We study the behavior of generic initial ideals with respect to fibre products. In our main result we determine the generic initial ideal of the fibre product with respect to the reverse lexicographic order. As an application we compute the symmetric algebraic shifted complex of two disjoint simplicial complexes as was conjectured by Kalai. This result is the symmetric analogue of a theorem of ...

Simple-homotopy for simplicial and CW complexes is a special kind of topological homotopy constructed by elementary collapses and expansions. In this paper we introduce graph homotopy for graphs and Graham homotopy for hypergraphs, and study the relation between these homotopies and the simplehomotopy for simplicial complexes. The graph homotopy is useful to describe topological properties of d...

Simplicial complexes consist of a set of vertices together with designated subsets. They can be thought of as embedded in Rn with the induced metric and topology, where n is large enough that the points can all be geometrically independent. The sets of points spanned by these designated subsets then form triangles or their higher or lower dimensional analogues according to some restrictions. By...

A tower is a sequence of simplicial complexes connected by simplicial maps. We show how to compute a filtration, a sequence of nested simplicial complexes, with the same persistent barcode as the tower. Our approach is based on the coning strategy by Dey et al. (SoCG 2014). We show that a variant of this approach yields a filtration that is asymptotically only marginally larger than the tower a...

We study the Fräıssé limit of the class of all finite simplicial complexes. Whilst the natural model-theoretic setting for this class uses an infinite language, a range of results associated with Fräıssé limits of structures for finite languages carry across to this important example. We introduce the notion of a local class, with the class of finite simplicial complexes as an archetypal exampl...

We give evidence for the difficulty of computing Betti numbers of simplicial complexes over a finite field. We do this by reducing the rank computation for sparse matrices with m non-zero entries to computing Betti numbers of simplicial complexes consisting of at most a constant times m simplices. Together with the known reduction in the other direction, this implies that the two problems have ...

This paper contains a survey of the A-theory of simplicial complexes and graphs, a combinatorial homotopy theory developed recently.The initialmotivation arises from the use of simplicial complexes as models for a variety of complex systems and their dynamics. This theory diverges from classical homotopy theory in several crucial aspects. It is related to prior work in matroid theory, graph the...

Abstract 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 [3]. We compare our data structu...

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...

In the computational geometry field, simplicial complexes have been used to describe an underlying geometric shape knowing a point cloud sampled on it. In this article, an adequate statistical framework is first proposed for the choice of a simplicial complex among a parametrized family. A least squares penalized criterion is introduced to choose a complex, and a model selection theorem states ...