B. Enriquez, V. V. Vershinin
We consider the Lie algebra associated with the descending central series filtration of the pure braid group of a closed surface of arbitrary genus. R. Bezrukavnikov gave a presentation of this Lie algebra over the rational numbers. We show that his presentation remains true for this Lie algebra itself, i.e. over integers.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
The group of planar (or flat) pure braids on $n$ strands, also known as the pure twin group, is the fundamental group of the configuration space $F_{n,3}(\mathbb{R})$ of $n$ labelled points in $\mathbb{R}$ no three of which coincide. The planar pure braid groups on 3, 4 and 5 strands are free. In this note we describe the planar pure braid group on 6 strands: it is a free product of the free group on 71 generators and 20 copies of the free abelian group of rank two.
In this article we prove theorem on Lifting for the set of virtual pure braid groups. This theorem says that if we know presentation of virtual pure braid group $VP_4$, then we can find presentation of $VP_n$ for arbitrary $n > 4$. Using this theorem we find the set of generators and defining relations for simplicial group $T_*$ which was defined in the previuos article of the authors. We find a decomposition of the Artin pure braid group $P_n$ in semi-direct product of free groups in the cabled generators.
A presentation for the planar pure braid group
本文构造了 $n$ 股平面纯辫群 $\overline{P}_n$ 的极小展示，其生成元与 Y.Baryshnikov 发现的上同调环生成元对偶，关系位于换位子群中，并给出了纯仙人掌群的显式非极小展示。
In this paper we first show that many braid groups of low genus surfaces have their centers as direct factors. We then give a description of centralizers and normalizers of prime order elements in pure mapping class groups of surfaces with spherical quotients using automorphism groups of fundamental groups of the quotient surfaces. As an application, we use these to show that the $p$-primary part of the Farrell cohomology groups of certain mapping class groups are elementary abelian groups. At the end we compute the $p$-primary part of the Farrell cohomology of a few pure mapping class groups.
This article is an exposition of certain connections between the braid groups, classical homotopy groups of the 2-sphere, as well as Lie algebras attached to the descending central series of pure braid groups arising as Vassiliev invariants of pure braids. Natural related questions are posed at the end of this article.
Two virtual link diagrams are homotopic if one may be transformed into the other by a sequence of virtual Reidemeister moves, classical Reidemeister moves, and self crossing changes. We recall the pure virtual braid group. We then describe the set of pure virtual braids that are homotopic to the identity braid.
Action of automorphisms of pure braid groups on homotopy groups of two-sphere
本文研究了纯辫群相关Delta-群结构的Moore复形，证明循环群与边界群在纯辫群所有自同构下不变，从而推广了前人结果，并得出纯辫群自同构在二维球面同伦群上存在诱导作用且计算了少量股数的作用。
By exploring simplicial structure of pure virtual braid groups, we give new connections between the homotopy groups of the 3-sphere and the virtual braid groups that are related to the theory of Brunnian virtual braids. The group structure of VP_n with n > 4 is determined by VP_3, VP_4 and virtual cablings given by iterated degeneracy operations on the generators and defining relations. The complete proofs will be published in the two forthcoming papers.
Milnor invariants of braids and welded braids up to homotopy
本文研究同伦下纯焊接辫子群的结构，通过将其商群分解为迭代半直积解决Andreadakis问题并给出生成元表示，同时关联Milnor不变量并讨论经典辫子的对应情况。
We compute the Lusternik-Schnirelmann category (LS-cat) and the higher topological complexity ($TC_s$, $s\geq2$) of the "no-$k$-equal" configuration space Conf$_k(\mathbb{R},n)$. This yields (with $k=3$) the LS-cat and the higher topological complexity of Khovanov's group PP$_n$ of pure planar braids on $n$ strands, which is an $\mathbb{R}$-analogue of Artin's classical pure braid group on $n$ strands. Our methods can be used to describe optimal motion planners for PP$_n$ provided $n$ is small.
We prove that, for $n\geq 3$, the minimal dimension of a model of the classifying space of the full braid group $B_n$, and of the pure braid group $P_n$, with respect to the family of virtually cyclic groups is $n$.
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.
We show that the fundamental group of the space of ordered affine-equivalent configurations of at least five points in the real plane is isomorphic to the pure braid group modulo its centre. In the case of four points this fundamental group is free with eleven generators.
By studying braid group actions on Milnor's construction of the 1-sphere, we show that the general higher homotopy group of the 3-sphere is the fixed set of the pure braid group action on certain combinatorially described group. We also give certain representation of higher differentials in the Adams spectral sequence for the homotopy groups of the 2-sphere.
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 $\mathrm{cd}([P_n,P_n])=n-2$.
