Jie Wu
The homology with coefficients in a field of the configuration spaces $C(M\times \bold R ^n,M_o\times \bold R ^n;X)$ is determined in this paper.
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Categorifying connected domination via graph überhomology
本文将图的 überhomology 解释为偏序集同调，证明 bold homology 范畴化连通支配多项式，并给出树的消失性与完全图的刻画。
A spectral sequence calculating the homology groups of some spaces of maps equivariant under compact group actions is described. For the main example, we calculate the rational homology groups of spaces of even and odd maps $S^m \to S^M$, $m<M$, or, which is the same, the stable homology groups of spaces of non-resultant homogeneous polynomial maps ${\mathbb R}^{m+1} \to {\mathbb R}^{M+1}$ of growing degrees. Also, we find the homology groups of spaces of ${\mathbb Z}_r$-equivariant maps of odd-dimensional spheres for any $r$. As a technical tool, we calculate the homology groups of configurat…
We show that the homology of ordered configuration spaces of finite trees with loops is torsion free. We introduce configuration spaces with sinks, which allow for taking quotients of the base space. Furthermore, we give a concrete generating set for all homology groups of configuration spaces of trees with loops and the first homology group of configuration spaces of general finite graphs. An important technique in the paper is the identification of the $E^1$-page and differentials of Mayer-Vietoris spectral sequences for configuration spaces.
Using factorization homology, we realize the rational homology of the unordered configuration spaces of an arbitrary manifold $M$, possibly with boundary, as the homology of a Lie algebra constructed from the compactly supported cohomology of $M$. By locating the homology of each configuration space within the Chevalley-Eilenberg complex of this Lie algebra, we extend theorems of Bödigheimer-Cohen-Taylor and Félix-Thomas and give a new, combinatorial proof of the homological stability results of Church and Randal-Williams. Our method lends itself to explicit calculations, examples of which we …
Borel-Moore Homology of 2-Particle Unordered Configuration Spaces of Graphs via Discrete Morse Theory
本研究通过离散莫尔斯理论定义图的2粒子无序构形空间的开正则胞腔复形上的莫尔斯函数，借助同构关系得出其Borel-Moore同调群由图的第一贝蒂数决定的结论。
We provide a short proof that the dimensions of the mod $p$ homology groups of the unordered configuration space $B_k(T)$ of $k$ points in a torus are the same as its Betti numbers for $p>2$ and $k\leq p$. Hence the integral homology has no $p$-power torsion. The same argument works for the punctured genus $g$ surface with $g>0$, thereby recovering a result of Brantner-Hahn-Knudsen via Lubin-Tate theory.
We show that the homology over a field of the space of free maps from the n-sphere to the n-fold suspension of X depends only on the cohomology algebra of X and compute it explicitly. We compute also the homology of the closely related labelled configuration space on the n-sphere with labels in X and of its completion, that depends only on the homology of X. In many but not all cases the homology of the configuration space coincides with the homology of the mapping space. In particular we obtain the homology of the unordered configuration spaces on a sphere.
Vertical configuration spaces and their homology
该研究提出欧氏空间中满足垂直性条件的点簇的有序与无序构型空间，计算了有序情形的同调并证明无序情形的同调稳定性，丰富了构型空间的同调理论研究。
Homology of configuration spaces in positive characteristic via point-set constructions
本文通过显式链复形计算正特征下流形（无序）构型空间的同调，将Knudsen定理提升至模型范畴层次，并构造新的谱序列收敛到这些同调群。
For a continuous angle-valued map defined on a compact ANR, a fixed field and any degree one proposes a refinement of the Novikov-Betti number and of the Novikov homology of the pair consisting of the ANR and the degree one integral cohomology class represented by the map. For each degree the first refinement is a configuration of points with multiplicity located in the punctured complex plane whose total cardinality is the Novikov-Betti number of the pair. The second refinement is a configuration of submodules of the Novikov homology whose direct sum is isomorphic to the Novikov homology and …
If $R$ is a commutative ring, $M$ a compact $R$-oriented manifold and $G$ a finite graph without loops or multiple edges, we consider the graph configuration space $M^G$ and a Bendersky-Gitler type spectral sequence converging to the homology $H_*(M^G, R)$. We show that its $E_1$ term is given by the graph cohomology complex $C_A(G)$ of the graded commutative algebra $A = H^*(M, R)$ and its higher differentials are obtained from the Massey products of $A$, as conjectured by Bendersky and Gitler for the case of a complete graph $G$. Similar results apply to the spectral sequence constructed fro…
The first homology of square-section braid groups
该研究计算矩形内n个开单位正方形无序构形空间的一阶同调，利用谱序列证明特定条件下其与点构形空间一阶同调自然同构，并给出h=2时的同调群秩结果。
Following ideas of Graeme Segal, we construct an equivariant con- figuration space that is a model of equivariant connective K-homology spec- trum for finite groups, as a consequence we obtain an induction structure for equivariant connective K-homology. We describe explicitly the homology with complex coefficients for the fixed points of this configuration space as a Hopf algebra.
