The amazing world of simplicial complexes

Israel Journal of Mathematics, 1994

We find a decomposition of simplicial complexes that implies and sharpens the characterization (due to Bj6rner and Kalai) of the f-vector and Betti numbers of a simplicial complex. It generalizes a result of Stanley, who proved the a~yclic case, and settles a conjecture of Stanley and Kalai.

Discrete Geometry for Computer Imagery, 2003

simplicial complexes are used in many application contexts to represent multi-dimensional, possibly non-manifold and nonuniformly dimensional, geometric objects. In this paper we introduce a new general yet compact data structure for representing simplicial complexes, which is based on a decomposition approach that we have presented in our previous work . We compare our data structure with the existing ones and we discuss in which respect it performs better than others.

1998

Advances in Applied Mathematics, 2001

This paper lays the foundations of a combinatorial homotopy theory, called A-theory, for simplicial complexes, which reflects their connectivity properties. A collection of bigraded groups is constructed, and methods for computation are given. A Seifert-Van Kampen type theorem and a long exact sequence of relative A-groups are derived. A related theory for graphs is constructed as well. This theory provides a general framework encompassing homotopy methods used to prove connectivity results about buildings, graphs, and matroids.

ArXiv, 2020

Simplicial complexes are a versatile and convenient paradigm on which to build all the tools and techniques of the logic of knowledge, on the assumption that initial epistemic models can be described in a distributed fashion. Thus, we can define: knowledge, belief, bisimulation, the group notions of mutual, distributed and common knowledge, and also dynamics in the shape of simplicial action models. We give a survey on how to interpret all such notions on simplicial complexes, building upon the foundations laid in prior work by Goubault and others.

arXiv (Cornell University), 2008

In the spirit of topological entropy we introduce new complexity functions for general dynamical systems (namely groups and semigroups acting on closed manifolds) but with an emphasis on the dynamics induced on underlying simplicial complexes. For expansive systems remarkable properties are observed. Known examples are revisited and new examples are presented. 2 From open covers to simplicial complexes Consider a compact and connected metric space, say (V, d V), and the totality of finite covers of V by open sets, that we denote by C V. One calls the members of C V open covers. If α and β are in C V , one says that α is finer than β if every element in α is contained in some element in β, and writes α ≻ β. We denote by α∩ β the refinement of α by β (or equivalently the refinement of β by α): its elements are intersections of one element from α and another from β. Given α in C V there is a canonical simplicial complex associated to α, known as the nerve 1 of α. Let △ k (α) denote the set consisting of all the k-simplices in such a complex, so that |△ k (α)| is the number of those. For example, if α is the trivial cover for V , then |△ k (α)| is equal to zero whenever k is bigger or equal than one. One says that α is irreducible if there is no β finer than α that admits a strict simplicial embedding from its nerve to the nerve of α, i.e. if there is no β finer than α so that the nerve of β is a proper sub-complex of the nerve of α. It is useful to be aware of: Lemma 2.1. For every k in N and α in C V the minimum of |△ k (β)| among those β's finer than α is obtained for irreducible β's. In particular, if α is irreducible, then the minimum mentioned above is obtained for α itself. The same is true for the sum k i=0 |△ i (β)|.

European Journal of Combinatorics

The theory of k-regular graphs is closely related to group theory. Every k-regular, bipartite graph is a Schreier graph with respect to some group G, a set of generators S (depending only on k) and a subgroup H. The goal of this paper is to begin to develop such a framework for k-regular simplicial complexes of general dimension d. Our approach does not directly generalize the concept of a Schreier graph, but still presents an extensive family of k-regular simplicial complexes as quotients of one universal object: the k-regular d-dimensional arboreal complex, which is itself a simplicial complex originating in one specific group depending only on d and k. Along the way we answer a question from [PR16] on the spectral gap of higher dimensional Laplacians and prove a high dimensional analogue of Leighton's graph covering theorem. This approach also suggests a random model for k-regular d-dimensional multicomplexes.

2021

Given an arbitrary hypergraph H, we may glue toH a family of hypergraphs to get a new hypergraph H having H as an induced subhypergraph. In this paper, we introduce three gluing techniques for which the topological and combinatorial properties (such as Cohen-Macaulayness, shellability, vertex-decomposability etc.) of the resulting hypergraph H is under control in terms of the glued components. This enables us to construct broad classes of simplicial complexes containing a given simplicial complex as induced subcomplex satisfying nice topological and combinatorial properties. Our results will be accompanied with some interesting open problems.