Related papers: Representation asymptotics in the homology of pure…
We give explicit formulas for the asymptotic Betti numbers of the unordered configuration spaces of an arbitrary finite graph over an arbitrary field.
We compute the homology of the matching complex $M(\Gamma)$, where $\Gamma$ is the complete hypergraph on $n\geq 2$ vertices, and analyse the $S_n$-representations carried by this homology. These results are achieved using standard…
In this paper we introduce the concept of characteristic number that are proven to be useful in the study of the combinatorics of graph cohomology. We claim that it is a good combinatorial counterpart for geometric Betti numbers. We then…
We study the homology of an explicit finite-index subgroup of the automorphism group of a partially commutative group, in the case when its defining graph is a tree. More concretely, we give a lower bound on the first Betti number of this…
We study the asymptotic behaviour of the cohomology of subgroups $\Gamma$ of an algebraic group $G$ with coefficients in the various irreducible rational representations of $G$ and raise a conjecture about it. Namely, we expect that the…
In this paper we consider the asymptotic behavior of invariants such as Betti numbers, minimal numbers of generators of singular homology, the order of the torsion subgroup of singular homology, and torsion invariants. We will show that all…
We will provide bounds on both the Betti numbers and the torsion part of the homology of hyperbolic orbifolds. These bounds are linear in the volume and are a direct consequence of an efficient simplicial model of the thick part, which we…
In this paper we study some algebraic properties of hypergraphs, in particula their Betti numbers. We define some different types of complete hypergraphs, which to the best of our knowledge, are not previously considered in the literature.…
Asymptotic normality is frequently observed in large combinatorial structures, rigorously established for many quantities such as cycles or inversions in random permutations, the number of prime factors of random integers, and various…
We investigate the cohomology of the level 4 subgroup of the braid group, namely, the kernel of the mod 4 reduction of the Burau representation at $t=-1$. This group is also equal to the kernel of the mod 2 abelianization of the pure braid…
We introduce a new notation for representing labeled regular bipartite graphs of arbitrary degree. Several enumeration problems for labeled and unlabeled regular bipartite graphs have been introduced. A general algorithm for enumerating all…
Persistent homology is a common technique in topological data analysis providing geometrical and topological information about the sample space. All this information, known as topological features, is summarized in persistence diagrams, and…
Let X be a k-dimensional simplicial complex such that the (k-j-2)-dimensional homology of the links of all j-dimensional simplices in X vanishes. An upper bound is given on the (k-1)-th Betti number of X. Examples based on sum complexes…
A multicomplex structure is defined from an ordered lattice of multigraphs. This structure will help us to observe the features of Persistent Homology in this context, its interaction with the ordering and the repercussions of the process…
We introduce a novel type of stabilization map on the configuration spaces of a graph, which increases the number of particles occupying an edge. There is an induced action on homology by the polynomial ring generated by the set of edges,…
If a (possibly finite) compact Lie group acts effectively, locally linearly, and homologically trivially on a closed, simply-connected four-manifold with second Betti number at least three, then it must be isomorphic to a subgroup of S^1 x…
The symmetric group acts on the power set and also on the set of square free polynomials. These two related representations are analyzed from the stability point of view. An application is given for the action of the symmetric group on the…
We define an algebraic setup of homology for hypergraphs, which defaults to simplicial homology in the case of graphs, and study its basic properties. As part of our study we define algebraic spanning trees of hypergraphs, along with…
The persistent Betti numbers are used in topological data analysis to infer the scales at which topological features appear and disappear in the filtration of a topological space. Most commonly by means of the corresponding barcode or…
We study homological representations of mapping class groups, including the braid groups. These arise from the twisted homology of certain configuration spaces, and come in many different flavours. Our goal is to give a unified general…