Karim A. Adiprasito, Eran Nevo, Jose A. Samper
We generalize the fundamental graph-theoretic notion of chordality for higher dimensional simplicial complexes by putting it into a proper context within homology theory. We generalize some of the classical results of graph chordality to this generality, including the fundamental relation to the Leray property and chordality theorems of Dirac.
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Vertex decomposable complexes of directed forests, conflict graphs and chordality
本文研究了有向森林的顶点可分解复形，通过构造冲突图并应用独立复形理论，得出了保证复形顶点可分解、壳性和序列Cohen-Macaulay性质的图论准则，并给出了相关多重有向图的分类。
Chordality, syzygies, and shellability for hypergraphic analogues of interval graphs
本文证明共区间超图与下闭复形在补集意义下等价，并证明下闭团簇是Woodroofe弦性的，从而回答Dochtermann和Engström的问题，表明其Alexander对偶复形是顶点可分解的，且具有由下闭顶点序诱导的壳化。
We study the geometric change of Chow cohomology classes in projective toric varieties under the Weil-McMullen dual of the intersection product with a Lefschetz element. Based on this, we introduce toric chordality, a generalization of graph chordality to higher skeleta of simplicial complexes with a coordinatization over characteristic 0, leading us to a far-reaching generalization of Kalai's work on applications of rigidity of frameworks to polytope theory. In contrast to "homological" chordality, the notion that is usually studied as a higher-dimensional analogue of graph chordality, we wil…
Generalizations of interval and proper interval graphs for simplicial complexes
本文将区间图和真区间图推广到单纯复形，给出组合与代数刻画，并应用于可排序性、Cohen-Macaulay 整环及禁子图性质。
The complex of $r$-co-connected subgraphs, chordality and Fröberg's theorem
本文研究了 r-连通子图复形的拓扑与代数性质，揭示了其与图类 co-chordality 的关系，并扩展了 Fröberg 定理。
The Hom-complexes were introduced by Lovasz to study topological obstructions to graph colorings. It was conjectured by Babson and Kozlov, and proved by Cukic and Kozlov, that Hom(G,K_n) is (n-d-2)-connected, where d is the maximal degree of a vertex of G. We give a short proof of the conjecture.
We study the question of realisability of iterated higher Whitehead products with a given form of nested brackets by simplicial complexes, using the notion of the moment-angle complex $Z_K$. Namely, we say that a simplicial complex $K$ realises an iterated higher Whitehead product $w$ if $w$ is a nontrivial element of $π_*(Z_K)$. The combinatorial approach to the question of realisability uses the operation of substitution of simplicial complexes: for any iterated higher Whitehead product $w$ we describe a simplicial complex $\partialΔ_w$ that realises $w$. Furthermore, for a particular form o…
As part of various obstruction theories, non-trivial Massey products have been studied in symplectic and complex geometry, commutative algebra and topology for a long time. We introduce a general approach to constructing non-trivial Massey products in the cohomology of moment-angle complexes, using homotopy theoretical and combinatorial methods. Our approach sets a unifying way of constructing higher Massey products of arbitrary cohomological classes and generalises all existing examples of non-trivial Massey products in moment-angle complexes. As a result, we obtain explicit constructions of …
For $r\geq 1$, the $r$-independence complex of a graph $G$ is a simplicial complex whose faces are subset $I \subseteq V(G)$ such that each component of the induced subgraph $G[I]$ has at most $r$ vertices. In this article, we determine the homotopy type of $r$-independence complexes of certain families of graphs including complete $s$-partite graphs, fully whiskered graphs, cycle graphs and perfect $m$-ary trees. In each case, these complexes are either homotopic to a wedge of equi-dimensional spheres or are contractible. We also give a closed form formula for their homotopy types.
The main result of this paper is a proof of the following conjecture of Babson & Kozlov: Theorem. Let G be a graph of maximal valency d, then the complex Hom(G,K_n) is at least (n-d-2)-connected. Here Hom(-,-) denotes the polyhedral complex introduced by Lovász to study the topological lower bounds for chromatic numbers of graphs. We will also prove, as a corollary to the main theorem, that the complex Hom(C_{2r+1},K_n) is (n-4)-connected, for $n\geq 3$.
