Kouyemon Iriye, Daisuke Kishimoto
Golodness of 2-dimensional simplicial complexes is studied through polyhedral products, and combinatorial and topoogical characterization of Golodness of surface triangulations is given. An answer to the question of Berglund is also given so that there is a 2-dimensional simplicial complex which is rationally Golod but is not Golod over $\mathbb{Z}/p$.
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Tight complexes are Golod
本文证明紧单纯复形具有 Golod 性质，并由此得出闭连通可定向流形三角剖分中 Golod 性与紧性等价。
We characterize two-dimensional Golod complexes combinatorially by vertex-breakability and topologically by the fat-wedge filtration of a polyhedral product. Applying the characterization, we consider a difference between Golodness over fields and rings, which enables us to give a two-dimensional simple Golod complex over any field such that the corresponding moment-angle complex is not a suspension.
The polyhedral product constructed from a collection of pairs of cones and their bases and a simplicial complex $K$ is studied by investigating its filtration called the fat wedge filtration. We give a sufficient condition for decomposing the polyhedral product in terms of the fat wedge filtration of the real moment-angle complex for $K$, which is a desuspension of the decomposition of the suspension of the polyhedral product due to Bahri, Bendersky, Cohen, and Gitler. We show that the condition also implies a strong connection with the Golodness of $K$, and is satisfied when $K$ is dual seque…
We prove that for a simplicial complex $K$ whose Taylor resolution for the Stanley-Reisner ring is minimal, the following four conditions are equivalent: (1) $K$ satisfies the strong gcd-condition; (2) $K$ is Golod; (3) the moment-angle complex $\mathcal{Z}_K$ is homotopy equivalent to a wedge of spheres; (4) the decomposition of the suspension of the polyhedral product $\mathcal{Z}_K(C\underline{X},\underline{X})$ due to Bahri, Bendersky, Cohen, and Gitler desuspends.
Golod and tight 3-manifolds
通过拓扑刻画维数≥3的紧邻流形三角剖分的多面体乘积，证明了单纯复形的Golod性和紧性在3-流形三角剖分中是等价的。
In this paper, we define and prove basic properties of complement polyhedral product spaces, dual complexes and polyhedral product complexes. Then we compute the universal algebra of polyhedral product complexes under certain split conditions and the Alexander duality isomorphism on certain polyhedral product spaces.
Representation stability in the sense of Church-Farb is concerned with stable properties of representations of sequences of algebraic structures, in particular of groups. We study this notion on objects arising in toric topology. With a simplicial $G$-complex $K$ and a topological pair $(X, A)$, a $G$-polyhedral product $(X, A)^K$ is associated. We show that the homotopy decomposition [2] of $Σ(X, A)^K$ is then $G$-equivariant after suspension. In the case of $Σ_m$-polyhedral products, we give criteria on simplicial $Σ_m$-complexes which imply representation stability of $Σ_m$-representations …
The construction of a simplicial complex given by polyhedral joins (introduced by Anton Ayzenberg), generalizes Bahri, Bendersky, Cohen and Gitler's $J$-construction and simplicial wedge construction. This article gives a cohomological decomposition of a polyhedral product over a polyhedral join for certain families of pairs of simplicial complexes. A formula for the Hilbert-Poincaré series is given, which generalizes Ayzenberg's formula for the moment-angle complex.
The polyhedral product is a space constructed from a simplicial complex and a collection of pairs of spaces, which is connected with the Stanley Reisner ring of the simplicial complex via cohomology. Generalizing the previous work Grbic and Theriault, Grujic and Welker, and the authors, we show a decomposition of polyhedral products for a large class of simplicial complexes including the ones whose Alexander duals are shellable or sequentially Cohen-Macaulay. This implies the property, called Golod, of the corresponding Stanley-Reisner rings proved by Herzog, Reiner and Welker.
Simplicial vs. cubical spheres, polyhedral products and the Nevo-Petersen conjecture
本文通过多面体乘积与神经复形方法分类 flag Murai 球面和极小非 Golod Bier 球面，证明前者满足 Nevo-Petersen 猜想，并引入立方 Bier 球面研究其组合与几何性质。
A generalization of the Davis-Januszkiewicz construction and applications to toric manifolds and iterated polyhedral products
复曲面流形的基本Davis-Januszkiewicz结构被重新解释，以便于推广。应用涉及单纯楔$J$ -建设和Ayzenberg的最近的身份所产生的组成单纯复形。
This paper aims to find the most general combinatorial conditions under which a moment-angle complex $(D^2,S^1)^K$ is a co-$H$-space, thus splitting unstably in terms of its full subcomplexes. In this way we study to which extent the conjecture holds that a moment-angle complex over a Golod simplicial complex is a co-$H$-space. Our main tool is a certain generalisation of the theory of labelled configuration spaces.
