Face vectors of flag complexes

Stellar theory for flag complexes

Refining a basic result of Alexander, we show that two flag simplicial complexes are piecewise linearly homeomorphic if and only if they can be connected by a sequence of flag complexes, each obtained from the previous one by either an edge subdivision or its inverse. For flag spheres we pose new conjectures on their combinatorial structure forced by their face numbers, analogous to the extremal examples in the upper and lower bound theorems for simplicial spheres. Furthermore, we show that our algorithm to test the conjectures searches through the entire space of flag PL spheres of any given …

Face Numbers of Certain Cohen-Macaulay Flag Complexes

We show that if a $d$-dimensional Cohen-Macaulay complex is, in a certain sense, sufficiently "close" to being balanced, then there is a $d$-dimensional balanced Cohen-Macaulay complex having the same $f$-vector. This in turn provides some partial evidence for a conjecture of Kalai on the $f$-vectors of Cohen-Macaulay flag complexes.

The EKR property for flag pure simplicial complexes without boundary

We prove that the family of facets of a pure simplicial complex of dimension up to three satisfies the Erdős-Ko-Rado property whenever it is flag and has no boundary ridges. We conjecture the same to be true in arbitrary dimension and give evidence for this conjecture. Our motivation is that complexes with these two properties include flag pseudo-manifolds and cluster complexes.

On face numbers of flag simplicial complexes

Denham, Suciu and Panov, Ray computed ranks of homotopy groups and Poincaré series of a moment-angle-complex $\mathcal Z(\mathcal K$) / Davis-Januzskiewicz space $DJ(\mathcal K)$ associated to a flag simplicial complex $\mathcal K$. In this note we revisit these results and interpret them as polynomial bounds on the face numbers of an arbitrary simplicial flag complex.

Extremal problems related to Betti numbers of flag complexes

We study the problem of maximizing Betti numbers of simplicial complexes. We prove an upper bound of 1.32^n for the sum of Betti numbers of any n-vertex flag complex and 1.25^n for the independence complex of a triangle-free graph. These findings imply upper bounds for the Betti numbers of various related classes of spaces, including the neighbourhood complex of a graph. We also make some related observations.

Flag complexes and homology

We prove several relations on the $f$-vectors and Betti numbers of flag complexes. For every flag complex $Δ$, we show that there exists a balanced complex with the same $f$-vector as $Δ$, and whose top-dimensional Betti number is at least that of $Δ$, thereby extending a theorem of Frohmader by additionally taking homology into consideration. We obtain upper bounds on the top-dimensional Betti number of $Δ$ in terms of its face numbers. We also give a quantitative refinement of a theorem of Meshulam by establishing lower bounds on the $f$-vector of $Δ$, in terms of the top-dimensional Betti n…

On the h-vectors of Cohen-Macaulay Flag Complexes

Starting from an unpublished conjecture of Kalai and from a conjecture of Eisenbud, Green and Harris, we study several problems relating h-vectors of Cohen-Macaulay, flag simplicial complexes and face vectors of simplicial complexes.

Eigenvalues and homology of flag complexes and vector representations of graphs

Let X(G) denote the flag complex of a graph G=(V,E) on n vertices. We study relations between the first eigenvalues of successive higher Laplacians of X(G). One consequence is the following result: Let λ_2(G) denote the second smallest eigenvalue of the Laplacian of G. If λ_2(G)> \frac{kn}{k+1} then the real k-th reduced cohomology group H^k(X(G)) is zero. Applications include a lower bound on the homological connectivity of the independent sets complex I(G), in terms of a new graph domination parameter Γ(G) defined via certain vector representations of G. This in turns implies a Hall type the…

Cohen-Macaulay graphs and face vectors of flag complexes

We introduce a construction on a flag complex that, by means of modifying the associated graph, generates a new flag complex whose $h$-factor is the face vector of the original complex. This construction yields a vertex-decomposable, hence Cohen-Macaulay, complex. From this we get a (non-numerical) characterisation of the face vectors of flag complexes and deduce also that the face vector of a flag complex is the $h$-vector of some vertex-decomposable flag complex. We conjecture that the converse of the latter is true and prove this, by means of an explicit construction, for $h$-vectors of Coh…

Small flag complexes with torsion

We classify flag complexes on at most 12 vertices with torsion in the first homology group. The result is moderately computer-aided. As a consequence we confirm a folklore conjecture that the smallest poset whose order complex is homotopy equivalent to the real projective plane (and also the smallest poset with torsion in the first homology group) has exactly 13 elements.

