Bruno Vallette
We define a poset of partitions associated to an operad. We prove that the operad is Koszul if and only if the poset is Cohen-Macaulay. In one hand, this characterisation allows us to compute the homology of the poset. This homology is given by the Koszul dual operad. On the other hand, we get new methods for proving that an operad is Koszul.
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The set of hypertrees on $n$ vertices can be endowed with a poset structure. J. McCammond and J. Meier computed the dimension of the unique non zero homology group of the hypertree poset. We give another proof of their result and use the theory of species to determine the action of the symmetric group on this homology group, which is linked with the anti-cyclic structure of the $\operatorname{Prelie}$ operad. We also compute the action on the Whitney homology of the poset. ----- L'ensemble des hyperarbres à $n$ sommets peut être muni d'un ordre partiel. J. McCammond et J. Meier ont calculé la …
We study homology groups of posets with functor coefficients and apply our results to give a novel approach to study Khovanov homology of knots and related homology theories.
We describe some open problems related to homology representations of subposets of the partition lattice, beginning with questions first raised in Stanley's work on group actions on posets.
Invariable generation of finite simple groups and rational homology of coset posets
本文证明每个有限单群都可由一个西洛子群与一个循环群不变生成，并据此推出任意有限群的陪集偏序集阶复形具有非平凡约化有理同调。
Filtered order complexes and magnitude homology of finite graded posets
研究通过秩函数定义的分次偏序集序复形的子复形族，探讨其拓扑性质与量值同调的关系，证明闭流形剖分和可壳复形的同调性质，以及几何半格中复形同伦等价于球面楔和。
Double, über and poset homology
研究双重上同调与über上同调在邻接复形上的差异，揭示三重双度数对复形邻接性的影响
We define semi-pointed partition posets, which are a generalisation of partition posets and show that they are Cohen-Macaulay. We then use multichains to compute the dimension and the character for the action of the symmetric groups on their homology. We finally study the associated incidence Hopf algebra, which is similar to the Fa{à} di Bruno Hopf algebra.
Möbius Homology
本文提出 Möbius 同调，将偏序集拓扑与 Möbius 反演直接联系，并范畴化持续图，为一般偏序集上的持续同调提供新不变量。
The decorated hypercube found in the construction of Khovanov homology for links is an example of a Boolean lattice equipped with a presheaf of modules. One can place this in a wider setting as an example of a coloured poset, that is to say a poset with a unique maximal element equipped with a presheaf of modules. In this paper we initiate the study of a bundle theory for coloured posets, producing for a certain class of base posets a Leray-Serre type spectral sequence. We then show how this theory finds application in Khovanov homology by producing a new spectral sequence converging to the Kh…
We introduce and study additive posets. We show that the top homology group (with coefficients in Z/2Z) of a finite dimensional CW-complex carries a structure of an additive poset invariant under subdivisions. Applications to CW-complexes and graphs are discussed.
The aim of this short note is to develop a (co)homology theory for topological spaces together with the specialisation preorder. A known way to construct such a (co)homology is to define a partial order on the topological space starting from the preorder, and then to consider some (co)homology for the obtained poset; however, we will prove that every topological space with the above preorder consists of two disjoint parts (one called 'poset part', and the other one called 'complementary part', which is not a poset in general): this suggests an improvement of the previous method that also takes…
We examine the lattice of all order congruences of a finite poset from the viewpoint of combinatorial algebraic topology. We will prove that the order complex of the lattice of all nontrivial order congruences (or order-preserving partitions) of a finite $n$-element poset $P$ with $n\geq 3$ is homotopy equivalent to a wedge of spheres of dimension $n-3$. If $P$ is connected, then the number of spheres is equal to the number of linear extensions of $P$. In general, the number of spheres is equal to the number of cyclic extensions of $P$.
In [1] we introduced the notion of 'structured space', i.e. a space which locally resembles various algebraic structures. In [2] and [3] we studied some cohomology theories related to these space. In this paper we continue in this direction: while in [2] we mainly focused on cohomologies arising from $f_s$ and $h$, and in [3] we considered cohomologies for generalisations of objects which involved structured spaces, here we deal with (co)homologies coming from the poset associated to a structured space via an equivalence relation defined at the end of Section 4 in [1]. More precisely, we will …
We classify flag complexes on at most 12 vertices with torsion in the first homology group. The result is moderately computer-aided. As a consequence we confirm a folklore conjecture that the smallest poset whose order complex is homotopy equivalent to the real projective plane (and also the smallest poset with torsion in the first homology group) has exactly 13 elements.