Polyhedral products over finite posets
本文在任意带基点偏序集上建立多面体乘积的一般同伦论构造，引入多面体张量积与多面体偏序集的 Stanley-Reisner 环，并证明其与上同调及单纯偏序集情形的同伦一致性。
We prove that certain conditions on multigraded Betti numbers of a simplicial complex $K$ imply existence of a higher Massey product in cohomology of a moment-angle-complex $\mathcal Z_K$, which contains a unique element (a strictly defined product). Using the simplicial multiwedge construction, we find a family $\mathcal{F}$ of polyhedral products being smooth closed manifolds such that for any $l,r\geq 2$ there exists an $l$-connected manifold $M\in\mathcal F$ with a nontrivial strictly defined $r$-fold Massey product in $H^{*}(M)$. As an application to homological algebra, we determine a wi…
If $K$ is a simplicial complex on $m$ vertices the flagification of $K$ is the minimal flag complex $K^f$ on the same vertex set that contains $K$. Letting $L$ be the set of vertices, there is a sequence of simplicial inclusions $L\to K\to K^f$. This induces a sequence of maps of polyhedral products $(\underline X,\underline A)^L\stackrel g\longrightarrow(\underline X,\underline A)^K\stackrel f\longrightarrow (\underline X,\underline A)^{K^f}$. We show that $Ωf$ and $Ωf\circΩg$ have right homotopy inverses and draw consequences. For a flag complex $K$ the polyhedral product of the form $(\unde…
Moment-angle complexes and polyhedral products for convex polytopes
设P是一般不单的凸多胞体。本文的重点是一个单纯复形K_P，它携带了关于P的组合类型的完整信息。在P是单纯复形的情况下，K_P与dP* 相同，其中P* 是一个极对偶多胞形。利用多面体P在非负正交中的标准嵌入，我们引入了多面体P的矩角空间Z_P.已知当P是单的时，空间Z_P同胚于多面体积（D^2，S^1）^{K_P}.当P不单时，证明了空间Z_P与空间（D^2，S^1）^{K_P}同伦等价.这允许对任何凸多胞体引入二阶贝蒂数。多胞形P的Stanley-Reisner环可以定义为单纯复形K_P的Stanley-Reisner环。所有这些考虑引出一个自然的问题：对于某个多胞形P，哪些单纯复形作为K_P出现？我们已经在这个方向上引进了多面体单纯复形的概念。它具有以下性质：多面体复形中的每个单形的链环要么可收缩，要么可收缩到一个子复形，同胚到一个球面。复形K_P是任意多面体P的多面体单纯复形。多面体复形中所谓面单纯形的链环也是多面体复形。这一事实足以将单纯复形K_P的面多项式与多胞形P的面多项式联系起来，给出一系列关于P的某些组合特征的不等式。其中两个不等式对每个P都是等式，并表示Euler-Poincare公式和一个关于旗f数的Bayer-Billera关系。在P是单的情况下，所有的不等式都是经典的Dehn-Sommerville关系。
On the graph products of simplicial groups and connected Hopf algebras
本文在域上研究单纯群与连通Hopf代数的图积，证明其分类空间可在多面体积框架下统一处理，并给出结构定理。
Steenrod operations on polyhedral products
该研究刻画了模2 Steenrod代数在多面体乘积等空间上同调中的作用，提出上链层公式及从单纯复形到多面体join的传播方法，建立其与矩角复形作用的关联，为无稳定同伦分解的情形提供研究思路。
Loop spaces of polyhedral products associated with the polyhedral join product
本文研究了与多面体连接运算相关的多面体积空间的循环空间，建立了其与有限类型球面乘积的同伦等价，并给出了必要条件。
We show that a relation among minimal non-faces of a fillable complex $K$ yields an identity of iterated (higher) Whitehead products in a polyhedral product over $K$. In particular, for the $(n-1)$-skeleton of a simplicial $n$-sphere, we always have such an identity, and for the $(n-1)$-skeleton of a $(n+1)$-simplex, the identity is the Jacobi identity of Whitehead products ($n=1$) and Hardie's identity of higher Whitehead products ($n\ge 2$).
