Hausdorff metric between simplicial complexes

Mayer-Vietoris sequence in cohomology of Lie algebroids on simplicial complexes

It is shown that the Mayer-Vietoris sequence holds for the cohomology of complexes of Lie algebroids which are defined on simplicial complexes and satisfy the compatibility condition concerning restrictions to the faces of each simplex. The Mayer-Vietoris sequence will be obtained as a consequence of the extension lemma for piecewise smooth forms defined on complexes of Lie algebroids.

Generalized equivariant homology on simplicial complexes

A careful account is given of generalized equivariant homology theories on the category of topological pairs acted on by a group. In particular, upon restriction to the category of equivariant simplicial complexes, the equivalence of equivariant simplicial homology (also known as Bredon homology), the second derived term of the Atiyah--Hirzebruch spectral sequence, and equivariant singular homology is demonstrated.

Homotopical decompositions of simplicial and Vietoris Rips complexes

Motivated by applications in Topological Data Analysis, we consider decompositions of a simplicial complex induced by a cover of its vertices. We study how the homotopy type of such decompositions approximates the homotopy of the simplicial complex itself. The difference between the simplicial complex and such an approximation is quantitatively measured by means of the so called obstruction complexes. Our general machinery is then specialized to clique complexes, Vietoris-Rips complexes and Vietoris-Rips complexes of metric gluings. For the latter we give metric conditions which allow to recov…

超图和单纯复形的同调临界点

Homological critical points for hypergraphs and simplicial complexes

本文定义并研究了持续超图与持续单纯复形及其映射的同调临界点，证明了子集关系的交换图，并给出参数配置空间的实例。

随机有根单纯复形的$\ell^2$-Betti数

$\ell^2$-Betti numbers of random rooted simplicial complexes

本文通过在有根单纯复形空间上引入幺模测度及链复形与迹函数，定义了随机有根单纯复形的$\ell^2$-Betti数，并证明其在Benjamini-Schramm收敛下具有连续性。

The Complement of Polyhedral Product Spaces and the Dual Simplicial Complexes

In this paper, we define and prove basic properties of complement polyhedral product spaces, dual complexes and polyhedral product complexes. Then we compute the universal algebra of polyhedral product complexes under certain split conditions and the Alexander duality isomorphism on certain polyhedral product spaces.

Spectra of combinatorial Laplace operators on simplicial complexes

We first develop a general framework for Laplace operators defined in terms of the combinatorial structure of a simplicial complex. This includes, among others, the graph Laplacian, the combinatorial Laplacian on simplicial complexes, the weighted Laplacian, and the normalized graph Laplacian. This framework then allows us to define the normalized Laplace operator $Δ_{i}^{up}$ on simplicial complexes which we then systematically investigate. We study the effects of a wedge sum, a join and a duplication of a motif on the spectrum of the normalized Laplace operator, and identify some of the comb…

双重同调与楔形可分解单纯复形

Double homology and wedge-decomposable simplicial complexes

本文证明楔形可分解单纯复形在特定双次数处具有双重同调群$\mathbb{Z}\oplus\mathbb{Z}$。

Large Random Simplicial Complexes, III; The Critical Dimension

In this paper we study the notion of critical dimension of random simplicial complexes in the general multi-parameter model described in our previous papers of this series. This model includes as special cases the Linial-Meshulam-Wallach model as well as the clique complexes of random graphs. We characterise the concept of critical dimension in terms of various geometric and topological properties of random simplicial complexes such as their Betti numbers, the fundamental group, the size of minimal cycles and the degrees of simplexes. We mention in the text a few interesting open questions.

Simplicial Polytope Complexes and Deloopings of $K$-theory

This paper is a continuation of our previous work in which we defined the notion of a polytope complex and its $K$-theory. In this paper we produce formulas for the delooping of a simplicial polytope complex and the cofiber of a morphism of simplicial polytope complexes. Along the way we also prove that the (classical and higher) scissors congruence groups of polytopes in a homogeneous $n$-manifold (with sufficient geometric data) are determined by its local properties.

