Simplicial homology manifolds are proposed as an interesting class of geometric objects, more general than topological manifolds but still quite tractable, in which questions about the microstructure of space-time can be naturally formulated. Their string orientations are classified by $H^3$ with coefficients in an extension of the usual group of D-brane charges, by cobordism classes of homology three-spheres with trivial Rokhlin invariant.
A careful account is given of generalized equivariant homology theories on the category of topological pairs acted on by a group. In particular, upon restriction to the category of equivariant simplicial complexes, the equivalence of equivariant simplicial homology (also known as Bredon homology), the second derived term of the Atiyah--Hirzebruch spectral sequence, and equivariant singular homology is demonstrated.
In this paper, an algorithm building the effective homology version of the pushout of the simplicial morphisms $f:X\rightarrow Y$ and $g:X\rightarrow Z$, where $X,Y$ and $Z$ are simplicial sets with effective homology is presented.
In this paper, we consider homology groups induced by the exterior algebra generated by a simplicial compliment of a simplicial complex $K$. These homology groups are isomorphic to the Tor-groups $\mathrm{Tor}_{i, J}^{\mathbf{k}[m]}(\mathbf{k}(K),\mathbf{k})$ of the face ring $\mathbf{k}(K)$, which is very useful and much studied in toric topology. By using $\check{C}ech$ homology theory and Alexander duality theorem, we prove that these homology groups have dualities with the simplicial cohomology groups of the full subcomplexes of $K$. Then we give a new proof of Hochster's theorem.
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 extend the notion of simplicial set with effective homology to diagrams of simplicial sets. Further, for a given finite diagram of simplicial sets $X \colon \mathcal{I} \to \mathsf{sSet}$ such that each simplicial set $X(i)$ has effective homology, we present an algorithm computing the homotopy colimit $\mathsf{hocolim} X$ as a simplicial set with effective homology. We also give an algorithm computing the cofibrant replacement $X^\mathsf{cof}$ of $X$ as a diagram with effective homology. This is applied to computing of equivariant cohomology operations.
A simplicial complement P is a sequence of subsets of [m] and the simplicial complement P corresponds to a unique simplicial complex K with vertices in [m]. In this paper, we defined the homology of a simplicial complement $H_{i,σ}(Λ^{*,*}[P], d)$ over a principle ideal domain k and proved that $H_{*,*}(Λ[P], d)$ is isomorphic to the Tor of the corresponding face ring k(K) by the Taylor resolutions. As applications, we give methods to compute the ring structure of Tor_{*,*}^{k[x]}(k(K), k)$, $link_{K}σ$, $star_{K}σ$ and the cohomology of the generalized moment-angle complexes.
The use of homological and homotopical devices, such as Tor and André-Quillen homology, have found substantial use in characterizing commutative algebras. The primary category setting has been differentially graded algebras and modules, but recently simplicial categories have also proved to be useful settings. In this paper, we take this point of view up a notch by extending some recent uses of homological algebra in characterizing Noetherian commutative algebras to characterizing simplicial commutative algebras having finite Noetherian homotopy through the use of simplicial homotopy theory. T…
In this paper we introduce the notion of a $d$-dimensional cycle which is a homological generalization of the idea of a graph cycle to higher dimensions. We examine both the combinatorial and homological properties of this structure and use these results to describe the relationship between the combinatorial structure of a simplicial complex and its simplicial homology. In particular, we show that over any field of characteristic 2 the existence of non-zero $d$-dimensional homology corresponds exactly to the presence of a $d$-dimensional cycle in the simplicial complex. We also show that $d$-d…
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 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.
Let $A$ be either a simplicial complex $K$ or a small category $\mathcal C$ with $V(A)$ as its set of vertices or objects. We define a twisted structure on $A$ with coefficients in a simplicial group $G$ as a function $$ δ\colon V(A)\longrightarrow \operatorname{End}(G), \quad v\mapsto δ_v $$ such that $δ_v\circ δ_w=δ_w\circ δ_v$ if there exists an edge in $A$ joining $v$ with $w$ or an arrow either from $v$ to $w$ or from $w$ to $v$. We give a canonical construction of twisted simplicial group as well as twisted homology for $A$ with a given twisted structure. Also we determine the homotopy t…
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…
In this paper, we study the André-Quillen homology of simplicial commutative $\ell$-algebras, $\ell$ a field, having certain vanishing properties. When $\ell$ has non-zero characteristic, we obtain an algebraic version of a theorem of J.-P. Serre and Y. Umeda that characterizes such simplicial algebras having bounded homotopy groups. We further discuss how this theorem fails in the rational case and, as an application, indicate how the algebraic Serre theorem can be used to resolve a conjecture of D. Quillen for algebras of finite type over Noetherian rings, which have non-zero characteristic.
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 $θ$…
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 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.
The simplicial volume is a homotopy invariant of oriented closed connected manifolds measuring the efficiency of representing the fundamental class by singular chains with real coefficients. Despite of its topological nature, the simplicial volume is linked to Riemannian geometry in various ways, e.g., by the proportionality principle. The proportionality principle of simplicial volume states that the simplicial volume and the Riemannian volume are proportional for oriented closed connected Riemannian manifolds sharing the same universal Riemannian covering. Thurston indicated a proof of the p…
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.
