We study d-dimensional simplicial complexes that are PL embeddable in $$\mathbb{R}^{d+1}$$ . It is shown that such a complex must satisfy a certain homological condition. The existence of this obstruction allows us to provide a systematic approach to deriving upper bounds for the number of top-dimensional faces of such complexes, particularly in low dimensions.

A numerical characterization is given of the so-called h-triangles of sequentially Cohen-Macaulay simplicial complexes. This result characterizes the number of faces of various dimensions and codimensions in such a complex, generalizing the classical Macaulay-Stanley theorem to the nonpure case. Moreover, we characterize the possible Betti tables of componentwise linear ideals. A key tool in our investigation is a bijection between shifted multicomplexes of degree at most d and shifted pure (d-1)-dimensional simplicial complexes.

Let x(1) , x(2) , . . . , x(n) be real numbers summing to zero, and let P+ be the family of all subsets J subset of [n] : = {1, 2, . . . ,n} such that Sigma(j is an element of J) x(j) > 0. Subset families arising in this way are the objects of study here.We prove that the order complex of P+, viewed as a poset under set containment, triangulates a shellable ball whose f-vector does not depend on the choice of x, and whose h-polynomial is the classical Eulerian polynomial. Then we study various components of the flag f-vector of P+ and derive some inequalities satisfied by them.It has been conjectured by Manickam, Miklos and Singhi in 1986 that ((n - 1)(k - 1)) is a lower bound for the number of k-element subsets in P+, unless n/k is too small. We discuss some related results that arise from applying the order complex and flag f-vector point of view.Some remarks at the end include brief discussions of related extensions and questions. For instance, we mention positive sum set systems arising in matroids whose elements are weighted by real numbers.

This paper is concerned with lower bounds for the connectivity of graphs (one-dimensional skeleta) of triangulations of compact manifolds. We introduce a structural invariant b_M for simplicial d-manifolds M taking values in the range 0 <= b_M <= d-1. The main result is that b_M influences connectivity in the following way: The graph of a d-dimensional simplicial compact manifold M is (2d - b_M)-connected. The parameter b_M has the property that b_M = 0 if the complex M is flag. Hence, our result interpolates between Barnette's theorem (1982) that all d-manifold graphs are (d+1)-connected and Athanasiadis' theorem (2011) that flag d-manifold graphs are 2d-connected. The definition of b_M involves the concept of banner triangulations of manifolds, a generalization of flag triangulations.

The purpose of this paper is to establish analogues of the classical Lefschetz Section Theorem for smooth tropical varieties. More precisely, we prove tropical analogues of the Section Theorems of Lefschetz, Andreotti–Frankel, Bott–Milnor–Thom, Hamm–Lê and Kodaira–Spencer, and the vanishing theorems of Andreotti–Frankel and Akizuki–Kodaira–Nakano. We start the paper by resolving a conjecture of Mikhalkin and Ziegler (2008) concerning positive sum systems of geometric lattices, which generalizes earlier work of Rota, Folkman and Björner. This translates to a crucial index estimate for the stratified Morse data at critical points of the tropical variety. Tropical geometry is a relatively new field in mathematics, based on early work of Bergman [Ber71] and Bieri–Groves [BG84]. Figuratively speaking, it arises by attempting to do algebraic geometry over the tropical max-plus semiring T = ([−∞,∞),max,+). Since tropical varieties are, in essence, polyhedral spaces obtained as limits of complex algebraic varieties [Ber71, GKZ94, Vir84], tropical geometry naturally connects the fields of algebraic geometry and combinatorics. Since its origins in the seventies, tropical geometry has been developed extensively [Gat06, RGST05, Spe05, SS09]. It has been applied to classical algebraic geometry [Gub07, Kat09], enumerative algebraic geometry [KT02, Mik05, Mik06, Shu05], mirror symmetry [Gro11, KS01], integrable systems [AMS12], and to several branches of applied mathematics, such as signals processing, mathematical biology, control theory and optimization, theoretical computer science and mathematical physics, cf. [Gro95, NGVR12, Pin98]. Several classical results and theories in algebraic geometry have natural analogues in tropical geometry, such as Brill–Noether theory and the Riemann–Roch, Torelli and Bézout Theorems, compare [CDPR12, RGST05]. Further motivated by tropical intersection theory and its relation to classical intersection theory of algebraic varieties [Kat12, Mik06], we here want to consider tropical analogues of one of the most central results in algebraic intersection theory, the Lefschetz Section Theorem (or Lefschetz Hyperplane Theorem). We attempt to give an almost complete picture of the Lefschetz Section Theorem in tropical geometry, and give tropical analogues of many of the classical Lefschetz theorems (and associated vanishing theorems). Along the way, we build on and generalize significant results in the topological theory of geometric lattices and matroids of Rota, Folkman, Björner and others. The Tropical Lefschetz Section Theorems. The classical Lefschetz Section Theorem comes in many different guises. Intuitively, Lefschetz theorems relate the topology of a complex algebraic variety X to the topology of the intersection of X with a hyperplane H transversal to X (or, alternatively, to an ample divisor D of X). Theorem (The classical Lefschetz Section theorem, [Lef50, AF59]). Let X denote any smooth projective algebraic n-dimensional variety in CP, and let H denote a generic hyperplane in CP. Then the inclusion Date: January 28, 2014. 2010 Mathematics Subject Classification. todo.