We consider families of simple polytopes $P$ and simplicial complexes $K$ well-known in polytope theory and convex geometry, and show that their moment-angle complexes have some remarkable homotopy properties which depend on combinatorics of the underlying complexes and algebraic properties of their Stanley--Reisner rings. We introduce infinite series of Golod and minimally non-Golod simplicial complexes $K$ with moment-angle complexes $\mathcal Z_K$ having free integral cohomology but not homotopy equivalent to a wedge of spheres or a connected sum of products of spheres respectively. We then…
Loops on polyhedral products and diagonal arrangements
本文建立了由一个单纯复形K（所谓的多面体积）定义的广义楔形的闭路空间同调与K相关联的某些对角排列的同调之间的联系。我们说明了这些结果，通过发现这些循环同调代数的某些K推广Panov-Ray，Papadima-Suciu，Lemaire的结果的介绍。最后，我们表明，在函子被施加到悬浮液的情况下，这种同调分裂来自稳定的同伦分裂的闭路空间。
We consider a certain class of simplicial complexes which includes the independence complexes of forests. We show that if a simplicial complex $K$ belongs to this class, then the polyhedral join $\mathcal{Z}^*_{K}(\underline{X}, \emptyset)$ is homotopy equivalent to a wedge sum of CW complexes of the form $Σ^r X_{i_1} * X_{i_2} * \cdots * X_{i_k}$, where $\underline{X}$ is a family $\{X_i\}_{i \in V(K)}$ of CW complexes and $Σ$ denotes the unreduced suspension. This result is applied to study the homotopy type of the independence complex of the lexicographic product $G[H]$ of a graph $H$ over …
We characterise integral Poincaré duality moment-angle complexes $\mathcal{Z}_{\mathcal{K}}$ in combinatorial terms of the Fan-Wang duality of the simplicial complex $\mathcal{K}$, and consequently in algebraic terms of the Gorenstein duality of the Stanley-Reisner ring $\mathbb{Z}[\mathcal{K}]$. We extend Poincaré duality results to certain polyhedral products using polyhedral join products of simplicial complexes.
On the fundamental group of certain polyhedral products
设$K$是有限单纯复形，$(X,A)$是一对空间.本文的目的是研究多面体积的基本群$Z_K(X,A)$，它表示$(X,A) = (D^2, S^1)$情形下Buchstaber-Panov的矩角复形，并将文献[2]中的定义推广到任意对。对于离散群$G$，给出了抽象单纯复形$K$使得$Z_K(\underline{BG})$表示的多面体积是Eilenberg-Mac Lane空间的充要条件. $Z_K(\underline{BG})$的基本群仅依赖于$K$的1-骨架.进一步的特殊例子多面体产品也进行了研究。最后，我们使用多面体积来研究一个与传递交换群相关的扩张问题，定义5.2中给出了这个问题。
A colimit decomposition for the loop homology of polyhedral products
本文通过余极限分解研究多面体积的环同调代数，将问题转化为1邻接的简单复形情况，并计算了不同类别的简单复形的Poincaré级数。
In this paper we present unpublished work by David Stone on polyhedral smash products. He proved that the polyhedral smash product of the CW-pair $(D^2, S^1)$ over a simplicial complex $K$ is homeomorphic to an iterated suspension of the geometric realization of $K$. Here we generalize his technique to the CW-pair $(D^{k+1}, S^{k})$, for an arbitrary $k$. We generalize the result further to a set of disks and spheres of different dimensions.
A polyhedral product is a natural subspace of a Cartesian product, which is specified by a simplicial complex K. The automorphism group Aut(K) of K induces a group action on the polyhedral product. In this paper we study this group action and give a formula for the fixed point set of the polyhedral product for any subgroup H of Aut(K). We use the fixed point data to compute examples of Bredon cohomolohgy for small non-Abelian groups such as $D_8$ and $Σ_4$.