Convex-Ear Decompositions and the Flag h-Vector

We prove a theorem allowing us to find convex-ear decompositions for rank-selected subposets of posets that are unions of Boolean sublattices in a coherent fashion. We then apply this theorem to geometric lattices and face posets of shellable complexes, obtaining new inequalities for their h-vectors. Finally, we use the latter decomposition to prove new inequalities for the flag h-vectors of face posets of Cohen-Macaulay complexes.

Generalized Macaulay representations and the flag $f$-vectors of generalized colored complexes

A colored complex of type $\mathbf{a} = (a_1, \dots, a_n)$ is a simplicial complex $Δ$ on a vertex set $V$, together with an ordered partition $(V_1, \dots, V_n)$ of $V$, such that every face $F$ of $Δ$ satisfies $|F \cap V_i| \leq a_i$. For each $\mathbf{b} = (b_1, \dots, b_n) \leq \mathbf{a}$, let $f_{\mathbf{b}}$ be the number of faces $F$ of $Δ$ such that $|F \cap V_i| = b_i$. The array of integers $\{f_{\mathbf{b}}\}_{\mathbf{b} \leq \mathbf{a}}$ is called the fine $f$-vector of $Δ$, and it is a refinement of the $f$-vector of $Δ$. In this paper, we generalize the notion of Macaulay repre…

On Equivariant flag $f$-vectors for balanced relative simplicial complexes

We study the equivariant flag $f$-vector and equivariant flag $h$-vector of a balanced relative simplicial complex with respect to a group action. When the complex satisfies Serre's condition $(S_{\ell}),$ we show that the equivariant flag $h$-vector, the equivariant $h$-vector, and the equivariant $f$-vector satisfy several inequalities. We apply these results to the study of $P$-partitions of double posets, and weak colorings of mixed graphs.

Decompositions of Augmented Bergman Complexes

We study the augmented Bergman complex of a closure operator on a finite set, which interpolates between the order complex of proper flats and the independence complex of the operator. In 2020, Braden, Huh, Matherne, Proudfoot, and Wang showed that augmented Bergman complexes of matroids are always gallery-connected, and recently Bullock, Kelley, Reiner, Ren, Shemy, Shen, Sun, Tao, and Zhang strengthened "gallery-connected" to "shellable" by providing two classes of shelling orders: "flag-to-basis" shellings and "basis-to-flag" shellings. We show that augmented Bergman complexes of matroids ar…

Reconstructible graphs, simplicial flag complexes of homology manifolds and associated right-angled Coxeter groups

In this paper, we investigate a relation between finite graphs, simplicial flag complexes and right-angled Coxeter groups, and we provide a class of reconstructible finite graphs. We show that if $Γ$ is a finite graph which is the 1-skeleton of some simplicial flag complex $L$ which is a homology manifold of dimension $n \ge 1$, then the graph $Γ$ is reconstructible.

Balanced complexes and complexes without large missing faces

The face numbers of simplicial complexes without missing faces of dimension larger than $i$ are studied. It is shown that among all such $(d-1)$-dimensional complexes with non-vanishing top homology, a certain polytopal sphere has the componentwise minimal $f$-vector; and moreover, among all such 2-Cohen--Macaulay (2-CM) complexes, the same sphere has the componentwise minimal $h$-vector. It is also verified that the $l$-skeleton of a flag $(d-1)$-dimensional 2-CM complex is $2(d-l)$-CM while the $l$-skeleton of a flag PL $(d-1)$-sphere is $2(d-l)$-homotopy CM. In addition, tight lower bounds …

Random Flag Complexes and Asymptotic Syzygies

We use the probabilistic method to construct examples of conjectured phenomenon about asymptotic syzygies. In particular, we use the Stanley-Reisner ideals of random flag complexes to construct new examples of Ein and Lazarsfeld's nonvanishing for asymptotic syzygies and of Ein, Erman, and Lazarsfeld's conjectural on the asymptotic normal distribution of Betti numbers.

