From Braids to Mapping Class

Groups

Pratyush Kumar Mishra

MP14008

A dissertation submitted for the partial fulfilment of

MS degree in Mathematical Science

Indian Institute of Science Education and Research Mohali

April 2017

Certificate of Examination

This is to certify that the dissertation titled “From Braids to Map-

ping Class Groups” submitted by Mr. Pratyush Kumar Mishra (Reg.

No. MP14008) for the partial fulfilment of MS degree programme in

Mathematical Sciences of the Institute, has been examined by the the-

sis committee duly appointed by the Institute. The committee finds

the work done by the candidate satisfactory and recommends that the

report be accepted.

Prof.Kapil Hari Paranjape Dr.Pranab Sardar Dr.Mahender Singh

(Supervisor)

Dated: April 21, 2017

Declaration

The work presented in this dissertation has been carried out by me un-

der the guidance of Prof. Kapil Hari Paranjape at the Indian Institute

of Science Education and Research Mohali.

This work has not been submitted in part or in full for a degree, a

diploma, or a fellowship to any other university or institute. Whenever

contributions of others are involved, every effort is made to indicate

this clearly, with due acknowledgement of collaborative research and

discussions. This thesis is a bonafide record of original work done by

me and all sources listed within have been detailed in the bibliography.

Pratyush Kumar Mishra

(Candidate)

Dated: April 21, 2017

In my capacity as the supervisor of the candidate’s project work, I cer-

tify that the above statements by the candidate are true to the best of

my knowledge.

Prof. Kapil Hari Paranjape

(Supervisor)

Acknowledgements

I would like to thank my supervisor Prof. Kapil Hari Paranjape, for

his invaluable guidance, support and help in understanding the material

presented in this thesis.

I am very thankful to Mr. Soumya Dey, a senior research fellow at

IISER Mohali for his continuous help in understanding many concepts

throughout the thesis work.

I would like to thank Dr. Pranab Sardar for clearing some of the

major doubts during the thesis work and understanding some of the

basic parts of the proofs.

Also, I am thankful to Dr. Mahender Singh for allowing me to audit

the course ‘Topics in Topology’, which helped me in clearing some of the

concepts in Higher Homotopy Theory and Dr. Chetan Balwe for helping

me with understanding some of the concepts during my last phase of the

project.

I would like to express my gratitude to all my teachers at IISER

Mohali Mathematics Department for being so supportive and helpful.

IISER, Mohali Department of Mathematics provided an excellent li-

brary and highly suggestive deadlines. I owe a debt of gratitude to the

generous souls who proofread this work in its more and less spell checked

stages.

I would like to express my gratitude to my parents for always encour-

aging me.

Pratyush Kumar Mishra

IISER Mohali

Preface

The central theme is Artin’s braid group, and the many ways that the notion of a

braid has proved to be important in low-dimensional topology.

It will be assumed that the reader is familiar with the very basic ideas of homo-

topy theory such as the ideas of homotopy equivalence, homomorphism, deformation

retractions and the notions of fundamental groups(and its computation) etc.

Chapter 1, as a preliminary develop the tools to be used in chapter 2 and 3 of the

thesis. The materials here are based on my understanding from the texts: ‘Algebraic

Topology’ by Allan Hatcher; ‘Combinatorial Group Theory’ by Magnus, Karrass and

Solitar; ‘Homotopy Theory’ by Sze-Tsen Hu.

Chapter 2 starts with definition of braid group and deals with the concepts of a

braid regarded as a group of motions of points in a manifold. Many algebraic and

structural properties of the braid groups of two manifolds are studied, and defining

relations are derived for the braid groups of E2 and S2. The materials presented in

this section is based on my understanding of Chapter 1, from the text ‘Braids, Links

and Mapping Class Groups’ by J.S. Birman [1]. The proof of the theorem 13 is based

on my understanding of the paper ‘Basic Results on Braids’, 2004 by J. Gonzalez

Meneses [22].

In Chapter 3, we will give some connections between braid groups and mapping

class group of the surfaces. Also we compute the mapping class group of the n-

punctured sphere. The contained of this chapter is based on my understanding of

section 4.1 and 4.2 of chapter 4 from the text ‘Braids, Links and Mapping Class

groups’ by Birman [1]. The proof of Lemma 7, is based on my understanding from

the text ‘A primer on Mapping class group’ by ’Benson Farb and Dan Margalit’.

Some of the figures are taken from Birman’s book [1] and some other from the

Internet.

I tried my best to give detailed explanations for each theorems and results which

were not that vivid in the original manuscript of Birman [1]. I mentioned the refer-

ences whenever required in the ‘Bibliography’.

Pratyush Kumar Mishra

IISER Mohali

Contents

1 Preliminaries 9

1.1 Covering Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.1.1 Lifting Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

1.1.2 The Classification of Covering Spaces . . . . . . . . . . . . . . . . 12

1.1.3 Deck Transformation and Group Actions . . . . . . . . . . . . . . 13

1.2 Graphs and Free Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

1.3 Higher Homotopy Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

1.3.1 Whitehead’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . 22

1.3.2 Cellular Approximation Theorem . . . . . . . . . . . . . . . . . . . 22

1.3.3 Freudenthal suspension theorem . . . . . . . . . . . . . . . . . . . 23

1.3.4 Fiber Bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

1.4 Homogeneous spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

1.5 Group Representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

1.6 Schreier-Reidemeister Method . . . . . . . . . . . . . . . . . . . . . . . . . 28

2 Braid Groups 31

2.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

2.2 Configuration spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

2.3 Braid groups of manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

2.4 The braid group of the plane . . . . . . . . . . . . . . . . . . . . . . . . . . 43

2.4.1 Braids as Automorphisms of free groups or Mapping Class groups 56

2.5 The braid group of the sphere . . . . . . . . . . . . . . . . . . . . . . . . . 58

2.6 Survey of 2-manifold braid groups . . . . . . . . . . . . . . . . . . . . . . . 59

2.7 Some examples where braiding appears in mathematics, unexpectedly . 60

2.7.1 Algebraic geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

7

3 Mapping Class Groups 63

3.1 Definition and Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

3.2 The natural homomorphism from M(g,n) to M(g,0) . . . . . . . . . . . . 64

3.3 The mapping class group of the n-punctured sphere . . . . . . . . . . . . 71

Chapter 1

Preliminaries

1.1 Covering Spaces

This section is based on the author’s understanding of Chapters 0,1,2 from the

Hatcher’s book on Algebraic Topology [2]. We will just state the definitions and

statements of the theorems(without proof), that we require for the rest of the section.

Definition: A covering space of a space X is a space X together with a surjective

map p ∶ X → X satisfying the following condition: There exists a open cover Uα of

X such that for each α, p−1(Uα) is a disjoint union of open sets in X, each of which

is mapped by p homeomorphically onto Uα.

Example: Take for example the map p ∶ R→ S1, defined by p(t) = (cos 2πt, sin 2πt).

It’s easy to check that it’s a covering map.

Example: One can also think of the map p ∶ S1 → S1, p(z) = zn which is known

as a n sheeted covering where we view z as the complex number with ∣z∣ = 1 and n is

any positive integer.

Simply-connected space: A path-connected space whose fundamental group is

trivial.

Proposition 1. A space X is simply-connected iff there is a unique homotopy class

of paths connecting any two points in X.

Example: It is easy to see that Rn is simply connected for all n.

A simply-connected covering space of X = S1 ∨ S1 is as in figure given below:

9

Covering Spaces Section 1.3 59

A simply-connected covering space of X can be constructed in the following way.

Start with the open intervals (−1,1) in the coordinate

axes of R2 . Next, for a fixed number λ , 0 < λ < 1/2, for

example λ = 1/3, adjoin four open segments of length

2λ , at distance λ from the ends of the previous seg-

ments and perpendicular to them, the new shorter seg-

ments being bisected by the older ones. For the third

stage, add perpendicular open segments of length 2λ2

at distance λ2 from the endpoints of all the previous

segments and bisected by them. The process is now

repeated indefinitely, at the nth stage adding open segments of length 2λn−1 at dis-

tance λn−1 from all the previous endpoints. The union of all these open segments is

a graph, with vertices the intersection points of horizontal and vertical segments, and

edges the subsegments between adjacent vertices. We label all the horizontal edges

a , oriented to the right, and all the vertical edges b , oriented upward.

This covering space is called the universal cover of X because, as our general

theory will show, it is a covering space of every other connected covering space of X .

The covering spaces (1)–(14) in the table are all nonsimply-connected. Their fun-

damental groups are free with bases represented by the loops specified by the listed

words in a and b , starting at the basepoint x0 indicated by the heavily shaded ver-

tex. This can be proved in each case by applying van Kampen’s theorem. One can

also interpret the list of words as generators of the image subgroup p∗(π1(X, x0)

)in π1(X,x0) =

⟨a,b

⟩. A general fact we shall prove about covering spaces is that

the induced map p∗ :π1(X, x0)→π1(X,x0) is always injective. Thus we have the at-

first-glance paradoxical fact that the free group on two generators can contain as a

subgroup a free group on any finite number of generators, or even on a countably

infinite set of generators as in examples (10) and (11).

Changing the basepoint vertex changes the subgroup p∗(π1(X, x0)

)to a conju-

gate subgroup in π1(X,x0) . The conjugating element of π1(X,x0) is represented by

any loop that is the projection of a path in X joining one basepoint to the other. For

example, the covering spaces (3) and (4) differ only in the choice of basepoints, and

the corresponding subgroups of π1(X,x0) differ by conjugation by b .

The main classification theorem for covering spaces says that by associating the

subgroup p∗(π1(X, x0)

)to the covering space p : X→X , we obtain a one-to-one

correspondence between all the different connected covering spaces of X and the

conjugacy classes of subgroups of π1(X,x0) . If one keeps track of the basepoint

vertex x0 ∈ X , then this is a one-to-one correspondence between covering spaces

p : (X, x0)→(X,x0) and actual subgroups of π1(X,x0) , not just conjugacy classes.

Of course, for these statements to make sense one has to have a precise notion of

when two covering spaces are the same, or ‘isomorphic.’ In the case at hand, an iso-

This covering space is called the universal cover of X because, it is a covering

space of every other connected covering space of X.

Universal cover: A covering space is a universal covering space if it is simply

connected.

Remark: We will use covering space theory to deduce some of the interesting

properties in Infinite group Theory like: the free group on two generators can contain

as a subgroup a free group on any finite number of generators, or even on a countably

infinite set of generators.

1.1.1 Lifting Properties

A lift of a map f ∶ Y → X is a map f ∶ Y → X such that pf = f . We will describe

three special lifting properties of covering spaces and see a few applications of these.

First we have the homotopy lifting property, or covering homotopy prop-

erty as,(we will also prove it as the construction is quite important)

Proposition 2. Given a covering space p ∶ X → X, a homotopy F ∶ Y × I → X, and

a map f0 ∶ Y → X lifting f0, then there exits a unique homotopy F ∶ Y × I → X of F

such that F ∣Y ×0 = f0.

Proof. Pick an open cover Uα of X such that p−1(Uα) can be decomposed as into a

disjoint union of open sets each of which is homeomorphic to Uα under p.

Step 1. Let y ∈ Y be given. We begin by constructing a lift F ∶ V × I → X, for

some neighborhood V of y. For each t ∈ I we may pick some neighborhood Vt and

some open interval It containing t such that F (Vt×It) ⊂ Uα. Cover I by finitely many

It and let V be the intersection of the corresponding Vt. Then, we may choose a finite

partition 0 = t0 < t1 < ⋅ ⋅ ⋅ < tn = 1 such that F (V × [ti, ti+1]) ⊂ Uαi for some index αi.

We now construct a lift F ∶ V × [0, ti] → X by induction on i. The base case

i = 0 is given by f0. Suppose F ∶ V × [0, ti] → X has been constructed. As, F (V ×

[ti, ti+1]) ⊂ Uαi , we may choose some set Ui ∈ X containing F (y, ti) such that Uαi is

homeomorphic to Uαi under p. Shrinking V if needed, by continuity we may assume

that F (V × ti) ⊂ Uαi . We may define F on the set V × [ti, ti+1] to be p−1(F ), where

p−1 denotes the homeomorphism p−1 ∶ Uαi → Uαi . By pasting lemma, the resulting

function F ∶ V × [0, ti+1] → X is continuous. This completes the induction furnishing

the map F ∶ V × I → X that lifts F ∣V ×I .