Quantitative Simplification of Filtered Simplicial Complexes

We introduce a new invariant defined on the vertices of a given filtered simplicial complex, called codensity, which controls the impact of removing vertices on persistent homology. We achieve this control through the use of an interleaving type of distance between fitered simplicial complexes. We study the special case of Vietoris-Rips filtrations and show that our bounds offer a significant improvement over the immediate bounds coming from considerations related to the Gromov-Hausdorff distance. Based on these ideas we give an iterative method for the practical simplification of filtered sim…

Characterizations of Line Simplicial Complexes

Let $G$ be a finite simple graph. The line graph $L(G)$ represents the adjacencies between edges of $G$. We define first the line simplicial complex $Δ_L(G)$ of $G$ containing Gallai and anti-Gallai simplicial complexes $Δ_Γ(G)$ and $Δ_{Γ'}(G)$ (respectively) as spanning subcomplexes. The study of connectedness of simplicial complexes is interesting due to various combinatorial and topological aspects. In Theorem 3.3, we prove that the line simplicial complex $Δ_L(G)$ is connected if and only if $G$ is connected. In Theorem 3.4, we establish the relation between Euler characteristics of line a…

Lusternik-Schnirelmann category of simplicial complexes and finite spaces

In this paper we establish a natural definition of Lusternik-Schnirelmann category for simplicial complexes via the well known notion of contiguity. This category has the property of being homotopy invariant under strong equivalences, and only depends on the simplicial structure rather than its geometric realization. In a similar way to the classical case, we also develop a notion of geometric category for simplicial complexes. We prove that the maximum value over the homotopy class of a given complex is attained in the core of the complex. Finally, by means of well known relations between sim…

Occupants in simplicial complexes

Let $M$ be a smooth manifold and $K\subset M$ be a simplicial complex of codimension at least 3. Functor calculus methods lead to a homotopical formula of $M\setminus K$ in terms of spaces $M\setminus T$ where $T$ is a finite subset of $K$. This is a generalization of the author's previous work with Michael Weiss where the subset $K$ is assumed to be a smooth submanifold of $M$ and uses his generalization of manifold calculus adapted for simplicial complexes.

用于拓扑数据分析的单纯复形豪斯多夫距离

Simplicial Hausdorff Distance for Topological Data Analysis

本文提出一种修正的单纯复形豪斯多夫距离，将几何邻近性与拓扑特征结合为扩展度量，并给出过滤复形版本及计算复杂度结果。

Erratum: Limit theorems for Betti numbers of random simplicial complexes

We correct the proofs of the main theorems in our paper "Limit theorems for Betti numbers of random simplicial complexes".

Maurer-Cartan elements in the Lie models of finite simplicial complexes

In a previous work, we have associated a complete differential graded Lie algebra to any finite simplicial complex in a functorial way. Similarly, we have also a realization functor from the category of complete differential graded Lie algebras to the category of simplicial sets. We have already interpreted the homology of a Lie algebra in terms of homotopy groups of its realization. In this paper, we begin a dictionary between models and simplicial complexes by establishing a correspondence between the Deligne groupoid of the model and the connected components of the finite simplicial complex…

Algebraic fundamental group and simplicial complexes

In this paper, we prove that the fundamental group of a simplicial complex is isomorphic to the algebraic fundamental group of its incidence algebra, and we derive some applications.

The homology of simplicial complement and the cohomology of the moment-angle complexes

A simplicial complement P is a sequence of subsets of [m] and the simplicial complement P corresponds to a unique simplicial complex K with vertices in [m]. In this paper, we defined the homology of a simplicial complement $H_{i,σ}(Λ^{*,*}[P], d)$ over a principle ideal domain k and proved that $H_{*,*}(Λ[P], d)$ is isomorphic to the Tor of the corresponding face ring k(K) by the Taylor resolutions. As applications, we give methods to compute the ring structure of Tor_{*,*}^{k[x]}(k(K), k)$, $link_{K}σ$, $star_{K}σ$ and the cohomology of the generalized moment-angle complexes.

关于具有极值总Betti数和总双阶Betti数的单纯复形

On Simplicial Complexes with Extremal Total Betti number and Total Bigraded Betti Number

本文研究了在每一维给定顶点数的情况下，确定哪些单纯复形的总Betti数和总双阶Betti数达到最大或最小值。

Steenrod coalgebras of simplicial complexes

In this paper, we extend earlier work by showing that if $X$ and $Y$ are simplicial complexes (i.e. simplicial sets whose simplices are determined by their vertices), a morphism $g:N(X)\to N(Y)$ of Steenrod coalgebras (normalized chain-complexes equipped with extra structure) induces one of their topological realizations $\hat{g}:|X|\to |Y|$. If $g$ is an isomorphism, then it induces an isomorphism between $X$ and $Y$, implying that they are homeomorphic.

