Asymptotic homology of graph braid groups

Homology of the family of hyperelliptic curves

Homology of braid groups and Artin groups can be related to the study of spaces of curves. We completely calculate the integral homology of the family of smooth curves of genus $g$ with one boundary component, that are double coverings of the disk ramified over $n = 2g + 1$ points. The main part of such homology is described by the homology of the braid group with coefficients in a symplectic representation, namely the braid group $\mathrm{Br}_n$ acts on the first homology group of a genus $g$ surface via Dehn twists. Our computations shows that such groups have only $2$-torsion. We also inves…

Homology computations for complex braid groups

Complex braid groups are the natural generalizations of braid groups associated to arbitrary (finite) complex reflection groups. We investigate several methods for computing the homology of these groups. In particular, we get the Poincaré polynomial with coefficients in a finite field for one large series of such groups, and compute the second integral cohomology group for all of them. As a consequence we get non-isomorphism results for these groups.

Embeddings of braid groups into mapping class groups and their homology

We construct several families of embeddings of braid groups into mapping class groups of orientable and non-orientable surfaces and prove that they induce the trivial map in stable homology in the orientable case, but not so in the non-orientable case. We show that these embeddings are non-geometric in the sense that the standard generators of the braid group are not mapped to Dehn twists.

纯图辫群同调中的表示渐近性

Representation asymptotics in the homology of pure graph braid groups

该研究针对图的有序构型空间同调问题，给出了任意域上渐近Betti数的显式公式，并在特征零情形下得到对称群不可约表示同调渐近重数的显式公式，推进了表示稳定性相关研究。

On the homology of the commutator subgroup of the pure braid group

We study the homology of $[P_n,P_n]$, the commutator subgroup of the pure braid group on $n$ strands, and show that $H_l([P_n,P_n])$ contains a free abelian group of infinite rank for all $1\leq l\leq n-2$. As a consequence we determine the cohomological dimension of $[P_n,P_n]$: for $n\geq 2$ we have $\mathrm{cd}([P_n,P_n])=n-2$.

正3辫、Khovanov同调与Garside理论

Positive 3-braids, Khovanov homology and Garside theory

本文利用Garside理论对正3辫共轭类进行分类，并计算其闭包Khovanov同调表的前四列和最低三行，进一步通过共轭最大化下确界以在线性时间内确定更多同调单元。

On the second homology of planar graph braid groups

We show that the second homology of the configuration spaces of a planar graph is generated under the operations of embedding, disjoint union, and edge stabilization by three atomic graphs: the cycle graph with one edge, the star graph with three edges, and the theta graph with four edges. We give an example of a non-planar graph for which this statement is false.

On Homology of Virtual Braids and Burau Representation

Virtual knots arise in the study of Gauss diagrams and Vassiliev invariants of usual knots. Virtual braids correspond naturally to virtual knots. We consider the group of virtual braids on n strings VB_n and its Burau representation, in particular we study their homological properties. We prove that the plus-construction of the classifying space of the virtual braid group on the infinite number of strings is an infinite loop space which is equivalent to a product of Q(S^0), S^1 and an infinite loop space Y. Connections with the K-functor of the integers are discussed.

Homology of the braid group with coefficients in the reduced Burau representation

The reduced Burau representation $V_n$ of the braid group $B_n$ is obtained from the action of $B_n$ on the homology of an infinite cyclic cover of the $n$-punctured disc. In this note, we calculate $H_*(B_n;V_n)$ as a module over the Laurent polynomial ring $\mathbb{Q}[t,t^{-1}]$.

