SURVEY ON BRAIDS

V. VERSHININ

Abstract. In the paper we give a survey on braid groups, their properties and generalizations.

Contents

1. Introduction 1 2. Historical Remarks 2 3. Definitions and General Properties 3 3.1. Systems of n curves in three-dimensional space and braid groups 3 3.2. Braid groups and configuration spaces 5 3.3. Braid groups as automorphism groups of free groups and the word problem 6 3.4. Commutator subgroup and other presentations 7 3.5. Presentaion of the pure braid group and Markov normal form 8 4. Garside Normal Form, Center and Conjugacy Problem 9 5. Ordering of Braids 10 6. Representations 11 6.1. Burau representation 11 6.2. Lawrence-Krammer representation 11 7. Generalizations of Braids 12 7.1. Configuration spaces of manifolds 12 7.2. Brieskorn-Artin braid groups 12 7.3. Braid groups of surfaces 14 7.4. Braid groups in handlebodies 14 7.5. Braids with singularities 16 8. Homological Properties 18 8.1. Configuration spaces and K(π, 1)-spaces 18 8.2. Cohomology of pure braid groups 19 8.3. Homology of braid groups 20 References 21

1. Introduction

Braid groups describe intuitive concept of classes of continuous deformations of braids, which are collections of interwining strands whose endpoints are fixed. Mathematically they can be considered from various points of view. The first intuitive approach is formalized naturally as isotopy classes of a collection of n connected curves (strings) in 3-dimensional space. This point of view is connected with the definition of braid group as the fundamental group of configuration space of n points on a plane. Also braids can be interpreted as a mapping class group of a punctured disc and as a subgroup of automorphism group of a free group (subsection 3.3).

2000 Mathematics Subject Classification. Primary 20F36; Secondary 20F38.

Key words and phrases. Braid.

The author was supported in part by the French-Russian Program of Research EGIDE (dossier No 04495UL).

The present survey is organized as follows. In Section 2 we make some historical remark. Definition and general properties are considered in Section 3. Configuration spaces appear in Subsection 3.2. Connections with groups of automorphisms of free groups are given in Subsection 3.3 Presentations of the braid group which appeared quite recently are observed in Subsection 3.4. Section 4 is devoted to F. A. Garside’s classical work [51] and Section 5 to that of P. Dehornoy on ordering for braids. Representations and in particular linearity are discussed in Section 6. In Section 7 various generalization of braids are presented. Homological properties are observed in Section 8.

The author is grateful to F. R. Cohen and Wu Jie for fruitful discussions on theory of braids and their applications. The author is also thankful to V. Kulikov who attracted his attention to the works of O. Zariski [96], [97] and to E. P. Volokitin for help and advices on the presentation of the paper.

2. Historical Remarks

Braids were rigorously defined by E. Artin [6] in 1925, although the roots of this natural concept are seen in the works of A. Hurwitz ([59], 1891), R. Fricke and F. Klein ([49], 1897) and even in the notebooks of K.-F. Gauss. E. Artin [6] gave the presentation of the braid group (see formulas (3.2) in Section 3) which is common now. Already in the book of Felix Klein [63] published in 1926 there appeared a chapter about braids. Essential topics about braids were also presented in the Reidemeister’s Knotentheorie [85] publisheded in 1932.

In 30-ies there appeared a series of papers of Werner Burau [20], [21], [22] where he in particular gave the presentation of the pure braid group (see subsection 3.5) and introduced the Burau representation (subsection 6.1). In his work [69] published in 1934 Wilhelm Magnus established relations between braid groups and the mapping class groups. At the same time there appeared the work of A. A. Markov [74], which together with Alexander Theorem [2] builds a bijection between links and equivalence classes of braids. It became an essential ingredient in study of links and knots (in the work of V. F. R. Jones [62], for example). In 1936-37 were published the works of O. Zariski [96], [97], where he discovered connections between braid groups and the fundamental group of the complement of discriminant of the general polynomial

f_{n}(t) = a_{0}t^{n} + a_{1}t^{n−1} + ... + a_{n−1}t + a_{n},

a point of view later rediscovered by V. I. Arnold [4]. Zariski also understood connections between braids and configuration spaces, gave the presentation of the braid group of the sphere, and studied the braid groups of Riemann surfaces. Amasingly and unfortunately these works of Zariski were not noticed by the specialists on braids and are not mentioned even in books and papers where the presentations of braids of surfaces are discussed.

In the begining of 60-ies R. Fox and L. Neuwirth [48] and E. Fadell and L. Neuwirth [42] studied configuration spaces. They turned out to be K(π, 1)-spaces and so give natural geometrical model of the classifying spaces for the braid groups. Later, V. I. Arnold [4] in this direction proved the first results on cohomologies of braids. The motivation for his study was a connection (which he discovered) with the problem of representing algebraic functions in several variables by superposing algebraic functions in fewer variables. Also, in 1969 V. I. Arnold completely described the cohomologies of pure braid groups [3].

In 1969 there appeared the publication of F. A. Garside’s work [51] where he suggests a new normal form of elements in the braid group and with its help gives a new solution of the word problem and also solves the conjugacy problem. In 1968 was published a two-page note of G. S. Makanin [73] where he sketches his algorithm for the solution of the conjugacy problem. The complete publication of Makanin’s work didn’t appear (as far as the author is aware).

In 70-ies the study of cohomologies of braids were continued independently and by different methods by D. B. Fuks [50] who determined mod 2 cohomologies, and F. R. Cohen [26], [27],

[28] who described these homologies with coefficients in ℤ and in ℤ/p as modules over the Steenrod algebra.

In 1984-85 independently N. V. Ivanov [60] and J. McCarthy [76] proved the ”Tits-alternative” for the mapping class groups of surfaces and as a consequence it is true for the braid groups. Namely, they proved that every subgroup of the mapping class group either contains an abelian subgroup of finite index, or contains a nonabelian free group.

The question of whether braid groups are linear attracted significant attention. It was realized that the Burau representation is faithful for Br_{3} [52], [72]. Then after a long break in 1991 J. A. Moody [78] proved that Burau representation is unfaithful for n ≥ 9. This bound was improved to n ≥ 6 by D. D. Long and M. Paton [68] and to n = 5 by S. Bigelow [9]. In 1999-2000 there appeared preprints of papers of D. Krammer [64], [65] and S. Bigelow [10] who proved that Br_{n} is linear for all n.

At the begining of nineties P. Dehornoy [34], [35], [36] proved that there exits a left order in braid groups.

Interesting generalizations of braids were introduced in the work of E. Brieskorn [16]. The configuration space can also be considered as the orbit space of the complement of the complexification of the arrangement of hyperplanes corresponding to the Coxeter group A_{n−1} = Σ_{n}. Generalizing this approach for any finite Coxeter group, E. Brieskorn defined the so-called generalized braid groups which are also called Artin groups.

Another way of generalization is to consider braid groups in 3-manifolds, possibly with a boundary. The simplest examples are braid groups in handlebodies. A. B. Sossinsky [88] was the first who studied them. Such a group can be interpreted as the fundamental group of the configuration space of a plane without g points where g is the genus of the handlebody. The generalized braid group of type C is isomorphic to the braid group in the solid torus.

Under the theory of Vassiliev invariants singular braids were introduced. The corresponding algebraic structures are the Baez–Birman monoid [8], [12] and the braid-permutation group by R. Fenn, R. Rimányi and C. Rourke [45], [46]. Various properties of these objects were studied [47], [98], [53], [54], [33], [61], [31], [55].

3. Definitions and General Properties

3.1. Systems of n curves in three-dimensional space and braid groups. First of all as it was already mentioned braids naturally arise as objects in 3-space. Let us consider two parallel planes P_{0} and P_{1} in ℝ^{3}, which contain two ordered sets of points A_{1}, ..., A_{n} ∈ P_{0} and B_{1}, ..., B_{n} ∈ P_{1}. These points are lying on parallel lines L_{A} and L_{B} respectively. The space between the planes P_{0} and P_{1} we denote by Π. Suppose that the point B_{i} is lying under the point A_{i}. Let us connect the set of points A_{1}, ..., A_{n} with the set of points B_{1}, ..., B_{n} by simple nonintersecting curves C_{1}, ..., C_{n} lying in the space Π and such that each curve meets only once each parallel plane P_{t} lying in the space Π (see Figure 3.1). This object is called a braid and the curves are called the strings of a braid. Usually braids are depicted by projections on

Figure 3.1

[figure: 1; i − 1; i; i + 1; i + 2; n; …; …] Figure 3.2

the plane passing through the lines L_{A} and L_{B}. This projection is supposed to be in general position so that there is only finite number of double points of intersection which are lying on pairwise different levels and intersections are transversal. The simplest braid σ_{i} (Fig. 3.2) corresponds to the transposition (i, i + 1).

