Daniel Farley
If G is a finite graph and n is a natural number, then the n-strand braid group of G is the fundamental group of the configuration space of n points on G. This article gives a complete computation of the integral cohomology rings of the n-strand braid groups in case G is a tree. Some of the argument builds on earlier joint work with Lucas Sabalka.
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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。
The purpose of this article is to describe the integral cohomology of the braid group B_3 and SL_2(Z) with local coefficients in a classical geometric representation given by symmetric powers of the natural symplectic representation. These groups have a description in terms of the so called "divided polynomial algebra". The results show a strong relation between torsion part of the computed cohomology and fibrations related to loop spaces of spheres.
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.
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.
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.
Let $Γ$ be a finite connected graph. The (unlabelled) configuration space $UC^n Γ$ of $n$ points on $Γ$ is the space of $n$-element subsets of $Γ$. The $n$-strand braid group of $Γ$, denoted $B_nΓ$, is the fundamental group of $UC^n Γ$. We use the methods and results of our paper "Discrete Morse theory and graph braid groups" to get a partial description of the cohomology rings $H^*(B_n T)$, where $T$ is a tree. Our results are then used to prove that $B_n T$ is a right-angled Artin group if and only if $T$ is linear or $n<4$. This gives a large number of counterexamples to Ghrist's conjecture…
The main purpose of this article is to give the integral cohomology of classical principal congruence subgroups in SL(2,Z) as well as their analogues in the third braid group with local coefficients in symmetric powers of the natural symplectic representation. The resulting answers (1) correspond to certain modular forms in characteristic zero, and (2) the cohomology of certain spaces in homotopy theory in characteristic p. The torsion is given in terms of the structure of a "p-divided power algebra". The work is an extension of the work in arXiv:1204.5390v1 as well as extensions of a classica…
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。
We study the cohomology of spaces of string links and braids in $\mathbb{R}^n$ for $n\geq 3$ using configuration space integrals. For $n>3$, these integrals give a chain map from certain diagram complexes to the deRham algebra of differential forms on these spaces. For $n=3$, they produce all finite type invariants of string links and braids.
Presentations of the braid group of the complex reflection group $G(d,d,n)$
本文证明了复反射群 $G(d,d,n)$ 的辫群是某轨道折叠商空间辫群的指数 $d$ 子群，并给出了与标记三角剖分相容的表示。
The word problem of a group is a very important question. The word problem in the braid group is of particular interest for topologists, algebraists and geometers. In previouse article we have looked at the braid group from a topological point of view, and thus using a new computerized representation of some elements of the fundamental group we gave a solution for its word problem. In this paper we will give an algorithm that will make it possible to transform the new presentation into a syntactic presentation. This will make it possible to computerize the group operation to sets of elements o…
There are well known relations between braid groups and symmetric groups, between Artin-Briskorn braid groups and Coxeter groups. Inverse braid monoid the same way is related to the inverse symmetric monoid. In the paper we show that similar relations exist between the inverse braid monoid of type $B$ and the inverse reflection monoid of type $B$. This gives a presentation of the last monoid.
Empirical properties of generating systems for complex reflection groups and their braid groups have been observed by Orlik-Solomon and Broué-Malle-Rouquier, using Shephard-Todd classification. We give a general existence result for presentations of braid groups, which partially explains and generalizes the known empirical properties. Our approach is invariant-theoretic and does not use the classification. The two ingredients are Springer theory of regular elements and a Zariski-like theorem.
The Cohomology of the Mod 4 Braid Group
本文构造了分类模4辫群扩张的2-上循环，证明其为已知上循环的模2约化，并给出扩张表示及level 4同余子群的正规生成集。
We study geometric presentations of braid groups for particles that are constrained to move on a graph, i.e. a network consisting of nodes and edges. Our proposed set of generators consists of exchanges of pairs of particles on junctions of the graph and of certain circular moves where one particle travels around a simple cycle of the graph. We point out that so defined generators often do not satisfy the braiding relation known from 2D physics. We accomplish a full description of relations between the generators for star graphs where we derive certain quasi-braiding relations. We also describ…
For a tree $T$ and a positive integer $n$, let $B_nT$ denote the $n$-strand braid group on $T$. We use discrete Morse theory techniques to show that the cohomology ring $H^*(B_nT)$ is encoded by an explicit abstract simplicial complex $K_nT$ that measures $n$-local interactions among essential vertices of $T$. We show that, in many cases (for instance when $T$ is a binary tree), $H^*(B_nT)$ is the exterior face ring determined by $K_nT$.
We design an algorithm writing down presentations of graph braid groups. Generators are represented in terms of actual motions of robots moving without collisions on a given graph. A key ingredient is a new motion planning algorithm whose complexity is linear in the number of edges and quadratic in the number of robots. The computing algorithm implies that 2-point braid groups of all light planar graphs have presentations where all relators are commutators.