Relations among higher Whitehead maps
本文定义了聚合同态积之间的广义高阶Whitehead映射，通过研究聚合同态积的同伦性质与单纯复形的组合性质之间的关系，描述了新的映射关系家族，并恢复和推广了已知的Whitehead积恒等式。
Polyhedral products and features of their homotopy theory
本文综述了多面体乘积及其同伦理论性质的研究发展，提供了简短历史概述，总结了主要结果并描述了其在多个数学领域的应用。
On Félix-Tanré rational models for polyhedral products
本文研究纤维包含的多面体乘积的Félix-Tanré有理模型，探究李群对多面体乘积的有理模型与有理同伦群，并证明环面流形相关偏商的三个等价条件。
The homotopy type of the polyhedral product for shifted complexes
本文证明了Bahri，Bendersky，Cohen和Gitler的一个猜想：如果K是n个顶点上的移位单纯复形，X_1，. X_n是空间，CX_i是X_i上的锥，则由K和对（CX_i，X_i）所确定的多面体积同伦等价于X_i的一个劈的悬置。这推广了两位作者在每个X_i是闭路空间的特殊情况下的早期工作。连接环面拓扑学，组合学和经典同伦理论。
Polyhedral coproducts
通过将多面体积的对偶化，引入多面体余积作为单纯复形面偏序集上的同伦极限，并研究其基本性质，特别是给出了空间楔和与笛卡尔积之间的函子性插值以及对偶于Bahri等人悬架分裂的环路空间分解。
The Goodwillie calculus of polyhedral products
本文通过分析 Denham-Suciu 纤维序列、Iriye-Kishimoto 分解与 Hilton-Milnor 分解，研究了厚楔滤过平凡时多面体积的 Goodwillie 演算，并证明了其在整性上收敛且在 $v_h$-周期同伦中发散的条件。
The polyhedral product functor: a method of computation for moment-angle complexes, arrangements and related spaces
本文给出了广义矩-角复形或{ \it部分积空间}的悬置的自然分解，这些悬置产生为下面描述的{ \it多面体积函子}。在某些子空间排列的补的特殊情况下，几何分解意味着Goresky-MacPherson \cite{goresky.macpherson}，Hochster \cite{hochster}，Baskakov \cite{baskakov}，Panov \cite{panov}和Buchstaber-Panov \cite{buchstaber.panov}中的同调分解。由于分裂是几何的，一个广义矩角复形的类似同调分解适用于任何同调理论。这种分解给出了有限单纯复形的Stanley-Reisner环的加法分解，推广了Porter \cite{porter}和Ganea \cite{ganea}的同伦理论结果.这里的工作精神遵循Denham-Suciu在\cite{denham.suciu}的精神。
The relative join operad and polyhedral products
该论文引入对称相对连接操作子，统一了Abramyan-Panov替换与Ayzenberg复合，证明其非有限生成，并揭示其对箭头的作用、对偶构造及在PL球-边界对上的子操作子结构。
The homotopy type of the moment-angle complex associated to the complex of injective words
本文研究利用多面体积 functor 构造有向图的拓扑空间，并探讨其同调类型，揭示了同调与组合学的紧密联系，通过计算注入词复合体的同调类型，展示了其由 h-向量决定。
Combinatorial groupoids, cubical complexes, and the Lovasz conjecture
为组合群胚的理论奠定了基础，使我们能够在单纯（多面体）复形、偏序集、图、多面体和其他组合对象的上下文中使用“完整性”、“并行传输”、“对称性”、“组合曲率”等概念。本文引入了一个新的立方复形的完整型不变量，并由此得到了一个立方复形不可嵌入立方格的组合“极值定理”。Hom复形和映射的并行运输被用作一种工具，用于将Babson-Kozlov-Lovasz图着色结果扩展到关于单纯复形和图的非退化映射（着色）的更一般的陈述。
Discrete Morse theory for moment-angle complexes of pairs (D^n,S^{n-1})
对于有限单纯复形K和CW-对（X，A），存在一个相应的CW-复形Z_K（X，A），称为多面体积。本文将离散的莫尔斯理论应用于n维球矩角复形Z_K（D^{n}，S^{n-1}）上的一个特殊CW结构。对于顶点可分解的n维单纯复形，证明了n维球矩角复形具有球楔的同伦型.作为一个推论，我们表明，一个足够高的悬挂的任何限制的单纯复形与顶点可分解的对偶同伦等价的楔形球。
On product of cocycles in a polyhedral complex
构造了多面体复形上链的乘积。乘法算法取决于参数的选择。该参数是环境空间上的线性泛函。上圈形成上链环的子环，上界形成上圈环的理想，而商环是上同调环。如果复形是单纯形的，则该算法简化为${\rm\breve{C}}$ ech的公知算法。类似的算法被用于多面体几何中，用于乘以在周围空间的外部代数中取值的余圈。因此，我们假设值环是超交换的。本文包括凸几何的基本定义、定理陈述及其应用。全文将提供给“消息报：数学”。
In toric topology, to a simplicial complex $K$ with $m$ vertices, one associates two spaces, the moment-angle complex $\mathcal{Z}_K$ and the Davis-Januszkiewicz space $DJ_K$. These spaces are connected by a homotopy fibration $\mathcal{Z}_K\to DJ_K\to(\mathbb{C}P^\infty)^m$. In this paper, we show that the map $\mathcal{Z}_K\to DJ_K$ is identified with a wedge of iterated (higher) Whitehead products for a certain class of simplicial complexes $K$ including dual shellable complexes. We will prove the result in a more general setting of polyhedral products.
