Dimensional Simplicial Complex Research Articles

Decay of scalar curvature on uniformly contractible manifolds with finite asymptotic dimension

AbstractGromov proved a quadratic decay inequality of scalar curvature for a class of complete manifolds. In this paper, we prove that for any uniformly contractible manifold with finite asymptotic dimension, its scalar curvature decays to zero at a rate depending only on the contractibility radius of the manifold and the diameter control of the asymptotic dimension. We construct examples of uniformly contractible manifolds with finite asymptotic dimension whose scalar curvature functions decay arbitrarily slowly. This shows that our result is the best possible. We prove our result by studying the index pairing between Dirac operators and compactly supported vector bundles with Lipschitz control. A key technical ingredient for the proof of our main result is a Lipschitz control for the topological K‐theory of finite dimensional simplicial complexes.

Wheels: A new criterion for non-convexity of neural codes

We introduce new geometric and combinatorial criteria that preclude a neural code from being convex, and use them to tackle the classification problem for codes on six neurons. Along the way, we give the first example of a code that is non-convex, has no local obstructions, and has simplicial complex of dimension two. We also characterize convexity for neural codes for which the simplicial complex is pure of low or high dimension.

On the dimension of dual modules of local cohomology and the Serre's condition for the unmixed Stanley–Reisner ideals of small height

In this paper, we focus on the dimension of dual modules of local cohomology of Stanley–Reisner rings to obtain a new vector. This vector contains important information on the Serre's condition (Sr) and the CMt property as well as the depth of Stanley–Reisner rings. We prove some results in this regard including lower bounds for the depth of Stanley–Reisner rings. Further, we give a characterization of (d−1)-dimensional simplicial complexes with codimension two which are (Sd−3) but they are not Cohen–Macaulay. By using this characterization, we obtain a condition to equality of projective dimension of the Stanley–Reisner rings and the arithmetical rank of their Stanley–Reisner ideals. Moreover, our characterization allows us to compute the h-vectors and give a negative answer to a known question regarding these vectors.

Linear bounds for constants in Gromov’s systolic inequality and related results

Let $M^n$ be a closed Riemannian manifold. Larry Guth proved that there exists $c(n)$ with the following property: if for some $r>0$ the volume of each metric ball of radius $r$ is less than $({r\over c(n)})^n$, then there exists a continuous map from $M^n$ to a $(n-1)$-dimensional simplicial complex such that the inverse image of each point can be covered by a metric ball of radius $r$ in $M^n$. It was previously proven by Gromov that this result implies two by now famous Gromov's inequalities: $Fill Rad(M^n)\leq c(n)vol(M^n)^{1\over n}$ and, if $M^n$ is essential, then also $sys_1(M^n)\leq 6c(n)vol(M^n)^{1\over n}$ with the same constant $c(n)$. Here $sys_1(M^n)$ denotes the length of a shortest non-contractible closed curve in $M^n$. We prove that these results hold with $c(n)=({n!\over 2})^{1\over n}\leq {n\over 2}$. We demonstrate that for essential Riemannian manifolds $sys_1(M^n) \leq n\ vol^{1\over n}(M^n)$. All previously known upper bounds for $c(n)$ were exponential in $n$. Moreover, we present a qualitative improvement: In Guth's theorem the assumption that the volume of every metric ball of radius $r$ is less than $({r\over c(n)})^n$ can be replaced by a weaker assumption that for every point $x\in M^n$ there exists a positive $\rho(x)\leq r$ such that the volume of the metric ball of radius $\rho(x)$ centered at $x$ is less than $({\rho(x)\over c(n)})^n$ (for $c(n)=({n!\over 2})^{1\over n}$). Also, if $X$ is a boundedly compact metric space such that for some $r>0$ and an integer $n\geq 1$ the $n$-dimensional Hausdorff content of each metric ball of radius $r$ in $X$ is less than $({r\over 4n})^n$, then there exists a continuous map from $X$ to a $(n-1)$-dimensional simplicial complex such that the inverse image of each point can be covered by a metric ball of radius $r$.