Let us introduce the following equivalence relation on the set of all braids with n strings and with fixed P_{0}, P_{1}, A_{i} and B_{i}. It is defined by homeomorphisms h : Π → Π, identical on P_{0} ∪ P_{1} and such that h(P_{t}) = P_{t}. Braids β and β′ are equivalent if there exists such a homeomorphism h that h(β) = β′. On the set Br_{n} of equivalence classes under the considered relation the structure of a group introduces as follows. We put a copy Π′ of the domain Π under the Π in such a way that P′_{0} coincides with P_{1} and each A_{i} coincides with B_{i} and it is possible to glue braids β and β′. This gluing gives a composition of braids ββ′ (Fig. 3.3). Unit element is the equivalence class containing a braid of n parallel intervals, the braid β^{−1} inverse to β is defined by reflection of β with respect to the plane P_{1/2}. A string C_{i} of a braid β connects the point A_{i} with the pont B_{k_{i}} defining the permutation S^{β}. If this permutation is identical then the braid β is called pure. The map β → S^{β} defines an epimorphism τ_{n} of the braid group Br_{n} on the permutation group Σ_{n}with the kernel consisting of all pure braids:

(3.1)  1 → P_{n} → Br_{n} ⟶^{τ_{n}} Σ_{n} → 1.

The following presentation of the braid group Br_{n} with generators σ_{i}, i = 1, ..., n − 1 and two types of relations:

(3.2)  \begin{cases}σ_{i}σ_{j} & = σ_{j} σ_{i}, if |i − j| > 1, \\ σ_{i}σ_{i+1}σ_{i} & = σ_{i+1}σ_{i}σ_{i+1} \end{cases}

is the algebraic expression of the fact that any isotopy of braids can be broken down into “elemntary moves” of two types that correspond to two types of relations.

Figure 3.3

[figure: ↔; ↔] Figure 3.4

Figure 3.5

If the symmetric group Σ_{n} is given by its canonical presentation with generators s_{i}, i = 1, ..., n − 1 and relations:

(3.3)  \begin{cases}s_{i}s_{j} & = s_{j} s_{i}, if |i − j| > 1, \\ s_{i}s_{i+1}s_{i} & = s_{i+1}s_{i}s_{i+1} \\ s_{i}^{2} = 1, \end{cases}

then the homomorphism τ_{n} is given by the formula

τ_{n}(σ_{i}) = s_{i}, i = 1, …, n − 1.

It is possible to consider brads as classes of equivalence of braid diagrams which are generic projections of three dimensional braids on a plane. The classes of equivalence are defined by the Reidemeister moves depicted at Figure 3.4.

3.2. Braid groups and configuration spaces. If we look at the Figure 3.1, then this picture can be interpreted as a graph of a loop in the configuration space of n points on a plane, that is the space of unordered sets of n points on a plane, see Figure 3.5. So, it is possible to interpret the braid group as the fundamental group of the configuration space. Formally it is done as follows. The symmetric group Σ_{m} acts on the Cartesian power (ℝ^{2})^{m} of the space ℝ^{2}:

(3.4)  w(y_{1}, ..., y_{m}) = (y_{w^{−1}(1)}, ..., y_{w^{−1}(m)}), w ∈ Σ_{m}.

Denote by F (ℝ^{2}, m) the space of m-tuples of pairwise different points in ℝ^{2}:

F (ℝ^{2}, m) = {p_{1}, ..., p_{m}) ∈ (ℝ^{2})^{m} : p_{i} ≠ p_{j} for i ≠ j}.

This is the space of regular points of this action. We call the orbit space of this action B(ℝ^{2}, m) = F (ℝ^{2}, m)/Σ_{m} the configuration space of n points on a plane. The braid group Br_{m} is the fundamental group of configuration space

Br_{m} = π_{1}(B(ℝ^{2}, m)).

The pure braid group P_{m} is the is the fundamental group of the space F (ℝ^{2}, m). The covering

p : F (ℝ^{2}, m) → B(ℝ^{2}, m)

defines the exact sequence:

(3.5)  1 → π_{1}(F (ℝ^{2}, m)) →^{p_{∗}} π_{1}(B(ℝ^{2}, m)) → Σ_{n} → 1.

which is equivalent to sequence (3.1).

It can be used for proving the canonical presentation of the braid group (3.2) as it is done, for example in the book of J. Birman [11].

Such considerations were made by R. Fox and L. Neuwirth [48].

3.3. Braid groups as automorphism groups of free groups and the word problem. Another important approach to the braid group bases on the fact that this group may be considered as a subgroup of the automorphism group of a free group.

Let F_{n} be the free group of rank n with the set of generators {x_{1}, …, x_{n}}. Assume further that Aut F_{n} is the automorphism group of F_{n}.

We have the standard inclusions of the symmetric group Σ_{n} and the braid group Br_{n} into Aut F_{n}. These inclusions may be described as follows. Let \overline{σ}_{i} ∈ Aut F_{n}, i = 1, 2, …, n − 1, be given by the formula which describes its action on generators:

(3.6)  \begin{cases}x_{i} & ↦ x_{i+1}, \\ x_{i+1} & ↦ x_{i+1}^{−1}x_{i}x_{i+1}, \\ x_{j} & ↦ x_{j}, j ≠ i, i + 1. \end{cases}

Let us define a map i of the generators σ_{i}, i = 1, …, n − 1 of the braid group Br_{n} to these automorphisms:

(3.7)  i(σ_{i}) = \overline{σ}_{i}.

Theorem 3.1. Formulas (3.7) define correctly a homomorphism

i : Br_{n} → Aut F_{n}.

which is a monomorphism.

Theorem 3.1 gives a solution of the word problem for the braid groups. It was done first by E. Artin [6].

The free group F_{n} is a fundamental group of a disc D_{n} without n points and the generator x_{i} corresponds to a loop going around the i-th point. The braid group Br_{n} is the mapping class group of a disc D_{n} with its boundary fixed [11] and so it acts on the fundamental group of D_{n}.

[figure: .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; x_{i}; x_{i+1}; x_{i+1}^{−1}x_{i}x_{i+1}] Figure 3.6

This action is described by the formulas (3.6) where x_{i} correspond to the canonical loops on D_{n} which form the generators of the fundamental group. Geometrically this action is depicted in the Figure 3.6.

3.4. Commutator subgroup and other presentations. Let us define a homomorphism from braid group to integers by taking the sum of exponents of the entries of the generators σ_{i} in the expression of any element of the group through these canonical generators:

deg : Br_{n} → ℤ, deg(b) = ∑_{j} m_{j}, where b = (σ_{i_{1}})^{m_{1}} … (σ_{i_{k}})^{m_{k}}.

Proposition 3.1. The homomorphism

deg : Br_{n} → ℤ

gives the abelianization of the braid group and the commutator subgroup Br′_{n} is characterized by the condition

b ∈ Br′_{n} if and only if deg(b) = 0.

Proof. Let a : Br_{n} → A be a homomorphism to any other abelian group A, then from the relations (3.2) we have:

a(σ_{i})a(σ_{i+1})a(σ_{i}) = a(σ_{i+1})a(σ_{i})a(σ_{i+1}).

the commutativity of A gives that a(σ_{i+1}) = a(σ_{i}). This means that the homomorphism deg is universal. □

Of course, there exist another presentations of the braid group. Let

σ = σ_{1}σ_{2} … σ_{n−1},

then the group Br_{n} is generated by σ_{1} and σ because

σ_{i+1} = σ^{i}σ_{1}σ^{−i}, i = 1, … n − 2.

The relations for the generators σ_{1} and σ are the following

(3.8)  \begin{cases}σ_{1}σ^{i}σ_{1}σ^{−i} & = σ^{i}σ_{1}σ^{−i}σ_{1} for 2 ≤ i ≤ n/2, \\ σ^{n} & = (σσ_{1})^{n−1}. \end{cases}

This was observed by Artin in the initial paper [6].

An interesting series of presentations was given by V. Sergiescu [87]. For every planar graph he constructed a presentation of the group Br_{n}, where n is the number of vertices of the graph, with generators corresponding to edges and relations reflecting the geometry of the graph. Artin’s presentation in this context corresponds to the graph consisting of the interval from 1 to n with the natural numbers (from 1 to n) as vertices and with segments between them as edges.

J. S. Birman; K. H. Ko; S. J. Lee [13] introduced the presentation with the generators a_{ts} with 1 < s < t ≤ n and relations

(3.9)  \begin{cases}a_{ts}a_{rq} & = a_{rq}a_{ts} for (t − r)(t − q)(s − r)(s − q) > 0, \\ a_{ts}a_{sr} & = a_{tr}a_{ts} = a_{sr}a_{tr} for 1 ≤ r < s < t ≤ n. \end{cases}

The generators a_{ts} are expressed by the canonical generators σ_{i} in the following form:

a_{ts} = (σ_{t−1}σ_{t−2} ⋯ σ_{s+1})σ_{s}(σ_{s+1}^{−1} ⋯ σ_{t−2}^{−1}σ_{t−1}^{−1}) for 1 < s < t ≤ n.

The set of generators for braid groups were even enlarged in the work of Jean Michel [77] as follows. Let | | : Σ_{n} → ℤ be the length function on the symmetric group with respect to the generators s_{i}: for x ∈ Σ_{n}, |x| is the smallest natural number k such that x is a product of k elements of the set {s_{1}, ..., s_{n−1}}. It is known ([15], Sect. 1, Ex. 13(b)) that two minimal expressions for an element of Σ_{n} are equivalent by using only of the relations (3.2). This implies that the canonical projection τ_{n} : Br_{n} → Σ_{n} has a unique set-theoretic section r : Σ_{n} → Br_{n} such that r(s_{i}) = σ_{i} for i = 1, ..., n−1 and r(xy) = r(x) r(y) whenever |xy| = |x|+|y|. Then the group Br_{n} admits a presentation by generators {r(x) | x ∈ Σ_{n}} and relations r(xy) = r(x) r(y) for all x, y ∈ Σ_{n} such that |xy| = |x| + |y|.