We use the Taylor resolution of a monomial ideal to compute the Tor algebra of the Stanley-Reisner ring of a simplicial complement of a simplicial complex.
We give a proof of the Jardine-Tillmann generalized group completion theorem. It is much in the spirit of the original homology fibration approach by McDuff and Segal, but follows a modern treatment of homotopy colimits, using as little simplicial technology as possible. We compare simplicial and topological definitions of homology fibrations.
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 $L_1$.
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.
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.
L. Avramov, following D. Quillen, posed a conjecture to the effect that if $R \to A$ is a homomorphism of Noetherian rings then the André-Quillen homology on the category of A-modules satisfies: $D_{s}(A|R;-) = 0$ for $s\gg 0$ implies $D_{s}(A|R;-) = 0$ for s>2. In an earlier paper, the author posed an extended version of this conjecture which considered A to be a simplicial commutative R-algebra with Noetherian homotopy such that the characteristic of $π_{0}A$ is non-zero. In addition, a homotopy characterization of such algebras was described. The main goal of this paper is to develop a stra…
Measure homology is a variation of singular homology designed by Thurston in his discussion of simplicial volume. Zastrow and Hansen showed independently that singular homology (with real coefficients) and measure homology coincide algebraically on the category of CW-complexes. It is the aim of this paper to prove that this isomorphism is isometric with respect to the l^1-seminorm on singular homology and the seminorm on measure homology induced by the total variation. This, in particular, implies that one can calculate the simplicial volume via measure homology -- as already claimed by Thurst…
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）
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 $c_d(\log |G|)^{1/d} \leq T_d(G) \leq C_d(\log |G|)^{1/d}.$
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…
Given a Poisson process on a $d$-dimensional torus, its random geometric simplicial complex is the complex whose vertices are the points of the Poisson process and simplices are given by the uCech complex associated to the coverage of each point. By means of Malliavin calculus, we compute explicitly the n$th$ order moment of the number of $k$-simplices. The two first order moments of this quantity allow us to find the mean and the variance of the Euler caracteristic. Also, we show that the number of any connected geometric simplicial complex converges to the Gaussian law when the intensity of …
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 …
\newcommand{\rhomi}[1]{\widetilde{H}_{#1}} \newcommand{\rbeti}[1]{β_{#1}} \newcommand{\kk}{\mathbf k} \newcommand{\dimk}{\dim_{\kk}} We show that algebraically shifting a pair of simplicial complexes weakly increases their relative homology Betti numbers in every dimension. More precisely, let $Δ(K)$ denote the algebraically shifted complex of simplicial complex $K$, and let $\rbeti{j}(K,L)=\dimk \rhomi{j}(K,L;\kk)$ be the dimension of the $j$th reduced relative homology group over a field $\kk$ of a pair of simplicial complexes $L \subseteq K$. Then $\rbeti{j}(K,L) \leq \rbeti{j}(Δ(K),Δ(L))$ …
In this paper, we study simplicial commutative algebras with finite André-Quillen homology. Here we restrict our focus to simplicial algebras having characteristic 2. Our aim is to find a generalization of results established by the author. Our goal is to replace the finiteness condition on homotopy with a weaker condition expressed in terms of nilpotency for the action of the homotopy operations. Coupled with the finiteness assumption on André-Quillen homology, this nilpotency condition provides a way to bound the height at which the homology vanishes. As a consequence, we establish a special…
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…
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…
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 Image-Computing Spectral Sequence computes the homology of the image of a finite map from the alternating homology of the multiple point spaces of the map. A related spectral sequence was obtained by Gabrielov, Vorobjob and Zell which computes the homology of the image of a closed map from the homology of $k$-fold fibred products of the map. We give new proofs of these results, in case the map can be triangulated. Thanks to work of Hardt, this holds for a very wide range of maps, and in particular for most of the finite maps of interest in singularity theory. The proof seems conceptually s…
We define a variant of Benjamini-Schramm convergence for finite simplicial complexes with the action of a fixed finite group G which leads to the notion of random rooted simplicial G-complexes. For every random rooted simplicial G-complex we define a corresponding $\ell^2$-homology and the $\ell^2$-multiplicity of an irreducible representation of G in the homology. The $\ell^2$-multiplicities generalize the $\ell^2$-Betti numbers and we show that they are continuous on the space of sofic random rooted simplicial G-complexes. In addition, we study induction of random rooted complexes and discus…
In many scientific and technological contexts we have only a poor understanding of the structure and details of appropriate mathematical models. We often, therefore, need to compare different models. With available data we can use formal statistical model selection to compare and contrast the ability of different mathematical models to describe such data. There is, however, a lack of rigorous methods to compare different models \emph{a priori}. Here we develop and illustrate two such approaches that allow us to compare model structures in a systematic way {by representing models in terms of si…
Hyperoctahedral homology for involutive algebras is the homology theory associated to the hyperoctahedral crossed simplicial group. It is related to equivariant stable homotopy theory via the homology of equivariant infinite loop spaces. In this paper we show that there is an E-infinity algebra structure on the simplicial module that computes hyperoctahedral homology. We deduce that hyperoctahedral homology admits Dyer-Lashof homology operations. Furthermore, there is a Pontryagin product which gives hyperoctahedral homology the structure of an associative, graded-commutative algebra. We also …