The NF-Number of a Simplicial Complex

Let [Formula: see text] be a simplicial complex on [Formula: see text]. The [Formula: see text]-complex of [Formula: see text] is the simplicial complex [Formula: see text] on [Formula: see text] for which the facet ideal of [Formula: see text] is equal to the Stanley–Reisner ideal of [Formula: see text]. Furthermore, for each [Formula: see text], we introduce the [Formula: see text]th [Formula: see text]-complex [Formula: see text], which is inductively defined as [Formula: see text] by setting [Formula: see text]. One can set [Formula: see text]. The [Formula: see text]-number of [Formula: see text] is the smallest integer [Formula: see text] for which [Formula: see text]. In the present paper we are especially interested in the [Formula: see text]-number of a finite graph, which can be regraded as a simplicial complex of dimension one. It is shown that the [Formula: see text]-number of the finite graph [Formula: see text] on [Formula: see text], which is the disjoint union of the complete graphs [Formula: see text] on [Formula: see text] and [Formula: see text] on [Formula: see text] for [Formula: see text] and [Formula: see text] with [Formula: see text], is equal to [Formula: see text]. As a corollary, we find that the [Formula: see text]-number of the complete bipartite graph [Formula: see text] on [Formula: see text] is also equal to [Formula: see text].

Completing and Extending Shellings of Vertex Decomposable Complexes

We say that a pure $d$-dimensional simplicial complex $\Delta$ on $n$ vertices is shelling completable if $\Delta$ can be realized as the initial sequence of some shelling of $\Delta_{n-1}^{(d)}$, the $d$-skeleton of the $(n-1)$-dimensional simplex. A well-known conjecture of Simon posits that any shellable complex is shelling completable. In this note we prove that vertex decomposable complexes are shelling completable. In fact we show that if $\Delta$ is a vertex decomposable complex then there exists an ordering of its ground set $V$ such that adding the revlex smallest missing $(d+1)$-subset of $V$ results in a complex that is again vertex decomposable. We explore applications to matroids, shifted complexes, as well as $k$-vertex decomposable complexes. We also show that if $\Delta$ is a $d$-dimensional complex on at most $d+3$ vertices then the notions of shellable, vertex decomposable, shelling completable, and extendably shellable are all equivalent.

A Robustness Study for the Extraction of Watertight Volumetric Models from Boundary Representation Data

Geometrically induced topology plays a major role in applications such as simulations, navigation, spatial or spatio-temporal analysis and many more. This article computes geometrically induced topology useful for such applications and extends previous results by presenting the unpublished used algorithms to find inner disjoint (d+1)-dimensional simplicial complexes from a set of intersecting d-dimensional simplicial complexes which partly shape their B-Reps (Boundary Representations). CityGML has been chosen as the input data format for evaluation purposes. In this case, the input data consist of planar segment complexes whose triangulated polygons serve as the set of input triangle complexes for the computation of the tetrahedral model. The creation of the volumetric model and the computation of its geometrically induced topology is partly parallelized by decomposing the input data into smaller pices. A robustness analysis of the implementations is given by varying the angular precision and the positional precision of the epsilon heuristic inaccuracy model. The results are analysed spatially and topologically, summarised and presented. It turns out that one can extract most, but not all, volumes and that the numerical issues of computational geometry produce failures as well as a variety of outcomes.

Topology of random -dimensional cubical complexes

Abstract We study a natural model of a random $2$ -dimensional cubical complex which is a subcomplex of an n-dimensional cube, and where every possible square $2$ -face is included independently with probability p. Our main result exhibits a sharp threshold $p=1/2$ for homology vanishing as $n \to \infty $ . This is a $2$ -dimensional analogue of the Burtin and Erdoős–Spencer theorems characterising the connectivity threshold for random graphs on the $1$ -skeleton of the n-dimensional cube. Our main result can also be seen as a cubical counterpart to the Linial–Meshulam theorem for random $2$ -dimensional simplicial complexes. However, the models exhibit strikingly different behaviours. We show that if $p> 1 - \sqrt {1/2} \approx 0.2929$ , then with high probability the fundamental group is a free group with one generator for every maximal $1$ -dimensional face. As a corollary, homology vanishing and simple connectivity have the same threshold, even in the strong ‘hitting time’ sense. This is in contrast with the simplicial case, where the thresholds are far apart. The proof depends on an iterative algorithm for contracting cycles – we show that with high probability, the algorithm rapidly and dramatically simplifies the fundamental group, converging after only a few steps.

