Braid groups
Luis Paris
These are the notes of a course given at a CIMPA School in Hanoi in January 2011.
1
Braids
Let n ≥ 1 be an integer, and let P1 , . . . , Pn be n distinct points in the plane R2 (except
mention of the contrary, we will always assume Pk = (k, 0) for all 1 ≤ k ≤ n). Define a
braid on n strands to be an n-tuple β = (b1 , . . . , bn ) of paths, bk : [0, 1] → R2 , such that
(a) bk (0) = Pk for all 1 ≤ k ≤ n;
(b) there exists a permutation χ = θ(β) ∈ Sn such that bk (1) = Pχ(k) for all 1 ≤ k ≤ n;
(3) bk (t) ̸= bl (t) for all k ̸= l and all t ∈ [0, 1].
Two braids α and β are said to be homotopic if there exists a continuous family {γs }s∈[0,1]
of braids such that γ0 = α and γ1 = β. Note that θ(α) = θ(β) if α and β are homotopic.
We represent graphically a homotopy class of braids as follows. For 1 ≤ k ≤ n, let Ik be
a copy of the interval [0, 1]. Take a braid β = (b1 , . . . , bn ) and define the geometric braid
β g : I1 ⊔ · · · ⊔ In → R × [0, 1]
by β g (t) = (bk (t), t) for all t ∈ Ik and all 1 ≤ k ≤ n. Let proj : R2 × [0, 1] → R × [0, 1] be
the projection defined by
proj(x, y, t) = (x, t) .
Up to homotopy, we can assume that proj◦β g is a smooth immersion with only transverse
double points that we call crossings. In each crossing we indicate graphically which strand
goes over the other. Such a representation of β is called a braid diagram of β.
positive
crossing
negative
crossing
Figure. Crossings in a braid diagram.
1
R2 × {0}
R2 × {1}
P3
P3
P2
P2
P1
P1
0
1
t
Figure. A braid diagram.
The product of two braids α = (a1 , . . . , an ) and β = (b1 , . . . , bn ) is defined to be the braid
α · β = (a1 bχ(1) , . . . , an bχ(n) ) ,
where χ = θ(α).
=
=
Figure. Product of two braids.
Let Bn denote the set of homotopy classes of braids on n strands. It is easily seen that
the above defined multiplication of braids induces an operation on Bn .
Proposition. Bn endowed with this operation is a group.
The group Bn of the above proposition is called the braid group on n strands. The identity
is the constant braid Id = (Id1 , . . . , Idn ), where, for 1 ≤ k ≤ n, Idk denotes the constant
path on Pk . The inverse of a braid β is its mirror.
β −1
β
Figure. Inverse of a braid.
Recall that if two braids α, α′ are homotopic, then θ(α) = θ(α′ ). Hence, the map θ from
the set of braids on n strands to Sn induces a map θ : Bn → Sn . It is easily checked that
this map is an epimorphism. Its kernel is called the pure braid group on n strands and is
denoted by PBn .
2
2
Presentation
Let σk be the braid illustrated in the next figure. One can easily verify that σ1 , . . . , σn−1
generate the braid group Bn and satisfy the relations
if |k − l| ≥ 2 ,
if |k − l| = 1 .
σk σl = σl σl