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.
Braid groups and mapping class groups have many features in common. Similarly to the notion of inverse braid monoid inverse mapping class monoid is defined. It concerns surfaces with punctures, but among given $n$ punctures several can be omitted. This corresponds to braids where the number of strings is not fixed. In the paper we give the analogue of the Dehn-Nilsen-Baer theorem, propose a presentation of the inverse mapping class monoid for a punctured sphere and study the word problem. This shows that certain properties and objects based on mapping class groups may be extended to the invers…
Integral cohomology of classical Artin groups: spherical types $A,C,D$ and affine types $\widetilde B,\widetilde C,\widetilde D$
本文利用代数Salvetti复形和Fuchs复形，描述了除仿射族$\widetilde A$外的无限经典族Artin群的整上同调代数加法结构。
In 1947, in the paper "Theory of Braids", Artin raised the question of whether isotopy and homotopy of braids on the disk coincide. Twenty seven years later, Goldsmith answered his question: she proved that in fact the group structures are different, exhibiting a group presentation and showing that the homotopy braid group on the disk is a proper quotient of the Artin braid group on the disk $B_{n}$, denoted by $\widehat{B}_{n}$. In this paper, we extend Goldsmith's answer to Artin for closed, connected and orientable surfaces different from the sphere. More specifically, we define the notion …
Let G be a compact connected Lie group with a maximal torus T\subsetG. In the context of Schubert calculus we obtain a canonical presentation for the integral cohomology ring H^{\ast}(G/T) of the complete flag manifold G/T. The result have been applied in [15] to construct the integral cohomology ring H^{\ast}(G) in terms of Schubert classes on G/T, and in [16] to determine the structure of the modp cohomology H^{\ast}(G;F_{p}) as a Hopf algebra over the Steenrod algebra.
Let $G$ be a compact connected Lie group and $T$ be its maximal torus. The homogeneous space $G/T$ is called the (complete) flag manifold. One of the main goals of the {\em equivariant Schubert calculus} is to study the $T$-equivariant cohomology $H^*_T(G/T)$ with regard to the $T$-action on $G/T$ by multiplication. There are three presentations known for $H^*_T(G/T)$; (1) the free $H^*(BT)$-module generated by the Schubert varieties (2) (with the rational coefficients) the {\em double coinvariant ring} of the Weyl group (3) the {\em GKM ring} associated to the Hasse graph of the Weyl group. E…
The Gysin braid for $S^3$-actions on manifolds
针对非半自由的 $\mathbb S^3$ 光滑作用，本文构造联系 $M$ 的上同调与轨道空间相交上同调的 Gysin 辫，并揭示其依赖于 $M^{\mathbb S^1}$ 相交上同调的奇异项。
We show that the group cohomology of the diffeomorphisms of the disk with $n$ punctures has the cohomology of the braid group of $n$ strands as the summand. As an application of this method, we also prove that there is no cohomological obstruction to lifting the "standard" embedding $\mathrm{Br}_{2g+2}\hookrightarrow \mathrm{Mod}_{g,2}$ to a group homomorphism between diffeomorphism groups.
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…
We use rational formality of configuration spaces and the bar construction to study the cohomology of the space of braids in dimension four or greater. We provide a diagram complex for braids and a quasi-isomorphism to the de Rham cochains on the space of braids. The quasi-isomorphism is given by a configuration space integral followed by Chen's iterated integrals. This extends results of Kohno and of Cohen and Gitler on the cohomology of the space of braids to a CDGA suitable for integration. We show that this integration is compatible with Bott-Taubes configuration space integrals for long l…
We give an explicit presentation for the integral cohomology ring of the complement of any arrangement of level sets of characters in a complex torus (alias "toric arrangement"). Our description parallels the one given by Orlik and Solomon for arrangements of hyperplanes, and builds on De Concini and Procesi's work on the rational cohomology of unimodular toric arrangements. As a byproduct we extend Dupont's rational formality result to formality over $\mathbb Z$. The data needed in order to state the presentation is fully encoded in the poset of connected components of intersections of the ar…
We produce combinatorial models for configuration space in a simplicial complex, and for configurations near a single point ("local configuration space.") The model for local configuration space is built out of the poset of poset structures on a finite set. The model for global configuration space relies on a combinatorial model for a simplicial complex with a deleted subcomplex. By way of application, we study the nodal curve $y^2 z = x^3 + x^2 z$, obtaining a presentation for its two-strand braid group, a conjectural presentation for its three-strand braid group, and presentations for its tw…