We investigate the (co)homological properties of two classes of Lie algebras that are constructed from any finite poset: the solvable class $\frak{gl}^\preceq$ and the nilpotent class $\frak{gl}^\prec$. We confirm the conjecture of Jollenbeck that says: every prime power $p^r\!\leq\!n\!-\!2$ appears as torsion in $H_\ast(\frak{nil}_n;\mathbb{Z})$, and every prime power $p^r\!\leq\!n\!-\!1$ appears as torsion in $H_\ast(\frak{sol}_n;\mathbb{Z})$. If $\preceq$ is a bounded poset, then the (co)homology of $\frak{gl}^\preceq$ is \emph{torsion-convex}, i.e. if it contains $p$-torsion, then it also …
We consider partitions of a set with $r$ elements ordered by refinement. We consider the simplicial complex $\bar{K}(r)$ formed by chains of partitions which starts at the smallest element and ends at the largest element of the partition poset. A classical theorem asserts that $\bar{K}(r)$ is equivalent to a wedge of $r-1$-dimensional spheres. In addition, the poset of partitions is equipped with a natural action of the symmetric group in $r$ letters. Consequently, the associated homology modules are representations of the symmetric groups. One observes that the $r-1$th homology modules of $\b…
Cohomology of non-finite CL-shellable posets
本文提出稳定函子概念，利用模型范畴和壳可剖分偏序集结构，证明特定维度上同调消失，并应用于Mackey函子、外幂同调及Bianchi群分解。
Hepworth, Willerton, Leinster and Shulman introduced the magnitude homology groups for enriched categories, in particular, for metric spaces. The purpose of this paper is to describe the magnitude homology group of a metric space in terms of order complexes of posets. In a metric space, an interval (the set of points between two chosen points) has a natural poset structure, which is called the interval poset. Under additional assumptions on sizes of $4$-cuts, we show that the magnitude chain complex can be constructed using tensor products, direct sums and degree shifts from order complexes of…
The homological information about a filtered simplicial complex over the poset of positive real numbers is often presented by a barcode which depicts the evolution of the associated Betti numbers. However, there is a wonderfully complex combinatorics associated with the homology classes of a filtered complex, and one can do more than just counting them over the index poset. Here, we show that this combinatorial information can be encoded by filtered matroids, or even better, by rooted forests. We also show that these rooted forests can be realized as cobordisms.
Multipath cohomology of directed graphs
本文研究有向图的多路上同调，将其定义为路径偏序集关于特定函子的同调，证明了其比高阶极限更可计算，并在树上非消失，且在相干定向多边形上恢复Hochschild同调，最后探讨了其与无向图着色同调的联系。
Lie-operads and operadic modules from poset cohomology
本文建立了一种一般形式体系，在具有自然递归结构的偏序集族上同调上产生算畴与算畴模结构，该框架超越了装饰划分与算畴Koszul对偶性，并包含元交换李算畴和Kontsevich树算畴等实例。
Operads with compatible CL-shellable partition posets admit a Poincaré-Birkhoff-Witt basis
该研究关联Koszul性与Cohen-Macaulay性的两种细化概念，证明相关偏序集具同构相容CL-壳化的基本集operad容许Poincaré-Birkhoff-Witt基，并给出逆命题反例，深化偏序集与operad的联系。
Linear extensions and directed clique counts via modular partitions
本文提出通过模分区结构递归计算偏序集线性扩展的新方法，并探讨其在有向图和神经回路中的应用。
Decomposing zero-dimensional persistent homology over rooted tree quivers
本文针对有根树偏序集上的零维持续同调，证明其对应表示范畴的加法闭包为有限型，给出分解为不可分解对象的二次时间算法，还利用Ringel树模完成代数刻画并明确不可分解对象的性质。
Let $X$ be 2n-dimensional compact manifold with a locally standard action of a compact torus. The orbit space $X/T$ is a manifold with corners. Suppose that all proper faces of $X/T$ are acyclic. In the paper we study the homological spectral sequence $E^*_{*,*}\Rightarrow H_*(X)$ corresponding to the filtration of $X$ by orbit types. When the free part of the action is not twisted, we describe the whole spectral sequence in terms of homology and combinatorial structure of $X/T$. In this case we describe the kernel and the cokernel of the natural map $k[X/T]/(l.s.o.p.) \to H_*(X)$, where $k[X/…
Let $P$ be a finite poset. We will show that for any reasonable $P$-persistent object $X$ in the category of finite topological spaces, there is a $P-$ weighted graph, whose clique complex has the same $P$-persistent homology as $X$.