Step 2. We prove the uniqueness in the case where Y is a single point y. Let

F , F ′ be two lifts of F ∶ y× I →X for which F (y, ti) = F ′(y, ti); once again we may

choose a finite partition 0 = t0 < t1 < ⋅ ⋅ ⋅ < tn = 1 such that F (t × [ti, ti+1]) ⊂ Uαi for

some index αi. We claim that F = F ′ on [0, ti] for all i; once again we proceed by

induction. The base case i = 0 follows by assumption. Suppose F = F ′ on [0, ti]; as

F (t × [ti, ti+1]), F ′(t × [ti, ti+1]) are connected and F (y, ti) = F ′(y, ti), both must

lie in the same open set Uαi ∈ X that is homeomorphic to Uαi under p. But as p∣Uαiis injective and pF = F = pF ′ , this implies F = F ′ on y × [ti, ti+1] completing the

induction.

Step 3. We now prove the theorem. First, we show uniqueness: if F ∶ Y × I → X

is a lift of F , then F ∣y×I is a lift of F ∣y×I , so by step 2, F is unique. Furthermore,

given two lifts F ∶ V × I → X , F ′ ∶ V × I → X constructed in Step 1, by Step 2, F

and F ′ must agree on V ∩ V ′. Therefore, by pasting together lifts F ∶ V × I → X for

each point y ∈ Y , one obtains a well defined lift F ∶ Y × I → X, completing the proof

of the proposition. ∎

Corollary 1. (The path lifting property) Let f ∶ I →X be a path such that f(0) = x0.

Given a point x0 ∈ p−1(x0), there exits a unique lift f ∶ I →X of f such that f(0) = x0.

In particular, every lift of a constant path is a constant.

Corollary 2. (The path homotopy lifting property): Let ft be a path homotopy in X.

Given a lift f0 of f0, there exists a unique lift ft in X of ft; this lift ft is also a path

homotopy.

Proposition 3. The map p∗ ∶ π1(X, x0) → π1(X,x0) induced by a covering space

p ∶ (X, x0) → (X,x0) is injective. The image subgroup p∗(π1(X, x0)) in π1(X,x0)

consists of the homotopy classes of loops in X based at x0 whose lifts to X starting

at x0 are loops.

Proposition 4. The number of sheets of a covering space p ∶ (X, x0) → (X,x0) with

X and X path-connected equals the index of p∗(π1(X, x0)) in π1(X,x0).

Proposition 5. (lifting criterion) Suppose given a covering space p ∶ (X, x0) →

(X,x0) and a map f ∶ (Y, y0) → (X,x0) with Y path-connected and locally-path-

connected. Then a lift f ∶ (Y, y0)→ (X, x0) of f exists iff f∗(π1(Y, y0)) ⊂ p∗(π1(X, x0)).

Next, we have unique lifting property:

Proposition 6. Given a covering space p ∶ X → X and a map f ∶ Y → X with two

lifts f1, f2 ∶ Y → X that agree at one point of Y, then if Y is connected, these two lifts

must agree on all of Y .

1.1.2 The Classification of Covering Spaces

Now we will classify all the different covering spaces of a fixed space X, which is at

least locally path-connected, path-connected and connected.

A space X semi-locally simply connected if for each point x ∈X has a neigh-

borhood U such that the inclusion-induced map π1(U,x)→ (X,x0) is trivial. This is

a necessary condition for X to have a simply-connected covering space.

Example: Consider the subspace A ⊂ R2, consisting of the circles of radius 1n

centered at the point ( 1n ,0) for n = 1,2, . . . ,. This is an example of a space that is not

semilocally simply-connected.

Proposition 7. Suppose X is path-connected, locally path-connected, and semilocally

simply-connected. Then for every subgroup H ⊂ π1(X,x0) there is a covering space

p ∶XH →X such that p∗(π1(XH , x0)) =H for a suitable chosen basepoint x0 ∈XH .

An Isomorphism between covering spaces p1 ∶ X1 → X and p2 ∶ X2 → X is a

homeomorphism f ∶ X1 → X2 such that p1 = p2f

Proposition 8. If X is path-connected and locally path-connected then two path-

connected covering spaces p1 ∶ X1 → X and p2 ∶ X2 → X are isomorphic via an

isomorphism f ∶ X1 → X2 taking a basepoint x1 ∈ p−1(x0) to a basepoint 2 ∈ p−12 (x0) iff

p1∗(π1(X1, x1)) = p2∗(π1(X2, x2)).

Here we state the classification theorem, which is the most awaited result also

known as Galois correspondence of covering spaces.

Proposition 9. Let X be path-connected, locally path-connected, and semilocally

simply-connected. Then there is a bijection between the set of basepoint-preserving iso-

morphisms classes of the path-connected covering spaces p ∶ (X, x0)→ (X,x0) and the

set of the subgroups of π1(X,x0), obtained by associating the subgroup p∗(π1(X, x0))

to the covering space (X, x0). If the basepoints are ignored then this correspondence

gives a bijection between isomorphism classes of the path-connected covering spaces

p ∶ X →X and conjugacy classes of the subgroups of π1(X,x0).

Remark: There is a partial ordering on the various path-connected covering

spaces of X, according to which one cover the others. If the basepoints are ignored,

this corresponds to the partial inclusion of the corresponding subgroups of π1(X), or

conjugacy classes of the subgroups.

1.1.3 Deck Transformation and Group Actions

For a covering space p ∶ X →X the isomorphisms X → X are called deck transformations

or covering transformations. These form a group G(X) under composition.

Example: For the covering space p ∶ R → S1 projecting an infinite helix onto

a circle, the deck transformations are the vertical translations taking the helix onto

itself, so G(X) ≈ Z.

A covering space p ∶ X →X is normal or regular if for each x ∈X and each pair

of lifts x, x′ of x, ∃ a deck transformation taking x to x′.

Example: The covering R→ S1 and S1 → S1(defined by z ↦ zn) are both normal.

Intuitively, a normal covering space is one with maximal symmetry.

The term ’normal’ is motivated by the following result:

Proposition 10. Let p ∶ (X, x0) → (X,x0) be a path-connected covering space of the

path-connected, locally path-connected space X, and H be the subgroup of p∗(π1(X, x0)) ⊂

π1(X,x0). Then:

(a). This covering space is normal iff H is a normal subgroup of π1(X,x0).

(b). G(X) ≈ N(H)/H where N(H) is the normalizer of H in π1(X,x0).

In particular, G(X) ≈ π1(X,x0)/H if X is a normal covering. Hence for the

universal cover X →X, we have G(X) ≈ π1(X)

The group of deck transformation is a special case of the general notion of ’groups

acting on spaces’. Given a group G and a space Y , then an action of G on Y is a

homeomorphism ρ from G to the group Homeo(Y ) of all homeomorphisms from Y

to itself.

Definition: A covering space action is the action of a group on a space Y such

that for each y ∈ Y has a neighborhood U such that, for any g1, g2 ∈ G,g1(U)∩g2(U) ≠

∅ implies g1 = g2.

The action of the deck transformation group G(X) on X. To see this, let U ⊂ X

project homeomorphically to U ⊂ X. If g1(U) ∩ g2(U) ≠ ∅ for some g1, g2 ∈ G(X),

then g1(x1) = g2(x2) for some x1, x2 ∈ U . Since x1 and x2 must lie in the same set

p−1(x), which intersects U in only one point, we must have x1 = x2. Then g−11 g2 fixes

this point, so g−11 g2 = Id and g1 = g2.

Given an action of a group on a space Y, we can form a space Y /G, the quotient

space of Y in which each point y is identified with all its images g(y) as g ranges over

G. The points of Y /G are thus the orbits Gy = g(y)∣g ∈ G in Y, and Y /G is called

the orbit space of the action. For example, for a normal covering space X →X, the

orbit space X/G(X)is just X.

Proposition 11. If an action of a group G on a space Y is a covering space action,

then:

(a) The quotient map p ∶ Y → Y /G, p(y) = Gy, is a normal covering space.

(b) G is the group of deck transformation of this covering space Y → Y /G if Y is

path-connected.

(c) G ≈ π1(Y /G)/p∗(π1(Y )) if Y is path-connected and locally-path-connected.

Sometimes these are called ‘properly discontinuous’ actions, but more often

this rather unattractive term means something weaker: Every point x ∈ X has a

neighborhood U such that U ∩g(U) ≠ ∅ for only finitely many g ∈ G. Many symmetry

groups have this proper discontinuity property without satisfying the covering space

action property, for example, the group of symmetries of the familiar tiling of R2 by

regular hexagons. The reason why the action of this group on R2 fails to satisfy the

covering space property is that there are fixed points: points y for which there is a

nontrivial element g ∈ G with g(y) = y.

An action without fixed points is called a free action.

It is easy to see that freeness implies covering space action. But the converse

is not true,

Example:(Covering space action doesn’t implies freeness) Consider the action of

Z on S1 in which a generator of Z acts by rotation through an angle α that is an

irrational multiple of 2π.

Example ∶ Let Y be the closed orientable surface of genus 11, an ‘11-hole torus’

as shown in the figure.(taken from Hatcher’s Algebraic Topology)

Covering Spaces Section 1.3 73

something weaker: Every point x ∈ X has a neighborhood U such that U ∩ g(U)is nonempty for only finitely many g ∈ G . Many symmetry groups have this proper

discontinuity property without satisfying (∗) , for example the group of symmetries

of the familiar tiling of R2 by regular hexagons. The reason why the action of this

group on R2 fails to satisfy (∗) is that there are fixed points: points y for which

there is a nontrivial element g ∈ G with g(y) = y . For example, the vertices of the

hexagons are fixed by the 120 degree rotations about these points, and the midpoints

of edges are fixed by 180 degree rotations. An action without fixed points is called a

free action. Thus for a free action of G on Y , only the identity element of G fixes any

point of Y . This is equivalent to requiring that all the images g(y) of each y ∈ Y are

distinct, or in other words g1(y) = g2(y) only when g1 = g2 , since g1(y) = g2(y)is equivalent to g−1

1 g2(y) = y . Though condition (∗) implies freeness, the converse

is not always true. An example is the action of Z on S1 in which a generator of Z acts

by rotation through an angle α that is an irrational multiple of 2π . In this case each

orbit Zy is dense in S1 , so condition (∗) cannot hold since it implies that orbits are

discrete subspaces. An exercise at the end of the section is to show that for actions

on Hausdorff spaces, freeness plus proper discontinuity implies condition (∗) . Note

that proper discontinuity is automatic for actions by a finite group.

Example 1.41. Let Y be the closed orientable surface of genus 11, an ‘11 hole torus’ as

shown in the figure. This has a 5 fold rotational symme-

try, generated by a rotation of angle 2π/5. Thus we have

the cyclic group Z5 acting on Y , and the condition (∗) is

obviously satisfied. The quotient space Y/Z5 is a surface

C2

C1

C 3C4

C5

C

p

of genus 3, obtained from one of the five subsurfaces of

Y cut off by the circles C1, ··· , C5 by identifying its two

boundary circles Ci and Ci+1 to form the circle C as

shown. Thus we have a covering space M11→M3 where

Mg denotes the closed orientable surface of genus g .

In particular, we see that π1(M3) contains the ‘larger’

group π1(M11) as a normal subgroup of index 5, with

quotient Z5 . This example obviously generalizes by re-

placing the two holes in each ‘arm’ of M11 by m holes and the 5 fold symmetry by

n fold symmetry. This gives a covering space Mmn+1→Mm+1 . An exercise in §2.2 is

to show by an Euler characteristic argument that if there is a covering space Mg→Mhthen g =mn+ 1 and h =m+ 1 for some m and n .

As a special case of the final statement of the preceding proposition we see that

for a covering space action of a group G on a simply-connected locally path-connected

space Y , the orbit space Y/G has fundamental group isomorphic to G . Under this

isomorphism an element g ∈ G corresponds to a loop in Y/G that is the projection of

This has a 5-fold rotational symmetry, generated by a rotation of angle 2π/5 Thus

we have the cyclic group Z5 acting on Y , and the condition for the covering space

action is satisfied(by Proposition 12 below). The quotient space Y /Z5 is a surface

of genus 3, obtained from one of the five subsurfaces of Y cut off by the circles

C1, . . .C5 by identifying its two boundary circles Ci and Ci+1 to form the circle C

as shown. Thus, we have a covering space M11 → M3 where Mg denotes the closed

orientable surface of genus g. In particular, we see that π1(M3) contains the ‘larger’

group π1(M11) as the normal subgroup of index 5, with quotient Z5. This example

generalizes by replacing the two holes in each ‘arm’ of M11 by m holes and 5−fold

symmetry by n−fold symmetry. This gives a covering space Mmn+1 →Mm+1.