Configuration Lie groupoids and orbifold braid groups
本文提出配置李群胚的两种定义，证明一类李群胚的Fadell-Neuwirth型纤维化定理，并利用纯轨道折叠辫群的多重几乎自由结构，推导出相应Artin群及轨道折叠辫群的Farrell-Jones猜想。
Pentagon equations, Delaunay triangulations and pure braid group invariant
本文通过Delaunay三角剖分构造$(2n+1)$阶矩阵，并由此定义了从$(n+3)$线纯辫群到$\text{GL}_{2n+1}(\mathbb{Q})$的同态，揭示了平面点运动与线性群表示之间的联系。
We prove Farber's conjecture on the stable topological complexity of configuration spaces of graphs. The conjecture follows from a general lower bound derived from recent insights into the topological complexity of aspherical spaces. Our arguments apply equally to higher topological complexity.
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。
In this paper, we generalize the tools that were introduced in [Dar19b] in order to study the Andreadakis problem for subgroups of IAn. In particular, we study the behaviour of the Andreadakis problem when we add inner automorphisms to a subgroup of IAn. We notably use this to show that the Andreadakis equality holds for the pure braid group on n strands modulo its center acting on the free group on n-1 generators , that is, for the (pure, based) mapping class group of the n-punctured sphere acting on its fundamental group.
Let B_n(RP^2)$ (respectively P_n(RP^2)) denote the braid group (respectively pure braid group) on n strings of the real projective plane RP^2. In this paper we study these braid groups, in particular the associated pure braid group short exact sequence of Fadell and Neuwirth, their torsion elements and the roots of the `full twist' braid. Our main results may be summarised as follows: first, the pure braid group short exact sequence 1 --> P_{m-n}(RP^2 - {x_1,...,x_n}) --> P_m(RP^2) --> P_n(RP^2) --> 1 does not split if m > 3 and n=2,3. Now let n > 1. Then in B_n(RP^2), there is a k-torsion ele…
Let G be the fundamental group of the complement of a K(G,1) hyperplane arrangement (such as Artin's pure braid group) or more generally a homologically toroidal group (as defined in the paper). The subgroup of elements in the complex K-theory of BG which arises from complex unitary representations of G is shown to be trivial. In the case of real K-theory, this subgroup is an elementary abelian 2-group, which is characterized completely in terms of the first two Stiefel-Whitney classes of the representation. Furthermore, an orthogonal representation of G gives rise to a trivial bundle if and o…
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…
Cohomology of quasi-abelianized braid groups
本研究针对广义辫群模纯辫群换位子群的商群有理上同调问题，通过图族同构类给出组合描述，建立了庞加莱对偶性并证明了无穷反射群序列的稳定性质。
The BNS invariants of the braid groups and pure braid groups of some surfaces
该研究计算并刻画了球面、实射影平面、环面和克莱因瓶的辫群与纯辫群的BNS不变量$Σ^1$，分析了其补集的拓扑结构，并推导得到克莱因瓶纯辫群自同构群中存在有限指数正规子群使得其内所有元素的Reidemeister数均为无穷。
We study for each $n$ a one-parameter family of complex-valued measures on the symmetric group $S_n$, which interpolate the probability of a monic, degree $n$, square-free polynomial in $\mathbb{F}_q[x]$ having a given factorization type. For a fixed factorization type, indexed by a partition $λ$ of $n$, the measure is known to be a Laurent polynomial. We express the coefficients of this polynomial in terms of characters associated to $S_n$-subrepresentations of the cohomology of the pure braid group $H^{\bullet}(P_n, \mathbb{Q})$. We deduce that the splitting measures for all parameter values…
The fundamental group of the real locus of the Deligne-Mumford compactification of the moduli space of rational curves with $n$ marked points, the pure cactus group, resembles the pure braid group in many ways. As it is the case for several "pure braid like" groups, it is known that its cohomology ring is generated by its first cohomology. In this note we survey what the $FI$-module theory developed by Church, Ellenberg and Farb can tell us about those examples. As a consequence we obtain uniform representation stability for the sequence of cohomology groups of the pure cactus group.
The purpose of this article is to describe connections between the loop space of the 2-sphere, Artin's braid groups, a choice of simplicial group whose homotopy groups are given by modules called Lie(n), as well as work of Milnor, and Habegger-Lin on "homotopy string links". The current article exploits Lie algebras associated to Vassiliev invariants in work of T. Kohno, and provides connections between these various topics. Two consequences are as follows: 1) the homotopy groups of spheres are identified as "natural" sub-quotients of free products of pure braid groups, and 2) an axiomatizatio…
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。
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。