We study the multiscale simplicial flat norm (MSFN) problem, which computes flat norm at various scales of sets defined as oriented subcomplexes of finite simplicial complexes in arbitrary dimensions. We show that the multiscale simplicial flat norm is NP-complete when homology is defined over integers. We cast the multiscale simplicial flat norm as an instance of integer linear optimization. F...

For a simplicial complex on {1, 2, . . . , n} we define enriched homology and cohomology modules. They are graded modules over k[x1, . . . , xn] whose ranks are equal to the dimensions of the reduced homology and cohomology groups. We characterize Cohen-Macaulay, l-Cohen-Macaulay, Buchsbaum, and Gorenstein∗ complexes , and also orientable homology manifolds in terms of the enriched modules. We ...

Suppose that K and L are simplicial complexes. Recall that a vertex map between these complexes is a function φ : V ert(K) → V ert(L) such that the vertices of a simplex in K map to the vertices of a simplex in L. Given such a φ, we can create a simplicial map f : |K| → |L| by linearly extending φ over each simplex. On the other hand, suppose we have an arbitrary continuous map g : |K| → |L|. T...

Simplicial Complexes Karol Pąk Institute of Computer Science University of Białystok Poland Summary. In this article we define the notion of abstract simplicial complexes and operations on them. We introduce the following basic notions: simplex, face, vertex, degree, skeleton, subdivision and substructure, and prove some of their properties. MML identifier: SIMPLEX0, version: 7.11.01 4.117.1046...

A foundation is laid for a theory of combinatorial groupoids, allowing us to use concepts like “holonomy”, “parallel transport”, “bundles”, “combinatorial curvature” etc. in the context of simplicial (polyhedral) complexes, posets, graphs, polytopes and other combinatorial objects. A new, holonomy-type invariant for cubical complexes is introduced, leading to a combinatorial “Theorema Egregium”...

We consider simplicial polytopes, and more general simplicial complexes, without missing faces above a fixed dimension. Sharp analogues of McMullen’s generalized lower bounds, and of Barnette’s lower bounds, are conjectured for these families of complexes. Some partial results on these conjectures are presented.

Scattered over the past few years have been several occurrences of simplicial complexes whose topological behavior characterize the Cohen–Macaulay property for quotients of polynomial rings by arbitrary (not necessarily squarefree) monomial ideals. It is unclear whether researchers thinking about this topic have, to this point, been aware of the full spectrum of related developments. Therefore,...

The notion of simplicial complexes is usually introduced at the graduate level, as part of the study of algebraic topology. So the main objective of this paper is to organize and present materials found in the literature so that it is accessible to students at the undergraduate level. We first generalize linear algebra to affine linear algebra. Then using these results, we define, and subsequen...

Abstract Exponential random graph models have attracted significant research attention over the past decades. These models are maximum-entropy ensembles subject to the constraints that the expected values of a set of graph observables are equal to given values. Here we extend these maximum-entropy ensembles to random simplicial complexes, which are more adequate and versatile constructions to m...

In the first lecture, we looked at concepts from point set topology, the branch of topology that studies continuity from an analytical point of view. This view does not have a computational nature: we cannot represent infinite point sets or their associated infinite open sets on a computer. Starting with this lecture, we will look at concepts from another major branch of topology: combinatorial...