We develop a framework for computing the homology of weighted simplicial complexes with coefficients in a discrete valuation ring. A weighted simplicial complex, $(X,v)$, introduced by Dawson [Cah. Topol. Géom. Différ. Catég. 31 (1990), pp. 229--243], is a simplicial complex, $X$, together with an integer-valued function, $v$, assigning weights to simplices, such that the weight of any of faces are monotonously increasing. In addition, weighted homology, $H_n^v(X)$, features a new boundary operator, $\partial_n^v$. In difference to Dawson, our approach is centered at a natural homomorphism $θ$ of weighted chain complexes. The key object is $H^v_{n}(X/θ)$, the weighted homology of a quotient of chain complexes induced by $θ$, appearing in a long exact sequence linking weighted homologies with different weights. We shall construct bases for the kernel and image of the weighted boundary map, identifying $n$-simplices as either $κ_n$- or $μ_n$-vertices. Long exact sequences of weighted homology groups and the bases, allow us to prove a structure theorem for the weighted simplicial homology with coefficients in a ring of formal power series $R=\mathbb{F}[[π]]$, where $\mathbb{F}$ is a field. Relative to simplicial homology new torsion arises and we shall show that the torsion modules are connected to a pairing between distinguished $κ_n$ and $μ_{n+1}$ simplices.
We study random simplicial complexes in the multi-parameter upper model. In this model simplices of various dimensions are taken randomly and independently, and our random simplicial complex $Y$ is then taken to be the minimal simplicial complex containing this collection of simplices. We study the asymptotic behavior of the homology of $Y$ as the number of vertices goes to $\infty$. We observe the following phenomenon asymptotically almost surely. The given probabilities with which the simplices are taken determine a range of dimensions $\ell \leq k \leq \ell'$ with $\ell' \leq 2\ell +1$, out…
The concepts of a dihedral and a reflexive module with $\infty$-simplicial faces are introduced. For each involutive $A_\infty$-algebra, the dihedral and the reflexive tensor modules with $\infty$-simplicial faces are constructed. On the basis of dihedral and reflexive modules with $\infty$-simplicial faces that defined by an involutive $A_\infty$-algebra the constructions of the dihedral and the reflexive homology of involutive $A_\infty$-algebras over any unital commutative rings are given. The conception of an involutive homotopy unital $A_\infty$-algebra is introduced. A long exact sequenc…
In this paper we prove results relating to four homology theories developed in the topology of digital images: a simplicial homology theory by Arslan et al which is the homology of the clique complex, a singular simplicial homology theory by Lee, a cubical homology theory by Jamil and Ali, and a new kind of cubical homology for digital images with $c_1$-adjacency which is easily computed, and generalizes a construction by Karaca & Ege. We show that the two simplicial homology theories are isomorphic, and that the two cubical theories are isomorphic, but that the simplicial theory is not isomor…
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 $n\to\infty$, the first homology group over $\mathbb Z$ vanishes at the very moment when all the edges are covered by triangular faces.