抽象单纯复形的单项极化与去极化

Monomial polarization and depolarization of abstract simplicial complexes

本研究将单项理想的极化与去极化运算迁移到抽象单纯复形，明确两类Koszul单纯复形的关系，提出保同调的非初等坍缩约简方法，并将其用于Alexander对偶的高效计算。

单纯GKM轨形复形的整等变K-理论与配边环

Integral equivariant $K$-theory and cobordism ring of simplicial GKM orbifold complexes

本文定义了单纯GKM轨形复形，引入了可构建与可分性质，并计算了可分单纯GKM轨形复形的整等变上同调环、等变K-理论环及等变配边环。

中间 regime 随机单纯复形的拓扑

Topology of medial regime random simplicial complexes

该研究分析中间 regime 随机单纯复形的拓扑，证明其具高度连通性与迭代悬挂同伦型，提出更灵活的高连通性组合判据，确定其拓扑复杂度上界为2且齐次类几乎必然等于2。

与二维单纯复形相关联的矩角复形的环路空间分解

Loop space decompositions of moment-angle complexes associated to two dimensional simplicial complexes

本文证明了二维单纯复形对应的矩角复形的环路空间可分解为球面、球面上的环路及Moore空间环路分解中出现的不可分解空间的有限型乘积，并给出局部化条件下仅由球面及其环路构成分解的判据。

Dimensional operators for mathematical morphology on simplicial complexes

In this work we study the framework of mathematical morphology on simplicial complex spaces. Simplicial complexes are widely used to represent multidimensional data, such as meshes, that are two dimensional complexes, or graphs, that can be interpreted as one dimensional complexes. Mathematical morphology is one of the most powerful frameworks for image processing, including the processing of digital structures, and is heavily used for many applications. However, mathematical morphology operators on simplicial complex spaces is not a concept fully developed in the literature. Specifically, we …

Revisiting function complexes and simplicial localisation

In this paper three results are established: firstly, that the homotopy function complexes of Dwyer and Kan can be defined as certain total right derived functors; secondly, that they functorially compute the homotopy type of the hom-spaces in the simplicial localisation; and thirdly, that they can be computed by fibrant replacements in a suitable left Bousfield localisation of the projective model structure on simplicial presheaves.

软随机单纯复形的中心极限定理

Central limit theorems for Soft random simplicial complexes

本文提出了建立在软随机图上的软随机单纯复形模型，并利用正态逼近定理证明了其 $k$-面数量与欧拉示性数的中心极限定理。

Composition of simplicial complexes, polytopes and multigraded Betti numbers

For a simplicial complex K on m vertices and simplicial complexes K1,...,Km a composed simplicial complex K(K1,...,Km) is introduced. This construction generalizes an iterated simplicial wedge construction studied by A. Bahri, M. Bendersky, F. R. Cohen and S. Gitler and allows to describe the combinatorics of generalized joins of polytopes P(P1,...,Pm) defined by G. Agnarsson. The composition defines a structure of an operad on a set of finite simplicial complexes. We prove the following: (1) a composed complex K(K1,...,Km) is a simplicial sphere iff K is a simplicial sphere and Ki are the bou…

On lower bounds of the sum of multigraded Betti numbers of simplicial complexes

We find some general lower bounds of the sum of certain families of multigraded Betti numbers of any simplicial complex with a vertex coloring.

Colorings of simplicial complexes and vector bundles over Davis-Januszkiewicz spaces

We show that coloring properties of a simplicial complex K are reflected by splitting properties of a bundle over the associated Davis-Januszkiewicz space whose Chern classes are given by the elementary symmetric polynomials in the generators of the Stanley-Reisner algebra of K.

Golodness and polyhedral products for two dimensional simplicial complexes

Golodness of 2-dimensional simplicial complexes is studied through polyhedral products, and combinatorial and topoogical characterization of Golodness of surface triangulations is given. An answer to the question of Berglund is also given so that there is a 2-dimensional simplicial complex which is rationally Golod but is not Golod over $\mathbb{Z}/p$.