On Betti numbers of flag complexes with forbidden induced subgraphs

We analyze the asymptotic extremal growth rate of the Betti numbers of clique complexes of graphs on n vertices not containing a fixed forbidden induced subgraph H. In particular, we prove a theorem of the alternative: for any H the growth rate achieves exactly one of five possible exponentials, that is, independent of the field of coefficients, the nth root of the maximal total Betti number over n-vertex graphs with no induced copy of H has a limit, as n tends to infinity, and, ranging over all H, exactly five different limits are attained. For the interesting case where H is the 4-cycle, the…

Simplicial Complexes are Game Complexes

Strong placement games (SP-games) are a class of combinatorial games whose structure allows one to describe the game via simplicial complexes. A natural question is whether well-known invariants of combinatorial games, such as "game value", appear as invariants of the simplicial complexes. This paper is the first step in that direction. We show that every simplicial complex encodes a certain type of SP-game (called an "invariant SP-game") whose ruleset is independent of the board it is played on. We also show that in the class of SP-games isomorphic simplicial complexes correspond to isomorphi…

The Hirsch conjecture holds for normal flag complexes

Using an intuition from metric geometry, we prove that any flag and normal simplicial complex satisfies the non-revisiting path conjecture. As a consequence, the diameter of its facet-ridge graph is smaller than the number of vertices minus the dimension, as in the Hirsch conjecture. This proves the Hirsch conjecture for all flag polytopes, and more generally, for all (connected) flag homology manifolds.

Regular Incidence Complexes, Polytopes, and C-Groups

Regular incidence complexes are combinatorial incidence structures generalizing regular convex polytopes, regular complex polytopes, various types of incidence geometries, and many other highly symmetric objects. The special case of abstract regular polytopes has been well-studied. The paper describes the combinatorial structure of a regular incidence complex in terms of a system of distinguished generating subgroups of its automorphism group or a flag-transitive subgroup. Then the groups admitting a flag-transitive action on an incidence complex are characterized as generalized string C-group…

Monomial ideals arising from flag complexes whose generic initial ideals do not depend on term orders

We will study monomial ideals $I$ in the exterior algebra as well as in the polynomial ring whose generic initial ideal is constant for all term orders up to permutations of variables. First, in the exterior algebra, we determine all graphs and all flag complexes whose exterior face ideal satisfies the above condition. Second, in the polynomial ring, it will be shown that the generic initial ideal $\gin_σ(I(G))$ of the edge ideal $I(G)$ of a graph $G$ is constant for all term orders $σ$ up to permutations of variables if and only if $G$ is a complete bipartite graph.

On Lusztig-Dupont homology of flag complexes

Let $V$ be an $n$-dimensional vector space over the finite field of order $q$. The spherical building $X_V$ associated with $GL(V)$ is the order complex of the nontrivial linear subspaces of $V$. Let $\mathfrak{g}$ be the local coefficient system on $X_V$, whose value on the simplex $σ=[V_0 \subset \cdots \subset V_p] \in X_V$ is given by $\mathfrak{g}(σ)=V_0$. Following the work of Lusztig and Dupont, we study the homology module $D^k(V)=\tilde{H}_{n-k-1}(X_V;\mathfrak{g})$. Our results include a construction of an explicit basis of $D^1(V)$, and the following twisted analogue of a result of …

Regular Polygonal Complexes in Space, II

Regular polygonal complexes in euclidean 3-space are discrete polyhedra-like structures with finite or infinite polygons as faces and with finite graphs as vertex-figures, such that their symmetry groups are transitive on the flags. The present paper and its predecessor describe a complete classification of regular polygonal complexes in 3-space. In Part I we established basic structural results for the symmetry groups, discussed operations on their generators, characterized the complexes with face mirrors as the 2-skeletons of the regular 4-apeirotopes in 3-space, and fully enumerated the sim…

How to construct a flag complex with a given face vector

A method that often works for constructing a flag complex with a specified face vector is given. This method can also be adapted to construct a vertex-decomposable (and hence Cohen-Macaulay) flag complex with a specified h-vector.

旗复形的第一至第七个Betti数的域独立性

Field independence of the first seven Betti numbers of flag complexes

本文证明旗复形第七个Betti数域独立，通过更强的拓扑结果τ(d)≥d+10，并推广到边理想的前七个Betti数。

随机旗复形同调的单侧锐阈值

One-sided sharp thresholds for homology of random flag complexes

本文研究随机旗复形同调与基本群的概率阈值，证明存在同调不消失概率介于零与一之间的中间区间，并给出基本群为自由群的锐阈值新界，同时构造出基本群既非自由群也不具有 Kazhdan 性质 (T) 的概率区间。

Cohen–Macaulay旗单纯复形的递归性质与来自f-向量的Lefschetz分解

Recursive properties of Cohen--Macaulay flag simplicial complexes and Lefschetz decompositions from $f$-vectors

