Sergei Ovchinnikov
The paper presents geometric models for the set WO of weak orders on a finite set. In particulary, WO is modeled as a set of vertices of a cubical subdivision of a permutahedron. This approach is an alternative to the usual representation of WO by means of weak order polytopes.
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The main goal of this paper is proving the fixed point theorem for finite groups acting on weakly systolic complexes. As corollaries we obtain results concerning classifying spaces for the family of finite subgroups of weakly systolic groups and conjugacy classes of finite subgroups. As immediate consequences we get new results on systolic complexes and groups. The fixed point theorem is proved by using a graph-theoretical tool - dismantlability. In particular we show that 1-skeleta of weakly systolic complexes, i.e. weakly bridged graphs, are dismantlable. On the way we show numerous characte…
We examine the topology of the clique complexes of the graphs of weakly and strongly separated subsets of the set $[n]=\{1,2,...,n\}$, which, after deleting all cone points, we denote by $\hatΔ_{ws}(n)$ and $\hatΔ_{ss}(n)$, respectively. In particular, we find that $\hatΔ_{ws}(n)$ is contractible for $n\geq4$, while $\hatΔ_{ss}(n)$ is homotopy equivalent to a sphere of dimension $n-3$. We also show that our homotopy equivalences are equivariant with respect to the group generated by two particular symmetries of $\hatΔ_{ws}(n)$ and $\hatΔ_{ss}(n)$: one induced by the set complementation action …
We characterize the cd-indices of Gorenstein* posets of rank 5, equivalently the flag f-vectors of Gorenstein* order complexes of dimension 3. As a corollary, we characterize the f-vectors of Gorenstein* order complexes in dimensions 3 and 4. This characterization rise a speculated intimate connection between the f-vectors of flag homology spheres and the f-vectors of Gorenstein* order complexes.
We reprove and generalize in a combinatorial way the result of A. Björner [J.\ Comb.\ Th.\ A {\bf 30}, 1981, pp.~90--100, Theorem 3.3], that order complexes of noncomplemented lattices are contractible, namely by showing that these simplicial complexes are in fact nonevasive, in particular collapsible.
Minimal displacement set for weakly systolic complexes
本文研究弱收缩复形的最小位移集，证明其具有收缩性并可等距嵌入复形，进而得到等距映射的椭圆或双曲二分类结论。
Assume sigma is a face of a Gorenstein* simplicial complex D. We investigate the question of whether the Weak Lefschetz Property of the Stanley-Reisner ring k[D] (over an infinite field k) is equivalent to the same property of the Stanley-Reisner ring k[D_sigma] of the stellar subdivision D_sigma. We prove that this is the case if the dimension of sigma is big compared to the codimension.
Linearly shellable complexes
本文定义线性可剥壳纯单纯复形，证明多类 Coxeter 复形满足该性质，并提出线性强可剥壳性概念。
In this paper we define and study for a finite partially ordered set P a class of simplicial complexes on the set P_r of r-element multichains from P. The simplicial complexes depend on a strictly monotone function from [r] to [2r]. We show that there exactly 2^r such functions which yield subdivisions of the order complex of P of which 2^{r-1} are pairwise different. Within this class are for example the order complexes of the interval and the zig-zag poset of P and the rth edgewise subdivision of the order complex of P. We also exhibit a large subclass for which our simplicial complexes are …
$h$-vector inequalities under weak maps
本文研究拟阵复形的 h-向量在弱映射下的单调性，证明平坦格有序复形的 h-向量逐分量非增，并推广到旗 h-向量及独立复形情形。
The $\widetilde{A}_n$ Coxeter groups are known to not be systolic or cocompactly cubulated for $n\geq 3$. We prove that these groups act geometrically on weakly modular graphs, a weak notion of nonpositive curvature generalizing the 1-skeleta of $\mathrm{CAT}(0)$ cube complexes and systolic complexes. To prove weak modularity we describe the canonical emeddings of the 1-skeleta of $\widetilde{A}_n$ Coxeter complexes into the Euclidean spaces $\mathbb{R}^{n+1}$. We also prove weak modularity for buildings of type $\widetilde{A}_3$.
In this paper we inititate the study of abstract simplicial complexes which are initial segments of qualitative probability orders. This is a natural class that contains the threshold complexes and is contained in the shifted complexes, but is equal to neither. In particular we construct a qualitative probability order on 26 atoms that has an initial segment which is not a threshold simplicial complex. Although 26 is probably not the minimal number for which such example exists we provide some evidence that it cannot be much smaller. We prove some necessary conditions for this class and make a…
Imposing a strong condition on the linear order of shellable complexes, we introduce strong shellability. Basic properties, including the existence of dimension-decreasing strong shelling orders, are developed with respect to nonpure strongly shellable complexes. Meanwhile, pure strongly shellable complexes can be characterized by the corresponding codimension one graphs. In addition, we show that the facet ideals of pure strongly shellable complexes have linear quotients.
