Related papers: Shellability and Regularity of Chain Complexes ove…
We introduce a notion of pure-minimality for chain complexes of modules and show that it coincides with (homotopic) minimality in standard settings, while being a more useful notion for complexes of flat modules. As applications, we…
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…
For any simplicial complex $X$ with a total ordering of its vertices, one can construct a chain complex $\mathbb{L}_\bullet(X)$ generated by necklaces of simplices in $X$, which computes the homology of the free loop space of the geometric…
We define a chain complex for generalized splines on graphs, analogous to that introduced by Billera and refined by Schenck-Stillman for splines on polyhedral complexes. The hyperhomology of this chain complex yields bounds on the…
This thesis addresses the theory of topological spaces and the foundations of persistence theory. We will discuss chain complexes and the associated simplicial homology groups, as well as their relationship with singular homology theory.…
The purpose of this paper is to introduce a version of singular homology based on smooth mappings of manifolds with corners. Although variants of such a theory exists in the literature, we felt that certain points were not adequately…
Various simplicial complexes can be associated with a graph. Box complexes form an important families of such simplicial complexes and are especially useful for providing lower bounds on the chromatic number of the graph via some of their…
We introduce the class of linearly shellable pure simplicial complexes. The characterizing property is the existence of a labeling of their vertices such that all linear extensions of the Bruhat order on the set of facets are shelling…
Complex networks can be understood as graphs whose connectivity deviates from those of regular or near-regular graphs, which are understood as being `simple'. While a great deal of the attention so far dedicated to complex networks has been…
Spencer cohomology theory studies the cohomology of chain complexes of modules over the ring of differential operators $\mathscr{D}$ of a smooth analytic space. In this paper we give a generalisation of Spencer cohomology suitable for…
A special subcomplex of the singular chain complex for a topological space, historically called oriented singular chain complex is used here with the new name "alternative" singular chain complex. It was already known that this subcomplex…
The idea of transversality is explored in the construction of cohomology theory associated to regularized sequences of multiple products of rational functions associated to vertex algebra cohomology of codimension one foliations on complex…
Symmetric homology is an analog of cyclic homology in which the cyclic groups are replaced by symmetric groups. The foundations for the theory of symmetric homology of algebras are developed in the context of crossed simplicial groups using…
Intersection homology is defined for simplicial, singular and PL chains and it is well known that the three versions are isomorphic for a full filtered simplicial complex. In the literature, the isomorphism, between the singular and the…
Motivated by questions in biological classification, we discuss some elementary combinatorial and computational properties of certain set systems that generalize hierarchies, namely, 'patchworks', 'weak patchworks', 'ample patchworks' and…
The main question we target is the following: If one fixes a topological type of a complex normal surface singularity then what are the possible analytic types supported by it, and/or, what are the possible values of the geometric genus? We…
Persistent homology was shown by Carlsson and Zomorodian to be homology of graded chain complexes with coefficients in the graded ring $\kk[t]$. As such, the behavior of persistence modules -- graded modules over $\kk[t]$ is an important…
We use the theory of lexicographic shellability to provide various examples in which the rank of the homology of a Rees product of two partially ordered sets enumerates some set of combinatorial objects, perhaps according to some natural…
We give a simple algebraic description of opetopes in terms of chain complexes, and we show how this description is related to combinatorial descriptions in terms of treelike structures. More generally, we show that the chain complexes…
Although random cell complexes occur throughout the physical sciences, there does not appear to be a standard way to quantify their statistical similarities and differences. The various proposals in the literature are usually motivated by…