Gregory Z. Arone, William G. Dwyer, Kathryn Lesh
We prove that the Bredon homology or cohomology of the partition complex with fairly general coefficients is either trivial or computable in terms of constructions with the Steinberg module. The argument involves developing a theory of Bredon homology and cohomology approximation.
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We study Bredon homology approximations for spaces with an action of a compact Lie group G. We show that if M is a coMackey functor satisfying mild p-locality conditions, then Bredon homology of a G-space X with coefficients in M is determined by fixed points of p-toral subgroups of G acting on X. As an application we prove a vanishing result for the Bredon homology of the complex L_n of direct-sum decompositions of complex n-space.
We define the orbit category for transitive topological groupoids and their equivariant CW-complexes. By using these constructions we define equivariant Bredon homology and cohomology for actions of transitive topological groupoids. We show how these theories can be obtained by looking at the action of a single isotropy group on a fiber of the anchor map, extending equivariant results for compact group actions. We also show how this extension from a single isotropy group to the entire groupoid action can be applied to the structure of principal bundles and classifying spaces.
Representation-graded Bredon homology of elementary abelian 2-groups
本文计算了所有初等阿贝尔2-群在常数模2 Mackey函子系数下的表示分阶Bredon同调环，给出了基于pre-Euler类和逆Thom类的极小展示，并由此推导出几何不动点环与局部化扭曲上同调环的计算结果。
Bredon homology and equivariant K-homology of SL(3,Z)
通过计算分类空间\underline{E} SL（3，Z）关于有限子群和表示环中系数的Bredon同调，得到了SL（3，Z）的等变K-同调.对于GL（3，Z）我们也得到了相应的结果。因此，我们的计算给出了这些群体的鲍姆-康纳斯猜想的拓扑侧。
In this paper we compute the Bredon homology of wallpaper groups with respect to the family of finite groups and with coefficients in the complex representation ring. We provide explicit bases of the homology groups in terms of irreducible characters of the representation rings of the stabilizers.
For Artin groups of dihedral type, we compute the Bredon homology groups of the classifying space for the family of virtually cyclic subgroups with coefficients in the K-theory of a group ring.
Representing Bredon cohomology with local coefficients
对于离散群G，我们将局部系数的Bredon上同调表示为等参交叉复形范畴中映射的同伦类。随后，我们构造了一个朴素的参数化G-谱，使得由它定义的上同调理论在限制于悬浮谱时，可以简化为具有局部系数的Bredon上同调。
Intermediaries in Bredon (Co)homology and Classifying Spaces
对于某些可缩的G-CW-复形和G的子群族F，我们构造了一个谱序列，它收敛于G的F-Bredon上同调，其E1-项由稳定子群的F-Bredon上同调给出.作为应用，我们得到了关于族F的分类空间的上同调维数和几何维数的几个推论。我们还引入了一个分层定义的群类，它包含所有可数的初等顺从群和可数的特征为零的线性群，并证明了如果群G在这个群类中，则G有有限的F-Bredon（余）同调维数当且仅当G有跳F-Bredon（余）同调.
A Functor Converting Equivariant Homology to Homotopy
在本文中，我们证明了经典的Dold-Thom定理的一个等变版本。结合有限群、CW-复形和Bredon意义下的协变系数系，利用余端构造法构造了一个拓扑阿贝尔群.证明了这个拓扑阿贝尔群的同伦群与CW-复形的Bredon等变同调自然同构。最后，我们提出了几个例子，这个结果。
Bredon homological stability for configuration spaces of $G$-manifolds
本文研究了G-流形的无序配置空间的Bredon同调稳定性，定义了equivariant稳定映射，并在Dedekind群条件下证明了同调稳定性结果。
Localization for coarse homology theories
本文引入Bredon风格等变粗同调理论，证明其满足局部化定理，并指出一般等变粗同调理论可被其逼近，进而得到Segal局部化定理的粗类比。
Equivariant Dold-Thom topological groups
设$M$是有限群$G$的协变系数系.在本文中，我们分析了几个拓扑阿贝尔群，其中一些新的，其同伦群同构的Bredon-伊尔曼$G$ -等变同调论与系数$M$。我们称这些群为等变的Dold-Thom拓扑群，并证明了它们在同伦下是唯一的。我们利用其中一个新群证明了Bredon-Illman同调满足无穷楔形公理，并对零阶等变同调作了一些计算.
Generalized equivariant homology on simplicial complexes
一个仔细的帐户给出广义等变同调论的范畴上的拓扑对作用于一组。特别地，在限制于等变单纯复形的范畴后，证明了等变单纯同调（也称为Bredon同调）、Atiyah-Hirzebruch谱序列的第二个导出项和等变奇异同调的等价性.