For $r\geq 1$, the $r$-matching complex of a graph $G$, denoted $M_r(G)$, is a simplicial complex whose faces are the subsets $H \subseteq E(G)$ of the edge set of $G$ such that the degree of any vertex in the induced subgraph $G[H]$ is at most $r$. In this article, we give a closed form formula for the homotopy type of the $(n-2)$-matching complex of complete graph on $n$ vertices. We also prove that the $(n-1)$-matching complex of complete bipartite graph $K_{n,n}$ is homotopy equivalent to a sphere of dimension $(n-1)^2-1$.
Shellability in Clique-Free Complexes of Graphs
本文研究了$t$-无团复形的壳性，证明了特定条件下的复形可分解且壳性，并引入了团附加操作以构造壳性复形，同时扩展了团尾巴的概念并建立了弦性与线性解的关系。
Vertex orders in higher dimensions
本文研究了单位区间复形和区间复形的高维推广，证明了强连通单位区间复形在单位区间序诱导下可壳化且顶点可分解，同时给出了高维中不可壳化的强连通区间复形实例。
Using the random complexes of Linial and Meshulam, we exhibit a large family of simplicial complexes for which, whenever affinely embedded into Euclidean space, the filling areas of simplicial cycles is greatly distorted. This phenomenon can be regarded as a higher order analogue of the metric distortion of embeddings of random graphs.
The paper establishes that the rank of a regular polygonal complex in 3-space E^3 cannot exceed 4, and that the only regular polygonal complexes of rank 4 in 3-space are the eight regular 4-apeirotopes.
We show how to find higher generating families of subgroups, in the sense of Abels and Holz, for groups acting on Cohen-Macaulay complexes. We apply this to groups with a BN-pair to prove higher generation by parabolic and Levi-subgroups and describe higher generating families of parabolic subgroups in Aut(F_n).
We investigate generalized notions of the nerve complex for the facets of a simplicial complex. We show that the homologies of these higher nerve complexes determine the depth of the Stanley-Reisner ring $k[Δ]$ as well as the $f$-vector and $h$-vector of $Δ$. We present, as an application, a formula for computing regularity of monomial ideals.
We consider s-chromatic polynomials of simplicial complexes, higher dimensional analogues of chromatic polynomials for graphs.
A Fröberg type theorem for higher secant complexes
本研究借助Green-Lazarsfeld的$N_{q+1,p}$性质，将Fröberg定理推广到单纯复形的高阶正割复形，并研究了对应极小次数高阶正割簇几何现象的组合性质。
We introduce the notion of k-hyperclique complexes, i.e., the largest simplicial complexes on the set [n] with a fixed k-skeleton. These simplicial complexes are a higher-dimensional analogue of clique (or flag) complexes (case k=2) and they are a rich new class of simplicial complexes. We show that Dirac's theorem on chordal graphs has a higher-dimensional analogue in which graphs and clique complexes get replaced, respectively, by simplicial matroids and k-hyperclique complexes. We prove also a higher-dimensional analogue of Stanley's reformulation of Dirac's theorem on chordal graphs.
