We define an (r,s)-coloring of an abstract simplicial complex to be a coloring using r colors of the vertices so that in any simplex at most s vertices have the same color. We translate the problem of finding an (r,s)-coloring of a given simplex to a topological problem and to a ring theoretic problem. The topological version involves vector bundles over Davis--Januszkiewicz spaces and the algebraic one involves factorizations in Stanley-Reisner rings.
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.
We generalize some homotopy calculation techniques such as splittings and matching trees that are introduced for the computations in the case of the independence complexes of graphs to arbitrary simplicial complexes, and exemplify their efficiency on some simplicial complexes, the devoid complexes of graphs, which are simplicial complexes parametrized by graphs. Additionally, we compute the homotopy type of dominance complexes of chordal graphs.
A vertex coloring of a simplicial complex $Δ$ is called a linear coloring if it satisfies the property that for every pair of facets $(F_1, F_2)$ of $Δ$, there exists no pair of vertices $(v_1, v_2)$ with the same color such that $v_1\in F_1\backslash F_2$ and $v_2\in F_2\backslash F_1$. We show that every simplicial complex $Δ$ which is linearly colored with $k$ colors includes a subcomplex $Δ'$ with $k$ vertices such that $Δ'$ is a strong deformation retract of $Δ$. We also prove that this deformation is a nonevasive reduction, in particular, a collapsing.
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…
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.
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…
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.
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 …
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 …
We define two different simplicial complexes, the common divisor simplicial complex and the prime divisor simplicial complex, from a set of integers, and explore their similarities. We will define a map between the two simplicial complexes, and use this map to show that for any set of integers, the fundamental groups of the resulting simplicial complexes are isomorphic.
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 $\mathbb{Z}/p$.
In this paper we develop further the multi-parameter model of random simplicial complexes. Firstly, we give an intrinsic characterisation of the multi-parameter probability measure. Secondly, we show that in multi-parameter random simplicial complexes the links of simplexes and their intersections are also multi-parameter random simplicial complexes. Thirdly, we find conditions under which a multi-parameter random simplicial complex is connected and simply connected.
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…
In this paper we introduce a new model of random simplicial complexes depending on multiple probability parameters. This model includes the well-known Linial - Meshulam random simplicial complexes and random clique complexes as special cases. Topological and geometric properties of a multi-parameter random simplicial complex depend on the whole combination of the probability parameters and the thresholds for topological properties are convex sets rather than numbers (as in all previously known models). We discuss the containment properties, density domains and dimension of the random simplicia…
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 $n$.
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…
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…
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…
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.
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.
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…
We define a variant of Benjamini-Schramm convergence for finite simplicial complexes with the action of a fixed finite group G which leads to the notion of random rooted simplicial G-complexes. For every random rooted simplicial G-complex we define a corresponding $\ell^2$-homology and the $\ell^2$-multiplicity of an irreducible representation of G in the homology. The $\ell^2$-multiplicities generalize the $\ell^2$-Betti numbers and we show that they are continuous on the space of sofic random rooted simplicial G-complexes. In addition, we study induction of random rooted complexes and discus…
Let $ρ:(D^2)^m\to I^m$ be the orbit map for the diagonal action of the torus $T^m$ on the unit poly-disk $(D^2)^m$, $I^m=[0,1]^m$ is the unit cube. Let $C$ be a cubical subcomplex in $I^m$. The moment-angle complex $\ma(C)$ is a $T^m$-invariant bigraded cellular decomposition of the subset $ρ^{-1}(C)\subset(D^2)^m$ with cells corresponding to the faces of $C$. Different combinatorial problems concerning cubical complexes and related combinatorial objects can be treated by studying the equivariant topology of corresponding moment-angle complexes. Here we consider moment-angle complexes defined …
Buchstaber invariant is a numerical characteristic of a simplicial complex, arising from torus actions on moment-angle complexes. In the paper we study the relation between Buchstaber invariants and classical invariants of simplicial complexes such as bigraded Betti numbers and chromatic invariants. The following two statements are proved. (1) There exists a simplicial complex U with different real and ordinary Buchstaber invariants. (2) There exist two simplicial complexes with equal bigraded Betti numbers and chromatic numbers, but different Buchstaber invariants. To prove the first theorem …
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.
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.
The random $2$-dimensional simplicial complex process starts with a complete graph on $n$ vertices, and in every step a new $2$-dimensional face, chosen uniformly at random, is added. We prove that with probability tending to $1$ as $n\to\infty$, the first homology group over $\mathbb Z$ vanishes at the very moment when all the edges are covered by triangular faces.
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…
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.
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…
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.
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.
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…
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…
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…
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.
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…
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.
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.
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.
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 …
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.
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