σk σl σk = σl σk σl
Pk+1
Pk
Figure. The braid σk .
σk
σl
σl
σk σk+1 σk
σk
σk+1 σk σk+1
Figure. Relations in Bn .
Theorem (Artin 1925, Magnus 1934). The group Bn has a presentation with generators
σ1 , . . . , σn−1 and relations
if |k − l| ≥ 2 ,
if |k − l| = 1 .
σk σl = σl σk
σk σl σk = σl σk σl
Theorem (Burau 1932, Markov 1945). For 1 ≤ k < l ≤ n, let
−1
−1
δk l = σl−1 · · · σk+1 σk2 σk+1
· · · σl−1
.
Then the pure braid group PBn has a presentation with generators
δk l ,
1 ≤ k < l ≤ n,
and relations
δr s δk l δr−1s = δk l
δr k δk l δr−1k = δk−1l δr−1l δk l δr l δk l
δr k δr l δr−1k = δk−1l δr l δk l
δr s δk l δr−1s = δs−1l δr−1l δs l δr l δk l δr−1l δs−1l δr l δs l
3
if 1 ≤ r < s < k < l ≤ n
or 1 ≤ k < r < s < l ≤ n ,
if 1 ≤ r < k < l ≤ n ,
if 1 ≤ r < k < l ≤ n ,
if 1 ≤ r < k < s < l ≤ n .
3
Configuration spaces
We identify R2 with C and Pk with k ∈ C for all 1 ≤ k ≤ n. For 1 ≤ k < l ≤ n we denote
by Hk,l the linear hyperplane of Cn defined by the equation zk = zl . The big diagonal of
Cn is defined to be
∪
Diagn =
Hk,l .
1≤k<l≤n
The space of ordered configurations of n points in C is defined to be
Mn = Cn \ Diagn .
This is the space of n-tuples z = (z1 , . . . , zn ) of complex numbers such that zk ̸= zl for
k ̸= l. The symmetric group Sn acts freely on Mn . The quotient
Nn = Mn /Sn
is called the space of configurations of n points in C. This is the space of unordered
n-tuples z = {z1 , . . . , zn } of complex numbers such that zk ̸= zl for k ̸= l.
Proposition. Let P0 = (1, 2, . . . , n) ∈ Mn . Then π1 (Mn , P0 ) = PBn .
Proof. For a pure braid β = (b1 , . . . , bn ) we set
φ(β) : [0, 1] →
Mn
t
7→ (b1 (t), . . . , bn (t)) .
Clearly, φ(β) is a loop based at P0 . Moreover, two pure braids α and α′ are homotopic
if and only if φ(α) and φ(α′ ) are homotopic. Thus φ induces a bijection φ∗ : PBn →
π1 (Mn , P0 ) which turns out to be a homomorphism.
For z ∈ Mn , we denote by [z] the element of Nn = Mn /Sn represented by z.
Proposition. π1 (Nn , [P0 ]) = Bn .
Proof. For a braid β = (b1 , . . . , bn ) we set
φ̂(β) : [0, 1] →
Nn
t
7→ [b1 (t), . . . , bn (t)] .
Clearly, φ̂(β) is a loop based at [P0 ]. It is easily checked that φ̂ induces a homomorphism
φ̂∗ : Bn → π1 (Nn , [P0 ]), and that the following diagram commutes
1
/ PBn
/ Bn
φ∗ ≃
1
/ π1 (Mn , P0 )
φ̂∗
/ π1 (Nn , [P0 ])
4
/ Sn
/1
Id
/ Sn
/1
The first row is exact by definition, and the second one is associated to the regular covering
Mn → Nn = Mn /Sn , so it is exact, too. We conclude by the five lemma that φ̂∗ is an
isomorphism.
Let f, g ∈ C[x] be two non-constant polynomials. Set
f = a0 xm + a1 xm−1 + · · · + am , a0 ̸= 0 ,
g = b0 xn + b1 xn−1 + · · · + bn , b0 ̸= 0 .
The Sylvester matrix of f and g is defined to be