We provide a bottom up construction of torsion generators for weighted homology of a weighted complex over a discrete valuation ring $R=\mathbb{F}[[π]]$. This is achieved by starting from a basis for classical homology of the $n$-th skeleton for the underlying complex with coefficients in the residue field $\mathbb{F}$ and then lifting it to a basis for the weighted homology with coefficients in the ring $R$. Using the latter, a bijection is established between $n+1$ and $n$ dimensional simplices whose weight ratios provide the exponents of the $π$-monomials that generate each torsion summand …
The purpose of this note is to point out that simplicial methods and the well-known Dold-Kan construction in simplicial homotopy theory can be fruitfully applied to convert link homology theories into homotopy theories. Dold and Kan prove that there is a functor from the category of chain complexes over a commutative ring with unit to the category of simplicial objects over that ring such that chain homotopic maps go to homotopic maps in the simplicial category. Furthermore, this is an equivalence of categories. In this way, given a link homology theory, we construct a mapping taking link diag…
The central problem in computational algebraic topology is the computation of the homotopy groups of a given space, represented as a simplicial set. Algorithms have been found which achieve this, but the running times depend on the size of the input simplicial set. In order to reduce this dependence on the simplicial set chosen, we describe in this paper a procedure which, given a prime $p$ and a finite, simply-connected simplicial set with finite integral homology, finds a $p$-locally equivalent simplicial set with size upper bounded by a function of dimension and homology. Using this in conj…
This survey describes some useful properties of the local homology of abstract simplicial complexes. Although the existing literature on local homology is somewhat dispersed, it is largely dedicated to the study of manifolds, submanifolds, or samplings thereof. While this is a vital perspective, the focus of this survey is squarely on the local homology of abstract simplicial complexes. Our motivation comes from the needs of the analysis of hypergraphs and graphs. In addition to presenting many classical facts in a unified way, this survey presents a few new results about how local homology ge…
We show that, for a simplicial complex, the supported cap product operation on Borel-Moore homology coincides with the supported cap product on simplicial homology. For this purpose we introduce the supported cap product for locally finite singular homology, and compare the cap product on the three homology theories.
For $d \geq 2$ and $G$ a finite abelian group, define $T_d(G)$ to be the minimum number of vertices $n$ so that there exists a simplicial complex $X$ on $n$ vertices which has the torsion part of $H_{d - 1}(X)$ isomorphic to $G$. Here we establish an upper bound on $T_d(G)$ which matches the known lower bound up to a constant factor. That is, we prove that for every $d \geq 2$ there exist constants $c_d$ and $C_d$ so that for any finite abelian group $c_d(\log |G|)^{1/d} \leq T_d(G) \leq C_d(\log |G|)^{1/d}.$
Weighted hypergraphs are generalizations of weighted simplicial complexes. In recent years, weighted Laplacians of weighted simplicial complexes have been studied. In 2016, as a generalization of the homology of simplicial complexes, the embedded homology of hypergraphs was constructed. In this paper, we generalize the weighted Laplacians of weighted simplicial complexes to weighted hypergraphs. We study the relations between the weighted Laplacians and the weighted embedded homology of weighted hypergraphs. We generalize the Hodge decompositions of weighted simplicial complexes to weighted hy…
The University of Edinburgh（爱丁堡大学）；University of Oxford（牛津大学）
Simplicial complexes can be viewed as high dimensional generalizations of graphs that explicitly encode multi-way ordered relations between vertices at different resolutions, all at once. This concept is central towards detection of higher dimensional topological features of data, features to which graphs, encoding only pairwise relationships, remain oblivious. While attempts have been made to extend Graph Neural Networks (GNNs) to a simplicial complex setting, the methods do not inherently exploit, or reason about, the underlying topological structure of the network. We propose a graph convol…
In this paper we prove results relating to two homotopy relations and four homology theories developed in the topology of digital images. We introduce a new type of homotopy relation for digitally continuous functions which we call "strong homotopy." Both digital homotopy and strong homotopy are natural digitizations of classical topological homotopy: the difference between them is analogous to the difference between digital 4-adjacency and 8-adjacency in the plane. We also consider four different digital homology theories: a simplicial homology theory by Arslan et al which is the homology of …
We introduce a filtration on the simplicial homology of a finite simplicial complex X using bi-colourings of its vertices. This yields two dual homology theories closely related to discrete Morse matchings on X. We give an explicit expression for the associated graded object of these homologies when X is the matching complex of the Tait graph of a plane graph $G$, in terms of subgraphs determined by certain matchings on the dual of G. We then use one of these homologies, in the case where X is a graph, to define a conjecturally optimal dissimilarity pseudometric for graphs; we prove various re…
In this paper, we study Forman's discrete Morse theory in the context of weighted homology. We develop weighted versions of classical theorems in discrete Morse theory. A key difference in the weighted case is that simplicial collapses do not necessarily preserve weighted homology. We work out some sufficient conditions for collapses to preserve weighted homology, as well as study the effect of elementary removals on weighted homology. An application to sequence analysis is included, where we study the weighted ordered complexes of sequences.