Constructible sheaves on simplicial complexes and Koszul duality

We obtain a linear algebra data presentation of the category of constructible with respect to perverse triangulation sheaves on a finite simplicial complex. We also establish Koszul duality between the above mentioned category and the category of perverse sheaves constructible with respect to the triangulation

Limits of Dense Simplicial Complexes

We develop a theory of limits for sequences of dense abstract simplicial complexes, where a sequence is considered convergent if its homomorphism densities converge. The limiting objects are represented by stacks of measurable [0,1]-valued functions on unit cubes of increasing dimension, each corresponding to a dimension of the abstract simplicial complex. We show that convergence in homomorphism density implies convergence in a cut-metric, and vice versa, as well as showing that simplicial complexes sampled from the limit objects closely resemble its structure. Applying this framework, we als…

单纯复形与单纯集大小的拓扑下界

Topological lower bounds on the sizes of simplicial complexes and simplicial sets

本文证明满足特定拓扑条件的 n 维空间（如 n 维环面）在任何三角剖分或可缩面单纯集表示中至少具有 2^n 个 n 维面，给出了单纯结构大小的拓扑下界。

关于单纯复形和图的Thomason型模型结构

Thomason-Type Model Structures on Simplicial Complexes and Graphs

本文研究了单纯复形和图上的Thomason型模型结构，证明其与Kan-Quillen模型结构相关，并展示这些结构的共纤生成性和恰当性。

Ample单纯复形

Ample simplicial complexes

本文研究 $r$-ample 单纯复形，证明了其连通性与顶点数下界，利用概率方法确立存在性，并构造了具有近最优顶点数的 iterated Paley 单纯复形。

Large simplicial complexes: Universality, Randomness, and Ampleness

The paper surveys recent progress in understanding geometric, topological and combinatorial properties of large simplicial complexes, focusing mainly on ampleness, connectivity and universality. In the first part of the paper we concentrate on $r$-ample simplicial complexes which are high dimensional analogues of the $r$-e.c. graphs introduced originally by Erd\H os and Réniy. The class of $r$-ample complexes is useful for applications since these complexes allow extensions of subcomplexes of certain type in all possible ways; besides, $r$-ample complexes exhibit remarkable robustness properti…

A lower bound on the number of homotopy types of simplicial complexes on $n$ vertices

For $n \in \mathbb{N}$, let $h(n)$ denote the number of simplicial complexes on $n$ vertices up to homotopy equivalence. Here we prove that $h(n) \geq 2^{2^{0.02n}}$ when $n$ is large enough. Together with the trivial upper bound of $2^{2^n}$ on the number of labeled simplicial complexes on $n$ vertices this proves a conjecture of Kalai that $h(n)$ is doubly exponential in $n$.

Invariant Representations of Embedded Simplicial Complexes

Analyzing embedded simplicial complexes, such as triangular meshes and graphs, is an important problem in many fields. We propose a new approach for analyzing embedded simplicial complexes in a subdivision-invariant and isometry-invariant way using only topological and geometric information. Our approach is based on creating and analyzing sufficient statistics and uses a graph neural network. We demonstrate the effectiveness of our approach using a synthetic mesh data set.

Generalized Random Simplicial Complexes

We consider a multi-parameter model for randomly constructing simplicial complexes. This model interpolates between random clique complexes and Linial-Meshulam random $k$-dimensional complexes, two models that have been extensively studied. While these models asymptotically exhibit nontrivial cohomology in only one or two dimensions, we show that in this generalized setting nontrivial cohomology can occur in several dimensions simultaneously. We establish upper and lower thresholds for the appearance of nontrivial cohomology in a particular dimension, and in some instances characterize the beh…

Cores of simplicial complexes

We introduce a method to reduce the study of the topology of a simplicial complex to that of a simpler one. We give some applications of this method to complexes arising from graphs. As a consequence, we answer some questions raised in [R.Ehrenborg, G.Hetyei, European J.Combin.27 (2006), no.6, 906-923] on the independence complex and the dominance complex. The techniques used come mainly from monomial ideal theory.

Limit theorems for Betti numbers of random simplicial complexes

There have been several recent articles studying homology of various types of random simplicial complexes. Several theorems have concerned thresholds for vanishing of homology, and in some cases expectations of the Betti numbers. However little seems known so far about limiting distributions of random Betti numbers. In this article we establish Poisson and normal approximation theorems for Betti numbers of different kinds of random simplicial complex: Erdős-Rényi random clique complexes, random Vietoris-Rips complexes, and random Čech complexes. These results may be of practical interest in to…

Enumerating the class of minimally path connected simplicial complexes

In 1983 Kalai proved an incredible generalisation of Cayley's formula for the number of trees on a labelled vertex set to a formula for a class of $r$-dimensional simplicial complexes. These simplicial complexes generalise trees by means of being homologically $\Bbb Q$-acyclic. In this text we consider a different generalisation of trees to the class of pure dimensional simplicial complexes that \textit{minimally connect a vertex set} (in the sense that the removal of any top dimensional face disconnects the complex). Our main result provides an upper and lower bound for the number of these mi…