 a0 0 · · · 0 b0 0 · · · 0 



.. 
..
.
.
.
.
 a1 a0
. . b1 b0
. . 
 .

..
..
..
 .
. 0
. 0 
. b1
a1

 .


..
..
..
..
Sylv(f, g) =  a
. a0 bn .
. b0 
 m .

 0 a

a
0
b
b
m
1
n
1 

 .. . .
.
.. . .
. 
..
..
 .
.
. ..
.
. .. 
.


 0 · · · 0 am 0 · · · 0 b n 
{z
}|
{z
}
|
n columns
m columns
The resultant of f and g is defined to be
Res(f, g) = det(Sylv(f, g)) .
The following is classical in algebraic geometry.
Theorem. Let f, g ∈ C[x] be two non-constant polynomials. Then f and g have a
common root if and only if Res(f, g) = 0.
Corollary. Let f ∈ C[x] be a polynomial of degree d ≥ 2. Then f has a multiple root if
and only if Res(f, f ′ ) = 0.
Definition. The number Res(f, f ′ ) is called the discriminant of f and is denoted by
Disc(f ).
Example. If f = ax2 + bx + c, then Disc(f ) = b2 − 4ac.
Let n ≥ 2 and let Cn [x] be the set of monic polynomials of degree n. In particular, Cn [x]
is isomorphic to Cn . The map Disc : Cn [x] → C is clearly a polynomial function, thus
D = {f ∈ Cn [x]; f has a multiple root} = {f ∈ Cn [x]; Disc(f ) = 0}
is an algebraic hypersurface called the n-th discriminant.
5
Proposition. Nn = Cn [x] \ D.
Proof. Let Φ : Mn → Cn [x] \ D be the map defined by
Φ(z1 , . . . , zn ) = (x − z1 ) · · · (x − zn ) .
Then Φ is surjective and we have Φ(u) = Φ(v) if and only if there exists χ ∈ Sn such
that v = χ(u). Thus Cn [x] \ D ≃ Mn /Sn = Nn .
Theorem (Fadell, Neuwirth 1962). Let p : Mn+1 → Mn be defined by
p(z1 , . . . , zn , zn+1 ) = (z1 , . . . , zn ) .
Then p is a locally trivial fiber bundle which admits a cross-section κ : Mn → Mn+1 .
Let b0 = (1, 2, . . . , n). Then the fiber p−1 (b0 ) is naturally homeomorphic to C\{1, 2, . . . , n}
whose fundamental group is the free group Fn of rank n. A cross-section of p is the map
κ : Mn → Mn+1 defined by
κ(z1 , . . . , zn ) = (z1 , . . . , zn , |z1 | + · · · + |zn | + 1) .
Corollary. Let n ≥ 2. Then there is a split exact sequence
1
/ Fn
/ PBn+1
m
p∗
κ∗
/ PBn
/1 .
Definition. A connected CW-complex X is called K(π, 1) if its universal cover is contractible. Equivalently, X is K(π, 1) if πk (X) = {0} for all k ≥ 2. In particular, a space
X is K(π, 1) if and only if some of its connected cover Y is K(π, 1).
Corollary The spaces Mn and Nn are K(π, 1).
Proof. We prove by induction on n that Mn is K(π, 1). Since Mn → Nn is a covering,
this implies that Nn is also K(π, 1). We have M1 = C which is contractible, thus K(π, 1).
So, we may assume n ≥ 2 plus the inductive hypothesis.
For k ≥ 2, the homotopy long exact sequence gives rise to the exact sequence
πk (C \ {1, 2, . . . , n − 1}) = {0} → πk (Mn ) → πk (Mn−1 ) = {0} ,
thus πk (Mn ) = {0}.
Corollary. Bn = π1 (Nn ) is torsion free.
Proof. It is known that, if X is a finite dimensional K(π, 1) manifold, then π1 (X) is
torsion free. Since Bn = π1 (Nn ) and Nn is a 2n dimensional K(π, 1) manifold, it follows
that Bn is torsion free.
6
4
Mapping class groups
Definition. Let Σ = Σg,b be an oriented surface of genus g with b boundary components,
where g ≥ 0 and b ≥ 0, and let P = {P1 , . . . , Pn } be a collection of n punctures in
the interior of Σ. Let Homeo+ (Σ, P) denote the group of homeomorphisms h : Σ → Σ
which preserve the orientation, pointwise fix the boundary of Σ, and verifies h(P) = P.