We investigate the cohomology rings of regular semisimple Hessenberg varieties whose Hessenberg functions are of the form $h=(h(1),n\dots,n)$ in Lie type $A_{n-1}$. The main result of this paper gives an explicit presentation of the cohomology rings in terms of generators and their relations. Our presentation naturally specializes to Borel's presentation of the cohomology ring of the flag variety and it is compatible with the representation of the symmetric group $\mathfrak{S}_n$ on the cohomology constructed by J. Tymoczko. As a corollary, we also give an explicit presentation of the $\mathfr…
This paper upbuilds the theoretical framework of orbit braids in $M\times I$ by making use of the orbit configuration space $F_G(M,n)$, which enriches the theory of ordinary braids, where $M$ is a connected topological manifold of dimension at least 2 with an effective action of a finite group $G$ and the action of $G$ on $I$ is trivial. Main points of our work include as follows. We introduce the orbit braid group $\mathcal{B}_n^{orb}(M,G)$, and show that it is isomorphic to a group with an additional endowed operation (called the extended fundamental group of $F_G(M,n)$), formed by the homot…
We prove that H^{d-1}(SL_n Z; Q) = 0, where d = n-choose-2 is the cohomological dimension of SL_n Z, and similarly for GL_n Z. We also prove analogous vanishing theorems for cohomology with coefficients in a rational representation of the algebraic group GL_n. These theorems are derived from a presentation of the Steinberg module for SL_n Z whose generators are integral apartment classes, generalizing Manin's presentation for the Steinberg module of SL_2 Z. This presentation was originally constructed by Bykovskii. We give a new topological proof of it.
In this paper we compute the homology of the braid groups, with coefficients in the module Z[q^+-1] given by the ring of Laurent polynomials with integer coefficients and where the action of the braid group is defined by mapping each generator of the standard presentation to multiplication by -q. The homology thus computed is isomorphic to the homology with constant coefficients of the Milnor fiber of the discriminantal singularity.
The Springer variety of type $A$ associated to a nilpotent operator on $\mathbb{C}^n$ in Jordan canonical form admits a natural action of the $\ell$-dimensional torus $T^{\ell}$ where $\ell$ is the number of the Jordan blocks. We give a presentation of the $T^{\ell}$-equivariant cohomology ring of the Springer variety through an explicit construction of an action of the $n$-th symmetric group on the $T^{\ell}$-equivariant cohomology group. The $T^{\ell}$-equivariant analogue of so called Tanisaki's ideal will appear in the presentation.
The main result of this note gives an explicit presentation of the $S^1$-equivariant cohomology ring of the $(n-k,k)$ Springer variety (in type $A$) as a quotient of a polynomial ring by an ideal $I$, in the spirit of the well-known Borel presentation of the cohomology of the flag variety.
The main result of this note is an efficient presentation of the $S^1$-equivariant cohomology ring of Peterson varieties (in type $A$) as a quotient of a polynomial ring by an ideal $\mathcal{J}$, in the spirit of the well-known Borel presentation of the cohomology of the flag variety. Our result simplifies previous presentations given by Harada-Tymoczko and Bayegan-Harada. In particular, our result gives an affirmative answer to a conjecture of Bayegan and Harada that the defining ideal $\mathcal{J}$ is generated by quadratics.
A combinatorial genesis of the right-angled relations in Artin's classical braid groups
本文通过组合方法研究Artin经典辫形群的右角关系，利用UC(n,p×q)空间的特性，推导出其基本群与Artin群的近似关系，并计算了相关空间的范畴和拓扑复杂度。
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 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.
Oriented Cohomology Rings of Some Moduli Spaces via Blowups
本文通过定向余弦理论的加法吹胀公式，研究了光滑方案沿光滑中心的吹胀的定向余弦环，并给出了del Pezzo表面、P^3沿扭曲三次曲线的吹胀以及P^5沿Veronese面的吹胀的显式例子，展示了如何用任意定向余弦理论解决经典枚举几何问题。
A presentation of symplectic Steinberg modules and cohomology of $\operatorname{Sp}_{2n}(\mathbb{Z})$
本文给出辛 Steinberg 模的表示，并据此证明 $\operatorname{Sp}_{2n}(\mathbb{Z})$ 及主极化阿贝尔簇模栈 $\mathcal{A}_n$ 在次数 $n^2-1$ 的有理上同调消失。
In the present paper we give a new method for converting virtual knots and links to virtual braids. Indeed the braiding method given in this paper is quite general, and applies to all the categories in which braiding can be accomplished. We give a unifying topological interpretation of virtuals and flats (virtual strings) and their isotopies via ribbon surfaces and abstract link diagrams. We also give reduced presentations for the virtual braid group, the flat virtual braid group, the welded braid group and several other categories of braids. The paper includes a discussion of the topological …
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