Nested set complexes appear as the combinatorial core of De Concini-Procesi arrangement models. We show that nested set complexes are homotopy equivalent to the order complexes of the underlying meet-semilattices without their minimal elements. For atomic semilattices, we consider the realization of nested set complexes by simplicial fans as proposed in math.AG/0305142 by the first author and S. Yuzvinsky. We show that in this case the nested set complexes in fact are homeomorphic to the mentioned order complexes.
We introduce decomposition complexes of posets, which generalize order complexes. The main advantage of our construction is that decomposition complexes are closed under taking products. Other special instances of this theory include nested set complexes as well as Bergman complexes.
A fundamental question for simplicial complexes is to find the lowest dimensional Euclidean space in which they can be embedded. We investigate this question for order complexes of posets. We show that order complexes of thick geometric lattices as well as several classes of finite buildings, all of which are order complexes, are hard to embed. That means that such d-dimensional complexes require (2d + 1)-dimensional Euclidean space for an embedding. (This dimension is in general always sufficient for any d-complex.) We develop a method to show non-embeddability for general order complexes of …
7-location, weak systolicity and isoperimetry
本文研究 7-位置且局部 5-大型复形的非正曲率性质，证明其最小面积圆盘图保持 7-位置与局部 5-大型性，并借助 CAT(0) 度量证明此类复形具有二次等周函数，同时建立局部弱收缩性与 7-位置局部 5-大型性的联系。
Vertex orders in higher dimensions
本文研究了单位区间复形和区间复形的高维推广，证明了强连通单位区间复形在单位区间序诱导下可壳化且顶点可分解，同时给出了高维中不可壳化的强连通区间复形实例。
A lattice framework for generalizing shellable complexes and matroids
该研究引入幂格概念统一推广多种格结构，定义P-复形推广可剥性概念，构造推广型拟阵并证明相关可剥性结论，还给出多重集格拟阵构造方法及一类Stanley-Reisner环的序列Cohen-Macaulay性质。
Shelling orders are a ubiquitous tool used to understand invariants of cell complexes. Significant effort has been made to develop techniques to decide when a given complex is shellable. However, empirical evidence shows that some shelling orders are better than others. In this article, we explore this phenomenon in the case of matroid independence complexes. Based on a new relation between shellability of dual matroid polytopes and independence complexes, we outline a systematic way to investigate and compare different shellings orders. We explain how our new tools recast and deepen various c…
Motivated by Gröbner basis theory for finite point configurations, we define and study the class of "standard complexes" associated to a matroid. Standard complexes are certain subcomplexes of the independence complex that are invariant under matroid duality. For the lexicographic term order, the standard complexes satisfy a deletion-contraction-type recurrence. We explicitly determine the lexicographic standard complexes for lattice path matroids using classical bijective combinatorics.
The absolute order is a natural partial order on a Coxeter group W. It can be viewed as an analogue of the weak order on W in which the role of the generating set of simple reflections in W is played by the set of all reflections in W. By use of a notion of constructibility for partially ordered sets, it is proved that the absolute order on the symmetric group is homotopy Cohen-Macaulay. This answers in part a question raised by V. Reiner and the first author. The Euler characteristic of the order complex of the proper part of the absolute order on the symmetric group is also computed.
Complexes of C-Unbalanced Subsets and Balanced-Subset Posets
研究了平衡子集的同调类型，证明其与特定维度的球面同调等价。
We exhibit an identity of abstract simplicial complexes between the well-studied complex of trees and the reduced minimal nested set complex of the partition lattice. We conclude that the order complex of the partition lattice can be obtained from the complex of trees by a sequence of stellar subdivisions. We provide an explicit cohomology basis for the complex of trees that emerges naturally from this context. Motivated by these results, we review the generalization of complexes of trees to complexes of $k$-trees by Hanlon, and we propose yet another, in the context of nested set complexes mo…
Bergman complexes are polyhedral complexes associated to matroids. Faces of these complexes are certain matroids, called matroid types, too. In order to understand the structure of these faces we decompose matroid types into direct summands. Ardila/Klivans proved that the Bergman Complex of a matroid can be subdivided into the order complex of the proper part of its lattice of flats. Beyond that Feichtner/Sturmfels showed that the Bergman complex can even be subdivided to the even coarser nested set complex. We will give a much shorter and more general proof of this fact. Generalizing formulas…
Let $P$ be a finite poset with an element $s$ such that (1) for all $x\in P$, either $s\vee x$ or $s\wedge x$ exists; and (2) for all $x,y\in P$ such that $x<y$, if $s\wedge x$ does not exist but $s\wedge y$ does exist, then $(s\wedge y)\vee x$ exists. Kozlov, the winner of the 2005 European Prize in Combinatorics ("for deep combinatorial results obtained by algebraic topology and particularly for the solution of a conjecture of Lovász"), conjectured in the 1998 Proceedings of the American Mathematical Society that the order complex of $P$ is non-evasive. We prove this conjecture.