Quillen homology of spectral Lie algebras with application to mod $p$ homology of labeled configuration spaces
本文建立计算谱 Lie 代数模 $p$ Quillen 同调的一般方法，证明 $\mathbb F_p$ 线性谱 Lie operad 非形式，并用于给出带标记构形空间同调的上界、小 $k$ 显式计算及奇偶维数对照。
The purpose of this note is to clarify some details in McDuff and Segal's proof of the group-completion theorem and to generalize both this and the homology fibration criterion of McDuff to homology with twisted coefficients. This will be used in forthcoming work to identify the limiting homology of "oriented" configuration spaces, which doubly cover the classical configuration spaces of distinct unordered points in a manifold.
To a pair (X,f), X compact ANR and f a continuous angle valued map defined on X, a fixed field and a nonnegative integer one assigns a finite configuration of complex numbers with multiplicities located in the punctured complex plane and a finite configuration of free modules over the ring of Laurent polynomials (with coefficients in the fixed field) indexed by the same complex numbers. This is done in analogy with the configuration of eigenvalues and of generalized eigenspaces of an invertible linear operator in a finite dimensional complex vector space. The configuration of complex numbers r…
We study stability patterns in the high dimensional rational homology of unordered configuration spaces of manifolds. Our results follow from a general approach to stability phenomena in the homology of Lie algebras, which may be of independent interest.
We provide a construction of free factorization algebras in algebraic geometry and link factorization homology of a scheme with coefficients in a free factorization algebra to the homology of its (unordered) configuration spaces. As an application, this construction allows for a purely algebro-geometric proof of homological stability of configuration spaces.
Persistent Homology of Configuration Spaces of Trees
本研究针对拓扑机器人学等领域的多参数持续模问题，分析星图与广义H图第二构型空间对应的2参数持续模，给出其不可分解直和项，推进相关拓扑结构的代数刻画。
Secondary Homological Stability for Unordered Configuration Spaces
本文证明了连通流形无序构型空间同调的次同调稳定性，通过构造新的链层级稳定化映射解决了闭流形情形下缺乏诱导稳定性映射及同调呈周期性的难题。
Recently, the homology and cohomology of non-k-overlapping discs, or, equivalently, no k-equal subspaces of Euclidean space, were calculated by Dobrinskaya and Turchin. We calculate the homology and cohomology of two classes of more general spaces: decreasing polychromatic configuration spaces and bicolored configuration spaces. Instead of all points behaving similarly, we allow for varying behavior between points.
Lie algebras and the (co)homology of configuration spaces
该研究综述了构形空间（上）同调与李代数（上）同调关联的数十年研究，分析了不同路径对原定理的解释并传达各视角概念核心，最后提出问题列表。
In this note we study the homology of configuration spaces of closed manifolds. We sharpen the eventual periodicity results of Nagpal and Cantero-Palmer, prove integral homological stability for configuration spaces of odd-dimensional manifolds and introduce a stabilization map on the homology with $Z[1/2]$-coefficients of configuration spaces of odd-dimensional manifolds.