拓扑对称同调与辫子同调

Topological symmetric and braid homologies

本文将拓扑对称同调识别为$\mathbb{E}_1$-代数上的自由$\mathbb{E}_\infty$-代数，将拓扑辫子同调识别为$\mathbb{E}_1$-代数上的自由$\mathbb{E}_2$-代数，并发展了计算工具，证明拓扑对称同调非Morita不变，同时计算了Thom谱的拓扑$\Delta\mathbf{G}$-同调。

球面与射影辫群的稳定性模式

Stability Patterns for Spherical and Projective Braid Groups

本文利用Cohen–Pakianathan的成果与$\text{FI}$同调理论，将稳定性模式推广到球面$S^2$和射影平面$\rp^2$上的辫群，证明其等变同调分别具有表示稳定性与稳定周期性。

Edge stabilization in the homology of graph braid groups

We introduce a novel type of stabilization map on the configuration spaces of a graph, which increases the number of particles occupying an edge. There is an induced action on homology by the polynomial ring generated by the set of edges, and we show that this homology module is finitely generated. An analogue of classical homological and representation stability for manifolds, this result implies eventual polynomial growth of Betti numbers. We calculate the exact degree of this polynomial, in particular verifying an upper bound conjectured by Ramos. Because the action arises from a family of …

Fox-Neuwirth 胞、量子洗牌代数与 B 型 Artin 群的同调

Fox-Neuwirth cells, quantum shuffle algebras, and the homology of type-B Artin groups

本文从辫子向量空间发展 B 型 Artin 群的辫子表示，将其同调与量子洗牌代数上双模的上同调联系起来，并完整刻画了特征 0 域上一维辫子表示系数下的同调。

Stability phenomena in the homology of tree braid groups

For a tree $G$, we study the changing behaviors in the homology groups $H_i(B_nG)$ as $n$ varies, where $B_nG := π_1($UConf$_n(G))$. We prove that the ranks of these homologies can be described by a single polynomial for all $n$, and construct this polynomial explicitly in terms of invariants of the tree $G$. To accomplish this we prove that the group $\bigoplus_n H_i(B_nG)$ can be endowed with the structure of a finitely generated graded module over an integral polynomial ring, and further prove that it naturally decomposes as a direct sum of graded shifts of squarefree monomial ideals. Follo…

广义rook-Brauer代数及其同调

Generalized rook-Brauer algebras and their homology

该研究通过引入带群标记的等变结构与辫子交叉结构推广rook-Brauer代数及其子代数，利用图代数同调方法证明了参数可逆时广义图代数同调与对应群同调的等价性，还得到了奇数边情形下无参数条件的同调等价与同调稳定性结果。

Families of superelliptic curves, complex braid groups and generalized Dehn twists

We consider the universal family $E_n^d$ of superelliptic curves: each curve $Σ_n^d$ in the family is a $d$-fold covering of the unit disk, totally ramified over a set $P$ of $n$ distinct points; $Σ_n^d\hookrightarrow E_n^d\to C_n$ is a fibre bundle, where $C_n$ is the configuration space of $n$ distinct points. We find that $E_n^d$ is the classifying space for the complex braid group of type $B(d,d,n)$ and we compute a big part of the integral homology of $E_n^d,$ including a complete calculation of the stable groups over finite fields by means of Poincarè series. The computation of the main …

二阶 Hochschild 同调的自然对称性

Natural symmetries of secondary Hochschild homology

本文识别了环面的框架微分同胚群为环面与三股辫群的半直积，证明其作用于二阶迭代 Hochschild 同调，并据此提出不稳定二阶循环结构的定义。

Robertson 猜想与图辫子群同调中的通用有限生成性

Robertson's conjecture and universal finite generation in the homology of graph braid groups

本文提出 Robertson 猜想的范畴化版本，证明其与范畴图子式猜想可推出图构形空间同调的有限原子图生成性，并由既有结果推出 cographs 范畴的 Noetherian 性。

Some computations of stable twisted homology for mapping class groups

In this paper, we deal with stable homology computations with twisted coefficients for mapping class groups of surfaces and of 3-manifolds, automorphism groups of free groups with boundaries and automorphism groups of certain right-angled Artin groups. On the one hand, the computations are led using semidirect product structures arising naturally from these groups. On the other hand, we compute the stable homology with twisted coefficients by FI-modules. This notably uses a decomposition result of the stable homology with twisted coefficients for pre-braided monoidal categories proved in this …