We assume Homeo+ (Σ, P) endowed with the compact-open topology. This is defined as
follows. For K, U ⊂ Σ we set
U(K, U ) = {f ∈ Homeo+ (Σ, P); f (K) ⊂ U } .
Then the sets U(K, U ), where K ranges over the compacts subsets of Σ and U ranges
over the open subsets, form a basis for the topology of Homeo+ (Σ, P).
Figure. Surface of genus g = 2 with n = 2 boundary components and 3 marked points.
Proposition. Homeo+ (Σ, P) is locally arcwise connected.
+
Definition. Let Homeo+
0 (Σ, P) denote the component of the identity in Homeo (Σ, P).
+
Then Homeo+
0 (Σ, P) is a normal subgroup of Homeo (Σ, P), and
π0 (Homeo+ (Σ, P)) = Homeo+ (Σ, P)/Homeo+
0 (Σ, P)
is a discrete group. It is called the mapping class group of (Σ, P) and is denoted by
M(Σ, P).
Theorem (Artin, 1925). Let P = {P1 , . . . , Pn } be a collection of n punctures in the
interior of the disk D. Then M(D, P) ≃ Bn .
The isomorphism Φ : M(D, P) → Bn can be described as follows. Let φ ∈ Homeo+ (D, P).
We know that π0 (Homeo+ (D)) = {1}. Thus, there exists a continuous path {φt }t∈[0,1] in
Homeo+ (D) such that φ0 = Id and φ1 = φ. Let β = (b1 , . . . , bn ) be the braid defined by
bk (t) = φt (Pk ) ,
1 ≤ k ≤ n and t ∈ [0, 1] .
Then Φ(φ) is the homotopy class of β.
The inverse isomorphism Φ−1 : Bn → M(D, P) is more complicated to describe, but the
images of the standard generators can be easily defined in terms of braid twists.
7
Definition. We come back to the situation where Σ is an oriented compact surface and
P = {P1 , . . . , Pn } is a collection of n punctures in the interior of Σ. Let Pk , Pl ∈ P, k ̸= l.
An essential arc joining Pk to Pl is defined to be an embedding a : [0, 1] → Σ such that
a(0) = Pk , a(1) = Pl , a((0, 1)) ∩ P = ∅, and a([0, 1]) ∩ ∂Σ = ∅. Two essential arcs a and a′
are said to be isotopic if there is a continuous family {at }t∈[0,1] of essential arcs such that
a0 = a and a1 = a′ . Isotopy of essential arcs is an equivalence relation that we denote by
a ∼ a′ .
Figure. An arc.
Definition. Let a be an essential arc joining Pk to Pl . Let D = {z ∈ C; |z| ≤ 1} be the
standard disk, and let A : D → Σ be an embedding such that
(a) a(t) = A(t − 12 ) for all t ∈ [0, 1];
(b) A(D) ∩ P = {Pk , Pl }.
Let T ∈ Homeo+ (Σ, P) be defined by
(T ◦ A)(z) = A(e2iπ|z| z) ,
z ∈ D,
and T is the identity outside the image of A. The braid twist along a is defined to be the
element τa ∈ M(Σ, P) represented by T , that is, the isotopy class of T .
Pk
a
Pl
T
Figure. Braid twist.
Proposition.
(1) The definition of τa does not depend on the choice of A : D → Σ.
(2) If a is isotopic to a′ , then τa = τa′ .
8
Now, we view the disk D as the disk in C of radius n+1
centered at
2
for 1 ≤ k ≤ n. Let ak : [0, 1] → D be the arc defined by
ak (t) = k + t ,
a1
1
a2
2
n+1
,
2
and we set Pk = k
t ∈ [0, 1] .
an−1
3
n
Figure. The standard generators of M(D, P) = Bn .
Proposition. The inverse isomorphism Φ−1 : Bn → M(D, P) is given by
Φ−1 (σk ) = τak ,
5
1 ≤ k ≤ n − 1.
Automorphisms of free groups
Definition. For a group G, we denote by Aut(G) the group of automorphisms of G. Let
Fn = F (x1 , . . . , xn ) be the free group of rank n. For 1 ≤ k ≤ n − 1, let τk : Fn → Fn be
the automorphism defined by