Given a finite graph G and a topological space Z, the graphical configuration space Conf(G, Z) is the space of functions V(G) -> Z so that adjacent vertices map to distinct points. We provide a homotopy decomposition of Conf(G, X x Y) in terms of the graphical configuration spaces in X and Y individually. By way of application, we prove a stabilization result for homology of configuration space in X x C^p as p goes to infinity. We also compute the homology of Conf(K_3,T)/T, the space of ordered triples of distinct points in a torus T of rank r, where configurations are considered up to transla…
Persistent bundles over configuration spaces and obstructions for regular embeddings
本文构造了硬球构型空间上的持久丛，利用其Stiefel-Whitney类和Chern类分别给出实与复正则嵌入的障碍，并展示了持久同调工具在球填充与独立复形几何实现等问题中的近似计算应用。
The aim of this paper is to make sample computations with the Salvetti complex of the "center of mass" arrangement introduced in [arXiv:math/0611732] by Cohen and Kamiyama. We compute the homology of the Salvetti complex of these arrangements with coefficients in the sign representation of symmetric groups on F_p in the case of four particles. We show, when p is an odd prime, the homology is isomorphic to the homology of the configuration space F(C,4) of distinct four points in the complex plane with the same coefficients. When p=2, we show the homology is different from that of F(C,4), hence …
Stability in the homology of configuration spaces of high dimensional manifolds
该研究针对高维流形配置空间的同调稳定性问题，拓展了已有结论，证明了上同调集中在偶度数的流形其配置空间奇同调稳定域与流形维数相关，且二阶上同调非平凡时偶同调稳定域最优且仅由配置点数决定。
On the quasi polynomiality of extremal homology of configuration spaces
本文研究流形构型空间的极值同调群，确定了最高阶非平凡拟多项式的精确次数，证明 Knudsen、Miller 和 Tosteson 的上界对每个流形都是最优的。
We propose a refinement of the Betti numbers and of the homology with coefficients in a field of a compact ANR in the presence of a continuous real valued function. The refinement of Betti numbers consists of finite configurations of points with multiplicities in the complex plane whose total cardinality are the Betti numbers and the refinement of homology consists of configurations of vector spaces indexed by points in complex plane, with the same support as the first, whose direct sum is isomorphic to the homology. When the homology is equipped with a scalar product these vector spaces are c…
The homology of configuration spaces of point-particles in manifolds has been studied intensively since the 1970s; in particular it is known to be stable if the underlying manifold is connected and open. Closely related to configuration spaces are moduli spaces of manifolds with marked points, and in [Tillmann, 2016] this relation was used to show that the homology of these moduli spaces is also stable as the number of marked points goes to infinity. Since a disc neighbourhood of a marked point may be viewed as the cone on its boundary sphere, we may think of marked points as inessential conic…
Homology and Euler characteristic of generalized anchored configuration spaces of graphs
本文研究图上 n 个带标记点的广义锚定构型空间，给出非树连通图情形的非交错欧拉示性数公式，并完全确定圆图上相应空间的同调群。
We produce explicit geometric representatives of nontrivial homology classes in the space of long knots in R^d, when d is even. We generalize results of Cattaneo, Cotta-Ramusino and Longoni to define cycles which live off of the vanishing line of a homology spectral sequence due to Sinha. We use configuration space integrals to show our classes pair nontrivially with cohomology classes due to Longoni.
Homology of configuration spaces of surfaces modulo an odd prime
本文研究了带边界紧致可定向曲面无序构型空间在奇素数域上的同调，将其描述为特定Pontryagin环上的双分次模并计算维数，同时证明了mod-$p$ Johnson核是映射类群作用在该同调上的核。
Homology Generators and Relations for the Ordered Configuration Space of a Star Graph
本文研究了星图有序配置空间的同调生成元与关系，证明了不同叶数下同调群的生成元和关系数，并指出图的有序配置空间同调无有限普遍呈现。
We give explicit formulas for the asymptotic Betti numbers of the unordered configuration spaces of an arbitrary finite graph over an arbitrary field.
We let the mapping class group $Γ_{g,1}$ of a genus $g$ surface $Σ_{g,1}$ with one boundary component act on the homology $H_*(F_{n}(Σ_{g,1});\mathbb{Q})$ of the $n^{th}$ ordered configuration space of the surface. We prove that the action is non-trivial when restricted to the $(n-1)^{st}$ stage of the Johnson filtration, for all $n\ge 1$ and $g\ge 2$. We deduce an analogous result for closed surfaces.