Simplicial complexes Alexander dual to boundaries of polytopes

In the paper we treat Gale diagrams in a combinatorial way. The interpretation allows to describe simplicial complexes which are Alexander dual to boundaries of simplicial polytopes and, more generally, to nerve-complexes of general polytopes. This technique and recent results of N.Yu.Erokhovets are combined to prove the following: Buchstaber invariant $s(P)$ of a convex polytope equals 1 if and only if $P$ is a pyramid. In general, we describe a procedure to construct polytopes with $s_R(P)>k$. The construction has purely combinatorial consequences. We also apply Gale duality to the study of …

Dynamical systems defined on simplicial complexes: symmetries, conjugacies, and invariant subspaces

We consider the general model for dynamical systems defined on a simplicial complex. We describe the conjugacy classes of these systems and show how symmetries in a given simplicial complex manifest in the dynamics defined thereon, especially with regard to invariant subspaces in the dynamics.

Topology of random simplicial complexes: a survey

This expository article is based on a lecture from the Stanford Symposium on Algebraic Topology: Application and New Directions, held in honor of Gunnar Carlsson, Ralph Cohen, and Ib Madsen.

Simplicial chromatic polynomials as Hilbert series of Stanley--Reisner rings

We find families of simplicial complexes where the simplicial chromatic polynomials defined by Cooper--de Silva--Sazdanovic \cite{CdSS} are Hilbert series of Stanley--Reisner rings of auxiliary simplicial complexes. As a result, such generalized chromatic polynomials are determined by $h$-vectors of auxiliary simplicial complexes. In addition to generalizing related results on graphs and matroids, the simplicial complexes used allow us to consider problems that are not necessarily analogues of those considered for graphs. Some examples include supports of cyclotomic polynomials, log concavity …

Generalized max-flows and min-cuts in simplicial complexes

We consider high dimensional variants of the maximum flow and minimum cut problems in the setting of simplicial complexes and provide both algorithmic and hardness results. By viewing flows and cuts topologically in terms of the simplicial (co)boundary operator we can state these problems as linear programs and show that they are dual to one another. Unlike graphs, complexes with integral capacity constraints may have fractional max-flows. We show that computing a maximum integral flow is NP-hard. Moreover, we give a combinatorial definition of a simplicial cut that seems more natural in the c…

Gigantic random simplicial complexes

We provide a random simplicial complex by applying standard constructions to a Poisson point process in Euclidean space. It is gigantic in the sense that - up to homotopy equivalence - it almost surely contains infinitely many copies of every compact topological manifold, both in isolation and in percolation.

Maps on random hypergraphs and random simplicial complexes

Let $L$ be a simplicial complex. In this paper, we study random sub-hypergraphs and random sub-complexes of $L$. By considering the minimal complex that a sub-hypergraph can be embedded in and the maximal complex that can be embedded in a sub-hypergraph, we define some maps on the space of probability functions on sub-hypergraphs of $L$. We study the compositions of these maps as well as their actions on the space of probability functions.

A Notion of Harmonic Clustering in Simplicial Complexes

We outline a novel clustering scheme for simplicial complexes that produces clusters of simplices in a way that is sensitive to the homology of the complex. The method is inspired by, and can be seen as a higher-dimensional version of, graph spectral clustering. The algorithm involves only sparse eigenproblems, and is therefore computationally efficient. We believe that it has broad application as a way to extract features from simplicial complexes that often arise in topological data analysis.

Law of large numbers for Betti numbers of homogeneous and spatially independent random simplicial complexes

The Linial-Meshulam complex model is a natural higher-dimensional analog of the Erdős-Rényi graph model. In recent years, Linial and Peled established a limit theorem for Betti numbers of Linial-Meshulam complexes with an appropriate scaling of the underlying parameter. The present paper aims to extend that result to more-general random simplicial complex models. We introduce a class of homogeneous and spatially independent random simplicial complexes, including the Linial-Meshulam complex model and the random clique complex model as special cases, and we study the asymptotic behavior of their…

{\tt Macaulay2} 中的抽象单纯复形

Abstract simplicial complexes in {\tt Macaulay2}

该研究为计算机代数系统 {\tt Macaulay2} 开发了 {\tt AbstractSimplicialComplexes.m2} 包，以哈希表形式的分次列表实现抽象单纯复形，支持链复形直接实现及随机复形构造，补充现有框架。