3.5. Presentaion of the pure braid group and Markov normal form. Let f(y_{1}, ..., y_{m}) be a word with (possibly empty) entries of y_{i}^{ϵ}, where y_{i} are some letters and ϵ may be ±1. If y_{i} are elements of a group G then f(y_{1}, ..., y_{m}) will be considered as the corresponding element of G.

Let us define the elements s_{i,j}, 1 ≤ i < j ≤ m, of the braid group Br_{m} by the formula:

s_{i,j} = σ_{j−1}...σ_{i+1}σ_{i}^{2}σ_{i+1}^{−1}...σ_{j−1}^{−1}.

These elements satisfy the following Burau relations ([20], [75], see also [67]):

(3.10) s_{i,j}s_{k,l} = s_{k,l}s_{i,j} for i < j < k < l and i < k < l < j, s_{i,j}s_{i,k}s_{j,k} = s_{i,k}s_{j,k}s_{i,j} for i < j < k,

s_{i,k}s_{j,k}s_{i,j} = s_{j,k}s_{i,j}s_{i,k} for i < j < k,
s_{i,k}s_{j,k}s_{j,l}s_{j,k}^{−1} = s_{j,k}s_{j,l}s_{j,k}^{−1}s_{i,k} for i < j < k < l.

W. Burau and later A. A. Markov proved that the elements s_{i,j} with the relations (3.10) give a presentation of the pure braid group P_{m} [75]. The following formula is a consequence of the Burau relations and also belongs to A. A. Markov:

(3.11)  [s_{i,l}, s_{j,k}^{ϵ}] = f(s_{1,l}, ..., s_{l−1,l}), ϵ = ±1, k < l.

Let us define the elements σ_{k,l}, 1 ≤ k ≤ l ≤ m by the formulas

σ_{k,k} = e,
σ_{k,l} = σ_{k}^{−1}...σ_{l−1}^{−1}.

Let P_{m}^{k} be the subgroup of P_{m} generated by the elements s_{i,j} with k < j.

Theorem 3.2. (A. A. Markov) (i) Every element of the group Br_{m} can be uniquely written in the form

(3.12)  f_{m}(s_{1,m}, ..., s_{m−1,m})...f_{j}(s_{1,j}, ..., s_{j−1,j})...f_{2}(s_{1,2})σ_{i_{m},m}...σ_{i_{j},j}...σ_{i_{2},2}.

(ii) The factor group P_{m}^{k}/P_{m}^{k−1} is the free group on free generators s_{i,k+1}, 1 ≤ i ≤ k.

The form (3.12) is called the Markov normal form, it also gives the solution of the word problem for the braid groups. of the word problem for the braid groups. ———- X-Sun-Data-Type: default X-Sun-Data-Description: default X-Sun-Data-Name: sec3.tex X-Sun-Content-Lines: 99 X-Sun-Charset: iso-8859-1

4. Garside Normal Form, Center and Conjugacy Problem

Essential role in Garside work [51] plays the monoide of positive braids, that is the monoide which has a presentation with generators σ_{i}, i = 1, ..., n and relations (3.2). In other words each element of this monoide can be represented as a word on the elements σ_{i}, i = 1, ..., n with no entrances of σ_{i}^{−1}. The fundamental word Δ in the braid group Br_{n+1} is defined by the formula:

Δ = σ_{1} … σ_{n}σ_{1} … σ_{n−1} … σ_{1}σ_{2}σ_{1}.

For a positive word W in σ_{i}, i = 1, ..., n we say that Δ is a factor of W or simply W contains Δ, if W ≐ AΔB with A and B being arbitrary positive words, probably empty, the symbol ≐ means that words are positively equal, i.e. as elements in the positive braid monoide. If W does not contain Δ we shall say W is prime to Δ.

Among positive words on the alphabet {σ_{1} … σ_{n}} let us introduce a lexicographical ordering with the condition that σ_{1} < σ_{2} < ⋯ < σ_{n}. For a positive word W the base of W is the smallest positive word which is positively equal to W. The base is uniquely determined. If a positive word A is prime to Δ, then for the base of A the notation \overline{A} will be used.

Theorem 4.1. (F. A. Garside) Every word W in Br_{n+1} can be uniquely written in the form Δ^{m}\overline{A}, where m is an integer.

The form of this theorem is called the Garside normal form and the index m is called the power of W. Garside normal form also gives a solution to the word problem in the braid group.

Theorem 4.2. (F. A. Garside) The necessary and sufficient condition that two words in Br_{n+1} are equal is that their Garside normal forms are identical.

The center of the braid group was fistly found by W.-L. Chow [25]. Namely, as it follows from the presentation of braid groups with two generators σ_{1} and σ and relations (3.8) given in the subsection 3.1 the element σ^{n} commutes with σσ_{1} and so with σ_{1}. Chow proved that it generates the center.

Garside normal form gives an elegant proof of the following theorem.

Theorem 4.3. (i) When n = 1 the center of the group Br_{n+1} is generated by Δ. (ii) When n > 1 the center of the group Br_{n+1} is generated by Δ^{2}.

Let α be a positive word such that Δ ≐ αX, where X is an arbitrary positive word, probably empty. For any word W in Br_{n+1}, the word α^{−1}W α, reduced to Garside normal form is called an α-transformation of W.

For any word W in Br_{n+1} with the Garside normal form Δ^{m}\overline{A} ≡ W_{1} consider the following chains of α-transformations: take all the α-transformations of W_{1} and let those which are of power ≥ m and which are distinct from each other be W_{2}, W_{3}, …, W_{t}. Now repeat the process for each of the words W_{2}, W_{3}, …, W_{t} in turn, denoting successively by W_{t+1}, W_{t+2}, …, any new words occurring, the condition being always that each new word must be of power ≥ m. Continue to repeat the process for every new distinct word arrising, as the sequence W_{1}, W_{2}, W_{t+2}, …, expands.

Proposition 4.1. The set W_{1}, W_{2}, W_{t+2}, …, is finite.

Suppose that in the set W_{1}, W_{2}, W_{t+2}, …, the highest power reached is s and that the words of power s form the subset V_{1}, V_{2}, . . .. Then this set V_{1}, V_{2}, … is called the summit set of W.

Theorem 4.4. (F. A. Garside) Two elements A and B of the group Br_{n+1} are conjugate if and only if their summit sets are identical.

In subsequent works of S. I. Adyan [1], W. Thurston [40], E. El-Rifai and H. R. Morton [38], J. S. Birman, K. H. Ko and S. J. Lee [13] this direction in the theory of braids was developed and Garside normal form was precised. In particular Thurston introduced so called right-greedy normal form and with the help of it proved that the braid group is biautomatic. J. S. Birman, K. H. Ko and S. J. Lee considered the word δ = a_{n(n−1)} ⋯ a_{32}a_{21} = σ_{n−1} ⋯ σ_{2}σ_{1}, as a fundamental and proved that every element in Br_{n} has a representative W = δ^{j}A_{1}A_{2} ⋯ A_{k} with positive A_{i} in a unique way in some sense. Based on this form they gave an algorithm for the word problem in B_{n} which runs in time (nm^{2}) for a given word of length m.

5. Ordering of Braids

A group G is said totally (or linearly) left (correspondingly right) ordered if it has a total order < invariant by left (right) multiplication, i.e. if a < b, then ca < cb for any a, b, c ∈ G. If this order is also invariant by right (left) multiplication, then the group G is called ordered.

For any left ordered group G denote by P the set of positive elements {x ∈ G : x > 1}, then the set of negative elements is defined by the formula: P^{−1} = {x ∈ G : x ∈ P }. The total character of an order on G is expressed by the partition

G = P ∐{1} ∐ P^{−1}.

The invariance of multiplication is expressed by the inclusion P^{2} ⊂ P, where P^{2} is formed by products of couples of elements of P. Conversely, if there exist a subset P of a group G with the properties:

G = P ∐{1} ∐ P^{−1}, P^{2} ⊂ P,

then G is left ordered by the order defined by: x < y if and only if x^{−1}y ∈ P. A group G then is ordered if and only if xP x^{−1} ⊂ P for all x ∈ G.

Let i ∈ {1, …, n} and a word w on the alphabet {σ_{1}, …, σ_{n}} is expressed in the form

w_{0}σ_{i}w_{1}σ_{i} … σ_{i}w_{r},

where the subwords w_{0}, …, w_{r} are the words on the letters σ_{j}^{±1} with j > i. Then such a word is called σ_{i}-positive. This means that all entries of σ_{i}^{±1} in the word w with i minimal must be positive. If all such entries are negative then a word w is called σ_{i}-negative. A braid of Br_{n+1} is called σ_{i}-positive (σ_{i}-negative) if there exists its expression as a word on the standard generators which is σ_{i}-positive (σ_{i}-negative). A braid is called σ-positive (σ-negative) if it exists a number i, such that it is σ_{i}-positive (σ_{i}-negative).

Theorem 5.1. (P. Dehornoy) Every braid in Br_{n+1} different from 1 is either σ-positive or σ-negative.

Corollary 5.1. For all n the braid group Br_{n+1} is left ordered.