Given a finite set $A\subset\mathbb{R}^d$, let Cov$_{r,k}$ denote the set of all points within distance $r$ to at least $k$ points of $A$. Allowing $r$ and $k$ to vary, we obtain a 2-parameter family of spaces that grow larger when $r$ increases or $k$ decreases, called the \emph{multicover bifiltration}. Motivated by the problem of computing the homology of this bifiltration, we introduce two closely related combinatorial bifiltrations, one polyhedral and the other simplicial, which are both topologically equivalent to the multicover bifiltration and far smaller than a \v Cech-based model con…
A unifying view toward polyhedral products through panel structures
本文引入带面空间上的多面体乘积概念，为多面体乘积与广义矩角复形提供统一视角，通过计算其稳定分解、上同调环及等变上同调环，推广了面环概念，既重诠释了已知结果也推导了新结论。
In this thesis, we study the structure of the polyhedral product $\mathcal{Z}_{\mathcal{K}}(D^1,S^0)$ determined by an abstract simplicial complex ${\mathcal{K}}$ and the pair $(D^1,S^0)$. We showed that there is natural embedding of the hypercube graph in $\mathcal{Z}_{\mathcal{K}_n}(D^1,S^0)$ where ${\mathcal{K}}_n$ is the boundary of an $n$-gon. This also provides a new proof of a known theorem about genus of the hypercube graph. We give a description of the invertible natural transformations of the polyhedral product functor. Then, we study the action of the cyclic group $\mathbb{Z}_n$ on …
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…
In this follow-up to [16], we continue developing the notion of a lego category and its many applications to stratifiable spaces and the computation of their Grothendieck classes. We illustrate the effectiveness of this construction by giving very short derivations of the class of a quotient by the "stratified action" of a discrete group [1], the class of a crystallographic quotient, the class of both a polyhedral product and a polyhedral (or simplicial) configuration space [8], the class of a permutation product [19] and, foremost, the class of spaces of $0$-cycles [11].
One-relator groups and algebras related to polyhedral products
本文通过组合条件刻画了旗单纯复形对应的右角Coxeter群交换子子群及矩角复形Pontryagin代数成为一关系对象（群或代数）的充要条件，并给出了同调刻画。
In this survey, we discuss two research areas related to Massey's higher operations. The first direction is connected with the cohomology of Lie algebras and the theory of representations. The second main theme is at the intersection of toric topology, homotopy theory of polyhedral products, and the homology theory of local rings, Stanley-Reisner rings of simplicial complexes.
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。
Iterated Whitehead products in the homotopy groups of polyhedral products
研究单纯复形 K 的 Davis-Januszkiewicz 空间 DJ(K) 的同伦群结构，通过 Whitehead 积生成拟李子代数 QL(K) 和 Pi-子代数 S(K)，证明 S(K) 等于整个同伦群当且仅当 K 是旗复形。
The simplicial wedge construction on simplicial complexes and simple polytopes has been used by a variety of authors to study toric and related spaces, including non-singular toric varieties, toric manifolds, intersections of quadrics and more generally, polyhedral products. In this paper we extend the analysis to include toric orbifolds. Our main results yield infinite families of toric orbifolds, derived from a given one, whose integral cohomology is free of torsion and is concentrated in even degrees, a property which might be termed \emph{integrally equivariantly formal}. In all cases, it …
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