本文研究旗单纯复形中f-向量的Lefschetz型Boolean分解，通过双重悬浮与净单边细分的复合映射给出递归构造，并指出平衡复形暗示了这些分解的代数版本。

Pontryagin algebras and the LS-category of moment-angle complexes in the flag case

For any flag simplicial complex $K$, we describe the multigraded Poincare series, the minimal number of relations and the degrees of these relations in the Pontryagin algebra of the corresponding moment-angle complex $Z_K$. We compute the LS-category of $Z_K$ for flag complexes and give a lower bound in the general case. The key observation is that the Milnor-Moore spectral sequence collapses at the second sheet for flag $K$. We also show that the results of Panov and Ray about the Pontryagin algebras of Davis-Januszkiewicz spaces are valid for arbitrary coefficient rings, and introduce the $(…

旗双重Cohen-Macaulay单纯复形的重复Lefschetz型分解及旗球面的gamma向量

Repeated Lefschetz-like decompositions for flag doubly Cohen--Macaulay simplicial complexes and gamma vectors of flag spheres

本文提出了旗双重Cohen-Macaulay单纯复形h多项式的重复Lefschetz型分解，通过迭代双重悬垂和边细分净非负集合，建立了旗球面gamma向量与其几何构造的直接联系，并利用布尔与非布尔区别衡量旗球面偏离交叉多胞体边界的程度。

旗复形面结构与分解

Flag complex face structures and decompositions

本文利用组合方法与切比雪夫展开研究了任意旗单纯球伽马向量的分量结构，揭示了其正性性质，并探讨了重心细分下复形 $f$-向量的分解与几何变换间的相互作用。

On $γ$-vectors satisfying the Kruskal-Katona inequalities

We present examples of flag homology spheres whose $γ$-vectors satisfy the Kruskal-Katona inequalities. This includes several families of well-studied simplicial complexes, including Coxeter complexes and the simplicial complexes dual to the associahedron and to the cyclohedron. In these cases, we construct explicit simplicial complexes whose $f$-vectors are the $γ$-vectors in question. In another direction, we show that if a flag $(d-1)$-sphere has at most $2d+2$ vertices its $γ$-vector satisfies the Kruskal-Katona inequalities. We conjecture that if $Δ$ is a flag homology sphere then $γ(Δ)$ …

The homotopy theory of polyhedral products associated with flag complexes

If $K$ is a simplicial complex on $m$ vertices the flagification of $K$ is the minimal flag complex $K^f$ on the same vertex set that contains $K$. Letting $L$ be the set of vertices, there is a sequence of simplicial inclusions $L\to K\to K^f$. This induces a sequence of maps of polyhedral products $(\underline X,\underline A)^L\stackrel g\longrightarrow(\underline X,\underline A)^K\stackrel f\longrightarrow (\underline X,\underline A)^{K^f}$. We show that $Ωf$ and $Ωf\circΩg$ have right homotopy inverses and draw consequences. For a flag complex $K$ the polyhedral product of the form $(\unde…

均衡单纯复形的$f$-向量、旗球面与几何Lefschetz分解

$f$-vectors of balanced simplicial complexes, flag spheres, and geometric Lefschetz decompositions

本文证明了均衡单纯复形的$f$-向量可展现类几何Lefschetz分解的布尔分解，基于Nevo--Petersen猜想将其与旗球面多项式的正定性及几何问题相联系，并指出分解次数未减半。

组合拉普拉斯算子的特征值界及其在随机复形上的应用

Eigenvalue bounds for combinatorial Laplacians and an application to random complexes

该研究通过矩阵论方法推导单纯复形组合拉普拉斯的新特征值界，拓展了旗复形与一般复形的相关结论，并将其应用于改进随机图邻域复形的上同调与连通性结果。

旗情形下循环同调的moment-angle复形

Loop homology of moment-angle complexes in the flag case

本文研究了旗情形下moment-angle复形的循环同调，通过同调方法描述了其生成元和关系，并证明了其在有理数上的coformal性质及同调群的计算方法。

Sharp vanishing thresholds for cohomology of random flag complexes

For every $k \ge 1$, the $k$th cohomology group $H^k(X, \Q)$ of the random flag complex $X \sim X(n,p)$ passes through two phase transitions: one where it appears, and one where it vanishes. We describe the vanishing threshold and show that it is sharp. Using the same spectral methods, we also find a sharp threshold for the fundamental group $π_1(X)$ to have Kazhdan's property (T). Combining with earlier results, we obtain as a corollary that for every $k \ge 3$ there is a regime in which the random flag complex is rationally homotopy equivalent to a bouquet of $k$-dimensional spheres.