6. Representations

6.1. Burau representation. Let us map the generators of the braid group Br_{n} to the following elements of the group GL_{n}ℤ[t, t^{−1}]

(6.1)  σ_{i} ↦ (\begin{matrix}E_{i−1} & 0 & 0 & 0 \\ 0 & 1 − t & t & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & E_{n−i−1} \end{matrix}) ,

where E_{i} is the unit i × i matrix. The formula (6.1) define correctly the representation of the braid group in GL_{n}ℤ[t, t^{−1}]:

r : Br_{n} → GL_{n}ℤ[t, t^{−1}],

which is called Burau representation [22].

Theorem 6.1. Burau representation is faithful for n = 3.

Theorem 6.2. (J. A. Moody; D. D. Long and M. Paton; S. Bigelow) Burau representation is unfaithful for n ≥ 5.

The case n = 4 remains open.

6.2. Lawrence-Krammer representation. Consider the ring K = ℤ[q^{±1}, t^{±1}] of Laurent polynomials in two variables q, t, and the free K-module

V = ⨁_{1≤i<j≤n} R x_{i,j}.

For k ∈ {1, 2, …, n − 1}, define the action of the braid generators σ_{k} on the basis of V by the formula:

(6.2)  σ_{k}(x_{i,j}) = \begin{cases}x_{i,j}, & k < i − 1 or j < k; \\ x_{i−1,j} + (1 − q)x_{i,j}, & k = i − 1; \\ tq(q − 1)x_{i,i+1} + qx_{i+1,j}, & k = i < j − 1; \\ tq^{2}x_{i,j}; & k = i = j − 1 \\ x_{i,j} + tq^{k−i}(q − 1)^{2}x_{k,k+1}, & i < k < j − 1; \\ x_{i,j−1} + tq^{j−i}(q − 1)x_{j−1,j}, & k = j − 1; \\ (1 − q)x_{i,j} + qx_{i,j+1}, & k = j. \end{cases}

Direct comutation shows that this defines a representation

ρ_{n} : Br_{n} → GL(V),

which was firstly defined by R. Lawrence [66] in topological terms and in the explicit form (6.2) by D. Krammer [65].

Theorem 6.3. (S. Bigelow [10], D. Krammer [65]) The representation

ρ_{n} : Br_{n} → GL (V)

is faithful for all n ≥ 1.

Remark 6.1. Actually, S Bigelow [10] proved this theorem for the representation ρ_{n} characterized in homological terms and D. Krammer [65] proved the following. Let K = ℝ[t^{±1}], q ∈ ℝ, and 0 < q < 1. Then the representation ρ_{n} defined by (6.2) is faithful for all n ≥ 1. This result implies Theorem 6.3: if a representation over ℤ[q^{±1}, t^{±1}] becomes faithful after assigning a real value to q, then it is faithful itself.

M. G. Zinno [99] established connection between Birman-Murakami-Wenzl algebra [14], [80] and the Lawrence-Krammer representation. Namely, he proved that the Lawrence-Krammer representation is identical to the irreducible representations of the Birman-Murakami-Wenzl algebra parametrized by Young diagrams of shapes (n − 2) and (1^{n−2}). This means that the Young diagram in the case considered consists of one row (respectively of one column) only, with n−2 boxes. It follows that Lawrence-Krammer representation is irreducible. X-Sun-Data-Name: sec6.tex X-Sun-Content-Lines: 374 X-Sun-Charset: us-ascii

7. Generalizations of Braids

7.1. Configuration spaces of manifolds. The notion of configuration space of Subsection 3.2 can be naturally generalized for a manifold as follows.

Let Y be a connected topological manifold and let W be a finite group acting on Y. A point y ∈ Y is called regular if its stabilizer {w ∈ W : wy = y} is trivial, i.e., consists only of the unit of the group W. The set Ỹ of all regular points is open. Suppose that it is connected and nonempty. The subspace ORB(Y, W) of the space of all orbits Orb(Y, W) consisting of the orbits of all regular points is called the space of regular orbits. There is a free action of W on Ỹ and the projection p : Ỹ → Ỹ /W = ORB(Y, W) defines a covering. Let us consider the initial segment of the long exact sequence of this covering:

(7.1)  1 → π_{1}(Ỹ , y_{0}) →^{p_{∗}} π_{1}(ORB(Y, W), p(y_{0})) → W → 1.

The fundamental group π_{1}(ORB(Y, W), p(y_{0})) of the space of regular orbits is called the braid group of the action of W on Y and is denoted by Br(Y, W). The fundamental group π_{1}(Ỹ , y_{0}) is called the pure braid group of the action of W on Y and is denoted by P (Y, W). The spaces Ỹ and ORB(Y, W) are path connected, so the pair of these groups is defined uniquely up to isomorphism and we may omit mentioning the base point y_{0} in the notations.

For any space Y the symmetric group Σ_{m} acts on the Cartesian power Y^{m} of the space Y by the formulas (3.4) We denote by F (Y, m) the space of m-tuples of pairwise different points in Y:

F (Y, m) = {p_{1}, ..., p_{m}) ∈ Y^{m} : p_{i} ≠ p_{j} for i ≠ j}.

This is the space of regular points of this action. In the case when Y is a connected topological manifold M without boundary and dim M ≥ 2, the space of regular orbits ORB(M^{m}, Σ_{m}) is open, connected and nonempty. We call ORB(M^{m}, Σ_{m}) the configuration space of the manifold M and denote by B(M, m). The braid group Br(M^{m}, Σ_{m}) is called the braid group on m strings of the manifold M and is denoted by Br(m, M). Analogously, we call the group P (M^{m}, Σ_{m}) the pure braid group on m strings of the manifold M and denote it by P (m, M). These definitions of braid groups were given by R. Fox and L. Neuwirth [48].

7.2. Brieskorn-Artin braid groups. The braid groups are included in the series of so called generalized braid groups (this was their name in the work of E. Brieskorn of 1971 [16]), or Artin groups (so they we called by E. Brieskorn and K. Saito in the paper of 1972 [19]). They were defined by E. Brieskorn [16], so we call them Brieskorn–Artin groups.

Let V be a finite dimensional real vector space (dim V = n) with Euclidean structure. Let W be a finite subgroup of GL(V) generated by reflections. Let M be the set of hyperplanes such that W is generated by orthogonal reflections with respect to M ∈ M. We suppose that for every w ∈ W and every hyperplane M ∈ M the hyperplane w(M) belongs to M.

The group W is generated by the reflections w_{i} = w_{i}(M_{i}), i ∈ I, satisfying only the following relations

(w_{i}w_{j})^{m_{i,j}} = e, i, j ∈ I,

where the natural numbers m_{i,j} = m_{j,i} form the Coxeter matrix of W from which the Coxeter graph Γ(W) of W is constructed [15]. We use the following notation of P. Deligne [37]: prod(m; x, y) denotes the product xyxy... (m factors). The generalized braid group (or Brieskorn–Artin group) Br(W) of W [16], [37] is defined as the group with generators {s_{i}, i ∈ I} and relations:

prod(m_{i,j}; s_{i}, s_{j}) = prod(m_{j,i}; s_{j}, s_{i}).

From this we obtain the presentation of the group W by adding the relations:

s_{i}^{2} = e; i ∈ I.

We see in Theorem 7.1 that this definition of the generalized braid group agrees with our general definition of a braid group of an action of a group W (Subsection 7.1). We denote by τ_{W} the canonical homomorphism from Br(W) to W. The classical braids on k strings Br_{k} are obtained by this construction if W is the symmetric group on k + 1 symbols. In this case m_{i,i+1} = 3, and m_{i,j} = 2 if j ≠ i, i + 1.

Classification of irreducible (with connected Coxeter graph) Coxeter groups is well known (see for example Theorem 1 , Chapter VI, §4 of [Bo]). It consists of the three infinite series: A, C (which is also denoted by B because in the corresponding classification of simple Lie algebras two different series B and C have this group as a Weyl group) and D as well as the exceptional groups E_{6}, E_{7}, E_{8}, F_{4}, G_{2}, H_{3}, H_{4} and I_{2}(p).

Now let us consider the complexification V_{C} of the space V and the complexification M_{C} of M ∈ M. Let Y_{W} = V_{C} − ⋃_{M∈M} M_{C}. The group W acts freely on Y_{W}. Let X_{W} = Y_{W} /W then Y_{W} is a covering over X_{W} corresponding to the group W. Let y_{0} ∈ A_{0} be a point in some chamber A_{0} and let x_{0} stand for its image in X_{W}. We are in the situation described in Subsection 7.1 in the definition of the braid group of the action of the group W. This braid group is defined as the fundamental group of the space of regular orbits of the action of W. In our case ORB(V_{C}, W) = X_{W} . So, the generalized braid group is π_{1}(X_{W} , x_{0}). For each j ∈ I, let ℓ′_{j} be the homotopy class of paths in Y_{W} starting from y_{0} and ending in w_{j}(y_{0}) which contains a polygon line with successive vertices: y_{0}, y_{0} + iy_{0}, w_{j}(y_{0}) + iy_{0}, w_{j}(y_{0}). The image ℓ_{j} of the class ℓ′_{j} in X_{W} is a loop with base point x_{0}.

Theorem 7.1. The fundamental group π_{1}(X_{W} , x_{0}) is generated by the elements ℓ_{j} satisfying the following relations:

prod(m_{j,k}; ℓ_{j}, ℓ_{k}) = prod(m_{k,j}; ℓ_{k}, ℓ_{j}).

