Clara Loeh, Marco Moraschini
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.
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The loop group and the cobar construction
本文证明 1-约化单纯集上 Adams 的 cobar 构造是 Kan 环路群归一化链的强形变收缩核，并将结论推广到所有 0-约化单纯集，以支持代数结构扰动的同调转移。
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…
Extremal simplicial distributions on cycle scenarios with arbitrary outcomes
本文通过同调方法和测量空间分解，研究了具有任意结果的循环场景中非信号多面体的极值单纯分布特性。
Let P be a closed triangulated manifold, dim P=n. We consider the group of simplicial 1-chains C_1(P) and the homology group H_1(P). We also use some nonnegative weighting function L on C_1(P). For any homological class from H_1(P) method proposed in article builds a cycle z with minimal weight L(z). The main idea is in using a simplicial scheme of space of the regular covering with automorphism group H_1(P). We construct this covering applying index function relative to any basis of group H_{n-1}(P).
Applying language models to algebraic topology: generating simplicial cycles using multi-labeling in Wu's formula
本文利用机器学习探究球面同伦群生成元的群论结构，在Wu公式的单纯群设定下，将生成单纯循环转化为从Dyck语言相关数据集交集中采样的问题，并提出并评估了采用多标记信息的语言建模方法及非神经基线。
A $d$-hypertree on $[n]$ is a maximal acyclic $d$-dimensional simplicial complex with full $(d-1)$-skeleton on the vertex set $[n]$. Alternatively, in the language of algebraic topology, it is a minimal $d$-dimensional simplicial complex $T$ (assuming full $(d-1)$-skeleton) such that $\tilde{H}_{d-1}(T;\mathbb{F})=0$. The $d$-hypertrees are a basic object in combinatorial theory of simplicial complexes. They have been studied; and yet, many of their structural aspects remain poorly understood. In this paper we study the boundaries $\partial_d T$ of $d$-hypertrees, and the fundamental $d$-cycle…
Topological methods for studying contextuality: $N$-cycle scenarios and beyond
本文提出基于 Fourier–Motzkin 消元几何解释和测量空间坍缩的拓扑方法，重新证明 Fine 定理并推广到粘合循环场景，同时用于检测语境顶点、推导新 Bell 不等式及研究单纯分布的幺半群结构。
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.
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…
In this paper, we introduce a new method to compute magnitude homology of general graphs. To each direct sum component of magnitude chain complexes, we assign a pair of simplicial complexes whose simplicial chain complex is isomorphic to it. First we states our main theorem specialized to trees, which gives another proof for the known fact that trees are diagonal. After that, we consider general graphs, which may have cycles. We also demonstrate some computation as an application.
Let G be an algebraic group and let X be a smooth integral scheme over a field k. In this paper we construct homology-type groups $H_i(X,G)$ by considering cycles in the simplicial scheme $BG\times X (an idea suggested by Andrei Suslin). We discuss the basic properties of these groups and construct a spectral sequence, beginning with the groups $H_i(Δ^j,G)$, which converges to the etale cohomology of the simplicial group BG. These groups are therefore connected with the study of Friedlander's generalized isomorphism conjecture. <p> We also compute some examples, focusing in particular on the c…
Simplicial Homology Groups
本文自成体系地介绍了有限单纯复形的单纯同调，通过交替和公式定义边缘算子并构建同调群，结合低维实例与矩阵计算展示具体求解过程，最终证明同调不变性以连接几何特征与代数不变量。
Given a simplicial complex with weights on its simplices, and a nontrivial cycle on it, we are interested in finding the cycle with minimal weight which is homologous to the given one. Assuming that the homology is defined with integer coefficients, we show the following : For a finite simplicial complex $K$ of dimension greater than $p$, the boundary matrix $[\partial_{p+1}]$ is totally unimodular if and only if $H_p(L, L_0)$ is torsion-free, for all pure subcomplexes $L_0, L$ in $K$ of dimensions $p$ and $p+1$ respectively, where $L_0$ is a subset of $L$. Because of the total unimodularity o…
Let $K$ be a finite, connected, abstract simplicial complex. The Morse complex of $K$, first introduced by Chari and Joswig, is the simplicial complex constructed from all gradient vector fields on $K$. We show that if $K$ is neither the boundary of the $n$-simplex nor a cycle, then $\mathrm{Aut}(\mathcal{M}(K))\cong \mathrm{Aut}(K)$. In the case where $K= C_n$, a cycle of length $n$, we show that $\mathrm{Aut}(\mathcal{M}(C_n))\cong \mathrm{Aut}(C_{2n})$. In the case where $K=\partialΔ^n$, we prove that $\mathrm{Aut}(\mathcal{M}(\partialΔ^n))\cong \mathrm{Aut}(\partialΔ^n)\times \mathbb{Z}_2$…
We will show that the singular cohomology groups of a smooth quasi-projective complex variety relative to a normal crossing divisor can be described in terms of delta-admissible chains. Roughly speaking, a delta-admissible chain is a simplicial semi-algebraic chain meeting the "faces" properly. As an application, we show that the Abel-Jacobi map for higher Chow cycles can be described via delta-admissible chains. As an example, we will describe the Hodge realization of the polylog cycles constructed by Bloch-Kriz in terms of the Abel-Jacobi map.