Finiteness theorems for matroid complexes with prescribed topology

There are finitely many simplicial complexes (up to isomorphism) with a given number of vertices. Translating this fact to the language of h-vectors, there are finitely many simplicial complexes of bounded dimension with h1=k for any natural number k. In this paper we study the question at the other end of the h-vector: Are there only finitely many (d−1)-dimensional simplicial complexes with hd=k for any given k? The answer is no if we consider general complexes, but we focus on three cases coming from matroids: (i) independence complexes, (ii) broken circuit complexes, and (iii) order complexes of geometric lattices. Surprisingly, the answer is yes in all three cases.

Randomized Construction of Complexes with Large Diameter

We consider the question of the largest possible combinatorial diameter among pure dimensional and strongly connected (d-1)-dimensional simplicial complexes on n vertices, denoted H_s(n, d). Using a probabilistic construction we give a new lower bound on H_s(n, d) that is within an O(d^2) factor of the upper bound. This improves on the previously best known lower bound which was within a factor of e^{varTheta (d)} of the upper bound. We also make a similar improvement in the case of pseudomanifolds.

Pure $\mathcal{O}$-Sequences arising from $2$-Dimensional PS Ear-Decomposable Simplicial Complexes

We show that the $h$-vector of a $2$-dimensional PS ear-decomposable simplicial complex is a pure $\mathcal{O}$-sequence. This provides a strengthening of Stanley's conjecture for matroid $h$-vectors in rank $3$. Our approach modifies the approach of combinatorial shifting for arbitrary simplicial complexes to the setting of $2$-dimensional PS ear-decomposable complexes, which allows us to greedily construct pure multicomplex corresponding to each PS ear-decomposable complex.

Randomly Weighted $d$-Complexes: Minimal Spanning Acycles and Persistence Diagrams

A weighted $d$-complex is a simplicial complex of dimension $d$ in which each face is assigned a real-valued weight. We derive three key results here concerning persistence diagrams and minimal spanning acycles (MSAs) of such complexes. First, we establish an equivalence between the MSA face-weights and death times in the persistence diagram. Next, we show a novel stability result for the MSA face-weights which, due to our first result, also holds true for the death and birth times, separately. Our final result concerns a perturbation of a mean-field model of randomly weighted $d$-complexes. The $d$-face weights here are perturbations of some i.i.d. distribution while all the lower-dimensional faces have a weight of $0$. If the perturbations decay sufficiently quickly, we show that suitably scaled extremal nearest face-weights, face-weights of the $d$-MSA, and the associated death times converge to an inhomogeneous Poisson point process. This result completely characterizes the extremal points of persistence diagrams and MSAs. The point process convergence and the asymptotic equivalence of three point processes are new for any weighted random complex model, including even the non-perturbed case. Lastly, as a consequence of our stability result, we show that Frieze's $\zeta(3)$ limit for random minimal spanning trees and the recent extension to random MSAs by Hino and Kanazawa also hold in suitable noisy settings.

The spectral dimension of simplicial complexes: a renormalization group theory

Simplicial complexes are increasingly used to study complex system structures and dynamics including diffusion, synchronization and epidemic spreading. The spectral dimension of the graph Laplacian is known to determine the diffusion properties at long time scales. Using the renormalization group here we calculate the spectral dimension of the graph Laplacian of two classes of non-amenable d dimensional simplicial complexes: the Apollonian networks and the pseudo-fractal networks. We analyse the scaling of the spectral dimension with the topological dimension d for and we point out that randomness such as the one present in Network Geometry with Flavor can diminish the value of the spectral dimension of these structures.

Balanced Triangulations on few Vertices and an Implementation of Cross-Flips

A $d$-dimensional simplicial complex is balanced if the underlying graph is $(d+1)$-colorable. We present an implementation of cross-flips, a set of local moves introduced by Izmestiev, Klee and Novik which connect any two PL-homeomorphic balanced combinatorial manifolds. As a result we exhibit a vertex minimal balanced triangulation of the real projective plane, of the dunce hat and of the real projective space, as well as several balanced triangulations of surfaces and 3-manifolds on few vertices. In particular we construct small balanced triangulations of the 3-sphere that are non-shellable and shellable but not vertex decomposable.