In this paper we develop the theory of weighted persistent homology. In 1990, Robert J. Dawson was the first to study in depth the homology of weighted simplicial complexes. We generalize the definitions of weighted simplicial complex and the homology of weighted simplicial complex to allow weights in an integral domain $R$. Then we study the resulting weighted persistent homology. We show that weighted persistent homology can tell apart filtrations that ordinary persistent homology does not distinguish. For example, if there is a point considered as special, weighted persistent homology can t…
We show that the Connes-Consani semi-norm on singular homology with real coefficients, defined via s-modules, coincides with the ordinary $\ell^1$-semi-norm on singular homology in all dimensions.
For G a Lie group acting on a symplectic manifold $(M,ω)$ preserving a pair of Lagrangians $L_0$, $L_1$, under certain hypotheses not including equivariant transversality we construct a G-equivariant Floer cohomology of $L_0$ and $L_1$.
The Leray spectral sequence of a map $f$ computes the homology of the domain of $f$ from the fibers of $f$. In this expository paper, we relate in full detail the Leray spectral sequence associated to a simplicial map $f$ to the Leray cosheaves of $f$. We then give applications to level set persistent homology and Reeb spaces.
To any finite simplicial complex X, we associate a natural filtration starting from Chari and Joswig's discrete Morse complex and abutting to the matching complex of X. This construction leads to the definition of several homology theories, which we compute in a number of examples. We also completely determine the graded object associated to this filtration in terms of the homology of simpler complexes. This last result provides some connections to the number of vertex-disjoint cycles of a graph.
The fiber of multiparameter persistent homology for simplicial complexes
Mathematical Institute, University of Oxford（牛津大学数学研究所）；Max Planck Institute of Molecular Cell Biology and Genetics（马克斯·普朗克分子细胞生物学与遗传学研究所）；Centre for Systems Biology Dresden（德累斯顿系统生物学中心）；Isaac Newton Institute for Mathematical Sciences, University of Cambridge（剑桥大学牛顿数学科学研究所）
We study the inverse problem for persistent homology: For a fixed simplicial complex $K$, we analyse the fiber of the continuous map $\mathrm{PH}$ on the space of filters that assigns to a filter $f: K \to \mathbb R$ the total barcode of its associated sublevel set filtration of $K$. We find that $\mathrm{PH}$ is best understood as a map of stratified spaces. Over each stratum of the barcode space, the map $\mathrm{PH}$ restricts to a (trivial) fiber bundle with fiber a polyhedral complex. Amongst other we derive a bound for the dimension of the fiber depending on the number of distinct endpoi…
A hypergraph can be obtained from a simplicial complex by deleting some non-maximal simplices. By [11], a hypergraph gives an associated simplicial complex. By [4], the embedded homology of a hypergraph is the homology of the infimum chain complex, or equivalently, the homology of the supremum chain complex. In this paper, we generalize the discrete Morse theory for simplicial complexes by R. Forman [5-7] and give a discrete Morse theory for hypergraphs. We use the critical simplices of the associated simplicial complex to construct a sub-chain complex of the infimum chain complex and a sub-ch…
We generalize some of the fundamental results of algebraic topology from topological spaces to Čech's closure spaces, also known as pretopological spaces. Using simplicial sets and cubical sets with connections, we define three distinct singular (relative) simplicial and six distinct singular (relative) cubical (co)homology groups of closure spaces. Using acyclic models we show that the three simplicial groups have isomorphic cubical analogues among the six cubical groups. Thus, we obtain a total of six distinct singular (co)homology groups of closure spaces. Each of these is shown to have a c…