We show that the order complex of the poset of all cosets of all proper subgroups of a finite group $G$ is never $\mathbb{F}_{2}$-acyclic and therefore never contractible. This settles a question of K. S. Brown.
In this note we construct a poset map from a Boolean algebra to the Bruhat order which unveils an interesting connection between subword complexes, sorting orders, and certain totally nonnegative spaces. This relationship gives a new proof of Björner and Wachs' result \cite{BW} that the proper part of Bruhat order is homotopy equivalent to the proper part of a Boolean algebra --- that is, to a sphere. We also obtain a geometric interpretation for sorting orders. We conclude with two new results: that the intersection of all sorting orders is the weak order, and the union of sorting orders is t…
The local weak limit of $k$-dimensional hypertrees
本文研究按同调群大小平方加权的随机$k$维超树，确定其在顶点数趋于无穷时的局部弱极限，并证明其与1-out $k$-复形一致。
The weak categorical quiver minor theorem and its applications: matchings, multipaths, and magnitude cohomology
本文将弱范畴次型定理从无向图扩展至箭图，研究其对多路复形同调的影响，证明多路与匹配复形在无环有向图上的函子交换性，并据此计算特定二部图的匹配复形同伦类型及分析量级上同调。
Two equivalent descriptions of opetopes: in terms of zoom complexes and of partial orders
本文提出用偏序集组织面的方式定义不一定正的opetope，并证明该定义与Kock、Joyal、Batanin和Mascari的constellation描述等价。
We prove that the Stanley--Reisner ideal of the Alexander dual of the subword complexes in Coxeter groups has linear quotients with respect to the lexicographical order of the minimal monomial generators. As a consequence, we obtain a shelling order on the facets of the subword complex. We relate some invariants of the subword complexes or of their dual with invariants of the word. For a particular class of subword complexes, we prove that the Stanley--Reisner ring is a complete intersection ring.
We will study monomial ideals $I$ in the exterior algebra as well as in the polynomial ring whose generic initial ideal is constant for all term orders up to permutations of variables. First, in the exterior algebra, we determine all graphs and all flag complexes whose exterior face ideal satisfies the above condition. Second, in the polynomial ring, it will be shown that the generic initial ideal $\gin_σ(I(G))$ of the edge ideal $I(G)$ of a graph $G$ is constant for all term orders $σ$ up to permutations of variables if and only if $G$ is a complete bipartite graph.
Boundary rigidity of systolic and Helly complexes
本文证明了有限弱系统复形和Helly复形可通过边界距离重建，且Helly复形和二维系统复形可通过多项式时间算法重建，解决了边界刚性问题的离散版本。
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。
Given a finite group G and a natural number n, we study the structure of the complex of nested sets of the associated Dowling lattice Q(G) and of its subposet of the G-symmetric partitions Q_G which was recently introduced by Hultman together with the complex of G-symmetric phylogenetic trees T_G. Hultman shows that T_G and Q_G are homotopy equivalent and Cohen-Macaulay, and determines the rank of their top homology. An application of the theory of building sets and nested set complexes by Feichtner and Kozlov shows that in fact T_G is subdivided by the order complex of Q_G. We introduce the c…
Function classes are collections of Boolean functions on a finite set, which are fundamental objects of study in theoretical computer science. We study algebraic properties of ideals associated to function classes previously defined by the third author. We consider the broad family of intersection-closed function classes, and describe cellular free resolutions of their ideals by order complexes of the associated posets. For function classes arising from matroids, polyhedral cell complexes, and more generally interval Cohen-Macaulay posets, we show that the multigraded Betti numbers are pure, a…
On the determinicity of paths on substitution complexes
本文研究由逐边粘合四边形构成的替换复形上路径的弱确定染色问题，以有限色着色使第四边在一般情形下由三顶点颜色唯一确定，服务于有限表示代数结构构造。
An affine monoid is an additive monoid which is cancellative, pointed and finitely generated. An affine monoid $Λ$ has the partial order defined by $λ\le λ+ μ$. The Frobenius complex is the order complex of an open interval of $Λ$ with respect to this partial order. The reduced homology of the Frobenius complex is related to the torsion group of the monoid algebra $K[Λ]$. In this paper, we pay attention to homotopy types of Frobenius complexes, and we express the homotopy types of the Frobenius complexes of $Λ$ in terms of those of $Λ_1$ and $Λ_2$ when $Λ$ is an affine monoid obtained by gluin…
Efficient set-theoretic algorithms for computing high-order Forman-Ricci curvature on abstract simplicial complexes
针对高阶 Forman-Ricci 曲率计算成本过高的问题，本文提出集合论表述以揭示瓶颈并加速单纯复形上的计算，同时给出伪代码、FastForman 实现和基准比较。
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…
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