An upper bound for topological complexity

In [11], a new approximating invariant TCD for topological complexity was introduced called D-topological complexity. In this paper, we explore more fully the properties of TCD and the connections between TCD and invariants of Lusternik–Schnirelmann type. We also introduce a new TC-type invariant TC˜ that can be used to give an upper bound for TC,(1)TC(X)≤TCD(X)+TC˜(X). This then entails a connectivity-dimension estimateTC(X)≤TCD(X)+⌈2dimX−kk+1⌉, where X is a finite dimensional simplicial complex with k-connected universal cover X˜. The above inequality is a refinement of an estimate given by Dranishnikov [5].

Spectral expansion of random sum complexes

Let [Formula: see text] be a finite abelian group of order [Formula: see text] and let [Formula: see text] denote the [Formula: see text]-simplex on the vertex set [Formula: see text]. The sum complex [Formula: see text] associated to a subset [Formula: see text] and [Formula: see text], is the [Formula: see text]-dimensional simplicial complex obtained by taking the full [Formula: see text]-skeleton of [Formula: see text] together with all [Formula: see text]-subsets [Formula: see text] that satisfy [Formula: see text]. Let [Formula: see text] denote the space of complex-valued [Formula: see text]-cochains of [Formula: see text]. Let [Formula: see text] denote the reduced [Formula: see text]th Laplacian of [Formula: see text], and let [Formula: see text] be the minimal eigenvalue of [Formula: see text]. It is shown that if [Formula: see text] and [Formula: see text] are fixed, and [Formula: see text] is a random subset of [Formula: see text] of size [Formula: see text], then [Formula: see text]

Lefschetz properties of balanced 3-polytopes

In this paper, we study Lefschetz properties of Artinian reductions of Stanley-Reisner rings of balanced simplicial $3$-polytopes. A $(d-1)$-dimensional simplicial complex is said to be balanced if its graph is $d$-colorable. If a simplicial complex is balanced, then its Stanley-Reisner ring has a special system of parameters induced by the coloring. We prove that the Artinian reduction of the Stanley-Reisner ring of a balanced simplicial $3$-polytope with respect to this special system of parameters has the strong Lefschetz property if the characteristic of the base field is not two or three. Moreover, we characterize $(2,1)$-balanced simplicial polytopes, i.e., polytopes with exactly one red vertex and two blue vertices in each facet, such that an analogous property holds. In fact, we show that this is the case if and only if the induced graph on the blue vertices satisfies a Laman-type combinatorial condition.

Dense power-law networks and simplicial complexes.

There is increasing evidence that dense networks occur in on-line social networks, recommendation networks and in the brain. In addition to being dense, these networks are often also scale-free, i.e., their degree distributions follow P(k)∝k^{-γ} with γ∈(1,2]. Models of growing networks have been successfully employed to produce scale-free networks using preferential attachment, however these models can only produce sparse networks as the numbers of links and nodes being added at each time step is constant. Here we present a modeling framework which produces networks that are both dense and scale-free. The mechanism by which the networks grow in this model is based on the Pitman-Yor process. Variations on the model are able to produce undirected scale-free networks with exponent γ=2 or directed networks with power-law out-degree distribution with tunable exponent γ∈(1,2). We also extend the model to that of directed two-dimensional simplicial complexes. Simplicial complexes are generalization of networks that can encode the many body interactions between the parts of a complex system and as such are becoming increasingly popular to characterize different data sets ranging from social interacting systems to the brain. Our model produces dense directed simplicial complexes with power-law distribution of the generalized out-degrees of the nodes.

On Partial Barycentric Subdivision

The $$l{\mathrm{th}}$$ partial barycentric subdivision is defined for a $$(d-1)$$ -dimensional simplicial complex $$\Delta $$ and studied along with its combinatorial and geometric aspects. We analyze the behavior of the f- and h-vector under the lth partial barycentric subdivision extending previous work of Brenti and Welker on the standard barycentric subdivision – the case $$l = d$$ . We discuss and provide properties of the transformation matrices sending the f- and h-vector of $$\Delta $$ to the f- and h-vector of its lth partial barycentric subdivision. We conclude with open problems.

Face numbers of sequentially Cohen–Macaulay complexes and Betti numbers of componentwise linear ideals

A numerical characterization is given of the h -triangles of sequentially Cohen–Macaulay simplicial complexes. This result determines the number of faces of various dimensions and codimensions that are possible 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 ≤ d and shifted pure (d–1) -dimensional simplicial complexes