The stable braid group and the determinant of the Burau representation

This article gives certain fibre bundles associated to the braid groups which are obtained from a translation as well as conjugation on the complex plane. The local coefficient systems on the level of homology for these bundles are given in terms of the determinant of the Burau representation. De Concini, Procesi, and Salvetti [Topology 40 (2001) 739--751] considered the cohomology of the n-th braid group B_n with local coefficients obtained from the determinant of the Burau representation, H^*(B_n;Q[t^{+/-1}]). They show that these cohomology groups are given in terms of cyclotomic fields. Th…

Subdivisional spaces and graph braid groups

We study the problem of computing the homology of the configuration spaces of a finite cell complex $X$. We proceed by viewing $X$, together with its subdivisions, as a subdivisional space--a kind of diagram object in a category of cell complexes. After developing a version of Morse theory for subdivisional spaces, we decompose $X$ and show that the homology of the configuration spaces of $X$ is computed by the derived tensor product of the Morse complexes of the pieces of the decomposition, an analogue of the monoidal excision property of factorization homology. Applying this theory to the co…

Braids, mapping class groups, and categorical delooping

Dehn twists around simple closed curves in oriented surfaces satisfy the braid relations. This gives rise to a group theoretic from the braid group to the mapping class group. We prove here that this map is trivial in stable homology with any trivial coefficients. In particular this proves an old conjecture of J. Harer. The main tool is categorical delooping. In an appendix we discuss geometrically defined homomorphisms from the braid to the mapping class group.

Representation stability for pure braid group Milnor fibers

We prove a representation stability result for the Milnor fiber associated to the pure braid group. Our result connects previous work of Simona Settepenella to representation stability in the sense of Church--Ellenberg--Farb, answering a question of Graham Denham. We also use our result to compute the stable integral homology of the Milnor fiber.

Equivariant factorization homology of global quotient orbifolds

We introduce equivariant factorization homology, extending the axiomatic framework of Ayala-Francis to encompass multiplicative invariants of manifolds equipped with finite group actions. Examples of such equivariant factorization homology theories include Bredon equivariant homology and (twisted versions of) Hochschild homology. Our main result is that equivariant factorization homology satisfies an equivariant version of tensor excision, and is uniquely characterised by this property. We also discuss applications to representation theory, such as constructions of categorical braid group acti…

The Salvetti Complex and the Little Cubes

We study how the combinatorial structure of the Salvetti complexes of the braid arrangements are related to homotopy theoretic properties of iterated loop spaces. We prove the skeletal filtrations on the Salvetti complexes of the braid arrangements give rise to the cobar-type Eilenberg-Moore spectral sequence converging to the homology of $Ω^2Σ^2 X$. We also construct a new spectral sequence that computes the homology of $Ω^{\ell}Σ^{\ell} X$ for $\ell>2$ by using a higher order analogue of the Salvetti complex. The $E^1$-term of the spectral sequence is described in terms of the homology of $X…

Splitting of the homology of the punctured mapping class group

Let $Γ_{g,1}^m$ be the mapping class group of the orientable surface $Σ_{g,1}^m$ of genus $g$ with one parametrised boundary curve and $m$ permutable punctures; when $m=0$ we omit it from the notation. Let $β_{m}(Σ_{g,1})$ be the braid group on $m$ strands of the surface $Σ_{g,1}$. We prove that $H_*(Γ_{g,1}^m;\mathbb{Z}_2)\cong H_*(Γ_{g,1};H_*(β_{m}(Σ_{g,1});\mathbb{Z}_2))$. The main ingredient is the computation of $H_*(β_{m}(Σ_{g,1});\mathbb{Z}_2)$ as a symplectic representation of $Γ_{g,1}$.