This theorem was proved by E. Brieskorn [17].

The word problem and the conjugacy problem for Brieskorn-Artin groups were solved by E. Brieskorn and K. Saito [19] and P. Deligne [37]. The biautomatic structure of these groups was established by R. Charney [23].

[figure: 1; 2; g; g + 1; g + n; ...; ...] Figure 7.1

7.3. Braid groups of surfaces. Braid groups of a sphere Br_{n}(S^{2}) also have simple geometric interpretation as a group of isotopy classes of braids lying in a layer between two concentric spheres. It has the presentation with generators δ_{i}, i = 1, ..., n − 1, and relations:

(7.2)  \begin{cases}δ_{i}δ_{j} & = δ_{j} δ_{i}, if |i − j| > 1, \\ δ_{i}δ_{i+1}δ_{i} & = δ_{i+1}δ_{i}δ_{i+1}, \\ δ_{1}δ_{2} … δ_{n−2}δ_{n−1}^{2}δ_{n−2} … δ_{2}δ_{1} & = 1. \end{cases}

This presentation was find by O. Zariski [96] in 1936 and then rediscovered by E. Fadell and J. Van Buskirk [43] in 1961.

Presentations of braid groups of all closed surfaces were obtained by G. P. Scott [86] but look rather complicated.

7.4. Braid groups in handlebodies. The subgroup Br_{1,n+1} of the braid group Br_{n+1} consisting of braids with the fixed first string can be interpreted also as the braid group in a solid torus. Here we study braids in a handlebody of the arbitrary genus g.

Let H_{g} be a handlebody of genus g. The braid group Br_{n}^{g} on n strings in H_{g} was first considered by A. B. Sossinsky [88]. Let Q_{g} denote a subset of the complex plain ℂ, consisting of g different points, Q_{g} = {z_{1}^{0}, ..., z_{g}^{0}}, say, z_{i}^{0} = i. The interior of the handlebody H_{g} may be interpreted as the direct product of the complex plain ℂ without g points: ℂ \ Q_{g}, and the open interval, for example, (−1, 2):

Ḣ_{g} = (ℂ \ Q_{g}) × (−1, 2).

The space F (ℂ\Q_{g}, n) can be interpreted as the complement of the arrangement of hyperplanes in ℂ^{g+n} given by the formulas:

H_{j,k} : z_{j} − z_{k} = 0 for all j, k;
H_{j}^{i} : z_{j} = z_{i}^{0} for i = 1, ..., g; j = 1, ..., n.

The braids in Br_{n}^{g} are considered as lying between the planes with coordinates z = 0 and z = 1 and connecting the points ((g + 1, 0), ..., (g + n, 0)). So Br_{n}^{g} can be considered as a subgroup of the classical braid group Br_{g+n} on g + n strings such that the braids from Br_{n}^{g} leave the first g strings unbraided. In this subsection we denote by σ̄_{j} the standard generators of the group Br_{g+n}. Let τ_{k}, k = 1, 2, ..., g, be the following braids:

τ_{k} = σ̄_{g}σ̄_{g−1}...σ̄_{k+1}σ̄_{k}^{2}σ̄_{k+1}^{−1}...σ̄_{g−1}^{−1}σ̄_{g}^{−1}.

Such a braid is depicted in Figure 7.1. The elements τ_{k}, k = 1, 2, ..., g, generate a free subgroup F_{g} in the braid group Br_{g+n}. It follows for example from the Markov normal form that the

[figure: −; −] Figure 7.2

elements τ_{k}, k = 1, 2, ..., g, together with the standard generators σ̄_{g+1}, ..., σ̄_{g+n−1} generate the group Br_{n}^{g}. So, the braid group in the handlebody Br_{n}^{g} can be considered as a subgroup of Br_{g+n}, generated by two subgroups: F_{g} and Br_{n}. Denote by σ_{1}, ..., σ_{n−1} the standard generators of Br_{n} considered as the elements of Br_{n}^{g}, σ_{i} = σ̄_{g+i}, i = 1, …, n − 1. So we have the presentation of Br_{n}^{g} with the generators τ_{k} and σ_{i} and relations [88], [92], [95]:

(7.3)  \begin{cases}σ_{i}σ_{j} & = σ_{j}σ_{i} if |i − j| > 1, \\ σ_{i}σ_{i+1}σ_{i} & = σ_{i+1}σ_{i}σ_{i+1}, \\ τ_{k}σ_{i} & = σ_{i}τ_{k} if k ≥ 1, i ≥ 2, \\ τ_{k}σ_{1}τ_{k}σ_{1} & = σ_{1}τ_{k}σ_{1}τ_{k}, k = 1, 2, ..., g; \\ τ_{k}σ_{1}^{−1}τ_{k+l}σ_{1} & = σ_{1}^{−1}τ_{k+l}σ_{1}τ_{k}, k = 1, 2, ..., g − 1; l = 1, 2, ..., g − k. \end{cases}

The relation of the fourth type in (7.3) is the relation of the braid group of type B (C). The relations of the fifth type in (7.3) describe the interaction between the generators of the free group and their closest neighbor σ_{1}. Geometrically this is seen in Figure 7.2. If we introduce the new generators θ_{k}, k = 1, 2, ..., g − 1; by the formulas:

θ_{k} = σ_{1}^{−1}τ_{k}σ_{1}

we obtain the “positive” presentation of the group B_{n}^{g} with generators of the types σ_{i}, τ_{k}, θ_{k} and relations:

(7.4)  \begin{cases}σ_{i}σ_{j} & = σ_{j}σ_{i} if |i − j| > 1, \\ σ_{i}σ_{i+1}σ_{i} & = σ_{i+1}σ_{i}σ_{i+1}, \\ τ_{k}σ_{i} & = σ_{i}τ_{k} if k ≥ 1, i ≥ 2, \\ τ_{k}σ_{1}τ_{k}σ_{1} & = σ_{1}τ_{k}σ_{1}τ_{k}, k = 1, 2, ..., g; \\ τ_{k}θ_{k+l} & = θ_{k+l}τ_{k}, k = 1, 2, ..., g − 1; l = 1, 2, ..., g − k. \\ σ_{1}θ_{k} & = τ_{k}σ_{1}, k = 1, 2, ..., g − 1. \end{cases}

There is an analog of the Markov Theorem 3.2 for the group Br_{n}^{g} [92], [95].

7.5. Braids with singularities. Let BP_{n} be the subgroup of Aut F_{n}, generated by both sets of the automorphisms σ_{i} of (3.6) and ξ_{i} of the following form:

\begin{cases}x_{i} & ↦ x_{i+1}, \\ x_{i+1} & ↦ x_{i}, \\ x_{j} & ↦ x_{j}, j ≠ i, i + 1, \end{cases}

This is the braid-permutation group. R. Fenn, R. Rimányi and C. Rourke proved [45], [46] that this group is given by the set of generators: {ξ_{i}, σ_{i}, i = 1, 2, ..., n − 1} and relations:

\begin{cases}ξ_{i}^{2} & = 1, \\ ξ_{i}ξ_{j} & = ξ_{j}ξ_{i}, if |i − j| > 1, \\ ξ_{i}ξ_{i+1}ξ_{i} & = ξ_{i+1}ξ_{i}ξ_{i+1}. \end{cases}

Thesymmetric group relations

\begin{cases}σ_{i}σ_{j} & = σ_{j}σ_{i}, if |i − j| > 1, \\ σ_{i}σ_{i+1}σ_{i} & = σ_{i+1}σ_{i}σ_{i+1}. \end{cases}

The braid group relations

\begin{cases}σ_{i}ξ_{j} & = ξ_{j}σ_{i}, if |i − j| > 1, \\ ξ_{i}ξ_{i+1}σ_{i} & = σ_{i+1}ξ_{i}ξ_{i+1}, \\ σ_{i}σ_{i+1}ξ_{i} & = ξ_{i+1}σ_{i}σ_{i+1}. \end{cases}

The mixed relations

R. Fenn, R. Rimányi and C. Rourke also gave a geometric interpretation of BP_{n} as a group of welded braids. At first they defined a welded braid diagram on n strings as a collection of n monotone arcs starting from n points at a horizontal line of a plane (the top of the diagram) and going down to n points at another horizontal line (the bottom of the diagram). The diagrams may have crossings of two types: 1) the same as usual braids as for example on the Figure 3.2 or 2) welds as depicted in Figure 7.3

Composition of welded braid diagrams on n strings is defined by stacking one diagram under the other. The diagram with no crossings or welds is the identity with respect to composition. So the set of welded braid diagrams on n strings forms a semigroup which is denoted by W D_{n}.

R. Fenn, R. Rimányi and C. Rourke defined the allowable moves on welded braid diagrams. They consist of usual Reidemester moves (Figure 3.4) and the specific moves depicted of Figures 7.4, 7.5, 7.6 The automorphisms of F_{n} which lie in BP_{n} can be characterized as follows.

[figure] Figure 7.3

[figure: ↔; ↔] Figure 7.4

[figure: ↔; ↔] Figure 7.5

[figure: ↔; ...; ...; ...; ...; ...; ...] Figure 7.6

Let π ∈ Σ_{n} be a permutation and w_{i}, i = 1, 2..., n, be words in F_{n}. Then the mapping

x_{i} ↦ w_{i}^{−1}x_{π(i)}w_{i}

determines an injective endomorphism of F_{n}. If it is also surjective, we call it an automorphism of permutation-conjugacy type. The automorphisms of this type comprise a subgroup of Aut F_{n} which is precisely BP_{n}.

