multiposet is developed. As a main result, the existence of a flat deformation to discrete algebras is obtained. This is applied to the multicone over partial flag varieties for Kac-Moody groups proving a deformation theorem to a union of toric varieties. In order to achieve the Cohen-Macaulayness of the multicone we show that Bruhat posets (defined as glueing of minimal representatives modulo parabolic subgroups of a Weyl group) are lexicographically shellable. Received: June 23, 2000     

Iterated Homology of Simplicial Complexes

We develop an iterated homology theory for simplicial complexes. Thistheory is a variation on one due to Kalai. For a simplicial complex of dimension d – 1, and each r = 0, ...,d, we define rth iterated homology groups of . When r = 0, this corresponds to ordinary homology. If is a cone over , then when r = 1, we get the homology of . If a simplicial complex is (nonpure) shellable, then its iterated Betti numbers give the restriction numbers, h _k,j , of the shelling. Iterated Betti numbers are preserved by algebraic shifting, and may be interpreted combinatorially in terms of the algebraically shifted complex in several ways. In addition, the depth of a simplicial complex can be characterized in terms of its iterated Betti numbers.     

Two Decompositions in Topological Combinatorics with Applications to Matroid Complexes

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 M-shellability, which is a generalization to pure posets of the property of shellability of simplicial complexes, and derive inequalities that the rank-numbers of M-shellable posets must satisfy. We also introduce a decomposition property for simplicial complexes called a convex ear-decomposition, and, using results of Kalai and Stanley on -vectors of simplicial polytopes, we show that -vectors of pure rank- simplicial complexes that have this property satisfy and for . 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 ear-decomposition called a PS ear-decomposition. This enables us to construct an associated M-shellable poset, whose set of rank-numbers is the -vector of the matroid complex. This results in a combinatorial proof of a conjecture of Hibi that the -vector of a matroid complex satisfies the above two sets of inequalities.