Frankl-Füredi-Kalai Inequalities on the $γ$-vectors of flag nestohedra

For any flag nestohedron, we define a flag simplicial complex whose $f$-vector is the $γ$-vector of the nestohedron. This proves that the $γ$-vector of any flag nestohedron satisfies the Frankl-Füredi-Kalai inequalities, partially solving a conjecture by Nevo and Petersen \cite{np}. We also compare these complexes to those defined by Nevo and Petersen in \cite{np} for particular flag nestohedra.

旗单纯复形的简单同伦与图的可缩收缩

Simple homotopy of flag simplicial complexes and contractible contractions of graphs

本文证明了图的 $\mathfrak{I}$-同伦等价与团复形的简单同伦等价等价，并利用顶点链接刻画了任意单纯复形的简单同伦。

具有固定面数的纯单纯复形与团复形

Pure Simplicial and Clique Complexes with a Fixed Number of Facets

本文研究纯单纯复形与团复形的结构和计数问题，给出仅依赖面列表的团复形判定充要条件，证明一类纯团复形可由面邻接矩阵唯一确定，并计数固定面数的纯单纯复形及给出纯团复形数量的上界。

Face numbers of generalized balanced Cohen-Macaulay complexes

A common generalization of two theorems on the face numbers of Cohen-Macaulay (CM, for short) simplicial complexes is established: the first is the theorem of Stanley (necessity) and Bjorner-Frankl-Stanley (sufficiency) that characterizes all possible face numbers of a-balanced CM complexes, while the second is the theorem of Novik (necessity) and Browder (sufficiency) that characterizes the face numbers of CM subcomplexes of the join of the boundaries of simplices.

Convex Hull of Face Vectors of Colored Complexes

In this paper we verify a conjecture by Kozlov (Discrete Comput Geom 18 (1997) 421--431), which describes the convex hull of the set of face vectors of $r$-colorable complexes on $n$ vertices. As part of the proof we derive a generalization of Turán's graph theorem.

Polyhedra, Complexes, Nets and Symmetry

Skeletal polyhedra and polygonal complexes in ordinary Euclidean 3-space are finite or infinite 3-periodic structures with interesting geometric, combinatorial, and algebraic properties. They can be viewed as finite or infinite 3-periodic graphs (nets) equipped with additional structure imposed by the faces, allowed to be skew, zig-zag, or helical. A polyhedron or complex is "regular" if its geometric symmetry group is transitive on the flags (incident vertex-edge-face triples). There are 48 regular polyhedra (18 finite polyhedra and 30 infinite apeirohedra), as well as 25 regular polygonal co…

The upper bound theorem for flag homology 5-manifolds

We prove that among all flag homology $5$-manifolds with $n$ vertices, the join of $3$ circles of as equal length as possible is the unique maximizer of all the face numbers. The same upper bounds on the face numbers hold for $5$-dimensional flag Eulerian normal pseudomanifolds.

The Cyclic Sieving Phenomenon for Faces of Generalized Cluster Complexes

The notion of cyclic sieving phenomenon is introduced by Reiner, Stanton, and White as a generalization of Stembridge's $q=-1$ phenomenon. The generalized cluster complexes associated to root systems are given by Fomin and Reading as a generalization of the cluster complexes found by Fomin and Zelevinsky. In this paper, the faces of various dimensions of the generalized cluster complexes in type $A_n$, $B_n$, $D_n$, and $I_2(a)$ are shown to exhibit the cyclic sieving phenomenon under a cyclic group action. For the cluster complexes of exceptional type $E_6$, $E_7$, $E_8$, $F_4$, $H_3$, and $H…

Induced equators in flag spheres

We propose a combinatorial approach to the following strengthening of Gal's conjecture: $γ(Δ)\ge γ(E)$ coefficientwise, where $Δ$ is a flag homology sphere and $E\subseteq Δ$ an induced homology sphere of codimension $1$. We provide partial evidence in favor of this approach, and prove a nontrivial nonlinear inequality that follows from the above conjecture, for boundary complexes of flag $d$-polytopes: $h_1(Δ) h_i(Δ) \ge (d-i+1)h_{i-1}(Δ) + (i+1) h_{i+1}(Δ)$ for all $0\le i\le d$.