The Baez–Birman monoid SB_{n} or singular braid monoid [8], [12] is defined as monoid with generators g_{i}, g_{i}^{−1}, a_{i}, i = 1, …, n − 1, and relations

g_{i}g_{j} = g_{j}g_{i}, if |i − j| > 1,

[figure: 1; i − 1; i; i + 1; i + 2; n; …; …] Figure 7.7
a_{i}a_{j} = a_{j}a_{i}, if |i − j| > 1,
a_{i}g_{j} = g_{j}a_{i}, if |i − j| ≠ 1,
g_{i}g_{i+1}g_{i} = g_{i+1}g_{i}g_{i+1},
g_{i}g_{i+1}a_{i} = a_{i+1}g_{i}g_{i+1},
g_{i+1}g_{i}a_{i+1} = a_{i}g_{i+1}g_{i},
g_{i}g_{i}^{−1} = g_{i}^{−1}g_{i} = 1.

In pictures g_{i} corresponds to canonical generator of the braid group and a_{i} represents an intersection of the ith and (i + 1)st strand as in Figure 7.7. More detailed geometric interpretation of the Baez–Birman monoid can be found in the article of J. Birman [12]. R. Fenn, E. Keyman and C. Rourke proved [44] that the Baez–Birman monoid embeds in a group SG_{n} which they called the singular braid group:

SB_{n} → SG_{n}.

So, in SG_{n} the elements a_{i} become invertible and all relations of SB_{n} remain true.

8. Homological Properties

8.1. Configuration spaces and K(π, 1)-spaces. Let (q_{i})_{i∈ℕ} be a fixed sequence of distinct points in the manifold M and put Q_{m} = {q_{1}, ..., q_{m}}. We use

Q_{m,l} = (q_{l+1}, ..., q_{l+m}) ∈ F (M \ Q_{l}, m)

as the standard base point of the space F (M \ Q_{l}, m). If k < m we define the projection

proj : F (M \ Q_{l}, m) → F (M \ Q_{l}, k)

by the formula: proj(p_{1}, ..., p_{m}) = (p_{1}, ..., p_{k}). The following theorems were proved by E. Fadell and L. Neuwirth [42].

Theorem 8.1. The triple proj : F (M \ Q_{l}, m) → F (M \ Q_{l}, k) is a locally trivial fiber bundle with fiber proj^{−1} Q_{k,l} homeomorphic to F (M \ Q_{k+l}, m − k).

Consideration of the sequence of fibrations

F (M \ Q_{m−1}, 1) → F (M \ Q_{m−2}, 2) → M \ Q_{m−2},
F (M \ Q_{m−2}, 2) → F (M \ Q_{m−3}, 3) → M \ Q_{m−3},
... ,
F (M \ Q_{1}, m − 1) → F (M, m) → M

leads to the following theorem.

Theorem 8.2. For any manifold M

π_{i}(F (M \ Q_{1}, m − 1)) = ⊕_{k=1}^{m−1}π_{i}(M \ Q_{k})

for i ≥ 2. If proj : F (M, m) → M admits a section then

proj_{i} π_{i}(F (M, m)) = ⊕_{k=0}^{m−1}π_{i}(M \ Q_{k}), i ≥ 2.

Corollary 8.1. If M is the Euclidean r-space, then

π_{i}(F (M, m)) = ⊕_{k=0}^{m−1}π_{i}(\underbrace{S^{r−1} ∨ ... ∨ S^{r−1}}_{k}), i ≥ 2.

Corollary 8.2. If M is the Euclidean 2-space, then F (ℝ^{2}, m) is the K(P_{m}, 1)-space and B(ℝ^{2}, m) is the K(Br_{m}, 1)-space.

Let X_{W} be the space defined in Subsection 7.2.

Theorem 8.3. The universal covering of X_{W} is contractible, and so X_{W} is a K(π; 1)-space.

This theorem for the groups of types C_{n}, G_{2} and I_{2}(p), was proved by E. Brieskorn [16] in much the same way as E. Fadell and L. Neuwirth [42] proved Theorems 8.1, 8.2 and Corollary 8.2. For the groups of types D_{n} and F_{4} E. Brieskorn used this method with minor modifications. In general case Theorem was proved by P. Deligne [37].

It follows from Theorem 8.2 that F (ℂ \ Q_{g}, n) and B(ℂ \ Q_{g}) are K(π, 1)-spaces, π_{1}B(ℂ \ Q_{g}) = Br_{n}^{g}, so, B(ℂ \ Q_{g}) can be considered as the classifying space of Br_{n}^{g}.

8.2. Cohomology of pure braid groups. Cohomologies of pure braid groups were first calculated by V. I. Arnold [3]. The map

ϕ : S^{n−1} → F (ℝ^{n}, 2),

described by the formula ϕ(x) = (x, −x), is a Σ_{2}-equivariant homotopy equivalence. Denote by A the generator of H^{n−1}(F (ℝ^{n}, 2), ℤ) that is mapped by ϕ^{∗} to the standard generator of H^{n−1}(S^{n−1}, ℤ). For i and j, such that 1 ≤ i, j ≤ m, i ≠ j, specify π_{i,j} : F (ℝ^{n}, m) → F (ℝ^{n}, 2) by the formula π_{i,j}(p_{1}, ..., p_{m}) = (p_{i}, p_{j}). Put

A_{i,j} = π_{i,j}^{∗}(A) ∈ H^{n−1}(F (ℝ^{n}, m), ℤ).

It follows that A_{i,j} = (−1)^{n}A_{j,i} and A_{i,j}^{2} = 0. For w ∈ Σ_{m} there is an action w(A_{i,j}) = A_{w^{−1}(i),w^{−1}(j)}, since π_{i,j}w = π_{w^{−1}(i),w^{−1}(j)}. Note also that under restriction to

F (ℝ^{n} \ Q_{k}, m − k) ≅ π^{−1}(Q_{k}) ⊂ F (ℝ^{n}, m),

the classes A_{i,j} with 1 ≤ i, j ≤ k go to zero since in this case the map π_{i,j} is constant on π^{−1}(Q_{k}).

Theorem 8.4. The cohomology group H^{∗}(F (ℝ^{n} \ Q_{k}, m − k), ℤ) is the free Abelian group with generators

A_{i_{1},j_{1}}A_{i_{2},j_{2}}...A_{i_{s},j_{s}},

where k < j_{1} < j_{2} < ... < j_{s} ≤ m and i_{r} < j_{r} for r = 1, dots, s.

The multiplicative structure and the Σ_{m}-algebra structure of H^{∗}(F (ℝ^{n}, m), ℤ) are given by the following theorem which is proved using the Σ_{3}-action on H^{∗}(F (ℝ^{n}, 3), ℤ).

Theorem 8.5. The cohomology ring H^{∗}(F (ℝ^{n}, m), ℤ) is multiplicatively generated by the squarezero elements

A_{i,j} ∈ H^{n−1}(F (ℝ^{n}, m), ℤ), 1 ≤ i < j ≤ m,

subject only to the relations

(8.1)  A_{i,k}A_{j,k} = A_{i,j}A_{j,k} − A_{i,j}A_{i,k} for i < j < k.

The Poincaré series for F (ℝ^{n}, m) is the product ∏_{j=1}^{m−1}(1 + jt^{n−1}).

Remark 8.1. In the case of ℝ^{2} = ℂ the cohomology classes A_{j,k} can be interpreted as the classes of cohomology of differential forms

ω_{j,k} = \frac{1}{2πi} \frac{dz_{j} − dz_{k}}{z_{j} − z_{k}} .

E. Brieskorn calculated the cohomologies of pure generalized braid groups [16] using ideas of V. I. Arnold for the classical case. Let V be a finite-dimensional complex vector space and H_{j} ∈ V, j ∈ I be the finite family of complex affine hyperplanes given by linear forms l_{j}. E. Brieskorn proved the following fact.

Theorem 8.6. The cohomology classes, corresponding to the holomorphic differential forms

ω_{j} = \frac{1}{2πi} \frac{dl_{j}}{l_{j}} ,

generate the cohomology ring H^{∗}(V \ ∪_{j∈I}H_{j}, ℤ). Moreover, this ring is isomorphic to the ℤsubalgebra generated by the forms ω_{j} in the algebra of meromorphic forms on V.

The cohomologies of pure generalized braid groups are described as follows.

Theorem 8.7. (i) The cohomology group H^{k}(P (W), ℤ) of the pure braid group P (W) with integer coefficients is a free abelian group, and its rank is equal to the number of elements w ∈ W of length l(w) = k, where l is the length considered with respect to the system of generators consisting of all reflections of W.

(ii) The Poincaré series for H^{∗}(P (W), ℤ) is the product ∏_{j=1}^{n}(1 + m_{j}t), where m_{j} are the exponents of the group W.

(iii) The multiplicative structure of H^{∗}(P (W), ℤ) coincides with the structure of algebra, generated by 1-forms described in the previous theorem.