Proposition 12. The action of a finite group G on a Hausdorff space X is always

properly discontinuous and hence a covering space action.

Proof. Let x0 ∈ X, and then the orbit x0 is given by Gx0 = x0, x1, . . . , xk. As X is

Hausdorff, ∃ neighborhoods Vk of xk and Vλ of xλ such that Vk ∩ Vλ for k ≠ λ. Since

the map, x z→ gx is continuous we have for each g ∈ G, ∃ an open neighborhood Wg

of x0 such that gWg ⊂ Vk if gx0 = xk(one can take Wg = V0 ∩ g−1Vk). Then we have

U = V0 ∩g∈GWg

is an open neighborhood of x0, and hence we get, gU ⊂ Vk if gx0 = xk, in particular

gU ∩U ≠ ∅⇐⇒ gx0 = x0.

∎

On a non-Hausdorff space, the action of a finite group need not be discontinuous.

As an extreme example, consider the following:

Example: Take an indiscrete space, then the ‘trivial action’ is the only discon-

tinuous action.

For a covering space action of a group G on a simply-connected locally path-

connected space Y , π1(Y /G) ≈ G, where Y /G is the orbit space.

1.2 Graphs and Free Groups

As all groups can be realized as fundamental groups of spaces, this opens the way for

using topology to study algebraic properties of groups.

Definition: A graph is a 1-dimensional CW complex, in other words, a space X

obtained from a discrete set X0 by attaching a collection of 1-cells eα.

Thus, X is obtained from the disjoint union of X0 with closed intervals Iα by

identifying the two endpoints of each Iα with points of X0. The points of X0 are the

vertices and the 1-cells the edges of X. Note that with this definition an edge does

not include its endpoints, so an edge is an open subset of X. The two endpoints of an

edge can be the same vertex, so its closure eα of an edge eα is homeomorphic either

to I or S1.

Since X has the quotient topology from the disjoint union X0 ⊍ Iα, a subset of X

is open(or closed) iff it intersects the closure eα of each edge eα is an open(or closed)

sets in eα. One says that X has the weak topology with respect to the subspace eα.

A basis for the topology of X consists of the open intervals in the edges together

with the path-connected neighborhoods of the vertices. A neigborhood of the latter

sort about a vertex v is the union of connected open neighborhoods Uα of v in eα for

all eα containing v. In particular, we see that X is locally-path-connected. Hence, a

graph is connected iff it it is path-connected.

Definition: A subgraph of a graph X is a subspace Y ⊂ X that is a union of

vertices and edges of X, such that eα ⊂ Y implies eα ⊂ Y . The latter condition just

says that Y is a closed subspace of X.

Definition: A tree is a contractible graph. By a tree in a graph X we mean a

subgraph that is a tree.

Definition: A tree in X maximal if it contains all the vertices of X.

Proposition 13. Every connected graph contains a maximal tree, and in fact any

tree in the graph is contained in a maximal tree.

Proposition 14. For a connected graph X with maximal tree T , π1(X) is a free

group with basis the classes [fα] corresponding to the edges eα of X − T .

Proof. As (X,T ) is a CW pair and hence has homotopy extension property where T

is contractible, then by a basic result from homotopy theory we have, the quotient

map q ∶X →X/A is a homotopy equivalence. The quotient X/T is a graph with only

one vertex, hence is a wedge sum of circles, whose fundamental group is free with

basis the loops given by the edges of X/T , which are the images of the loops fα in

X. ∎

Corollary 3. A graph is a tree iff it is simply connected.

Here is a very useful fact about graphs:

Proposition 15. Every covering space of a graph is also a graph, with vertices and

edges the lifts of the vertices and edges in the base graph.

Now, we can apply all the theory that we established so far about graphs and their

fundamental groups to prove a basic important fact of group theory:

Proposition 16. Every subgroup of a free group is free.

Proof. Given a free group F, choose a graph X with π1(X) ≈ F , for example a wedge

of circles corresponding to a basis for F . For each subgroup G of F there is (by

Proposition 7), a covering space p ∶ X → X with p∗(π1(X)) = G, hence π1(X) ≈ G

since p∗ is injective by Proposition 3. Since X is a graph by the preceding proposition,

the group G ≈ π1(X) is free by Proposition 13. ∎

1.3 Higher Homotopy Groups

This subsection is based on the author’s understanding of Chapter 4 of the book

[2]. Let In be the n−dimensional unit cube, the product of n copies of the interval

[0,1]. The boundary ∂In of In is the subspace consisting of points with at least one

coordinate equal to 0 ot 1. For a space X with basepoint x0 ∈ X, define πn(X,x0)

to be the set of homotopy classes of maps f ∶ (In, ∂In) → (X,x0), where homotopies

ft are required to satisfy ft(∂In) = x0 for all t. The definition extends to the case

n = 0 by taking I0 to be a point and ∂I0 to be empty, so π0(X,x0) is just the set of

path-components of X.

When n ≥ 2, a sum operation in πn(X,x0), generalizing the composition operation

in π1, defined by

(f + g)(s1, s2, . . . , sn) =

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

f(2s1, s2, . . . , sn), s1 ∈ [0, 12]

g(2s1 − 1, s2, . . . , sn), s1 ∈ [12 ,1]

Clearly, this sum is well-defined on homotopy classes. Since only the first coor-

dinate is involved in the sum operation, the same arguments as for π1 shows that

πn(X,x0) is a group, with identity element the constant map sending In to x0 and

with inverses given by −f(s1, s2, . . . , sn) = f(1 − s1, s2, . . . , sn).

The additive notation for the group operation is used because π1(X,x0) is abelian

for n ≥ 2.

Maps (In, ∂In) → (X,x0) are the same as maps of the quotient In/∂In = Sn to x

taking the basepoint s0 = ∂In/∂In to x0. This means that we can view πn(X,x0) as

homotopy classes of maps (Sn, s0)→ (X,x0), where homotopies are through maps of

the same form (Sn, s) → (X,x0). In this interpretation of πn(X,x0), the sum f + g is

composition SncÐ→ Sn ∨Sn f∨gÐÐ→X where c collapses the equator Sn−1 in Sn to a point

and we choose the basepoint s0 to lie in this Sn−1.

We will show next that if X is path-connected, different choices of the basepoint

x0 always produce isomorphic groups πn(X,x0), just as for π1, so one is justified in

writing πn(X) for πn(X,x0) in these case. Given a path γ ∶ I → X from x0 = γ(0) to

another basepoint x1 = γ(1), we may associate to each map f ∶ (In, ∂In) → (X,x1) a

new map γf ∶ (In, ∂In)→ (X,x0) by shrinking the domain of f to a smaller concentric

cube in In, then inserting path γ on each radial segment in the shell between this

smaller cube and ∂In. When n = 1 the map γf is the composition of the three

paths γ, f, and the inverse of γ, so the notation γf conflicts with the notation for

composition of paths. Since we are mainly interested in the case n > 1, when n = 1, it

is clear. Here are three basic properties:

(1). γ(f + g) ≃ γf + γg.

(2). (γη)f ≃ γ(ηf)

(3). 1.f ≃ f , where 1 denotes the constant path.

For (1), we first deform f and g to be constant on the right and left halves of In.

respectively, producing maps we may call f+0 and 0+g, then we excise a progressively

wider symmetric middle slab of γ(f + 0) + γ(0 + g) untill it becomes γ(f + g).

An explicit formula for this homotopy is :

ht(s1, s2, . . . , sn) =

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

γ(f + 0)((2 − t)s1, s2, . . . , sn) s1 ∈ [0, 12]

γ(0 + g)((2 − t)s1 + t − 1, s2, . . . , sn) s1 ∈ [12 ,1]

Thus, we have γ(f + g) ≃ γ(f + 0) + γ(0 + g) ≃ γf + γg.

If we define a change-of-basepoint transformation βγ ∶ πn(X,x1) → πn(X,x0) by

β[F ] = [γf], then (1) shows that βγ is a homomorphism, while (2) and (3) imply that

βγ is an isomorphism with inverse βγ where γ is the inverse path of γ, γ(s) = γ(1−s).

Thus, if X is path-connected, different choice of basepoint x0 yield isomorphic groups

πn(X,x0), which is simply πn(X).

Now, let us restrict attention to loops γ at the basepoint x0. Since βγη = βγβη,

the association [γ]→ βγ defines a homomorphism from π1(X,x0) to Aut(πn(X,x0)),

the group of automorphisms of πn(X,x0). This is called the action of π1 on πn,

each element of π1 acting as an automorphisms [f] → [γf] of πn. When n = 1 this is

the action of π1 on itself by inner automorphisms. When n > 1, the action makes the

abelian group πn(X,x0) into a module over the group ring Z[π1(X,x0)]. Elements

of Z[π1] are finite sums ∑i niγi with ni ∈ Z and γi ∈ π1, multiplication is defined by

distributivity and the multiplication in π1. The module structure on πn is given by

(∑i niγi)α = ∑i ni(γiα) for α ∈ πn. Sometimes people say πn is a π−module rather

than a Z[π1]−module.

A space with trivial π1 action on πn is called ‘n-simple’, and ‘simple’ means ‘n-

simple for all n’. Here, we will call a space abelian if it has trivial action of π1 on all

homotopy groups πn, since when n = 1 this is the condition that π1 be abelian.

Homotopy groups behaves very nicely w.r.t covering spaces:

Proposition 17. A covering space projection p ∶ (X, x0) → (X,x0) induces isomor-

phisms p∗ ∶ πn(X, x0)→ πn(X,x0) for all n ≥ 2.

Proof. To prove surjectivity of p∗ we apply the lifting criterion in Propostion 5, which

implies that every map (Sn, s0)→ (X,x0) lifts to (X, x0) provided that n ≥ 2 so that

Sn is simply connected. Injectivity of p∗ is immediately follows from the covering

homotopy property, just as in Proposition 3. ∎

Note that πn(X,x0) = 0 for n ≥ 2 if X has a contractible universal cover. This

applies for example to S1. In general, the n−dimensional torus T n, the product of

n−circles, has universal cover Rn, so πi(T n) = 0 for i > 1. This is marked in contrast

to the homology groups Hi(T n) which are non-zero for all i ≤ n. We called the spaces

with πn = 0 for all n ≥ 2 as aspherical.

The homotopy groups behaves in a very simple manner with respect to products:

Proposition 18. For a product ΠαXα of an arbitrary collection of path-connected

spaces Xα there are isomorphisms πn(ΠαXα) ≈ Παπn(Xα) for all n.

Proof. A map f ∶ Y → ΠαXα can be regarded as a collection of maps fα ∶ Y → Xα.

Taking Y to be Sn and Sn × I gives the result. ∎

Generalizations of the homotopy groups πn(X,x0) are the relative homotopy

groups πn(X,A,x0) for a pair (X,A) with a basepoint x0 ∈ A, which are quite useful.

To define these, regard In−1 as the face of In with last coordinate sn = 0 and let Jn−1 be

the closure of ∂In−In−1, the union of the remaining faces of In. Then πn(X,A,x0) for

n ≥ 1 is defined to be the set of homotopy classes of maps (In, ∂In, Jn−1)→ (X,A,x0),

with homotopies through maps of the same form. Note that πn(X,x0, x0) = πn(X,x0),

hence absolute homotopy groups are a special case of relative homotopy groups.

A sum operation is defined in πn(X,A,x0) by the same formulas as for πn(X,x0),

except that the coordinates sn now plays a special role and is no longer available

for the sum operation. Thus πn(X,A,x0) is a group for n ≥ 2, and this group

is abelian for n ≥ 3. For n = 1 we have I1 = [0,1], I0 = 0, and J0 = 1, so

π1(X,A,x0) is the set of homotopy classes of paths in X from a varying point in A

to the fixed basepoint x0 ∈ A. In general, this is not a group in any natural way.