Soergel Bimodules, the Steenrod Algebra and Triply Graded Link homology

Soergel bimodules and their Hochschild homology are known to be important in the context of link homology. In this article we observe that Soergel bimodules may be naturally identified as the cohomology of well-defined objects in the category of spectra. This allows us to define forms of Soergel bimodules over the integers after inverting 2, indexed by elements in Braid groups associated to compact Lie groups of adjoint type. Reducing the Soergel bimodules modulo an odd prime, we endow the bimodules with an action of the reduced Steenrod algebra. This action extends to an action of the reduced…

曲面辫群和多曲面群的渐近维数和上同调维数

Asymptotic and cohomological dimension of surface braid groups and poly-surface groups

本文确定了所有曲面辫群（包括非定向和无限类型曲面相关群）的渐近维数，以及无 torsion 多有限生成曲面群的渐近维数。证明了这两类群的虚拟上同调维数与渐近维数相同，并展示了这些群在 Bieri-Eckmann 意义下是虚拟对偶群。

Symmetric Categories as a Means of Virtualization for Braid Groups and Racks

In this paper we propose, firstly, a categorification of virtual braid groups and groupoids in terms of "locally" braided objects in a symmetric category (SC), and, secondly, a definition of self-distributive structures (SDS) in an arbitrary SC. SDS are shown to produce braided objects in a SC. As for examples, we interpret associativity and the Jacobi identity in a SC as generalized self-distributivity, thus endowing associative and Leibniz algebras with a "local" braiding. Our "double braiding" approach provides a natural interpretation for virtual racks and twisted Burau representation. Hom…

Fox-Neuwirth 胞腔、量子洗牌代数与结式的特征和

Fox-Neuwirth cells, quantum shuffle algebras, and character sums of the resultant

本文利用拓扑方法计算多重穿孔平面辫群同调并证明混合辫群同调的消没范围，从而给出给定次数首一无平方因子多项式对上结式特征和的上界，回答了 Ellenberg 和 Shusterman 在二次情形下的问题。

Weighted sheaves and homology of Artin groups

In this paper we expand the theory of weighted sheaves over posets, and use it to study the local homology of Artin groups. First, we use such theory to relate the homology of classical braid groups with the homology of certain independence complexes of graphs. Then, in the context of discrete Morse theory on weighted sheaves, we introduce a particular class of acyclic matchings. Explicit formulas for the homology of the corresponding Morse complexes are given, in terms of the ranks of the associated incidence matrices. We use such method to perform explicit computations for the new affine cas…

Lawrence-Bigelow representations, bases and duality

We study homological representations of mapping class groups, including the braid groups. These arise from the twisted homology of certain configuration spaces, and come in many different flavours. Our goal is to give a unified general account of the fundamental relationships (non-degenerate pairings, embeddings, isomorphisms) between the many different flavours of homological representations. Our motivating examples are the Lawrence-Bigelow representations of the braid groups, which are of central importance in the study of the braid groups themselves, as well as their connections with quantu…

Diagrams for primitive cycles in spaces of pure braids and string links

The based loop space of a configuration space of points in a Euclidean space can be viewed as a space of pure braids in a Euclidean space of one dimension higher. We continue our study of such spaces in terms of Kontsevich's CDGA of diagrams and Chen's iterated integrals. We construct a power series connection which yields a Hopf algebra isomorphism between the homology of the space of pure braids and the cobar construction on diagrams. It maps iterated Whitehead products to trivalent trees modulo the IHX relation. As an application, we establish a correspondence between Milnor invariants of B…

图辫群同调中所有有限阶挠元

Torsion of every finite order in the homology of graph braid groups

本文确定了图辫群同调中所有有限阶挠元的结构，给出了显式代表元、最小图条件及无挠性判定，并证明了奇挠元首次出现的具体位置。