8.3. Homology of braid groups. To study the cohomologies of the classical braid groups H^{∗}(Br_{n}, ℤ), V. I. Arnold [4] interpreted the space K(Br_{n}, 1) ≅ B(ℝ^{2}, n) as the space of monic complex polynomials of degree n without multiple roots

P_{n}(t) = t^{n} + z_{1}t^{n−1} + ... + z_{n−1}t + z_{n}.

Using this idea he proved Theorems of finiteness, of recurrence and of stabilization. Homologies with coefficients in ℤ/2 were calculated by D. B. Fuks in the following theorems [50].

Theorem 8.8. The homology of the braid group on the infinite number of strings with coefficients in ℤ/2 as a Hopf algebra is isomorphic to the polynomial algebra on infinitely many generators a_{i}, i = 1, 2, ...; deg a_{i} = 2^{i} − 1:

H_{∗}(Br_{∞}, ℤ/2) ≅ ℤ/2[a_{1}, a_{2}, ..., a_{i}, ...]

with the coproduct given by the formula:

Δ(a_{i}) = 1 ⊗ a_{i} + a_{i} ⊗ 1.

Theorem 8.9. The canonical inclusion Br_{n} → Br_{∞} induces a monomorphism in homology with coefficients in ℤ/2. Its image is the subcoalgebra of the polynomial algebra ℤ/2[a_{1}, a_{2}, ..., a_{i}, ...] with ℤ/2-basis consisting of monomials

a_{1}^{k_{1}}...a_{l}^{k_{l}} such that ∑_{i} k_{i}2^{i} ≤ n.

Theorem 8.10. The canonical homomorphism Br_{n} → BO_{n}, 1 ≤ n ≤ ∞ induces a monomorphism (of Hopf algebras if n = ∞)

H_{∗}(Br_{n}, ℤ/2) → H_{∗}(BO_{n}, ℤ/2).

F. R. Cohen calculated the homology of braid groups with coefficients ℤ/p, p > 2 also as modules over the Steenrod algebra [26], [27], [28].

Later V. V. Goryunov [56], [57] applied the methods of Fuks and expressed the cohomologies of the generalized braid groups of types C and D in terms of the cohomologies of the classical braid groups.

References

[1] S. I. Adyan, Fragments of the word Δ in the braid group. (Russian) Mat. Zametki 36 (1984), no. 1, 25–34.

[2] J. W. Alexander, A lemma on systems of knotted curves. Proc. Nat. Acad. Sci. USA, 9, 93-95.

[3] V. I. Arnold, The cohomology ring of colored braids, Mat. Zametki, 1969, 5, 227–231 (Russian). English transl. in Math. Notes, 1969, 5, 138-140.

[4] V. I. Arnold, On some topological invariants of algebraic functions, Trudy Moscov. Mat. Obshch. 1970, 21, 27–46 (Russian), English transl. in Trans. Moscow Math. Soc., 1970, v. 21, 30–52.

[5] V. I. Arnold, Topological invariants of algebraic functions II, Functional Anal. Appl. 1970, v. 4, No 2, 1–9 (Russian).

[6] E. Artin, Theorie der Zöpfe, Abh. Math. Semin. Univ. Hamburg, 1925, v. 4, 47–72.

[7] E. Artin, Theory of braids, Ann. Math. 1947, v. 48, No 1, 101–126.

[8] J. C. Baez, Link invariants of finite type and perturbation theory. Lett. Math. Phys. 26 (1992), no. 1, 43–51.

[9] S. Bigelow, The Burau representation is not faithful for n = 5. Geom. Topol. 3 (1999), 397–404 (electronic).

[10] S. Bigelow, Braid groups are linear, J. Amer. Math. Soc. 14 (2001), no. 2, 471–486.

[11] J. S. Birman, Braids, Links, and Mapping Class Groups, Ann. Math. Stud., No 82, 1974.

[12] J. S. Birman, New points of view in knot theory, Bull. Amer. Math. Soc. 1993, 28, No 2, 253–387.

[13] J. S. Birman; K. H. Ko; S. J. Lee, A new approach to the word and conjugacy problems in the braid groups. Adv. Math. 139 (1998), no. 2, 322–353

[14] J. S. Birman and H. Wenzl, Braids, link polynomials and a new algebra. Trans. Amer. Math. Soc. 313 (1989), no. 1, 249–273.

[15] N. Bourbaki, Groupes et algèbres de Lie, Chaps. 4–6, Masson, Paris, 1981.

[16] E. Brieskorn, Sur les groupes de tresses [d’après V. I. Arnol’d]. (French) Séminaire Bourbaki, 24ème anne (1971/1972), Exp. No. 401, pp. 21–44. Lecture Notes in Math., Vol. 317, Springer, Berlin, 1973.

[17] E. Brieskorn, Die Fundamentalgruppe des Raumes der regulären Orbits einer endlichen komplexen Spiegelungsgruppe. (German) Invent. Math. 12 (1971), 57–61.

[18] E. Brieskorn, Automorphic sets and braids and singularities. Braids (Santa Cruz, CA, 1986), 45–115, Contemp. Math., 78, Amer. Math. Soc., Providence, RI, 1988.

[19] E. Brieskorn and K. Saito, Artin-Gruppen und Coxeter-Gruppen. (German) Invent. Math. 17 (1972), 245–271.

[20] W. Burau, Über Zopfinvarianten. (German) Abh. Math. Semin. Hamb. Univ. 9, 117-124 (1932).

[21] W. Burau, Über Verkettungsgruppen. (German) Abh. Math. Semin. Hamb. Univ. 11, 171-178 [22] W. Burau, Über Zopfgruppen und gleichsinnig verdrillte Verkettungen. (German) Abh. Math.

Semin. Hamb. Univ. 11, 179-186 (1935).

[23] R. Charney, Ruth Artin groups of finite type are biautomatic. Math. Ann. 292 (1992), no. 4, 671–683.

[24] R. Charney, Geodesic automation and growth functions for Artin groups of finite type. Math. Ann. 301 (1995), no. 2, 307–324.

[25] W.-L. Chow, On the algebraical braid group, Ann. Math. 1948, 49, No 3, 654–658.

[26] F. Cohen, Cohomology of braid spaces Bull. Amer. Math. Soc. 1973, 79, No 4 763–766.

[27] F. Cohen, Homology of Ω^{n+1}Σ^{n+1}X and C_{n+1}X, n > 0 Bull. Amer. Math. Soc. 1973, 79, No 6, 1236–1241.

[28] F. Cohen, Braid orientations and bundles with flat connections Invent. Math. 1978, 46, 99–110.

[29] F. Cohen, Artin’s braid groups, classical homotopy theory, and other curiosities, Braids (Contemp. Math. 78, 1988), 167–206.

[30] F. Cohen; T. Lada; J. P. May, The homology of iterated loop spaces. Lecture Notes in Mathematics, Vol. 533. Springer-Verlag, Berlin-New York, 1976. vii+490 pp.

[31] R. Corran, A normal form for a class of monoids including the singular braid monoids. J. Algebra. 223 (2000), no. 1, 256–282.

[32] H. S. M. Coxeter, Factor groups of the braid group. Proc. fourth Canad. math. Congr. Banff, 1957, (1959) 95-122.

[33] O. Dasbach and B. Gemein, A faithful representation of the singular braid monoid on three strands. Knots in Hellas ’98 (Delphi), 48–58, Ser. Knots Everything, 24, World Sci. Publishing, River Edge, NJ, 2000.

[34] P. Dehornoy, Deux propriétés des groupes de tresses. (French) [Two properties of braid groups] C. R. Acad. Sci. Paris Sr. I Math. 315 (1992), no. 6, 633–638.

[35] P. Dehornoy, Braid groups and left distributive operations. Trans. Amer. Math. Soc. 345 (1994), no. 1, 115–150.

[36] P. Dehornoy, Braids and self-distributivity. Progress in Mathematics, 192. Birkhuser Verlag, Basel, 2000. xx+623 pp.

[37] P. Deligne, Les immeubles des groupes de tresses généralisés. (French) Invent. Math. 17 (1972), 273–302.

[38] E. El-Rifai and H. R. Morton, Algorithms for positive braids. Quart. J. Math. Oxford Ser. (2) 45 (1994), no. 180, 479–497.

[39] M. Epple, Orbits of asteroids, a braid, and the first link invariant, Math. Intell. 1998, 20, No 1, 45-52.

[40] D. B. A. Epstein; J. W. Cannon; D. E. Holt; S. V. F.Levy; M. S. Paterson; W. P. Thurston, Word processing in groups. Jones and Bartlett Publishers, Boston, MA, 1992. xii+330 pp.

[41] E. Fadell; S. Y. Husseini, Geometry and topology of configuration spaces. Springer Monographs in Mathematics. Springer-Verlag, Berlin, 2001. xvi+313 pp.

[42] E. Fadell and L. Neuwirth, Configuration spaces, Math. Scand. 1962, 10, Fasc. I, 111–118.

[43] E. Fadell and J. Van Buskirk, The braid groups of E^{2} and S^{2}. Duke Math. J. 29 1962 243–257.

[44] R. Fenn; E. Keyman; C. Rourke, The singular braid monoid embeds in a group. J. Knot Theory Ramifications 7 (1998), no. 7, 881–892.

[45] R. Fenn; R. Rimányi; C. Rourke, Some remarks on the Braid-Permutation Group. Topics in Knot Theory. Kluwer Academic Publishers, 1993, 57–68.