Bredon cohomology is a cohomology theory that applies to topological spaces equipped with the group actions. For any group G, given a real linear representation V , the configuration space of V has a natural diagonal G-action. In the paper we study this group action on the configuration space and give a decomposition of the homology Bredon coefficient system of the configuration space and apply this to compute rational Bredon cohomology of the configuration space for small nonabelian group G.
On homological Real trace methods
该研究开发了同调实迹方法以理解实拓扑周期同调与实拓扑负循环同调的连续模2 Bredon同调，推广了Bruner和Rognes的工作，并利用该方法完成了实配边相关的同调计算。
The homology coalgebra and cohomology algebra of generalized moment-angle complexes
本文计算了广义矩角复形域上的同调余代数和上同调代数，并给出了余矩角复形的对偶定理。
Let $C_2$ be the cyclic group of order two. We show that the $RO(C_2)$-graded Bredon cohomology of a finite Rep($C_2$)-complex is free as a module over the cohomology of a point when using coefficients in the constant Mackey functor $\underline{\mathbb{F}_2}$. This paper corrects some errors in Kronholm's proof of this freeness theorem. It also extends the freeness result to finite type complexes, those with finitely many cells of each fixed-set dimension. We give a counterexample showing the theorem does not hold for locally finite complexes.
We investigate strictly developable simple complexes of groups with arbitrary local groups, or equivalently, group actions admitting a strict fundamental domain. We introduce a new method for computing the cohomology of such groups. We also generalise Bestvina's construction to obtain a polyhedral complex equivariantly homotopy equivalent to the standard development of the lowest possible dimension. As applications, for a group acting chamber transitively on a building of type $(W,S)$, we show that its Bredon cohomological dimension is equal to the virtual cohomological dimension of $W$ and gi…
Bredon sheaf cohomology
本文提出Bredon层上同调这一新的等变上同调理论，用以描述有限群作用下局部紧Hausdorff空间等变层的代数K-理论与连续函数C*-代数的等变E-理论，建立其性质并证明强唯一性定理，推广了Efimov与Clausen的相关结果。
Double homology and wedge-decomposable simplicial complexes
本文证明楔形可分解单纯复形在特定双次数处具有双重同调群$\mathbb{Z}\oplus\mathbb{Z}$。
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…
The homology of simpicial complements and the cohomology of moment-angle complexes
我们利用单项理想的泰勒分解来计算单纯复形的单纯补的Stanley-Reisner环的Tor代数。
Iterated bar complexes and E_n-homology with coefficients
第一作者在前一篇文章中证明了交换代数的n重bar构造可以推广到E_n-代数，并且利用这种迭代bar构造可以计算具有平凡系数的E_n-同调。本文将这个结果推广到系数在对称A-双模中的交换代数A的E_n-同调和E_n-上同调。
Persistent Homology with Selective Rips complexes detects geodesic circles
本文提出利用持久同调和新型选择性Rips复形检测测地空间中的测地圆与瓶颈环路，并基于局部绕数概念证明其能终止1维同调类或产生2维同调类。
Differential calculus for free algebra and constrained homology
本文研究自由代数上的微分学，将单纯复形与独立超图实现为超图的不变迹，并由此构造单纯复形滤过和独立超图滤过的约束持续（上）同调。
Stochastic Homology. Reduction Formulas for Computing Stochastic Betti Numbers of Maximal Random Complexes with Discrete Probabilities. Computation and Applications
给定一个链复形，唯一的修改是复形的每个单元都有一个指定的概率分布。我们称之为随机复合体，在实践中应该理解的是，我们有一个经典的链复形，其细胞根据某些概率分布出现和消失。本文主要研究随机复形的随机同调问题，其中单形具有独立的离散分布。
A Morse Mayer-Vietoris Sequence for Cosheaf Homology on Simplicial Complexes
该研究针对有限单纯复形上余层同调，构造迈耶-菲托里斯短正合序列并结合离散莫尔斯理论得到莫尔斯链复形的对应序列，建立其与标准序列的拟同构，还涉及范畴论视角的相关探讨。
WI-posets, graph complexes and Z_2-equivalences
在图复形的Z_2-同伦类型的研究中，我们引入WI-偏序集作为中间对象。事实证明，（几乎）所有与图相关联的图复合体都可以被视为同一对象的化身，只要它们的Z_2-同伦类型被关注。其中的应用是证明每个有限的自由Z_2-复形是一个图复形，以及评价圈C_n中独立集的复形Ind（C_n）的Z_2-同伦类型。本文的主要工具是Quillen纤维定理和Z_2-复形的Z_2-等价的Bredon准则。
The homology of simplicial complement and the cohomology of the moment-angle complexes
单纯补P是[m]的子集的序列，并且单纯补P对应于顶点在[m]中的唯一单纯复形K。本文定义了主理想整环k上的单纯补$H_{i,σ}(Λ^{*,*}[P], d)$的同调，并利用Taylor分解证明了$H_{*,*}(Λ[P], d)$同构于相应面环k（K）的Tor.作为应用，我们给出了计算Tor_{*,*} ^{k[x]}（k（K），k）$, $ link_{K} σ $, $ star_{K} σ$的环结构和广义矩角复形的上同调的方法.