Just as elements of πn(X,x0) can be regarded as homotopy classes of maps (Sn, s0)→

(X,x0), there is also an alternative definition of πn(X,A,x0) as the set of homotopy

classes of maps (Dn, Sn−1, s0) → (X,A,x0), since collapsing Jn−1 to a point converts

(In, ∂In, Jn−1) into (Dn, Sn−1, s0). From this viewpoint, addition is done via the map

c ∶Dn →Dn ∨Dn collapsing Dn−1 ⊂Dn to a point.

A useful and conceptually illuminating reformation of what it means for an element

of πn(X,A,x0) to be trivial is given by the following compression criteria:

A map f ∶ (Dn, Sn−1, s0) → (X,A,x0) represents zero in πn(X,A,x0) iff it is

homotopic rel Sn−1 to a map with image contained in A.

For if we have such a homotopy to a map g, then [f] = [g] in πn(X,A,x0), and

[g] = 0 via the homotopy obtained by composing g with a deformation retraction of

Dn onto s0. Conversely, if [f] = 0 via a homotopy F ∶Dn × I →X, then by restricting

F to a familly of n−disks in Dn × I starting with Dn × 0 and ending with the disk

Dn ×1∪Sn−1 × I, all the disks in the family having the same boundary, then we get

a homotopy from f to a map into A, stationary on Sn−1.

A map φ ∶ (X,A,x0) → (Y,B, y0) induces maps φ∗ ∶ πn(X,A,x0) → πn(Y,B, y0)

which are homomorphisms for all n ≥ 2 and have properties analogous to those in

the absolute case: (φψ)∗ = φ∗ψ∗, Id∗ = Id, and φ∗ = ψ∗ if φ ≃ ψ through maps

(X,A,x0)→ (Y,B, y0).

The most useful property of the relative groups πn(X,A,x0) is that they fit into

a long exact sequence

⋅ ⋅ ⋅→ πn(A,x0)i∗Ð→ πn(X,x0)

j∗Ð→ πn(X,A,x0)∂Ð→ πn−1(A,x0)→ ⋅ ⋅ ⋅→ π0(X,x0)

Here i and j are the inclusion (A,x0) (X,x0) and (X,x0, x0) (X,A,x0).

The map ∂ is just the restrictions of the map (In, ∂In, Jn−1) → (X,A,x0) to In−1,

or by restricting maps (Dn, Sn−1, s0) → (X,A,x0) to Sn−1. The map ∂, is called the

boundary map, is a homnomorphism when n > 1.

Theorem 1. This Sequence is exact.

Near the end of the sequence, where group structures are not defined, exactness

still makes sense: The image of one map is the kernel of the next, those elements

mapping to the homotopy class of the constant map.

Next, we will state three important theorems (without proofs). The first one is:

1.3.1 Whitehead’s Theorem

Theorem 2. If a map f ∶ X → Y between connected CW complexes induces isomor-

phisms f∗ ∶ πn(X) → πn(Y ) for all n, then f is a homotopy equivalence. In case f is

the inclusion of a subcomplex X Y , the conclusion is stronger: X is a deformation

retract of Y .

1.3.2 Cellular Approximation Theorem

This is the second important theorem, which states that:

Theorem 3. Every map f ∶X → Y of CW complexes is homotopic to a cellular map.

If f is already cellular on a subcomplex A ⊂ X, the homotopy may be taken to be

stationary on A.

For maps between CW complexes it turns out to be sufficient for many purposes in

homotopy theory to require just that cells map to cells of the same or lower dimension.

such a map f ∶X → Y , satisfying f(Xn) ⊂ Y n for all n, is called a cellular map. So,

from the cellular approximation theorem we can conclude that arbitrary maps can

always be deformed to be cellular.

Corollary 4. πn(Sk) = 0 for n < k.

Proof. Is Sn and Sk are given their usual CW structures, with the 0−cells as base-

points, then every basepoint-preserving map Sn → Sk can be homotoped, fixing the

basepoint, to be cellular, and hence constant if n < k. ∎

1.3.3 Freudenthal suspension theorem

Theorem 4. The map πi(Sn) → πi+1(Sn+1) is an isomorphism for i < 2n − 1 and a

surjection for i = 2n−1. More generally this holds for the suspension πi(X)→ πi+1SX

whenever X is an (n − 1)-connected CW complex.

Corollary 5. πn(Sn) ≈ Z, generated by the identity map, for all n ≥ 1. In particular,

the degree map πn(Sn)→ Z is an isomorphism.

1.3.4 Fiber Bundles

We know that a ‘short exact sequence of spaces’ A X X/A gives rise to a long

exact sequence of homology groups, but not to a long exact sequence of homotopy

groups as because of failure of excision. However, there is a different sort of ‘short

exact sequence of spaces’ that does give a long exact sequence of homotopy groups.

This sort of short exact sequence F → EpÐ→ B, called a fiber bundle, is prominent

from the type A X → X/A in that it has more homogeneity: All the subspace

p−1(b) ⊂ E, which are called fibers, are homeomorphic. For example, E could be the

product F ×B with p ∶ E → B the projection. General fiber bundles can be thought

of as twisted products. Well known examples are the Mobius band, which is a twisted

annulus with line segments as fibers, and the Klein bottle, which is a twisted torus

with circles as fibers.

The topological homogeneity of all the fibers of a fiber bundle is rather like the

algebraic homogeneity in a short exact sequence of groups 0 → K → GpÐ→ H → 0

where the ‘fibers’ p−1(h) are the cosets of K in G. In a few fiber bundles F → E → B

the space E is actually a group, F is a subgroup(through seldom a normal subgroup),

and B is the space of left or right cosets. One of the nicest such examples is the Hoff

bundle S1 → S3 → S2 where S3 is the group of quaternions of unit norm and S1 is

the subgroup of unit complex numbers. For this bundle, the long exact sequence of

homotopy groups takes the form:

⋅ ⋅ ⋅→ πi(S1)→ πi(S3)→ πi(S2)→ πi−1(S1)→ πi−1(S3)→ . . .

The exact sequence gives an isomorphism π2(S2) ≈ π1(S1) since the two adjacent

terms π2(S3) and π1(S3) are zero by cellular approximation(see Theorem 3 above).

Thus we have a direct homotopy-theoretic proof that π2(S2) ≈ Z. Also, since πi(S1) =

0 for i > 1 by Proposition 16, the exact sequence implies that there are isomorphisms

πi(S3) ≈ πi(S2) for all i ≥ 3, so in particular π3(S2) ≈ π3(S3), and by Corollary 5 the

latter group is Z.

After these preliminary remarks, let us begin by defining the property that leads

to a long exact sequence of homotopy groups. A map p ∶ E → B is said to have

the homotopy lifting property with respect to a space X if, given a homotopy

gt ∶X → B and a map g0 ∶X → E lifting g0, so pg0 = g0, then ∃ a homotopy gt ∶X → E

lifting gt. This can be seen as a special case of the lift extension property for a pair

(Z,A), which states that every map Z → B has a lift Z → E extending a given lift

defined on the subspace A ⊂ Z. The case (Z,A) = (X × I,X × 0) is the homotopy

lifting property.

Long Exact Sequence of Homotopy Groups

A fibration is a map p ∶ E → B having the homotopy lifting property w.r.t all spaces

X. For example, B × F → B is a fibration since one can always choose lifts of the

form gt(x) = (gt(x), h(x)) where g0(x) = (g0(x), h(x)).

Theorem 5. Suppose p ∶ E → B has the homotopy lifting property with respect to

disks Dk for all k ≥ 0. Choose basepoints b0 ∈ B and x0 ∈ F = p−1(b0). Then the

map p∗ ∶ πn(E,F,x0) → πn(B, b0) is an isomorphism for all n ≥ 1. Hence if B is

path-connected, there is a long exact sequence

⋅ ⋅ ⋅→ πn(F,x0)→ πn(E,x0)p∗Ð→ πn(B, b0)→ πn−1(F,x0)→ ⋅ ⋅ ⋅→ π0(E,x0)→ 0

The proof actually use a relative form of the homotopy lifting property. The map

p ∶ E → B is said to have the homotopy lifting property for a pair (X,A) if each

homotopy ft ∶X → B lifts to a homotopy gt ∶X → E starting with a given lift g0 and

extending a given lift gt ∶ A → E. In other words, the homotopy lifting property for

(X,A) is the lift extension property for (X × I,X × 0 ∪A × I).

Since the pairs (Dk×I,Dk×0) and (Dk×I,Dk×0∪∂Dk×I) are homeomorphic,

we see that the homotopy lifting property for Dk is equivalent to the homotopy lifting

property for (Dk, ∂Dk) . From this we can conclude that the homotopy lifting property

for disks is equivalent to the homotopy lifting property for all CW pairs (X,A). For

induction over the skeleta of X it suffices to construct a lifting gt one cell of X −A

at a time. Composing with the characteristic map Φ ∶ Dk → X of a cell then gives a

reduction to the case (X,A) = (Dk, ∂Dk). A map p ∶ E → B satisfying the homotopy

lifting property for disks is known as Serre fibration.

A fiber bundle structure on a space E, with fiber F , consists of a projection

map p ∶ E → B such that each point of B has a neighborhood U for which there is

a homeomorphism h ∶ p−1(U) → U × F making the diagram (below) commute, where

the unlabeled map is projection onto the first factor.

p−1(U) U × F

U

p

h

Commutativity of the diagram means that h carries each fiber Fb = p−1(b) homeo-

morphically onto the copy b×F of F . Thus the fibers Fb are arranged locally as in

the product B × F , though not necessarily globally. An h as above is called a local

trivialization of the bundle. Since the first coordinate of h is just p, h, is determined

by its second coordinate, a map p−1(U)→ F which is a homeomorphism on each fiber

Fb.

The fiber bundle structure is determined by the projection map p ∶ E → B, but to

indicate what the fiber is we sometimes write a fiber bundle as F → E → B, a ’short

exact sequence of spaces’. The space B is called the base space of the bundle, and

E is called the total space.

Example: A fiber bundle in which we have a discrete fiber space is a covering

space. Conversely, a covering space in which the cardinality of all the fibers are same,

for example a covering space over a connected base space, is a fiber bundle with

discrete fiber.

Proposition 19. A fiber bundle F → EpÐ→ B has the homotopy extension property

for disks.

1.4 Homogeneous spaces

This section is based on the author’s understanding of Chapters 3, exercises, problem

section D, pp 99, from the book ’Homotopy Theory’ by Sze-Tsen Hu [31]. Here the

exercises in section D are solved by the author and hence converted into propositions.

Let E be a topological group and let F be a closed subgroup of E. Define an equiva-

lence relation in E as follows: two elements a, b of E are said to be equivalent iff there

is an element † ∈ F such that a† = b. Thus the elements of E are divided into disjoint

equivalence classes called the left cosets of F in E. The left coset containing a ∈ E

is obviously the closed set aF of E. And hence we obtain a quotient space B = E/F

whose elements are left cosets of F in E and a natural projection

p ∶ E → B

which maps a ∈ E onto the left coset aF ∈ B.B = E/F is called the quotient space of

E by F ; this B will be called simply a homogeneous space.

Here are some of the important properties of the homogeneous space.

Proposition 20. B = E/F is a Hausdorff space.

Proof. As topological groups are automatically Hausdorff by definition (one can use

the property that X is Hausdorff iff ∆X = (x,x) is closed, by considering the map

G ×G → G that takes (x, y) → xy−1), Hence E is Hausdorff. Now, It just follows by

simple computation that E/F is Hausdorff. ∎

Proposition 21. The natural projection p is an open map.

Proof. Note that p−1(p(U)) = ∪g∈GUg = ∪u∈UuF hence p−1(p(U)) is open whenever U

is open. But then by the definition of quotient map p(U) is open and we are done! ∎

Proposition 22. E is a fiber bundle over B relative to p iff there is a local cross-

section of B in E; by this we mean a cross section κ ∶ V → E defined on an open

neighborhood V of the point b0 in B.

