Related papers: Simplicial Homology Groups
Hypergraphs, as a generalization of simplicial complexes, have long been a subject of interest in their geometric interpretation. The subdivision of simplicial complexes can, to some extent, provide insights into the geometry of simplicial…
We introduce a combinatorial characterization of simpliciality for arrangements of hyperplanes. We then give a sharp upper bound for the number of hyperplanes of such an arrangement in the projective plane over a finite field, and present…
We study when co-evolving (or adaptive) higher-order networks defined on directed hypergraphs admit a simplicial description. Binary and triadic couplings are modelled by time-dependent weight tensors. Using representation theory of the…
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…
We present definitions of homology groups associated to a family of amalgamation functors. We show that if the generalized amalgamation properties hold, then the homology groups are trivial. We compute the group H_2 for strong types in…
The theory of abelian categories proved very useful, providing an axiomatic framework for homology and cohomology of modules over a ring and, in particular, of abelian groups. For many years, a similar categorical framework has been lacking…
Symmetry is usually defined via transformations described by a (higher) group. But a symmetry really corresponds to an algebra of local symmetric operators, which directly constrains the properties of the system. In this paper, we point out…
In the last years complex networks tools contributed to provide insights on the structure of research, through the study of collaboration, citation and co-occurrence networks. The network approach focuses on pairwise relationships, often…
The paper is devoted to an approach to the bounded cohomology theory based on the theories of simplicial sets and Postnikov systems. In particular, the main results of the bounded cohomology theory of topological spaces are extended to…
We provide a foundation for working with homological and homotopical methods in categorical algebra. This involves two mutually complementary components, namely (a) the strategic selection of suitable axiomatic frameworks, some well known…
We modify a previous result, which showed that certain diagrams of spaces are essentially simplicial monoids, to construct diagrams of spaces which model simplicial groups. Furthermore, we show that these diagrams can be generalized to…
This work concerns finite free complexes over commutative noetherian rings, in particular over group algebras of elementary abelian groups. The main contribution is the construction of complexes such that the total rank of their underlying…
The homological scaffold leverages persistent homology to construct a topologically sound summary of a weighted network. However, its crucial dependency on the choice of representative cycles hinders the ability to trace back global…
In this paper we look at the notion of cohomological triviality of fibrations of homogeneous spaces of affine algebraic groups defined over $\mathbb{C}$ and use topological methods, primarily the theory of covering spaces. This is made…
Complex braid groups are the natural generalizations of braid groups associated to arbitrary (finite) complex reflection groups. We investigate several methods for computing the homology of these groups. In particular, we get the Poincar\'e…
{\tt AbstractSimplicialComplexes.m2} is a computer algebra package written for the computer algebra system {\tt Macaulay2} \cite{M2}. It provides new infrastructure to work with abstract simplicial complexes and related homological…
In this paper we introduce a method which allows us to study properties of the random uniform simplicial complex. That is, we assign equal probability to all simplicial complexes with a given number of vertices and then consider properties…
Given a manifold with corners $X$, we associates to it the corner structure simplicial complex $\Sigma_X$. Its reduced K-homology is isomorphic to the K-theory of the $C^*$-algebra $\mathcal{K}_b(X)$ of b-compact operators on $X$. Moreover,…
Complex networks can be used to represent and model an ample diversity of abstract and real-world systems and structures. A good deal of the research on these structures has focused on specific topological properties, including node degree,…
Hypergraphs and simplical complexes both capture the higher-order interactions of complex systems, ranging from higher-order collaboration networks to brain networks. One open problem in the field is what should drive the choice of the…