We calculate a certain homological obstruction introduced by De Concini, Procesi and Salvetti in their study of the Schwartz genus of the fibration from the space of ordered configuration of points in the plane to the space of the unordered configurations. We show that their obstruction group vanishes in almost all, but not all, the hitherto unknown cases. It follows that if $n$ is not a power of a prime, or twice the power of a prime, then the genus is less than $n$. The case of $n=2p^k$ where $p$ is an odd prime remains undecided for some $p$ and $k$.
Homology of configuration spaces of hard squares in a rectangle
本文推广“15 Puzzle”，研究矩形中硬方块有序构型空间的拓扑，通过构建同伦等价的立方体胞腔复形，利用离散Morse理论证明同调消没定理，并给出非平凡圈显式族与参数插值方法。
In this expository paper we give an elementary, hands-on computation of the homology of the little disks operad, showing that the homology of a $d-fold loop space is a Poisson algebra. One aim is to familiarize a greater audience with Euclidean configuration spaces, using tools accessible to second-year graduate students. We also give a brief introduction to the theory of operads. New results include identifying the pairing between homology and cohomology of these spaces as a pairing of graphs and trees, and treating the cooperad structure on cohomology.
Let M be an open, connected manifold. A classical theorem of McDuff and Segal states that the sequence of configuration spaces of n unordered, distinct points in M is homologically stable with coefficients in Z: in each degree, the integral homology is eventually independent of n. The purpose of this note is to prove that this phenomenon also holds for homology with twisted coefficients. We first define an appropriate notion of finite-degree twisted coefficient system for configuration spaces and then use a spectral sequence argument to deduce the result from the untwisted homological stabilit…
We compute the cycle index sum of the symmetric group action on the homology of the configuration spaces of points in a Euclidean space with the condition that no $k$ of them are equal.
Consider the space of continuous functions on a geometric tree $X$ whose persistent homology gives rise to a finite generic barcode $D$. We show that there are exactly as many path connected components in this space as there are merge trees whose barcode is $D$. We find that each component is homotopy equivalent to a configuration space on $X$ with specialized constraints encoded by the merge tree. For barcodes $D$ with either one or two intervals, our method also allows us to compute the homotopy type of this space of functions.
We present a mathematical framework for describing the topology of configuration spaces for particles on one-connected graphs. In particular, we compute the homology groups over integers for different classes of one-connected graphs. Our approach is based on some fundamental combinatorial properties of the configuration spaces, Mayer-Vietoris sequences for different parts of configuration spaces and some limited use of discrete Morse theory. As one of the results, we derive a closed-form formulae for ranks of the homology groups for indistinguishable particles on tree graphs. We also give a de…
Representation stability in the (co)homology of vertical configuration spaces
研究欧几里得空间中垂直配置空间，借助FI$_G$ -模理论等证明其（上）同调群关于$S_k \wr S_n$作用表示稳定，可表为特定诱导表示，还给出无序空间有理（上）同调稳定性新证及改进范围。
This paper consists of two related parts. In the first part we give a self-contained proof of homological stability for the spaces C_n(M;X) of configurations of n unordered points in a connected open manifold M with labels in a path-connected space X, with the best possible integral stability range of 2* \leq n. Along the way we give a new proof of the high connectivity of the complex of injective words. If the manifold has dimension at least three, we show that in rational homology the stability range may be improved to * \leq n. In the second part we study to what extent the homology of the …
We study topology of configuration spaces of planar linkages having one leg of variable length. Such telescopic legs are common in modern robotics where they are used for shock absorbtion and serve a variety of other purposes. Using a Morse theoretic technique, we compute explicitly, in terms of the metric data, the Betti numbers of configuration spaces of these mechanisms.