Simplicial simple-homotopy of flag complexes in terms of graphs

A flag complex can be defined as a simplicial complex whose simplices correspond to complete subgraphs of its 1-skeleton taken as a graph. In this article, by introducing the notion of s-dismantlability, we shall define the s-homotopy type of a graph and show in particular that two finite graphs have the same s-homotopy type if, and only if, the two flag complexes determined by these graphs have the same simplicial simple-homotopy type (Theorem 2.10, part 1). This result is closely related to similar results established by Barmak and Minian (Adv. in Math., 218 (2008), 87-104) in the framework …

Dense flag triangulations of 3-manifolds via extremal graph theory

We characterize f-vectors of sufficiently large three-dimensional flag Gorenstein* complexes, essentially confirming a conjecture of Gal [Discrete Comput. Geom., 34 (2), 269--284, 2005]. In particular, this characterizes f-vectors of large flag triangulations of the 3-sphere. Actually, our main result is more general and describes the structure of closed flag 3-manifolds which have many edges. Looking at the 1-skeleta of these manifolds we reduce the problem to a certain question in extremal graph theory. We then resolve this question by employing the Supersaturation Theorem of Erdos and Simon…

Cohen-Macaulay lexsegment complexes in arbitrary codimension

We characterize pure lexsegment complexes which are Cohen-Macaulay in arbitrary codimension. More precisely, we prove that any lexsegment complex is Cohen-Macaulay if and only if it is pure and its one dimensional links are connected, and, a lexsegment flag complex is Cohen-Macaulay if and only if it is pure and connected. We show that any non-Cohen-Macaulay lexsegment complex is a Buchsbaum complex if and only if it is a pure disconnected flag complex. For $t\ge 2$, a lexsegment complex is strictly Cohen-Macaulay in codimension $t$ if and only if it is the join of a lexsegment pure disconnect…

Face numbers of barycentric subdivisions of cubical complexes

The $h$-polynomial of the barycentric subdivision of any $n$-dimensional cubical complex with nonnegative cubical $h$-vector is shown to have only real roots and to be interlaced by the Eulerian polynomial of type $B_n$. This result applies to barycentric subdivisions of shellable cubical complexes and, in particular, to barycentric subdivisions of cubical convex polytopes and answers affirmatively a question of Brenti, Mohammadi and Welker.

Several Convex-Ear Decompositions

In this paper we give convex-ear decompositions for the order complexes of several classes of posets, namely supersolvable lattices with non-zero Mobius functions and rank-selected subposets of such lattices, rank-selected geometric lattices, and rank-selected face posets of shellable complexes which do not include the top rank. These decompositions give us many new inequalities for the h-vectors of these complexes. In addition, our decomposition of rank-selected face posets of shellable complexes allows us to prove inequalities for the flag h-vector of face posets of Cohen-Macaulay complexes.

Regular Polygonal Complexes in Space, I

A polygonal complex in euclidean 3-space is a discrete polyhedron-like structure with finite or infinite polygons as faces and finite graphs as vertex-figures, such that a fixed number r of faces surround each edge. It is said to be regular if its symmetry group is transitive on the flags. The present paper and its successor describe a complete classification of regular polygonal complexes in 3-space. In particular, the present paper establishes basic structure results for the symmetry groups, discusses geometric and algebraic aspects of operations on their generators, characterizes the comple…

New light on Bergman complexes by decomposing matroid types

Bergman complexes are polyhedral complexes associated to matroids. Faces of these complexes are certain matroids, called matroid types, too. In order to understand the structure of these faces we decompose matroid types into direct summands. Ardila/Klivans proved that the Bergman Complex of a matroid can be subdivided into the order complex of the proper part of its lattice of flats. Beyond that Feichtner/Sturmfels showed that the Bergman complex can even be subdivided to the even coarser nested set complex. We will give a much shorter and more general proof of this fact. Generalizing formulas…

Bisimplicial complexes and asphericity

We present a discrete Morse-theoretic method for proving that a regular CW complex is homeomorphic to a sphere. We use this method to define bisimplices, the cells of a class of regular CW complexes we call bisimplicial complexes. The 1-skeleta of bisimplices are complete bipartite graphs making them suitable in constructing higher dimensional skeleta for bipartite graphs. We show that the flag bisimplicial completion of a finite bipartite bi-dismantlable graph is collapsible. We use this to show that the flag bisimplicial completion of a quadric complex is contractible and to construct a comp…