Realization of permutation modules as homology of CW-complexes
本文研究置换模作为CW复形同调的实现问题，对任意群和特定次数的置换模给出部分解答，并完整解答了Kahn问题。
Distributed Persistent Homology for 2D Alpha Complexes
本文提出面向二维 alpha 复形的分布式持续同调算法，通过网格覆盖与 Mayer-Vietoris 谱序列实现并行计算，并给出 C++/Open MPI 实现和实验验证。
We introduce weighted versions of the classical Čech and Vietoris-Rips complexes. We show that a version of the Vietoris-Rips Lemma holds for these weighted complexes and that they enjoy appropriate stability properties. We also give some preliminary applications of these weighted complexes.
On $(n-2)$-connected $2n$-dimensional Poincaré complexes with torsion-free homology
该文研究具有无挠同调的$(n-2)$-连通$2n$维Poincaré复形，证明其可分解为两类Poincaré复形的连通和，在附加条件下可进一步分解，还应用结果分类了2-连通8维此类复形的同伦型。
A notion of homotopy for directed graphs and their flag complexes
本文为有向图及其旗复形引入同伦等价关系，证明其诱导同调同构，并据此建立过滤有向图持续同调的稳定性定理，同时指出与路径同调的差异。
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.
Complete digraphs are referred to in the combinatorics literature as tournaments. We consider a family of semi-simplicial complexes, that we refer to as "tournaplexes", whose simplices are tournaments. In particular, given a digraph $\mathcal{G}$, we associate with it a "flag tournaplex" which is a tournaplex containing the directed flag complex of $\mathcal{G}$, but also the geometric realisation of cliques that are not directed. We define several types of filtrations on tournaplexes, and exploiting persistent homology, we observe that flag tournaplexes provide finer means of distinguishing g…
Mayer Path Homology
本文引入Mayer路径同调，通过在路径复形上引入N- nilpotent微分，构建新的定向路径复形同调理论，揭示了更敏感的定向网络模因结构。
We consider a fixed basis of a finitely generated free chain complex as a finite topological space and we present a sufficient condition for the singular homology of this space to be isomorphic with the homology of the chain complex.
Iterated bar complexes of E-infinity algebras and homology theories
我们在前一篇文章中证明了E-无穷代数的棒复形继承了自然的E-无穷代数结构。因此，一个定义明确的迭代棒结构B^n（A）可以与E-无穷操作数上的任何代数相关联。在交换代数A的情况下，我们的迭代棒构造可简化为A的标准迭代棒复形。本文的第一个目的是给出E-无限代数的迭代杆复形的一个直接有效的定义。利用这个有效的定义，我们证明了n重杆复形B^n（A）是E_n-算子上代数范畴的扩张.证明了n重杆复形决定了E_n-算子上一类代数的同调论。对于n无穷大，我们得到了无穷棒结构的同调与通常的平凡系数Gamma-同调之间的同构。
We introduce Reeb complexes in order to capture how generators of homology flow along sections of a real valued continuous function. This intuition suggests a close relation of Reeb complexes to established methods in topological data analysis such as levelset zigzags and persistent homology. We make this relation precise and in particular explain how Reeb complexes and levelset zigzags can be extracted from the first pages of respective spectral sequences with the same termination.
Discrete Differential Calculus on Simplicial Complexes and Constrained Homology
本文在有限顶点集上建立离散微分演算，并据此分别构造单纯复形的约束同调群与独立超图的约束上同调群。
We consider a $q$-analogue of abstract simplicial complexes, called $q$-complexes, and discuss the notion of shellability for such complexes. It is shown that $q$-complexes formed by independent subspaces of a $q$-matroid are shellable. Further, we explicitly determine the homology of $q$-complexes corresponding to uniform $q$-matroids. We also outline some partial results concerning the determination of homology of arbitrary shellable $q$-complexes..