Proof. one way is clear! that is if E is a fiber bundle over B then there exists a

local cross section κ ∶ V → E by just defining κ(x) = z0 (a constant section) where

z0 ∈ p−1(x) ≈ F . To prove the other way, If one is given a local cross-section of B in E

say κ ∶ V → p−1(V ) then first we have to show that any local cross section of B in E

is a ’g’ translate of this given one. consider the following commutative diagram:

V p−1(V )

g(V ) p−1(g(V )) = gp−1(V )

κ

Lg Lg

κg

So, κg(x) = gκ(g−1x) which takes a ’g’ translate of V to a ’g’ translate of the fiber

in E. Now, to prove that that E is a fiber bundle over the base space B with fiber F,

we have to find a homeomorphism from p−1(U) to U × F , so consider the maps

φ ∶ p−1(U)→ U × F

defined as φ(x) = (p(x), µ(p(x))−1.x) and

φ−1 ∶ U × F → p−1(U)

defined as φ−1(b, f) = µ(b)f where µ is a local cross section. Now it is easy to see that

this both are inverses of each other and we obtain a homeomorphism. This completes

the proof of Proposition 22. ∎

1.5 Group Representation

This section is based on the author’s understanding from Wikipedia.

Definition: A representation of a group G on a vector space V over a field K is

a group homomorphism from G to GL(V), the general linear group on V. That is, a

representation is a map

µ ∶ G→ GL(V )

such that

µ(g1g2) = µ(g1)µ(g2),∀g1, g2 ∈ G

Here V is known as the representation space & the dimension of V is known as

the dimension of the representation.

Suppose V is of finite dimension n it is common to choose a basis for V and identify

GL(V ) with GL(n,K), the group of n × n invertible matrices over the field K.

∗ If G is a topological group and V is a topological vector space, a representa-

tion(continuous) of G on V is a representation µ such that φ ∶ G × V → V

given by φ(g, v) = µ(g)(v) is continuous.

∗ The Ker µ is given by:

kerµ = g ∈ G∣µ(g) = Id

A faithful representation is one in which the homomorphism G → GL(V ) is

injective; in other words, one whose ker = e.

∗ For any two vector spaces V and W over some field K, µ ∶ G → GL(V ) and

π ∶ G → GL(W ) are said to be equivalent if ∃ a vector space isomorphism

α ∶ V →W such that ∀ g ∈ G,

α µ(g) α−1 = π(g)

Example:Consider the complex number u = exp(2πi/3) which has the property

u3 = 1. The cyclic group C3 = 1, u, u2 has a representation µ on C2 given by:

µ(1) =⎡⎢⎢⎢⎢⎢⎣

1 0

0 1

⎤⎥⎥⎥⎥⎥⎦

µ(u) =⎡⎢⎢⎢⎢⎢⎣

1 0

0 u

⎤⎥⎥⎥⎥⎥⎦

µ(u2) =⎡⎢⎢⎢⎢⎢⎣

1 0

0 u2

⎤⎥⎥⎥⎥⎥⎦This representation is faithful because µ is a one-to-one map.

1.6 Schreier-Reidemeister Method

This section is based on the author’s understanding of the book [21], chapter 2, pp.

86-103. The Schreier-Reidemeister Method yields a presentation for a subgroup H of

a group G when H is of finite index in G and G is finitely generated.

We will just sketch this method and will not prove it. More detailed explanation

and proofs can be found from the book [21].

Schreier Set of Generators: For any group G, a generating set S is said to be

a Schreier set of generators if given any s ∈ S, any initial segment of s also belongs to

S.

Let G has presentation:

< a1, a2, . . . , an∣R1 = 1,R2 = 1, . . . ,Rm = 1 >

Schreier Set of Coset representative: Let us denote it by Λ, for any subgroup

H < G, we have a Schreier set of coset representative for G/H(whose proof can be

found in Magnus, Karrass and Solitar)

Now we will construct the generating set for H.

Define Sλ,a as :

Sλ,a = (λa)(λa)−1∣λ ∈ Λ, a ∈ a1, a2, . . . , an

where

¯∶ G→ Λ

such that (λa) = representative of λaH in Λ.

Now SΛ,a generates H. (proof can be found in chapter 2, section 2.3, [21]).

Now we will obtain relations for the subgroup H. A Rewriting Process φ is

defined as follows:

Take the ith relation from the group G say Ri, let us suppose that

Ri = aε1j1aε2j2. . . aεkjk , (εi = ±1)

Case I: Look at the ith letter aji , if its index εi = +1 then the ith letter of the

corresponding relator of H will be Sλi,aji where

λi = aε1j1aε2j2. . . aεi−1ji−1

Case II: If its index εi = −1, then letter of the corresponding relator of H will be

S−1λi,aji

where

λi = aε1j1aε2j2. . . aεiji

Then the corresponding relation of H will be

Ri = Sε1λ1,aj1Sε2λ2,aj2

. . . Sεkλk,ajk= 1

Also, there are few more relations for the subgroup H of G, that are defined as

follows: for every λ ∈ Λ, consider λRiλ−1,∀Ri = 1, and rewrite them using the above

rewriting process as above.

φ(λRiλ−1) = 1

will be the set of all other relators of H for all i and λ ∈ Λ

Hence, we obtain a set of generators and relations for H in G

Chapter 2

Braid Groups

This chapter is based on the author’s understanding of the book [1]. The main theme

in this chapter is the concept of a braid group, and the many ways that the notion of

a braid has been important in low dimensional topology. Here, we will be interested

mostly in the possibility of applying braid theory to the study of surface mappings.

Our objective in this chapter will be to develop the main structural and algebraic

properties of braid groups on connected manifolds. (We will limit our braid groups to

groups of motions of points; No further generalizations of motions of a sub-manifold of

dimension > 0 in a manifold will not be treated.) In this setting the “classical” braid

groups Bn defined at first by Artin appears as the “full braid group of the Euclidean

plane E2.”

Section 2.1 is deals with definitions. The problem of how to nicely define a braid, in

order to capture the important properties of “weaving patterns” and so to study them

mathematically, is a very basic one. It is a tribute to Artin’s extraordinary insight

as a Mathematician that the definition he gave in 1925 (see [4]) for equivalence of

geometric braids could ultimately be broadened and generalized in many different

directions without destroying the main features of the theory. A discussion of several

such generalizations can be seen in section 2.1; for generalization to higher dimensions

see [5] and [6]; for other generalizations, see [7] and [8].

Section 2.2 contains a development of the main properties of “configuration spaces,”

which were introduced by E. Fadell and L. Neuwirth in late 1960’s (see [9]). We will

use configuration spaces as a tool for finding defining relations in the braid groups of

surfaces. Here we followed the same method of approach as done by Birman, because

31

it gives a particular geometric insight into the algebraic structure of the classical braid

group as a sequence of semi-direct products of free groups. This same structure is

displayed by other methods in [Magnus 1934; Markoff 1945; Chow 1948] see [10], [11]

and [12] respectively.

Then in Section 2.3, we review the most important properties of braid groups

on manifolds other then E2 and S2. In Theorem 8 we will see that braid groups

of manifolds M of dimension n > 2 are not of very much interest. Theorems 9 and

10 establishes the relationships between Artin’s classical braid groups on E2 and the

braid groups of other closed 2-manifolds.

In section 2.4, we study the braid group of E2. In Theorem 11, we find generators

and defining relations for full braid group Bn of E2. Corollaries 6 and 7 relate to the

algebraic structure of Bn as a sequence of semi-direct products of free groups, and lead

to solutions of the ”word-problem” in Bn. In corollary 8, we establish that Bn has a

faithful representation as a subgroup of the automorphism group of a free group. This

subgroup is characterized in Theorem 12, by giving necessary and sufficient conditions

for an automorphism of a free group of rank n to be in Bn. Corollaries 9 identifies

Z(Bn). Finally, in Theorem 13 we establish another interpretation of Bn as the group

of topologically-induced automorphisms of the fundamental group of an n-punctured

disc, where admissible maps are required to keep the boundary of the fixed point wise,

this is the main motivational result of this thesis, that is M(Dn) = Bn.

Section 2.5 discusses the braid group of S2, which will play an important role

later, in relation to the theory of surface mappings. In Section 2.6, we give a list of

references for further results on braid groups of closed 2-manifolds.

In Section 2.7, we see some examples where braiding appears in mathematics,

unexpectively.

From now onwards, all the spaces will be assumed to be ‘nice’ that is they are

path-connected, locally path-connected, semi locally simply connected.

2.1 Definitions

Lets begin not with the classical braid group, but with a somewhat more general

concept of a braid as a motion of points in a manifold. And then the definition

will be shown to reduce to the classical case when one takes the manifold to be the

Euclidean plane E2.

Let M be a manifold of dimension ≥ 2. let∏ni=1M denote the n-fold product space,

and let F0,nM denote the subspace

F0,nM = (z1, . . . , zn) ∈n

∏i=1

M ∣zi ≠ zj, i ≠ j (2.1)

The meaning of the subscript “0” in the symbol F0,n will become clear later(when we

define configuration space). The fundamental group π1F0,nM of the space F0,nM is

the pure(or unpermuted) braid group with n strings of the manifold M .

Two points z and z′ of F0,nM are said to be equivalent if the coordinates (z1, z2, . . . , zn)

of z differ from the coordinates (z′1, z′2, . . . , z′n) of z′ by a permutation. Let B0,nM de-

note the quotient space of F0,nM under this equivalence relation. The fundamental

group π1B0,nM of the space B0,nM is called the full braid group of M , or more

simply, the braid group of M . By Proposition 12, it follows that the action of Σn,

the permutation group on n symbol on the space F0,nM is properly discontinuous, and

hence by Proposition 11,(a) the natural quotient projection map φ ∶ F0,nM → B0,nM

is a regular covering projection.

The classical braid group of Artin (see [4] and [13]) is the braid group π1B0,nE2,

where E2 denotes the Euclidean plane. Artin’s geometric definition of π1B0,nE2 can

be recovered from the definition above as follows:

Choose a base point z0 = (z01 , . . . , z

0n) ∈ F0,nE2 and a point z0 ∈ B0,nE2 such that

p(z0) = z0. Any element in π1B0,nE2 = π1(B0,nE2, z0) is represented by a loop

α ∶ I → B0,nE2,where α(0) = α(1) = z0

, which lifts uniquely to a path

α ∶ I → F0,nE2starting at α(0) = z0

If α(t) = (α1(t), α2(t), . . . , αn(t)), t ∈ I, then each of the coordinate functions αi

defines (via its graph) an arc ℘i = (αi(t), t) in E2 × I. Since α(t) ∈ F0,nE2 the arcs

℘1, . . . ,℘n (as at any particular time t, any two αi can’t be different) are disjoint.

Their union ℘ = ℘1 ∪ . . .℘n is called a geometric braid. The arcs ℘i is called the ith

braid string. This figure is taken from pp 6 from Birman’s book [1].

A geometric braid is a representative of a path class in the fundamental group

π1B0,nE2. Thus, if ℘ and ℘′ are geometric braids, then ℘ ∼ ℘′(i.e. they represent

the same element of π1B0,nE2) if the paths α and α′ which defines these braids are

homotopic relative to the base point (z01 , . . . , z

0n) in the space F0,nE2. Thus we require

the existence of a continuous mapping

F ∶ I × I → F0,nE2

with

F (t,0) = (F1(t,0), . . . , Fn(t,0)) = (α1(t), . . . , αn(t))

F (t,1) = (F1(t,1), . . . , Fn(t,1)) = (α1′(t), . . . , αn′(t))

F (0, s) = (F1(0, s), . . . , Fn(0, s)) = (z01 , . . . , z

0n)

F (1, s) = (F1(1, s), . . . , Fn(1, s)) = (z0µ1 , . . . , z

0µn)

where (µ1, . . . , µn) is a permutation of the array (1, . . . , n). The homotopy F

defines a continuous sequence of geometric braids

℘(s) = ℘1(s) ∪ . . .℘n(s), s ∈ I

where

℘i(s) = (Fi(t, s), t) such that ℘(0) = ℘ and ℘(1) = ℘′

The figures below(taken from Internet), are pictures of geometric braids which are

equivalent to the “trivial” braid and braids that generates the braid group.(which we

will prove latter)

There are various stronger and weaker forms of equivalence between geometric

braids defined by many mathematicians, and we mention several of these briefly:

i). Let ℘ and ℘′ be geometric braids. Note that ℘,℘′ ⊆ E2 × I. Then, we write

℘ ≈ ℘′ if there is an isotopic deformation Gs of E2×I which is the identity on E2×0

and on E2 × 1 for each s ∈ [0,1] and which has the property:

For each s ∈ [0,1], the image set ℘(s) of ℘ under Gs is a geometric braid, that

is, ℘(s) meets each plane E2 × t0, t0 ∈ I, in precisely n points, and moreover

℘(0) = ℘, ℘(1) = ℘′

It was proved by Artin (see [13]) that ℘ ≈ ℘′ iff ℘ ∼ ℘′. Thus a braid homotopy

may always be “extended” to E2 × I, in the sense defined above.

ii). If we think of our braids strings ℘1,℘2, . . . ,℘n as being made of elastic, one

might imagine a more general type of equivalence in which the strings could be

stretched or deformed in the region E2 × I without requiring that ℘(s) meet each

plane E2 × t0, t0 ∈ I, in precisely n points. In this situation, it might happen, for

example that some intermediate set ℘1(s0) ∪ ⋅ ⋅ ⋅ ∪ ℘n(s0) is as illustrated in figure

below:

(Note that this intermediate set is not a geometric braid.) More precisely, under

this more general notion, ℘ ≡ ℘′ if there is an isotopy Gs which is exactly like that

defined in i) above except that ℘s need not satisfy the property (). Again, Artin

established [13] that ℘ ≡ ℘′ iff ℘ ∼ ℘′.(This is the first hint of a relationship between

the concepts of equivalence of braids and equivalence of links)

iii). D.Goldsmith [6] has defined a concept of “homotopy” of braids by defining two

geometric braids to be homotopic if one can be deformed to the other by simultaneous

homotopies of the individual paths (αi(t), t) in E2 × I, fixing the end points, and

subject to the restriction that a string may intersect itself, but not any other string.