The concept of Cohen-Macaulay complexes emerged in the mid-1970s and swiftly became the focal point of an attractive and richly connected new area of mathematics, at the crossroads of combinatoics, commutative algebra and topology. As the main architect of these developments, Richard Stanley has made fundamental contributions over many years. This paper contains some brief mathematical discussions related to the Cohen-Macaulay property, and some personal memories. The characterization of Gorenstein* and homotopy Gorenstein* complexes and the relevance in that connection of the Poincaré conjecture is discussed. Another topic is combinatorial aspects of a recent result on the homotopy Cohen-Macaulayness of certain subsets of geometric lattices, motivated by questions in tropical geometry.

The purpose of this paper is to establish analogues of the classical Lefschetz Section Theorem for smooth tropical varieties. More precisely, we prove tropical analogues of the section theorems of Lefschetz, Andreotti-Frankel, Bott-Milnor-Thom, Hamm-Lê and Kodaira-Spencer, and the vanishing theorems of Andreotti-Frankel and Akizuki-Kodaira-Nakano. We start the paper by resolving a conjecture of Mikhalkin and Ziegler (2008) concerning the homotopy types of certain filtrations of geometric lattices, generalizing several known properties of full geometric lattices. This translates to a crucial index estimate for the stratified Morse data at critical points of the tropical variety; it can also by itself be interpreted as a Lefschetz-type theorem for matroids.

Let \Delta be a finite building (or, more generally, a thick spherical and locally finite building). The chamber graph G(\Delta), whose edges are the pairs of adjacent chambers in \Delta, is known to be q-regular for a certain number q=q(\Delta). Our main result is that G(\Delta) is q-connected in the sense of graph theory. Similar results are proved for the chamber graphs of Coxeter complexes and for order complexes of geometric lattices.

Let (W, S) be a crystallographic Coxeter group (this includes all finite and affine Weyl groups), and let J subset of S. Let W-J denote the set of minimal coset representatives modulo the parabolic ...

A Markov chain is considered whose states are orderings of an underlying fixed tree and whose transitions are local "random-to-front" reorderings, driven by a probability distribution on subsets of the leaves. The eigenvalues of the transition matrix are determined using Brown's theory of random walk on semigroups.

The starting point is the known fact that some much-studied random walks on permutations, such as the Tsetlin library, arise from walks on real hyperplane arrangements. This paper explores similar walks on complex hyperplane arrangements. This is achieved by involving certain cell complexes naturally associated with the arrangement. In a particular case this leads to walks on libraries with several shelves. We also show that interval greedoids give rise to random walks belonging to the same general family. Members of this family of Markov chains, based on certain semigroups, have the property that all eigenvalues of the transition matrices are non-negative real and given by a simple combinatorial formula. Background material needed for understanding the walks is reviewed in rather great detail.

We introduce (weighted) Segre and Rees products for posets and show that these constructions preserve the Cohen-Macaulay property over a field $k$ and homotopically. As an application we show that the weighted Segre product of two affine semigroup rings that are Koszul is again Koszul. This result generalizes previous results by Crona on weighted Segre products of polynomial rings. We also give a new proof of the fact that the Rees ring of a Koszul affine semigroup ring is again Koszul. The paper ends with a list of some open problems in the area.

The blocker A * of an antichain A in a finite poset P is the set of elements minimal with the property of having with each member of A a common predecessor. The following is done: (1) The posets P for which A** = A for all antichains are characterized. (2) The blocker A * of a symmetric antichain in the partition lattice is characterized. (3) Connections with the question of finding minimal size blocking sets for certain set families are discussed.

It is shown that the neighborhood complexes of a family ofvertex critical subgraphs of Kneser graphs—the stable Knesergraphs introduced by L. Schrijver—are spheres up to homotopy.Furthermore, it is shown that the neighborhood complexes of asubclass of the stable Kneser graphs contain the boundaries ofassociahedra (simplicial complexes encoding triangulations of apolygon) as a strong deformation retract.

In this paper we study combinatorial and topological properties of the intersection lattices of these subspace arrangements. Expressions for their Möbius functions and characteristic polynomials are derived. Lexicographic shellability is established in the case of , 1 ≤ < , which allows computation of the homology of its intersection lattice and the cohomology groups of the manifold =ℝ\∪. For instance, it is shown that (M) is torsion-free and is nonzero if and only if for some , 0 ≤ ≤[]. Torsion-free cohomology follows also for the complement in ℂ of the complexification , 1 ≤ < .

A multicomplexM is a collection of monomials closed under divisibility. For suchM we construct a cell complex ΔM whosei-dimensional cells are in bijection with thef i monomials ofM of degreei+1. The bijection is such that the inclusion relation of cells corresponds to divisibility of monomials. We then study relations between the numbersf i and the Betti numbers of ΔM. For squarefree monomials the construction specializes to the standard geometric realization of a simplicial complex.

To decide whether two permutations are comparable in Bruhat order of $S_n$ with the well-known tableau criterion requires $\binom{n}{2}$ comparisons of entries in certain sorted arrays. We show that to decide whether $x\le y$ only $d_1+d_2+...+d_k$ of these comparisons are needed, where $\{d_1,d_2,...,d_k\} = \{i|x(i)>x(i+1)\}$. This is obtained as a consequence of a sharper version of Deodhar's criterion, which is valid for all Coxeter groups.