The poset of cancellations induced by gradient dynamics in a filtered Lefschetz complex
本文通过组合梯度动力学方法，将持久同调中的过滤视为离散Morse函数，引入深度偏序集概念，并给出一种两遍矩阵约简算法来构建该偏序集，从而实现对拓扑特征的简化。
Computing Projective Implicit Representations from Poset Towers
本文研究了如何通过偏序塔中的链复形计算投影隐表示，该表示在保持同调的同时将simplicial数据转化为代数输入，用于全局计算持久同调。
Operadic Kazhdan-Lusztig-Stanley theory
该研究提出依赖偏序集族P、由偏序集链支配的新型类操作子结构P-operad，给出其与经典偏序集概念相关的实例，并证明该框架可范畴化几何格的Kazhdan–Lusztig多项式及其核，为其系数正性提供新证明。
Homological freeness criterion for operadic modules and application to Cohen-Macalayness of posets
本文提出Koszul operad上operadic模同调自由准则的变体，应用于装饰划分偏序集，证明其增广Cohen-Macaulay并计算同调，回答了相关开放问题。
The partition poset complex and the Goodwillie derivatives of the identity in spaces
该研究通过上单纯形对象配对构造空间恒等函子导数的典范同伦协调 operad 结构，证明其与 Ching 描述的结构一致，还揭示谱上交换余代数的导出本原元为该 operad 上的代数。
The commutative and homological algebra of modules over posets is developed, as closely parallel as possible to the algebra of finitely generated modules over noetherian commutative rings, in the direction of finite presentations, primary decompositions, and resolutions. Interpreting this finiteness in the language of derived categories of subanalytically constructible sheaves proves two conjectures due to Kashiwara and Schapira concerning sheaves with microsupport in a given cone. The motivating case is persistent homology of arbitrary filtered topological spaces, especially the case of multi…
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.
Ordered set partition posets
本文研究了有序集合划分格子的子格结构，探讨了块大小受限的拓扑和组合性质，推导了递归原子顺序及对称群作用的公式，并揭示了与k-Catalan数相关的区间结构。
We consider a sheaf of exterior algebras on a simplicial poset $S$ and introduce a notion of homological characteristic function. Two natural objects are associated with these data: a graded sheaf $\mathcal{I}$ and a graded cosheaf $\widehatΠ$. When $S$ is a homology manifold, we prove the isomorphism $H^{n-1-p}(S;\mathcal{I})\cong H_{p}(S;\widehatΠ)$ which can be considered as an extension of the Poincare duality. In general, there is a spectral sequence $E^2_{p,q}\cong H^{n-1-p}(S;\mathcal{U}_{n-1+q}\otimes \mathcal{I})\Rightarrow H_{p+q}(S;\widehatΠ)$, where $\mathcal{U}_*$ is the local hom…
Koszul complexes and relative homological algebra of functors over posets
本文研究偏序集上向量空间取值函子的相对同调代数，提出在特定条件下通过对函子族分次构建Koszul复形，以同调维数直接计算相对Betti图，避免显式求极小相对分解。
A cosheaf is the dual notion of a sheaf, but we cannot define its homology as the formal dual of sheaf cohomology, in general, because of the lack of the cosheafification. A cellular cosheaf is a contravariant functor from the face poset of a CW complex to the category of abelian groups. We show that given a cellular cosheaf $F$, there is a natural way to associate a cosheaf $\widehat{F}$, for which we can define homology as the formal dual of sheaf cohomology, such that the Borel-Moore homology of $F$ is isomorphic to the homology of $\widehat{F}$ whenever the underlying CW complex of $F$ is …
Let $\mathcal{L}$ be the noncrossing partition lattice associated to a finite Coxeter group $W$. In this paper we construct explicit bases for the top homology groups of intervals and rank-selected subposets of $\mathcal{L}$. We define a multiplicative structure on the Whitney homology of $\mathcal{L}$ in terms of the basis, and the resulting algebra has similarities to the Orlik-Solomon algebra. As an application, we obtain four chain complexes which compute the integral homology of the Milnor fibre of the reflection arrangement of $W$, the Milnor fibre of the discriminant of $W$, the hyperpl…
Stable homology isomorphisms for the partition and Jones annular algebras
本文证明琼斯环形代数同调在梯度1/2直线以下与循环群同调同构，分划代数同调在梯度1直线以下与对称群同调同构（加强已有结果），且同构区间超代数稳定范围，还得琼斯环形代数奇链与可逆参数结果。
Homology of spectral minimal partitions
本文通过定义修正拉普拉斯算子并借助代数拓扑中的相对同调工具，研究了非二分谱最小分割，证明其 Courant-sharp 特征函数的节点分割在特定拓扑类中最小，同时统一恢复了二分情形的已知结果。
The homology of the partition algebras
本文研究了划分代数的同调，证明其与对称群同调在随节点数增加的度数范围内同构，并在参数 $δ$ 可逆时在所有度数上同构，丰富了代数同调稳定性文献。
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