Note that if ℘ ∼ ℘′ then ℘ and ℘′ are also equivalent under Goldsmith’s rule, but the

converse need not be true. In fact, Goldsmith has exhibited non-trivial elements of the

group π1B0,3E2 which are homotopic to the identity element of π1B0,3E2. She goes on

to define a ”homotopy braid group,” Bn, and finds a group presentation for Bn which

exhibits Bn as a quotient group of the group π1B0,nE2. We note that Goldsmith’s

results were suggested by J.Milnor’s work on homotopy of links and isotopy of links

see [14].

iv) The concept of a braid group has been generalized by D.M Dahm [5] and by

D. Goldsmith [6] to a group of motions of a submanifold in a manifold. We now give

Goldsmith’s definition of that group. Let N be a subspace contained in the interior

of a manifold M. Denote by H(M) the group of autohomeomorphisms of M with

the compact open topology, where if M has boundary ∂M , all homeomorphisms are

required to fix ∂M pointwise. Denote the identity map of M by 1M ∶ M → M . A

motion of N in M is a path αt, in H beginning at α0 = 1M and ending at α1, where

α1(N) = N . The motion is said to be stationary motion of N in M if αt(N) = N

for all t ∈ [0,1]. To compose two motions, translate the second by multiplication in

the group H(M) so that its initial point coincides with the endpoint of the first, and

multiply as in the groupoid of paths. Now define the inverse f−1 of a motion f to be

the inverse of the path f in H(M), translated so that its initial point is 1M .

Eventually, let motions f and g be equivalent if the path f−1g is homotopic relative

to its endpoint to a stationary motion. The group of motions of N in M is the set of

equivalence classes of motions of N in M, with multiplication induced by composition

of motions. From this point of view, the group of motions of an interior point in a

manifold M is the group π1M , and the group of motions of n distinct points is the

pure braid group of M .

Dahm[5] studies the group of motions of n disjoint circles in S3, and Goldsmith

[6], studies the group of motions of torus links in S3.

We will now see the geometric interpretation of the product of two braids is imme-

diate; which suffice from Figure 1, the configuration of arcs between any two consecu-

tive dotted horizontal levels can be considered to be geometric braids (Ω1,Ω2,Ω3,Ω4)

of which the entire braids Ω = Ω1Ω2Ω3Ω4 is the product.

Geometric intuition suggests that an arbitrary braid is equivalent to a braid that

is a product of simples braids of the type illustrated in Figure 2. The equivalence

classes of these elementary braids will be denoted by the symbol σi and σ−1i . In the

example of Figure 1,

Ω = σ1σ22σ

−13

Intuitively, σ1, σ2, . . . , σn−1 generate the group π1B0,nE2, a fact which will be

proved later.

The following relations in π1B0,nE2 are obvious from Figure 2:

σiσj = σjσi if ∣i − j∣ ≥ 2; 1 ≤ i, j ≤ n − 1 (2.2)

σiσi+1σi = σi+1σiσi+1 1 ≤ i ≤ n − 2 (2.3)

It will be proved below that (2.2) and (2.3) comprise a set of defining relations in

π1B0,nE2. Our proof, which allows us at the same time to compute defining relations

for the braid groups of arbitrary 2-manifolds, will make use of the concept of the

“configuration space” of a manifold. Other proofs(for the special case M = E2) can

be found in [4], [13], [10], [15] and [16].

2.2 Configuration spaces

Proposition 23. The natural projection map φ ∶ F0,nM → B0,nM is a regular covering

space projection. The group of covering transformations is the full symmetric group

Σn on n letters. Therefore there is a canonical isomorphism

π1B0,nM/π1F0,nM ≈ Σn (2.4)

Proof. It suffices to show that the action of the permutation group on F0,nM is a

covering space action, which follows directly from Proposition 12. And the second

part of the proof follows from the Proposition 11. ∎

Since the map φ is known explicitly, it follows from Proposition 23 that it is not

difficult to analyze π1B0,nM once π1F0,nM is known. Hence, the remainder of the

section will be concentrated to the group π1F0,nM .

Let Qm = q1, . . . , qm be a set of distinguished points of M . The configuration

space Fm,nM of M is defined as the space F0,n(M −Qm). Note that the topological

type of Fm,nM does not depend on the choice of the particular points Qm, since it

is possible always find an isotopy of M which deforms any one such point set Qm

into any other Q′

m. Note that Fm,1M = M −Qm.(One may, similarly, define spaces

Bm,nM = B0,n(M −Qm), however we will only be interested in B0,nM .)

There is a keen relationship between the configuration spaces Fm,nM and F0,nM .

The key observation is the following theorem:

Theorem 6 (Fadell and Neuwirth, 1962). . Let π ∶ Fm,nM → Fm,rM be defined by

π(z1, . . . , zn) = (z1, . . . , zr), 1 ≤ r < n. (2.5)

Then π exhibits Fm,nM as a locally trivial fibre space over the base space Fm,rM ,

with fibre Fm+r,n−rM .

Proof. let us first consider, for some base point (z01 , . . . , z

0r) in Fm,rM , the fibre

π−1(z01 , . . . , z

0r):

π−1(z01 , . . . , z

0r) = (z0

1 , . . . , z0r , yr+1, . . . , yn),where z0

1 , . . . , z0r , yr+1, . . . , yn

are distinct points in M −Qm . (2.6)

define Qm+r = Qm ∪ z01 , . . . , z

0r, then

Fm+r,n−rM = (yr+1, . . . , yn)∣yi ≠ yj∀i ≠ j, where yi ∈M −Qm , (2.7)

and there is a natural homeomorphism

h:Fm+r,n−r(M)Ð→ π−1(z01 , . . . , z

0r) (2.8)

defined by h(yr+1, . . . , yn) = (z01 , . . . , z

0r , yr+1, . . . , yn). Now we will only proof the local

triviality of π, only for the case when r = 1 for notational and descriptive convenience.

Similarly, the other cases can be carried out. Fix a point x0 ∈ M − Qm = Fm,1M =

Fm,rM . And add another point qm+1 to the set Qm to form Qm+1 and then pick a

homeomorphism α ∶ M → M , fixed on Qm, such that α(qm+1) = x0. Let U denote

a neighborhood of x0 in M −Qm which is homeomorphic to an open ball, and let U

denote the closure of U. Define a map θ ∶ U × U Ð→ U with the following properties.

Setting θz(y) = θ(z, y) we require:

i) θ ∶ U → U is a homeomorphism which fixes ∂U .

ii) θz(z) = x0.

Such a choice is always possible as there exists a homeomorphism from a closed

unit disk Dn in Rn for n ≥ 2 to a closed unit disk Dn in Rn which takes a point a

to b in Dn keeping the boundary fixed. By i), θ can be extended to θ ∶ U ×M → M

defining θ(z, y) = y for y ∉ U . The required local product maps

U × Fm+1,n−1M π−1(U)φ

φ−1

is given by

φ(z, z2, . . . , zn) = (z, θ−1z α(z2), . . . , θ−1

z α(zn)) (2.9)

φ−1(z, z2, . . . , zn) = (z,α−1θz(z2), . . . , α−1θz(zn)) (2.10)

∎

Two important consequences of Theorem 6 now follow:

Proposition 24. If π2(M−Qm) = π3(M−Qm) = 0 for each m ≥ 0, then π2(F0,nM) = 0

Proof. The exact homotopy sequence of the fibration π ∶ Fm,nM → Fm,1M =M −Qm

of Theorem 6, has homotopy extension property(HEP) for disks by Proposition 19

and hence gives an long exact sequence of homotopy groups, by Theorem 5

⋅ ⋅ ⋅→ π3(M −Qm)→ π2Fm+1,n−1M → π2Fm,nM → π2(M −Qm)→ . . .

Since π2(M −Qm) = π3(M −Qm) = 0, it follows that π2Fm+1,n−1M and π2Fm,nM are

isomorphic. An inductive argument shows that

π2F0,nM ≈ π2F1,n−1M ≈ π2F2,n−2M ⋅ ⋅ ⋅ ≈ π2Fn−1,1M = π2(M −Qn−1) = 0 (2.11)

This completes the proof. ∎

Let π be the projection map from F0,nM to F0,n−1M defined by (2.5). Let

(z01 , . . . , z

0n) be the base point for π1F0,nM . Let Fn−1,1M =M−Qn−1 =M−z0

1 , . . . , z0n−1.

Define j to be the inclusion map from Fn−1,1M to F0,nM as

j(zn) = (z01 , . . . , z

0n−1, zn) zn ∈M − z0

1 , . . . , z0n−1 (2.12)

Theorem 7. If π2(M −Qm) = π3(M −Qm) = π0(M −Qm) = 1 for every m ≥ 0, then

the following sequence of groups and homomorphism is exact:

1→ π1(Fn−1,1M,z0) j∗Ð→ π1(F0,nM, (z01 , . . . , z

0n))

π∗Ð→ π1(F0,n−1M, (z01 , . . . , z

0n−1))→ 1

(2.13)

where π∗ and j∗ are the homomorphism induced by the mappings π and j.

Proof. As p ∶ F0,nM → F0,n−1M is a fiber bundle over the base space F0,n−1M with

fiber Fn−1,1M . And hence by Proposition 19, it has homotopy extension property for

disks. Now, we can use Theorem 5, to obtain a long exact homotopy sequence of

fibration. The identity terms in the sequence (2.13) reflect the equalities π2F0,n−1M =

1, established in Proposition 21, and π0Fn−1,1M = π0(M − Qn−1) = 1. Hence, The

sequence (2.13) is exact. ∎

The exact sequence (2.13) will be used later, in conjugation with Proposition 23,

to determine group presentation for π1B0,nE2 and π1B0,nS2.

2.3 Braid groups of manifolds

The following Theorem indicates that the most interesting braid groups are those on

2-dimensional manifolds:

Theorem 8. (Birman, 1969a,pp. 42-44). Let M be a closed, smooth manifold of

dimension n. Then for each integer k the inclusion map ik ∶ F0,nM → ∏nM induces

a homomorphism

(ik)∗ ∶ πkF0,nM →∏n

πkM (2.14)

which is surjective if dim M > k and also injective if dim M > 1 + k

Proof. We will not prove this theorem. The proof can be obtain from [18] or [5]. ∎

Among the braid groups on 2−dimensional manifolds, Artin’s classical braid group

π1B0,nE2 and Artin’s pure braid group π1F0,nE2 hold central positions. This state-

ment is justified by the remarks which follow by Theorem 9 and 10 below, which are

based on material in [18] and in [20].