For random graphs, the containment problem considers the probability that a binomial random graph $G(n,p)$ contains a given graph as a substructure. When asking for the graph as a topological minor, i.e., for a copy of a subdivision of the given graph, it is well-known that the (sharp) threshold is at $p=1/n$. We consider a natural analogue of this question for higher-dimensional random complexes $X^k(n,p)$, first studied by Cohen, Costa, Farber and Kappeler for $k=2$. Improving previous results, we show that $p=Θ(1/\sqrt{n})$ is the (coarse) threshold for containing a subdivision of any fixed…
Shellability of Higher Independence Complexes of Graphs
本文研究图的高阶独立复形的可壳性分类问题，针对块图、树等图族，基于直径和禁用子图建立可壳性充分条件，并通过星-团粘附、团加须等图操作给出构造性方法，推广了经典独立复形的相关结果。
Using the combinatorics of the underlying simplicial complex $K$, we give various upper and lower bounds for the Lusternik-Schnirelmann (LS) category of moment-angle complexes $\zk$. We describe families of simplicial complexes and combinatorial operations which allow for a systematic description of the LS category. In particular, we characterise the LS category of moment-angle complexes $\zk$ over triangulated $d$-manifolds $K$ for $d\leq 2$, as well as higher dimension spheres built up via connected sum, join, and vertex doubling operations. %This characterisation is given in terms of the co…
We present a generalization of Brouwer's conjectural family of inequalities -- a popular family of inequalities in spectral graph theory bounding the partial sum of the Laplacian eigenvalues of graphs -- for the case of abstract simplicial complexes of any dimension. We prove that this family of inequalities holds for shifted simplicial complexes, which generalize threshold graphs, and give tighter bounds (linear in the dimension of the complexes) for simplicial trees. We prove that the conjecture holds for the the first, second, and last partial sums for all simplicial complexes, generalizing…
Higher Order Bipartiteness vs Bi-Partitioning in Simplicial Complexes
本文将二部图基于奇环的刻画推广到可去定向单纯复形，刻画其下对偶图的环条件，并揭示奇环数量与最大 Hodge Laplacian 特征值及高维二部划分的关系。
In 1993 Stanley showed that if a simplicial complex is acyclic over some field, then its face poset can be decomposed into disjoint rank $1$ boolean intervals whose minimal faces together form a subcomplex. Stanley further conjectured that complexes with a higher notion of acyclicity could be decomposed in a similar way using boolean intervals of higher rank. We provide an explicit counterexample to this conjecture. We also prove a version of the conjecture for boolean trees and show that the original conjecture holds when this notion of acyclicity is as high as possible.
Cheeger type inequalities for high dimensional simplicial complexes
本文从组合学角度研究高维单纯复形上的离散Cheeger型不等式，将图论中的经典Cheeger不等式推广到高维情形。
We study the expected topological properties of Cech and Vietoris-Rips complexes built on i.i.d. random points in R^d. We find higher dimensional analogues of known results for connectivity and component counts for random geometric graphs. However, higher homology H_k is not monotone when k > 0. In particular for every k > 0 we exhibit two thresholds, one where homology passes from vanishing to nonvanishing, and another where it passes back to vanishing. We give asymptotic formulas for the expectation of the Betti numbers in the sparser regimes, and bounds in the denser regimes. The main techn…
We introduce constructible directed complexes, a combinatorial presentation of higher categories inspired by constructible complexes in poset topology. Constructible directed complexes with a greatest element, called atoms, encompass common classes of higher-categorical cell shapes, including globes, cubes, oriented simplices, and a large sub-class of opetopes, and are closed under lax Gray products and joins. We define constructible polygraphs to be presheaves on a category of atoms and inclusions, and extend the monoidal structures. We show that constructible directed complexes are a well-be…
Higher-dimensional book-spaces
针对Taylor关于高维单纯复形能否具有拓扑模格而非分配格结构的问题，本文对每个n≥2构造了无限族紧单纯复形，给出肯定回答。
Let $Φ$ be a finite root system of rank $n$ and let $m$ be a nonnegative integer. The generalized cluster complex $Δ^m (Φ)$ was introduced by S. Fomin and N. Reading. It was conjectured by these authors that $Δ^m (Φ)$ is shellable and by V. Reiner that it is $(m+1)$-Cohen-Macaulay, in the sense of Baclawski. These statements are proved in this paper. Analogous statements are shown to hold for the positive part $Δ^m_+ (Φ)$ of $Δ^m (Φ)$. An explicit homotopy equivalence is given between $Δ^m_+ (Φ)$ and the poset of generalized noncrossing partitions, associated to the pair $(Φ, m)$ by D. Armstro…
We give examples of finite, simplicial $2$-complexes that do not PL embed in $\mathbb{R}^4$ and exhibit, for each such complex, a family of PL immersions into $\mathbb{R}^4$ that hide the obstruction to embedding in "higher and higher order Milnor invariants". In addition, we show that the embedding obstructions defined by Krushkal in [8] vanish for our examples. We also answer a question in a paper of Avramidi-Okun-Schreve.