[46] R. Fenn; R. Rimányi; C. Rourke, The braid-permutation group. Topology 36 (1997), no. 1, 123– 135.

[47] R. Fenn; D. Rolfsen; J. Zhu, Centralisers in the braid group and singular braid monoid. Enseign. Math. (2) 42 (1996), no. 1-2, 75–96.

[48] R. Fox and L. Neuwirth, The braid groups, Math. Scand. 1962, 10, Fasc. I, 119–126.

[49] R. Fricke, F. Klein, Vorlesungen über die Theorie der Automorphen Functionen. Bd. I. Gruppentheoretischen Grundlagen. Teubner, Leipzig, 1897 (Johnson Repr. Corp., N. Y., 1965).

[50] D. B. Fuks, Cohomology of the braid group mod 2, Functional. Anal. i Prilozh. 1970, 4, No 2, 62–75 (Russian), English transl. in Functional Anal. Appl. 4 (1970), 143–151.

[51] F. A. Garside, The braid group and other groups, Quart. J. Math. Oxford Ser. 1969, 20, 235–254 [52] B. J. Gassner, On braid groups. Abh. Math. Sem. Univ. Hamburg, 25, 1961, 10–22.

[53] B. Gemein, Singular braids and Markov’s theorem. J. Knot Theory Ramifications 6 (1997), no. 4, 441–454.

[54] B. Gemein, Representations of the singular braid monoid and group invariants of singular knots. Topology Appl. 114 (2001), no. 2, 117–140.

[55] J. González-Meneses, Presentations for the monoids of singular braids on closed surfaces. Comm. Algebra 30 (2002), no. 6, 2829–2836.

[56] V. V. Goryunov, The cohomology of braid groups of series C and D and certain stratifications. (Russian) Funktsional. Anal. i Prilozhen. 12 (1978), no. 2, 76–77.

[57] V. V. Goryunov, Cohomology of braid groups of series C and D. (Russian) Trudy Moskov. Mat. Obshch. 42 (1981), 234–242.

[58] V. Hansen, Braids and coverings: selected topics. London Mathematical Society Student Text, 18. Cambridge etc.: Cambridge University Press. x, 191 p. (1989).

[59] A. Hurwitz, Über Riemannische Flächen mit gegebenen Verzweigungspunkten, Math. Ann. 1891, 39, 1–61.

[60] N. V. Ivanov, Algebraic properties of the Teichmüller modular group. (Russian) Dokl. Akad. Nauk SSSR 275 (1984), no. 4, 786–789.

[61] A. Járai Jr., On the monoid of singular braids. Topology Appl. 96 (1999), no. 2, 109–119.

[62] V. F. R. Jones, Hecke algebra representations of braid groups and link polynomials, Ann. Math. 1987, 126, 335–388.

[63] F. Klein, Vorlesungen über höhere Geometrie. 3. Aufl., bearbeitet und herausgegeben von W. Blaschke. (German) VIII + 405 S. Berlin, J. Springer (Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen Bd. 22) (1926).

[64] D. Krammer, The braid group B_{4} is linear, Invent. Math. 142 (2000), no. 3, 451–486.

[65] D. Krammer, Braid groups are linear, Ann. of Math. (2) 155 (2002), no. 1, 131–156.

[66] R. J. Lawrence, Homological representations of the Hecke algebra. Comm. Math. Phys. 135 (1990), no. 1, 141–191.

[67] V. Ya. Lin, Artinian braids and groups and spaces connected with them, Itogi Nauki i Tekhniki (Algebra, Topologiya, Geometriya) 1979, 17, 159–227 (Russian). English transl. in J. Soviet Math. 18 (1982) 736–788.

[68] D. D. Long and M. Paton, The Burau representation is not faithful for n ≥ 6. Topology 32 (1993), no. 2, 439–447.

[69] W. Magnus, Über Automorphismen von Fundamentalgruppen berandeter Flächen. (German) Math. Ann. 109, 617-646 (1934).

[70] W. Magnus Braid groups: a survey. Proc. 2nd internat. Conf. Theory of Groups, Canberra 1973, Lect. Notes Math. 372, 463-487 (1974)

[71] W. Magnus; A. Karrass; D. Solitar, Combinatorial group theory. Presentations of groups in terms of generators and relations. 2nd rev. ed. Dover Books on Advanced Mathematics. New York: Dover Publications, Inc. XII, 444 p. (1976).

[72] W. Magnus and A. Peluso, On knot groups. Comm. Pure Appl. Math. 20, 1967, 749–770.

[73] G. S. Makanin, The conjugacy problem in the braid group, (Russian) Dokl. Akad. Nauk SSSR. 182, (1968) 495–496.

[74] A. A. Markoff, Über die freie Äquivalenz der geschlossenen Zöpfe, Math. Sb. 1936, 16, 73–78.

[75] A. A. Markoff, Foundations of the Algebraic Theory of Tresses, Trudy Mat. Inst. Steklova, No 16, 1945 (Russian, English summary).

[76] J. McCarthy, A ”Tits-alternative” for subgroups of surface mapping class groups. Trans. Amer. Math. Soc. 291 (1985), no. 2, 583–612.

[77] J. Michel, A note on words in braid monoids. J. Algebra 215 (1999), no. 1, 366–377.

[78] J. A. Moody, The Burau representation of the braid group B_{n} is unfaithful for large n. Bull. Amer. Math. Soc. (N.S.) 25 (1991), no. 2, 379–384.

[79] S. Moran,The mathematical theory of knots and braids. An introduction. North-Holland Mathematics Studies, 82. North-Holland Publishing Co., Amsterdam, 1983. xii+295 pp.

[80] J. Murakami, The Kauffman polynomial of links and representation theory. Osaka J. Math. 24 (1987), no. 4, 745–758.

[81] K. Murasugi; B. Kurpita, A study of braids. Mathematics and its Applications, 484. Kluwer Academic Publishers, Dordrecht, 1999. x+272 pp.

[82] T. Ohtsuki, Quantum invariants. A study of knots, 3-manifolds, and their sets. Series on Knots and Everything, 29. World Scientific Publishing Co., Inc., River Edge, NJ, 2002. xiv+489 pp.

[83] E. Ossa, On the cohomology of configuration spaces. Algebraic topology: new trends in localization and periodicity (Sant Feliu de Guxols, 1994), 353–361, Progr. Math., 136, Birkhuser, Basel, 1996.

[84] V. V. Prasolov and A. B. Sossinsky, Knots, links, braids and 3-manifolds. An introduction to the new invariants in low-dimensional topology. Translations of Mathematical Monographs, 154. American Mathematical Society, Providence, RI, 1997. viii+239 pp.

[85] K. Reidemeister, Knotentheorie. (Erg. d. Math. u. ihrer Grenzgebiete. 1, No.1) Berlin: Julius Springer. VI, 74 S., 114 Fig. (1932). Englis translation: Knot theory. BCS Associates, Moscow, Idaho, 1983. xv+143 pp.

[86] G. P. Scott, Braid groups and the group of homeomorphisms of a surface. Proc. Cambridge Philos. Soc. 68, 1970, 605–617.

[87] V. Sergiescu, Graphes planaires et présentations des groupes de tresses, Math. Z. 1993, 214, 477– 490.

[88] A. B. Sossinsky, Preparation theorems for isotopy invariants of links in 3-manifolds. Quantum groups (Leningrad, 1990), 354–362, Lecture Notes in Math., 1510, Springer, Berlin, 1992.

[89] I. Sysoeva, Dimension n representations of the braid group on n strings. J. Algebra 243 (2001), no. 2, 518–538.

[90] V. Turaev, Faithful linear representations of the braid groups. Séminaire Bourbaki, Vol. 1999/2000. Astérisque No. 276 (2002), 389–409.

[91] V. A. Vassiliev, Complements of discriminants of smooth maps: topology and applications. Translations of Mathematical Monographs, 98. American Mathematical Society, Providence, RI, 1992. vi+208 pp.

[92] V. V. Vershinin, On braid groups in handlebodies. Sib. Math. J. 39, No.4, 645-654 (1998); translation from Sib. Mat. Zh. 39, No.4, 755-764 (1998).

[93] V. V. Vershinin, On homological properties of singular braids. Trans. Amer. Math. Soc. 350 (1998), no. 6, 2431–2455.

[94] V. V. Vershinin, Generalizations of braids from a homological point of view. Sib. Adv. Math. 9, No.2, 109-139 (1999).

[95] V. V. Vershinin, Braid groups and loop spaces. Russ. Math. Surv. 54, No.2, 273-350 (1999); translation from Usp. Mat. Nauk 54, No.2, 3-84 (1999).

[96] O. Zariski, On the Poincare group of rational plane curves, Am. J. Math. 1936, 58, 607-619.

[97] O. Zariski, The topological discriminant group of a Riemann surface of genus p, Am. J. Math. 1937, 59, 335-358.

[98] J. Zhu, On singular braids. J. Knot Theory Ramifications, 6 (1997), no. 3, 427–440.

[99] M. G. Zinno, On Krammer’s representation of the braid group. Math. Ann. 321 (2001), no. 1, 197–211.

Département des Sciences Mathématiques, Université Montpellier II, Place Eugéne Bataillon, 34095 Montpellier cedex 5, France

E-mail address: vershini@math.univ-montp2.fr

Sobolev Institute of Mathematics, Novosibirsk, 630090, Russia

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