Choose (z01 , . . . , z

0n) be the base point for the group π1Fm,nE2, as before, and

regard E2 as an open disc in M which contains the n points z01 , . . . , z

0n and also the

distinguished set Qm. Let Pn = (p1, . . . , pn) be a fixed n−point set for each n. Then

we may view Fm,nM as the set of embeddings of Pn in (M −Qm). From this point of

view, the space Fm,nE2 may be identified with a subset of Fm,nM by composing any

map from Pn to Fm,nE2 with the inclusion map E2 ⊆M . Let em,n ∶ Fm,nE2 → Fm,nM

be the required identification. Then the induced map e∗m,n ∶ π1Fm,nE2 → π1Fm,nM

takes any n−string braid on (E2 −Qm) and considers it as a braid in (M −Qm).

Theorem 9. If M is any compact surface except S2 or P 2, then ker e∗0,n = 1.

Proof. ([18]; see also Goldberg, 1973 [20] for a different proof). The homomorphism

e∗m,n together with the exact sequences of Theorem 7, yield a commutative diagram:

1 π1(E2 −Qn−1) π1F0,nE2 π1F0,n−1E2 1

1 π1(M −Qn−1) π1F0,nM π1F0,n−1M 1

e∗n−1,1 e∗0,n e∗0,n−1

Note that e∗n−1,1 is injective for each n(using Van-kampan Theorem, just think

E2 −Qn−1 as D2 −Qn−1). Now we can use the diagram inductively to prove that e∗0,n

is injective as well. To be precise, e∗0,1 is injective since π1F0,1E2 = π1E2 = 1. This

begins the induction. Suppose inductively that e∗0,n−1 is injective. Then the strong

5-lemma implies that e∗0,n is injective. This completes the induction and the proof of

the proposition.

Five Lemma: In a commutative diagram of abelian groups as below, if the two

rows are exact and α,β, δ and ε are isomorphism, then γ is an isomorphism as well.

A B C D E

A′ B′ C ′ D′ E′

i

α

j

β γ

k l

δ ε

i′ j′ k′ l′

∎

Focusing on the braid group π1F0,nM of a compact 2−manifold M , one readily

recognize two distinct types of phenomena which are displayed by representatives of

the elements of the group π1F0,nM .

i) There is “classical braiding,” which may be thought of as taking place in the

open disc E2 ⊂M .

ii) There is a wander of the individual strands about on the surface M . The next

theorem(which we will state without proof) says that, in effect for a closed surface

M ≠ S2 or P 2, nothing else happens.

Theorem 10. (Goldberg,1973). Let M be a closed surface different from S2 or P 2.

Let i ∶ F0,nM → ∏nM be the inclusion map. Then in the following sequence of (not

necessarily abelian) groups

1→ π1F0,nE2e∗0,nÐÐ→ π1F0,nM

i∗Ð→n

∏i=1

π1M → 1

the kernel of each homomorphism is equal to the normal closure of the image of the

previous homomorphism in the sequence.(the normal closure of a subset of a group is

the smallest normal subgroup that contains the subset).

Proof. The proof can be found on the Goldberg’s paper [20]. ∎

2.4 The braid group of the plane

From Theorem 8, 9 and 10, clearly, the groups π1B0,nE2 and π1F0,nE2 need special

attention. So, in this section we will consider only the case M = E2. And, we will

adopt abbreviations B0,n and F0,n for the spaces B0,nE2 and F0,nE2.

In this section the short exact sequence of Theorem 7 will be used inductively to

show that π1F0,n is constructed in nice ways from the building blocks π1(E2−Qi),1 ≤

i ≤ n − 1(Corollary 6), finding simultaneously generators and relations for the groups

π1B0,n and π1F0,n(Theorem 11 and Lemma 1). From the structure of the group

π1F0,n(uncovered in Corollary 6) a unique normal form will be developed, in Corollary

7, for elements in π1B0,n. This leads to a solution to the word problem in π1B0,n. In

Corollary 8 it will be proved that π1B0,n has a faithful representation as a group of

automorphisms of a free group of rank n. In Theorem 12 the particular subgroup

of the automorphism group of a free group which is so obtained is characterized

algebraically, Theorem 13 gives a new nice geometric meaning to the group π1B0,n,

which is the main motivation of this thesis.

Theorem 11. (Artin, 1925). The group π1B0,n admits a presentation with generators

σ1, . . . , σn−1 and defining relations

σiσj = σjσi if ∣i − j∣ ≥ 2,1 ≤ i, j ≤ n − 1.(2.15)

σiσi+1σi = σi+1σiσi+1 if 1 ≤ i ≤ n − 2. (2.16)

Proof. (The proof given here is due to Fadell and Van Buskirk, 1962 [17]). Let Bn

be the abstract group with the presentation of Theorem 11. Until we establish the

isomorphism between Bn and π1B0,n, we will use the symbols σ1, . . . , σn−1 for elements

of π1B0,n with τ ∶ Bn → π1B0,n defined by the pictures in figure 2. (Expecting the result

of Theorem 11, we used the symbols σ1 and σ−11 in Figure 2.) we now give an equivalent

definition which is more precise. Recall the covering projection φ ∶ F0,n → B0,n. Choose

the point φ((1,0), . . . , (n,0)) = z0 as base point for the group π1B0,n. Lift loops based

at φ((1,0), . . . , (n,0)) in B0,n to paths in F0,n with initial point ((1,0), . . . , (n,0)) = z0.

Then the generator σi ∈ π1B0,n is represented by the path l(t) in F0,n given by

l(t) = ((1,0), . . . , (i − 1,0), li(t), li+1(t), (i + 2,0), . . . , (n,0)), (2.17)

where li(t) = (i + t,−√t − t2) and li+1(t) = (i + 1 − t,

√t − t2). That is, l(t) is constant

on all but the ith and i + 1st strings and interchanging those two in a very nice way.

The proof of Theorem 11 will be by induction on n, and will use the relationship

already developed in Proposition 23 between π1B0,n and π1F0,n. Let

µ ∶ π1(B0,n, z0)→ Σn

be defined as follows: Let α ∈ π1B0,n be represented by a loop

β ∶ (I,0,1)→ (B0,n, z0)

and let β = (β1, . . . , βn) ∶ (I,0)→ (F0,n, z0) be the unique lift of β. Define

µ(α) =⎡⎢⎢⎢⎢⎢⎣

β1(0), . . . , βn(0)

β1(1), . . . , βn(1)

⎤⎥⎥⎥⎥⎥⎦∈ Σn (2.18)

The kernel of these homomorphism µ is the pure braid group π1F0,n. With respect

to the homomorphism µ is the homomorphism

µ ∶ Bn → Σn

from the abstract group Bn to the symmetric group Σn on n letters defined as follows:

µ(σi) = (i, i + 1) 1 ≤ i ≤ n − 1 (2.19)

Let Pn = Kerµ

Lemma 1. The homomorphism i ∶ Bn → π1B0,n is an isomorphism onto π1B0,n if i∣Pnis an isomorphism onto π1F0,n.

Proof. It is easy to see that the homomorphism µ is clearly surjective, since the

transpositions µ(σi)∣1 ≤ i ≤ n − 1 generate Σn. Therefore, we have a commutative

diagram as shown below:

1 Pn Bn Σn 1

1 π1F0,n π1B0,n Σn 1

in=i∣Pn i

µ

1

µ

with exact rows. Applying the “Five Lemma,” we obtain the desired result if we just

show that in = i∣Pn is an isomorphism. Hence, we complete the proof of Lemma 1. ∎

To show that i∣Pn is an isomorphism onto, we next find a presentation for Pn.

Lemma 2. The group Pn admits a presentation with generators

Aij = σj−1σj−2 . . . σi+1σi2σ−1

i+1 . . . σ−1j−2σ

−1j−1 (1 ≤ i < j ≤ n) (2.20)

and defining relations

A−1r,sAi,jAr,s =

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

Ai,j if r < s < i < j or i < r < s < j,

Ar,jAi,jA−1r,j if r < s = i < j,

Ai,jAs,jAi,jA−1s,jA

−1i,j if i = r < s < j,

Ar,jAs,jA−1r,jA

−1s,jAi,jAsjArjA

−1sjA

−1rj if r < i < s < j.

(The geometric braid Aij is shown below: figure taken from [3])

Proof. Note that [Bn ∶ Pn]= n!. We may choose as coset representatives for Pn in

Bn any set of n! words in the generators of Bn whose images under the mapping µ

range over all of Σn. Certainly, a set of right coset representatives are the collection

of all products of the form ∏nj=2Mj,kj ; j ≥ kj ≥ 1 where Mj,i = σj−1σj−2 . . . σiifj ≠ i,

or 1 if j = i. For example, coset representatives for P3 in B3 are the set M22M33 =

1,M22M32 = σ2,M22M31 = σ2σ1,M21M33 = σ1,M21M32 = σ1σ2, and M21M31 = σ1σ2σ1.

This is a Schreier Set(see section 1.6 for details); that is, any initial segment of a

coset representative is again a coset representative. Hence we may apply the Schreier-

Reidemeister method(see section 1.6 for the method) to obtain a group presentation

for Pn.

An alternate/different method of proving Lemma 2, which is conceptually more

complex than that described above, but mechanically easier to handle, is given by

Chow, 1948, see [12]. ∎

We will now complete the proof of Theorem 11. The group Pn−1 can be regarded as

the subgroup of Pn which is generated by Aij ∣1 ≤ i < j ≤ n − 1. Note that a natural

homomorphism η ∶ Pn → Pn−1 is obtained by the η(Aij) = Aij if 1 ≤ i < j ≤ n − 1,

while η(Ain) = 1,1 ≤ i < n. Thus Ker η is the normal closure in Pn of the elements

A1n,A2n, . . . ,An−1,n. Using the relation (2.20) we can show that in fact the subgroup

Un of Pn which is generated by A1n,A2n, . . . ,An−1,n is normal in Pn, hence Un = Ker

η.

Corresponding to the homomorphism η ∶ Pn → Pn−1 we have the homomorphism

π∗ ∶ π1F0,n → π1F0,n−1 of Theorem 7. By Theorem 7, we also know that kerπ∗ =

π1Fn−1,1 = π1(E2 −Qn−1), which is a free group of rank n − 1.

It is easy to see that the following diagram is commutative:

1 Un Pn Pn−1 1

1 π1Fn−1,1 π1F0,n π1F0,n−1 1

in∣Un in

η

in−1

π∗

with exact rows. In the bottom row(using the notation used in Theorem 7) the base

point for π1F0,n is (z01 , . . . , z

0n), so that z0

n is the base point for π1Fn−1,1 = π1(E2 −

z01 , . . . , z

0n−1). Now, from equation (2.20) and our picture definition of the geometric

braids σi = in(σn) one may identify the image in(Ajn) of the generator Ajn of Un

as being represented by a loop based at z0n which encircles the point z0

j once and

separates it from z01 , . . . , z

0j−1, z

0j+1, . . . , z

0n−1. Clearly the image set in(Ajn)∣1 ≤ j < n

is a free basis for the free group π1Fn−1,1. By the Hopfian property(A Group G is

said to have hopfian property if every epimorphism G → G is an isomorphism) for

finitely generated free groups, it then follows that Un must also be free and that in∣Unis an isomorphism onto. Now observe that P1 = 1 and π1F0,1 = 1. Therefore i1 is an

isomorphism. Assume inductively therefore that in−1 is an isomorphism. Then, since

in∣Un is an isomorphism for each n, in is an isomorphism by the five lemma. This

completes the proof of Theorem 11. ∎

Corollary 6. Pn = Un ⋊ Pn−1

Proof. Follows from the definition of Un (A homomorphism G → H that is identity

on H and whose kernel is N , then we write G = N ⋊H, that is G is semidirect product

of N and H). ∎

Definition: Theorem 11 gives an isomorphism i ∶ Bn → π1B0,n which takes the

abstract group Bn with the presentation of Theorem 11 onto the Artin braid group

π1B0,n of the plane E2. The two groups will now be identified and notation for the two

groups used interchangeably. Similarly, the group Pn will be identified with π1F0,n.

In particular, elements of Bn(respectively Pn) will be called the braids(respectively

pure braids) and Bn (respectively Pn) will be called the braid group(respectively pure

braid group) of the plane. The coset representatives for Pn in Bn which are defined

by equation (2,21) below is called the permutation braids. The relation (2,2) and

(2.3) are called the braid relations.

Corollary 7. Every element β ∈ Bn can be written uniquely in the form

β = β2β3 . . . βnπβ, (2.21)

where πβ is a permutation braid and each βj belongs to the free subgroup Uj defined

in the proof of Theorem 11.