 xk 7→ x−1
k xk+1 xk
xk+1 7→
xk
τk :

xl 7→
xl
if l ̸= k, k + 1
Proposition. The mapping σk 7→ τk , 1 ≤ k ≤ n − 1, determines a homomorphism
ρ : Bn → Aut(Fn ).
Definition. The above homomorphism ρ : Bn → Aut(Fn ) is called the Artin representation.
Theorem (Artin 1925).
(1) The Artin representation ρ : Bn → Aut(Fn ) is faithful.
(2) An automorphism α ∈ Aut(Fn ) belongs to Im ρ if and only if
α(xn · · · x2 x1 ) = xn · · · x2 x1 ,
and there exists a permutation χ ∈ Sn such that α(xk ) is conjugate to xχ(k) for all
1 ≤ k ≤ n.
9
Definition. A group G is called residually finite if for all g ∈ G \ {1} there exists a
homomorphism φ : G → H such that H is finite and φ(g) ̸= 1. A group G is called
Hopfian if every epimorphism φ : G → G is an isomorphism.
Theorem (Folklore).
(1) Ig G is residually finite, then any subgroup of Aut(G) is residually finite, too.
(2) If G is Hopfian, then any subgroup of Aut(G) is Hopfian, too.
(3) Fn is residually finite and Hopfian.
Corollary. The braid group Bn is residually finite and Hopfian.
Recall the fiber boundle
p:
Mn+1
→
Mn
(z1 , . . . zn , zn+1 ) 7→ (z1 , . . . , zn )
Let Sn act on Mn and on Mn+1 . The second action is on the first n coordinates, that is,
χ(z1 , . . . , zn , zn+1 ) = (zχ−1 (1) , . . . , , zχ−1 (n) , zn+1 ) ,
for χ ∈ Sn .
Then p : Mn+1 → Mn induces a map p̄ : Mn+1 /Sn → Mn /Sn = Nn which turns out to be
a locally trivial fiber bundle. The fiber is again homeomorphic to C \ {1, 2, . . . , n}, and
p̄ : Mn+1 /Sn → Nn has also a cross-section κ̄ : Nn → Mn+1 /Sn . So, from the homotopy
long exact sequence of a fiber bundle we obtain the following split exact sequence
1
/ Fn
/ π1 (Mn+1 /Sn )
n
p̄∗
/ π1 (Nn ) = Bn
/1
κ̄∗
where Fn = π1 (C \ {1, . . . , n}), which is a free group of rank n. The above split exact
sequence induces a homomorphism ρ̃ : Bn → Aut(Fn ).
Proposition. The homomorphism ρ̃ : Bn → Aut(Fn ) coicides with the Artin representation, up to conjugation.
We take a set P = {P1 , . . . Pn } of n punctures in the interior of D, and adopt the point of
view Bn = M(D, P). Let P0 ∈ ∂D be a basepoint. Remark that π1 (D \ P, P0 ) ≃ Fn . The
action of Homeo+ (D, P) on D\P induces a homomorphis ψ̃ : Homeo+ (D, P) → Aut(π1 (D\
P, P0 )) = Aut(Fn ). It is easily shown that ψ̃ is continuous, thus Homeo+
0 (D, P) < Ker ψ̃,
hence ψ̃ induces a homomorphism ψ : M(D, P) = Bn → Aut(Fn ).
Proposition. The homomorphism ψ : Bn → Aut(Fn ) coincides with the Artin representation, up to conjugation.
10
6
Garside groups
Definition. A monoid M is called atomic if there exists a function ν : M → N such that
• ν(α) = 0 if and only if α = 1;
• ν(αβ) ≥ ν(α) + ν(β) for all α, β ∈ M .
Such a function ν is called a norm on M . An element α ∈ M is called an atom if it is
indecomposable, that is, if α = βγ, then either β = 1 or γ = 1.
Lemma. Let M be an atomic monoid. A subset S ⊂ M generates M if and only if it
contains all the atoms. In particular, M is finitely generated if and only if it contains
finitely many atoms.
Definition. Let M be an atomic monoid. We define on M two partial orders ≤L and
≤R as follows.
• Set α ≤L β if there exists γ ∈ M such that αγ = β.
• Set α ≤R β if there exists γ ∈ M such that γα = β.
The orders ≤L and ≤R are called the left and right divisibily orders, respectively.