Homological obstructions for regular embeddings of graphs
本文研究了图k-正则嵌入的同调障碍，通过嵌入同调和独立复形的同调建立了同态关系，并探讨了相关正合序列和图示的构造。
The fiber of multiparameter persistent homology for simplicial complexes
本文研究固定有限单纯复形上n-滤子的多参数持续同调逆问题，通过赋予相关空间分层结构证明MPH映射的性质，给出纤维维数上界并推广了已知单参数结果。
We introduce the weighted path homology on the category of weigh\-ted directed hypergraphs and describe conditions of homotopy invariance of weighted path homology groups. We give several examples that explain the nontriviality of the introduced notions.
Hypergraphs are mathematical models for many problems in data sciences. In recent decades, the topological properties of hypergraphs have been studied and various kinds of (co)homologies have been constructed (cf. [3, 4, 12]). In this paper, generalising the usual homology of simplicial complexes, we define the embedded homology of hypergraphs as well as the persistent embedded homology of sequences of hypergraphs. As a generalisation of the Mayer-Vietoris sequence for the homology of simplicial complexes, we give a Mayer-Vietoris sequence for the embedded homology of hypergraphs. Moreover, as…
Smith Theory for algebraic varieties
我们展示了如何史密斯理论的群作用CW-复合物使用Bredon上同调的方法可以适应工作的代数品种。
The Persistent Homology of Dual Digital Image Constructions
研究灰度数字图像的两种立方复形构造的持续同调差异，通过对偶滤过胞腔复形理论推导出转换方法，使软件无需额外低级代码即可支持两种构造。
We reformulate the inequalities among self-closeness numbers of spaces in cofibrations making use of homology dimension and show that the self-closeness number of a space is less than or equal to the homology dimension of the space. Then we prove a relation of self-closeness numbers and the connectivity for manifolds satisfying Poincaré duality. On the other hand we determine the self-closeness numbers of the real projective spaces, lens spaces and a cell complex defined by Mimura and Toda. Moreover, making use of the models of Sullivan and Quillen, we show several properties of self-closeness…
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…
Vertex-Minimal Triangulation of Complexes with Homology
研究了具有非平凡同调的复杂体的最小顶点数，并在强连通性约束下提出新的解决方法。
Hypergraph is a topological model for networks. In order to study the topology of hypergraphs, the homology of the associated simplicial complexes and the embedded homology have been invented. In this paper, we give some algorithms to compute the homology of the associated simplicial complexes and the embedded homology of hypergraphs as well as some heuristics for efficient computations.
Cycle-free chessboard complexes and symmetric homology of algebras
棋盘复形及其相关复形是拓扑组合学中经常出现的重要课题之一。与之密切相关的“无圈棋盘复形”最近由奥尔特和菲多罗维奇引入，作为计算代数的循环同调的对称类似物的工具。我们研究了这些复合物的连通性，并证明了一个结果，证实了加强奥尔特和Fiedorowicz猜想。
Exact cohomology sequences with integral coefficients for torus actions
利用Atiyah在等变K-理论中应用的方法，Bredon得到了（足够好的）T-空间T=（S^1）^n的等变骨架的相对上同调（具有有理系数）的精确序列，其中BT的上同调具有自由等变上同调。在这里，我们证明了具有连通各向同性群的有限T-CW复形，对于这些复形，类似的结果在积分系数下成立。
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.
We show that counterexamples of Iyengar and Walker to the algebraic version of Gunnar Carlsson's conjecture on the rank of the homology of a free complex can be extended to examples over any finite group with many choices of the complex.
Examples of cohomology manifolds which are not homologically locally connected
Bredon构造了一个关于奇异同调的非同调局部连通的2维紧致上同调流形。在本文中，我们构造无穷多个这样的例子（这是另外的度量化空间）在所有剩余的维度$n \ge 3$。
The threshold function for vanishing of the top homology group of random $d$-complexes
对于正整数$n$和$d$，以及概率函数$0\leq p(n)\leq 1$，我们让$Y_{n,p,d}$表示$n$顶点上所有至多$d$维单纯复形的概率空间，这些复形包含完整的$(d-1)$维骨架，并且其$d$ -单纯形以概率$p(n)$出现.本文对所有的$d\geq 1$，确定了$Y_{n,p,d}$中顶同调群消失的门限函数.
Stability of persistent path homology of path complexes
该研究证明了路径复形的持久路径同调的稳定性，进而推导超图、序列超图的对应稳定性，恢复了Chowdhury与Mémoli关于有向图的已知结果。
Persistent Homology for Random Fields and Complexes
我们讨论并回顾了应用代数拓扑学领域的最新进展，如持久同源性和条形码。特别是，我们讨论了这些是如何与了解更多关于流形学习随机点云数据，由随机顶点确定的单纯复形的代数结构，以及最详细的，随机场的漂移集的代数拓扑结构。
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