Obstructions to shellability, partitionability, and sequential Cohen-Macaulayness

For a property P of simplicial complexes, a simplicial complex Γ is an obstruction to P if Γ itself does not satisfy P but all of its proper restrictions satisfy P. In this paper, we determine all obstructions to shellability of dimension ?2, refining the previous work by Wachs. As a consequence we obtain that the set of obstructions to shellability, that to partitionability and that to sequential Cohen-Macaulayness all coincide for dimensions ?2. We also show that these three sets of obstructions coincide in the class of flag complexes. These results show that the three properties, hereditary-shellability, hereditary-partitionability, and hereditary-sequential Cohen-Macaulayness are equivalent for these classes.     

Shellability and homology of q-complexes and q-matroids

Journal of Algebraic Combinatorics - We consider a q-analogue of abstract simplicial complexes, called q-complexes, and discuss the notion of shellability for such complexes. It is shown that...     

Nonconstructible Simplicial Balls and a Way of Testing Constructibility

Constructibility of simplicial complexes is a notion weaker than shellability. It is known that shellable pseudomanifolds are homeomorphic to balls or spheres but simplicial complexes homeomorphic to balls or spheres need not be shellable in general. Constructible pseudomanifolds are also homeomorphic to balls or spheres, but the existence of nonconstructible balls was not known. In this paper we study the constructibility of some nonshellable balls and show that some of them are not constructible, either. Moreover, we give a necessary and sufficient condition for the constructibility of three-dimensional simplicial balls, whose vertices are all on the boundary. Received September 29, 1997, and in revised form August 3, 1998 and August 11, 1998.     

On the Topology of the Combinatorial Flag Varieties

We prove that the simplicial complex Ω _n of chains of matroids (with respect to the ordering by the quotient relation) on n elements is shellable. This follows from a more general result on shellability of the simplicial complex of W -matroids for an arbitrary finite Coxeter group W , and generalises the well-known results by Solomon—Tits and Bj?rner on spherical buildings. Received January 16, 2000, and in revised form October 7, 2000, and April 16, 2001. Online publication December 21, 2001.     

Decompositions of simplicial balls and spheres with knots consisting of few edges

Constructibility is a condition on pure simplicial complexes that is weaker than shellability. In this paper we show that non-constructible triangulations of the d-dimensional sphere exist for every . This answers a question of Danaraj and Klee [10]; it also strengthens a result of Lickorish [16] about non-shellable spheres. Furthermore, we provide a hierarchy of combinatorial decomposition properties that follow from the existence of a non-trivial knot with “few edges” in a 3-sphere or 3-ball, and a similar hierarchy for 3-balls with a knotted spanning arc that consists of “few edges.” Received March 15, 1999 / in final form August 19, 1999 / Published online July 3, 2000     

A note on: Quasi-linear quotients and shellability of pure simplicial complexes

The purpose of this note is to point out a careless error in the algebraic criterion of shellability of a pure simplicial complex Δ given in [1 Anwar, I., Raza, Z. (2015). Quasi-linear quotients and shellability of pure simplicial complexes. Commun. Algebra 43:4698–4704.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]].     

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.

     

Resolutions of Stanley-Reisner rings and Alexander duality

Associated to any simplicial complex Δ on n vertices is a square-free monomial ideal I_Δ in the polynomial ring A = k[x₁, …, x_n], and its quotient k[Δ] = A/I_Δ known as the Stanley-Reisner ring. This note considers a simplicial complex Δ^* which is in a sense a canonical Alexander dual to Δ, previously considered in [1, 5]. Using Alexander duality and a result of Hochster computing the Betti numbers dim_k Tor_i^A (k[Δ],k), it is shown (Proposition 1) that these Betti numbers are computable from the homology of links of faces in Δ^*. As corollaries, we prove that I_Δ has a linear resolution as A-module if and only if Δ^* is Cohen-Macaulay over k, and show how to compute the Betti numbers dim_k Tor_i^A (k[Δ],k) in some cases where Δ^* is wellbehaved (shellable, Cohen-Macaulay, or Buchsbaum). Some other applications of the notion of shellability are also discussed.     