Inferring Ambient Cycles of Point Samples on Manifolds with Universal Coverings
本文研究如何利用流形的万有覆盖和点云度量数据，判断单纯复形中的边环是否对应环境流形的非平凡同调类，并在群胚与覆盖单值化框架下给出构造性方法。
A directed persistent homology theory for dissimilarity functions
本文提出一种针对有向单纯复形、可检测奇数维持有向环的持续同调理论，建立其与经典持续同调的联系并证明稳定性。
Persistent Cycle Representatives and Generalized Landscapes for Codimension 1 Persistent Homology
本文提出了一种基于持久同调的广义景观方法，通过计算体积最优的循环代表，区分具有相似持久同调但不同形状的点云。
Given an arbitrary non-zero simplicial cycle and a generic vector coloring of its vertices, there is a way to produce a graded Poincare duality algebra associated with these data. The procedure relies on the theory of volume polynomials and multi-fans. This construction includes many important examples, such as cohomology of toric varieties and quasitoric manifolds, and Gorenstein algebras of triangulated homology manifolds, introduced by Novik and Swartz. In all these examples the dimensions of graded components of such duality algebras do not depend on the vector coloring. It was conjectured…
We present a distributed algorithm to compute the first homology of a simplicial complex. Such algorithms are very useful in topological analysis of sensor networks, such as its coverage properties. We employ spanning trees to compute a basis for algebraic 1-cycles, and then use harmonics to efficiently identify the contractible and homologous cycles. The computational complexity of the algorithm is $O(|P|^ω)$, where $|P|$ is much smaller than the number of edges, and $ω$ is the complexity order of matrix multiplication. For geometric graphs, we show using simulations that $|P|$ is very close …
We initiate the study of persistent homology of random geometric simplicial complexes. Our main interest is in maximally persistent cycles of degree-$k$ in persistent homology, for a either the \cech or the Vietoris--Rips filtration built on a uniform Poisson process of intensity $n$ in the unit cube $[0,1]^d$. This is a natural way of measuring the largest "$k$-dimensional hole" in a random point set. This problem is in the intersection of geometric probability and algebraic topology, and is naturally motivated by a probabilistic view of topological inference. We show that for all $d \ge 2$ a…
Avalanche homology of digraphs via sandpile dynamics
提出基于沙堆模型动力学的雪崩同调理论，计算有向路径和循环的雪崩复形同伦型，并引入持续雪崩同调。
Let $X$ be a $2n$-manifold with a locally standard action of a compact torus $T^n$. If the free part of action is trivial and proper faces of the orbit space $Q$ are acyclic, then there are three types of homology classes in $X$: (1) classes of face submanifolds; (2) $k$-dimensional classes of $Q$ swept by actions of subtori of dimensions $<k$; (3) relative $k$-classes of $Q$ modulo $\partial Q$ swept by actions of subtori of dimensions $\geqslant k$. The submodule of $H_*(X)$ spanned by face classes is an ideal in $H_*(X)$ with respect to the intersection product. It is isomorphic to $(\mathb…
Cycle representatives of persistent homology classes can be used to provide descriptions of topological features in data. However, the non-uniqueness of these representatives creates ambiguity and can lead to many different interpretations of the same set of classes. One approach to solving this problem is to optimize the choice of representative against some measure that is meaningful in the context of the data. In this work, we provide a study of the effectiveness and computational cost of several $\ell_1$-minimization optimization procedures for constructing homological cycle bases for pers…
Finding a cycle of lowest weight that represents a homology class in a simplicial complex is known as homology localization (HL). Here we address this NP-complete problem using parameterized complexity theory. We show that it is W[1]-hard to approximate the HL problem when it is parameterized by solution size. We have also designed and implemented two algorithms based on treewidth solving the HL problem in FPT-time. Both algorithms are ETH-tight but our results shows that one outperforms the other in practice.
Efficient algorithms for optimal homology problems and their applications
本文提出一种基于最小费用流的多尺度单纯平坦范数（MSFN）多项式时间求解算法，建立了其对偶性与最优性条件，并利用多尺度平坦范数计算平面几何网络距离，较 Hausdorff 距离更准确且能定位差异区域。
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