Secondary representation stability and the ordered configuration space of the once-punctured torus
本文研究了一次穿孔环面的有序配置空间的二次表示稳定性，证明其同调生成元序列既非自由也非稳定零，并首次展示正亏格表面中二次稳定性现象的非平凡性。
The aim of this article is to provide space level maps between configuration spaces of graphs that are predicted by algebraic manipulations of cellular chains. More explicitly, we consider edge contraction and half-edge deletion, and identify the homotopy cofibers in terms of configuration spaces of simpler graphs. The construction's main benefit lies in making the operations functorial - in particular, graph minors give rise to compatible maps at the level of fundamental groups as well as generalized (co)homology theories. As applications we provide a long exact sequence for half-edge deletio…
By considering labeled configurations of ``bounded multiplicity'', one can construct a functor that fits between homology and stable homotopy. Based on previous work, we are able to give an equivalent description of this labeled construction in terms of loop space functors and symmetric products. This yields a direct generalization of the May-Milgram model for iterated loop spaces, and answers questions of Carlsson and Milgram posed in the handbook. We give a classifying space formulation of our results hence extending an older result of Segal. We finally relate our labeled construction to a t…
We explicitly study the extremal stability of configuration spaces of complex projective spaces of any dimension, and show that the homology groups are vanish in extremal stable range. As a consequence, we give an affirmative answer of the question of Knudsen, Miller and Tosteson.
Odd-primary torsion in the homology of unordered configurations on the torus
研究环面\(\Sigma_1\)上无序点配置空间\(B_k(\Sigma_1)\)同调中的奇素数挠率。通过证明不同\(k\)值和奇素数\(p\)下\(H_*(B_k(\Sigma_1);\mathbb{Z})\)的\(p\) - 挠率情况，得出特定条件下的相关结论。
Configuration spaces on a wedge of spheres and Hochschild-Pirashvili homology
该研究探讨球面楔点构型空间的紧支集有理上同调的对称群与$Out(F_g)$作用表示，证明其为多项式函子并计算合成因子，还给出曲线模空间上同调对称群作用的超指数下界。
For a given bundle $ξ\colon E \to M$ over a manifold, configuration-section spaces on $ξ$ parametrise finite subsets $z \subseteq M$ equipped with a section of $ξ$ defined on $M \smallsetminus z$, with prescribed "charge" in a neighbourhood of the points $z$. These spaces may be interpreted physically as spaces of fields that are permitted to be singular at finitely many points, with constrained behaviour near the singularities. As a special case, they include the Hurwitz spaces, which parametrise branched covering spaces of the $2$-disc with specified deck transformation group. We prove that …
Heisenberg homology on surface configurations
受经典辫群表示启发，本文研究带边界可定向曲面上无序构型的同调，利用离散海森堡群表示定义非交换局部系统，构造了扭曲与非扭曲的映射类群表示，并证明其保持半双线性形式。
Computing stable homology representations of graph configuration spaces
本文利用计算机代数计算了若干图族（完全图、完全二部图、皇冠图、完全三部图）的k=2配置空间的稳定同调表示，并给出了k=3时完全图的稳定重数。
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。
Configuration spaces of labeled points on a circle with two anchors
本文研究圆上含两个固定锚点的 $n$ 个标记点构型空间，利用离散 Morse 理论确定其 Betti 数，并给出同调与上同调基的显式组合描述。
Using the language of twisted skew-commutative algebras, we define \emph{secondary representation stability}, a stability pattern in the {\it unstable} homology of spaces that are representation stable in the sense of Church, Ellenberg, and Farb. We show that the rational homology of configuration spaces of ordered points in noncompact manifolds satisfies secondary representation stability. While representation stability for the homology of configuration spaces involves stabilizing by introducing a point ``near infinity,'' secondary representation stability involves stabilizing by introducing …
We propose a new method of computing cohomology groups of spaces of knots in $\R^n$, $n \ge 3$, based on the topology of configuration spaces and two-connected graphs, and calculate all such classes of order $\le 3.$ As a byproduct we define the higher indices, which invariants of knots in $\R^3$ define at arbitrary singular knots. More generally, for any finite-order cohomology class of the space of knots we define its principal symbol, which lies in a cohomology group of a certain finite-dimensional configuration space and characterizes our class modulo the classes of smaller filtration.
Multiplicative Chow-Künneth decomposition and homology splitting of configuration spaces
本文构造了光滑固有簇点构型空间上同调的分裂，证明了超椭圆Torelli群作用平凡及Leray谱序列立即退化，并据此对$M_{2,n}$和$H_{g,n}$的上同调进行了详细计算。
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