Shelling Coxeter-like complexes and sorting on trees

In their work on ‘Coxeter-like complexes’, Babson and Reiner introduced a simplicial complex ΔT associated to each tree T on n nodes, generalizing chessboard complexes and type A Coxeter complexes. They conjectured that ΔT is (n−b−1)-connected when the tree has b leaves. We provide a shelling for the (n−b)-skeleton of ΔT, thereby proving this conjecture. In the process, we introduce notions of weak order and inversion functions on the labellings of a tree T which imply shellability of ΔT, and we construct such inversion functions for a large enough class of trees to deduce the aforementioned conjecture and also recover the shellability of chessboard complexes Mm,n with n?2m−1. We also prove that the existence or nonexistence of an inversion function for a fixed tree governs which networks with a tree structure admit greedy sorting algorithms by inversion elimination and provide an inversion function for trees where each vertex has capacity at least its degree minus one.     

Graphic matroids,shellability and the Poincare Conjecture

In this paper we introduce a theory of edge shelling of graphs. Whereas the standard notion of shelling a simplicial complex involves a sequential removal of maximal simplexes, edge shelling involves a sequential removal of the edges of a graph. A necessary and sufficient condition for edge shellability is given in the case of 3-colored graphs, and it is conjectured that the result holds in general. Questions about shelling, and the dual notion of closure, are motivated by topological problems. The connection between graph theory and topology is by way of a complex ΔG associated with a graph G. In particular, every closed 2- or 3-manifold can be realized in this way. If ΔG is shellable, then G is edge shellable, but not conversely. Nevertheless, the condition that G is edge shellable is strong enough to imply that a manifold ΔG must be a sphere. This leads to completely graph-theoretic generalizations of the classical Poincaré Conjecture.     

General lexicographic shellability and orbit arrangements

We introduce a new poset property which we call EC-shellability. It is more general than the more established concept of EL-shellability, but it still implies shellability. Because of Theorem 3.10, EC-shellability is entitled to be called general lexicographic shellability. As an application of our new concept, we prove that intersection lattices Π_λ of orbit arrangementsA _λ are EC-shellable for a very large class of partitions λ. This allows us to compute the topology of the link and the complement for these arrangements. In particular, for this class of λs, we are able to settle a conjecture of Björner [B94, Conjecture 13.3.2], stating that the cohomology groups of the complement of the orbit arrangements are torsion-free. We also present a class of partitions for which Π_λ is not shellable, along with other issues scattered throughout the paper.     

A simplicial approach to factorization systems and Kurosh-Amitsur radicals

Regarding categories as simplicial sets via the nerve functor, we extend the notion of a factorization system from morphisms in a category, to 1-simplexes in an arbitrary simplicial set. Applied to what we call the simplicial set of short exact sequences, it gives the notion of Kurosh-Amitsur radical. That is, we present a unified approach to factorization systems and radicals.     

Simplicial ramified covering maps

In this paper we define the concept of a ramified covering map in the category of simplicial sets and we show that it has properties analogous to those of the topological ramified covering maps. We show that the geometric realization of a simplicial ramified covering map is a topological ramified covering map, and we also consider the relation with ramified covering maps in the category of simplicial complexes.     

Simplicial ramified covering maps

In this paper we define the concept of a ramified covering map in the category of simplicial sets and we show that it has properties analogous to those of the topological ramified covering maps. We show that the geometric realization of a simplicial ramified covering map is a topological ramified covering map, and we also consider the relation with ramified covering maps in the category of simplicial complexes.     

Face vectors of simplicial cell decompositions of manifolds

In this paper, we study face vectors of simplicial posets that are the face posets of cell decompositions of topological manifolds without boundary. We characterize all possible face vectors of simplicial posets whose geometric realizations are homeomorphic to the product of spheres. As a corollary, we obtain the characterization of face vectors of simplicial posets whose geometric realizations are odd-dimensional manifolds without boundary.     

Deciding nonconstructibility of 3-balls with spanning edges and interior vertices

Constructibility is a combinatorial property of simplicial complexes. In general, it requires a great deal of time to decide whether a simplicial complex is constructible or not. In this paper, we consider sufficient conditions for nonconstructibility of simplicial 3-balls to investigate efficient algorithms for the decision problem.     

Note on binary simplicial matroids

Using an earlier characterization of simplicial hypergraphs we obtain a characterization of binary simplicial matroids in terms of the existence of a special base.     

Integral foliated simplicial volume of aspherical manifolds

Simplicial volumes measure the complexity of fundamental cycles of manifolds. In this article, we consider the relation between the simplicial volume and two of its variants — the stable integral simplicial volume and the integral foliated simplicial volume. The definition of the latter depends on a choice of a measure preserving action of the fundamental group on a probability space.