Definition. An ordered set (E ≤) is called a lattice if:
(a) For all a, b ∈ E there exists c ∈ E such that a ≤ c, b ≤ c, and, if a ≤ c′ and b ≤ c′ ,
then c ≤ c′ ;
(b) For all a, b ∈ E there exists d ∈ E such that d ≤ a, d ≤ b, and, if d′ ≤ a and d′ ≤ b,
then d′ ≤ d.
The above elements c and d are denoted by c = a ∨ b and d = a ∧ b. Note that ∨ and ∧
are operations on E.
A monoid M is called a Garside monoid if
(a) M is atomic and finitely generated;
(b) M is cancelative (that is, if αβγ = αβ ′ γ, then β = β ′ , for all α, β, β ′ , γ ∈ M );
(c) (M, ≤L ) and (M, ≤R ) are lattices;
(d) there exists an element ∆ ∈ M , called a Garside element, such that the sets L(∆) =
{α ∈ M ; α ≤L ∆} and R(∆) = {α ∈ M ; α ≤R ∆} are equal and generate M .
11
If M is a Garside monoid, then the lattice operations of (M, ≤L ) (resp. (M, ≤R )) are
denoted by ∨L and ∧L (resp. ∨R and ∧R ).
Definition. Let M be a monoid. The group of fractions of M is defined to be the group
G(M ) presented with the generating set M and the relations α · β = γ if αβ = γ in M .
Lemma. The group of fractions G(M ) of a monoid M has the universal property that,
if φ : M → H is a homomorphism and H is a group, then there exists a unique homomorphism φ̂ : G(M ) → H such that φ = φ̂ ◦ ι, where ι : M → G(M ) is the natural
homomorphism.
Remark. The latter homomorphism ι : M → G(M ) is not injective in general.
Lemma. If M is a Garside monoid, then ι : M → G(M ) is injective.
Definition. A Garside group is defined to be the group of fractions of a Garside monoid.
Definition. The positive braid monoid is the monoid Bn+ having a presentation (as a
monoid) with generators
σ1 , . . . , σn−1 ,
and relations
if |k − l| ≥ 2 ,
if |k − l| = 1 .
σk σl = σl σl
σk σl σk = σl σk σl
Theorem (Garside, 1969). Bn+ is a Garside monoid. In particular, Bn is a Garside group.
Let M be a Garside monoid, let G = G(M ) be the group of fractions of M , and let ∆ be
a fixed Garside element of M . Define the set of simple elements to be
S = {a ∈ M ; a ≤L ∆} = {a ∈ M ; a ≤R ∆} .
By definition, S is finite and generates M .
Let α ∈ M . Then α can be uniquely written in the form
α = a1 a2 · · · al ,
where a1 , a2 , . . . , al ∈ S, and
ai = ∆ ∧L (ai ai+1 · · · al ) for all 1 ≤ i ≤ l .
Such an expression of α is called the normal form of α.
Let α ∈ G. Then α can be written in the form α = β −1 γ, where β, γ ∈ M . We can also
assume that β ∧L γ = 1. In that case β and γ are unique. Let β = b1 b2 · · · bp be the
normal form of β and let γ = c1 c2 · · · cq be the normal form of γ. Then the expression
−1 −1
α = b−1
p · · · b2 b1 c1 c2 · · · cq
12
is called the normal form of α.
Definition. Let G be a Garside group, let S be the set of simple elements, and let γ ∈ G.
The length of γ with respect to S is the minimal possible length of an expression of γ as
a product of elements of S and their inverses.
Theorem. Let G be a Garside group and let S be the set of simple elements of G.
−1
(1) Let b1 , . . . , bp , c1 , . . . , cq ∈ S. Then b−1
p · · · b1 c1 · · · cq is a normal form if and only
if bi bi+1 , cj cj+1 and b−1
1 c1 are normal forms for all i ∈ {1, . . . , p − 1} and j ∈
{1, . . . , q − 1}.
−1
(2) Let γ ∈ G and let γ = b−1
p · · · b1 c1 · · · cq be the normal form of γ. Then p + q is the
length of γ.
13