We introduce a generalization of the celebrated Lovász theta number of a graph to simplicial complexes of arbitrary dimension. Our generalization takes advantage of real simplicial cohomology theory, in particular combinatorial Laplacians, and provides a semidefinite programming upper bound of the independence number of a simplicial complex. We consider properties of the graph theta number such as the relationship to Hoffman's ratio bound and to the chromatic number and study how they extend to higher dimensions. Like in the case of graphs, the higher dimensional theta number can be extended t…
Uniform spanning trees on finite graphs and their analogues on infinite graphs are a well-studied area. On a Cayley graph of a group, we show that they are related to the first $\ell^2$-Betti number of the group. Our main aim, however, is to present the basic elements of a higher-dimensional analogue on finite and infinite CW-complexes, which relate to the higher $\ell^2$-Betti numbers. One consequence is a uniform isoperimetric inequality extending work of Lyons, Pichot, and Vassout. We also present an enumeration similar to recent work of Duval, Klivans, and Martin.
Graphs with given k vertices generate an (acyclic) simplicial complex. We describe the homology of its quotient complex, formed by all connected graphs, and demonstrate its applications to the topology of braid groups, knot theory, combinatorics, and singularity theory. The multidimensional analogues of this complex are indicated, which arise naturally in the homotopy theory, higher Chern-Simons theory and complexity theory.
Combinatorial Complexes: Bridging the Gap Between Cell Complexes and Hypergraphs
该研究指出超图与胞腔复形分别侧重集合型多体关系和层级式内部-边界关系，提出可融合两类关系的组合复形概念，并通过数值实验验证其在学习任务中的建模优势。
We discuss two combinatorical ways of generalizing the definition of expander graphs and Ramanujan graphs, to quotients of buildings of higher dimension. The two possible definitions are equivalent for affine buildings, giving the notion of an Lp-expander complex. We calculate explicit spectral gaps on many combinatorical operators, on any Lp-expander complex. We associate with any complex a natural "zeta function", generalizing the Ihara-Hashimoto zeta function of a finite graph. We generalize a well known theorem of Hashimoto, showing that a complex is Ramanujan if and only if the zeta funct…
We define $k$-dimensional digraphs and initiate a study of their spectral theory. The $k$-dimensional digraphs can be viewed as generating graphs for small categories called $k$-graphs. Guided by geometric insight, we obtain several new series of $k$-graphs using cube complexes covered by Cartesian products of trees, for $k \geq 2$. These $k$-graphs can not be presented as virtual products, and constitute novel models of such small categories. The constructions yield rank-$k$ Cuntz-Krieger algebras for all $k\geq 2$. We introduce Ramanujan $k$-graphs satisfying optimal spectral gap property, a…
Expander graphs in general, and Ramanujan graphs in particular, have been of great interest in the last three decades with many applications in computer science, combinatorics and even pure mathematics. In these notes we describe various efforts made in recent years to generalize these notions from graphs to higher dimensional simplicial complexes.
For graphs there exists a strong connection between spectral and combinatorial expansion properties. This is expressed, e.g., by the discrete Cheeger inequality, the lower bound of which states that $λ(G) \leq h(G)$, where $λ(G)$ is the second smallest eigenvalue of the Laplacian of a graph $G$ and $h(G)$ is the Cheeger constant measuring the edge expansion of $G$. We are interested in generalizations of expansion properties to finite simplicial complexes of higher dimension (or uniform hypergraphs). Whereas higher dimensional Laplacians were introduced already in 1945 by Eckmann, the generali…
Fröberg's Theorem, vertex splittability and higher independence complexes
该研究将Fröberg定理推广至高次无平方单项式理想，通过r-独立构造r-均匀超图，证明共弦图对应边理想具顶点可分裂性，计算了分次Betti数，并给出线性消解存在性的拓扑证明。
Higher resonance schemes and Koszul modules of simplicial complexes
本文研究有限单纯复形外Stanley-Reisner代数的共振概型与高阶Koszul模，证明共振概型既约，计算Hilbert级数并给出正则性与投射维数界，推广直角Artin群Chen秩公式。
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…
We consider higher-dimensional generalizations of the normalized Laplacian and the adjacency matrix of graphs and study their eigenvalues for the Linial-Meshulam model $X^k(n,p)$ of random $k$-dimensional simplicial complexes on $n$ vertices. We show that for $p=Ω(\log n/n)$, the eigenvalues of these matrices are a.a.s. concentrated around two values. The main tool, which goes back to the work of Garland, are arguments that relate the eigenvalues of these matrices to those of graphs that arise as links of $(k-2)$-dimensional faces. Garland's result concerns the Laplacian; we develop an analogo…
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