Proof. Since the permutation braids form a complete set of coset representatives for

Pn in Bn, β = βnπβ for some βn ∈ Pn and permutation braid πβ. By corollary 6,

βn = βn−1βn, (n > 2) for some βn ∈ Un and βn−1 ∈ Pn−1. By induction,

β = β2β3 . . . βnπβ

where βi ∈ Ui, for i = 3, . . . , n and β2 ∈ P2 = U2. Let β2 = β2. This completes the proof

of existence. Clearly, πβ is unique. The uniqueness of each βi(i = 2, . . . , n) derives by

induction, from the properties of semi-direct products of groups.

Since each βi lies in a free group on known free generators, it is possible algo-

rithmically calculate standard representatives for β2, β3, . . . , βn, and πβ in the given

generators for Bn. This solves the word problem in Bn. ∎

The procedure for putting a braid word into the normal form (2.22) is called

“combing the braid”. Note that each entry βj in (2.22) is a product of the free

generators A1j, . . . ,Aj−1,j of the free group Uj. Artin discourages any attempt to

carry out this procedure of combing a braid, experimentally on a living person, fearing

that it would “only lead to violent protests and discrimination against mathematics”

(Artin, 1947, p. 126) see [13].

Corollary 8. The braid group Bn has a faithful representation as a group of (right)

automorphisms of a free group Fn =< x1, . . . , xn >, of rank n. The representation is

induced by a mapping ξ from Bn to AutFn defined by:

(σi)ξ ∶

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

xi → xixi+1x−1i

xi+1 → xi

xj → xj if j ≠ i, i + 1.

(2.22)

The restriction of ξ to the pure braid group Pn maps the generator Ars of Pn to the

automorphism:

(Ars)ξ ∶ xi →

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

xi if s < i or i < r,

xrxix−1r if s = i,

xixsxix−1s x

−1i if r = i,

xrxsx−1r x

−1s xixsxrx

−1s x

−1r if r < i < s.

(2.23)

Proof. To see that ξ induces a representation is simply a matter of checking that

relations are preserved.

In the usual way ξ induces a representation of Bn as a group of automorphisms of

the commutator factor group Fn = Fn/[Fn, Fn] of Fn. Let us denote the generators of

Fn induced by x1, . . . , xn, we find from (2.23) that the automorphism of Fn induced

by σi maps xi → xi+1, xi+1 → xi and xj → xj if j ≠ i. Clearly, the automorphism of

Fn induced by the elements of Pn are trivial, and this is a faithful representation of

the permutation group Bn/Pn ≃ Σn. Hence it follows from the 5-lemma that ξ will be

faithful if ξ∣Pn is faithful.

We will now show that the representation defined by ξ arises naturally. Recall

that Pn+1 = Pn.Un+1(regarding Pn as a subgroup of Pn+1). Note that we saw Un+1 is

a free subgroup of Pn+1 of rank n. Also, we have observed that Un+1 ⊴ Pn+1; hence

Pn acts by conjugation (φ(σ) = ρσ for σ ∈ Pn) as a group of automorphisms of Un+1.

Define an isomorphism from Un+1 to Fn by sending Aj,n+1 to xj for each j = 1, . . . , n.

Comparing equations (2.20) and (2.24) we obtain a commutative diagram

Pn AutUn+1

AutFn

ξ

φ(σ)=ρσ

≈

Thus kernel ξ is precisely the subgroup of all elements of Pn which commute with

Un+1, where elements of both Pn and Un+1 are now regarded as elements of Pn+1.

Suppose now, that β ∈ Kerξ, with β ≠ 1. By Corollary 7 we may write β in

the form β = β2 . . . βi−1βi, where i is the largest integer such that βi ≠ 1, but βi+1 =

βi+2 = ⋅ ⋅ ⋅ = βn = 1. Now, β commutes with each element of Un+1, hence β commutes

with Ai,n+1. By equation (2.20), the elements Ai,n+1 depends only on σi, . . . , σn. Note

that each βj(2 ≤ j ≤ i) belongs to the free subgroup Uj of Pn+1 freely generated

by A1,j, . . . ,Aj−1,j, and hence (by equation (2.20)) βj depends only on σ1, . . . , σj−1.

Therefore the condition that β commutes with Ai,n+1 implies:

βiAi,n+1β−1i = Ai,n+1 (i = 1, . . . , n) (2.24)

(which follows by induction) We will now show that (2,25) implies that βi is 1, giving

the sought-for contradiction, so that kernel ξ = 1.

To motivate the algebraic manipulations which follow, we remark that the elements

A1,i, . . . ,Ai−1,i,Ai,i+1, . . . ,Ai,n+1 generate a free subgroup of Pn+1 which is naturally

isomorphic to the subgroup Un+1 freely generated by A1,n+1,A2,n+1, . . . ,An,n+1, for

each i = 2, . . . , n − 1. This is clear because it is quite arbitrary how we assign indices

to the braid “strings”. We will now establish this fact algebraically, in order to be

able to use the fact that equation (2.25) is a statement that a pair of elements in a

free group commute, and hence to conclude that βi and Ai,n+1 are powers of the same

element.

Let π = σnσn−1 . . . σi. Using braid relations (2.2) and (2.3) and equation (2.20) we

claim that

πAi,n+1π−1 = An,n+1 (i = 1, . . . , n) (2.25)

πAk,iπ−1 = Ak,n+1 (k = 1, . . . , i − 1) (2.26)

To establish equation (2.26), note first that

Ai,n+1 = σn . . . σi+1σ2i σ

−1i+1 . . . σ

−1n = σ−1

i . . . σ−1n−1σ

2nσn−1 . . . σi

. The easiest way to see this is to inspect the geometric braid Ai,n+1 (imagine the

figure), and to observe that when the n + 1st string is pulled taut, with the ith string

loose, the geometric braid defined by the left expression for Ai,n+1 goes over to the

geometric braid defined by the expression on the right. This can also be established

algebraically, as a consequence of equation (2.2) and (2.3). Using the expression on

the right above for Ai,n+1, equation (2.26) then follows easily, as follows:

πAi,n+1π−1 = (σnσn−1 . . . σi)(σnσn−1 . . . σi+1σ2i σ

−1i+1 . . . σ

−1n )(σ−1

i σ−1i+1 . . . σ

−1n−1σ

−1n )

= (σnσn−1 . . . σi)(σ−1i . . . σ−1

n−1σ2nσn−1 . . . σi)(σ−1

i σ−1i+1 . . . σ

−1n )

= σ2n = An,n+1 (2.27)

And Equation (2.27) is an immediate consequence of the definitions of Ak,i and of

Ak,n+1 as given by equation (2.20) as follows:

πAk,iπ−1 = (σnσn−1 . . . σi)(σi−1σi−2 . . . σ

2kσ

−1k+1 . . . σ

−1i−1)(σ−1

i σ−1i+1 . . . σ

−1n ) = Ak,n+1(by definition)

Transforming equation (2.25) by π, and applying (2.26), we obtain:

(πβiπ−1)An,n+1(πβ−1i π

−1) = An,n+1 (2.28)

But, by equation (2.27), the elements πβiπ−1 belongs to the free group Un+1, and if two

elements in a free group commute, then they must each be powers of some element in

that group. Since An,n+1 is a generator of Un+1, it then follows that πβiπ−1 = Asn,n+1

for some integer s. But then βi = π−1Asn,n+1π, and by equation (2.26) this is precisely

Asi,n+1. We now have established the sought-for contradiction, for βi belongs to the

free group Ui, and the only way that Asi,n+1 can be in Ui is if s = 0 giving βi = 1.

But, then β = 1, hence kerξ = 1. So, we complete the proof of corollary 8 for Pn and

therefore we can extend it to Bn by proposition 23. ∎

From now onwards we will use the symbol Bn to mean not only the abstract

group of Theorem 11, and the geometric braid group π1B0,nE2, but also its realiza-

tion as a group of right automorphisms of Fn. That is, we will replace the symbols

(Bn)ξ, (Pn)ξ, (Un)ξ, (σi)ξ, (Aij)ξ by Bn, Pn, Un, σi,Aij respectively.

Corollary 9. If n ≥ 3 the center of Bn is the infinite cyclic subgroup generated by

(σ1σ2 . . . σn−1)n = (A12)(A13A23) . . . (A1nA2n . . .An−1,n)

[Chow,1948]

Proof. As a consequence of equation (2.20), we will prove at first that

i) The element (A12)(A13A23) . . . (A1nA2n . . .An−1,n) ∈ Z(Pn).

ii) The element (A1nA2n . . .An−1,n) ∈ Centralizer of Pn−1 in Pn, where Pn−1 is

regarded as the subgroup of Pn which is generated by Ars; 1 ≤ r < s ≤ n − 1.

In order to prove i)(this part of the proof is taken from the book “Braid Group”

by Kassel, Turaev see [3]), First, let us verify that:

∆2n = ((σ1σ2 . . . σn−1)(σ1σ2 . . . σn−2) . . . (σ1σ2)σ1)2 = (σ1σ2 . . . σn−1)n

We will verify it for n = 3 and then by induction we can prove it in general,

for n = 3

(σ1σ2)3 = (σ1σ2σ1)(σ2σ1σ2) = (σ1σ2σ1)(σ1σ2σ1) = (σ1σ2σ1)2 = ∆23

Hence by induction we can prove it for any n ∈ N.

The braid ∆n can be obtained from the trivial braid by a half twist achieved by

keeping the top of the braid fixed and turning over the row of the lower ends by an

angle π. See figure below for a diagram of ∆5.1.3 Pure braid groups 23

Fig. 1.11. The braid Δ5

Proof. The braid Δn can be obtained from the trivial braid 1n by a half-twistachieved by keeping the top of the braid fixed and turning over the row of thelower ends by an angle of π. See Figure 1.11 for a diagram of Δ5. The braidθn = Δ2

n can be obtained from the trivial braid 1n by a full twist achieved bykeeping the top of the braid fixed and turning over the row of the lower endsby an angle of 2π. We have

π(Δn) = (n, n− 1, . . . , 1) ∈ Sn .

Hence θn ∈ Pn. It is a simple exercise to compute θn inductively from ι(θn−1),where ι : Pn−1 → Pn is the natural inclusion. Namely, θn = ι(θn−1)γ, where

γ = γn = A1,nA2,n · · ·An−1,n ∈ Pn ;

see Figure 1.12 for a diagram of γ5.

Fig. 1.12. The braid γ5

Sliding a crossing along the strands of the diagram of Δn from top tobottom, one easily obtains for all i = 1, 2, . . . , n− 1,

σi Δn = Δn σn−i . (1.8)

The braid θn = ∆2n can be obtained from the trivial braid by a full twist achieved

by keeping the top of the braid fixed and turning over the row of the lower ends by

an angle of 2π. We have, for n = 2k,

π(∆n) = (1 n)(2 n − 2) . . . (n k − 1)

and for n = 2k + 1

π(∆n) = (1 n)(2 n − 2) . . . (k − 1 k + 1)(k)

Hence θn ∈ Pn Now, It is easy to see geometrically(by geometric meaning of a braid)

that

σi∆n = ∆nσn−i ∀i = 1,2 . . . , n − 1

This implies that θn commutes with all the generators of Bn:

σiθn = σi∆n∆n = ∆nσn−i∆n = ∆n∆nσi = θnσi

Hence, θn ∈ Z(Bn).

Now, for ii), It is just a matter of verification as above.

Now, to prove Corollary 9, Let us take β ∈ Z(Bn). As the center of the symmetric

group Σn is trivial for n ≥ 3, β ∈ Ker µ, where µ is the homomorphism µ ∶ Bn → Σn,

that is β ∈ Pn. By Corollary 6 it then follows that β has a unique representation

β = βn−1βn where βn ∈ Un, βn−1 ∈ Pn−1. The condition that β ∈ Z(Bn) then implies:

βn−1βnAinβ−1n β

−1

n−1 = Ain 1 ≤ i ≤ n − 1 (2.29)

Multiplying together all the n − 1 equations obtained by putting i = 1,2, . . . , n − 1 in

equation (2.30), we get

βn(A1nA2n . . .An−1,n)β−1n = β−1

n−1(A1nA2n . . .An−1,n)βn−1

Since βn−1 ∈ Pn−1, condition ii) above then implies:

βn(A1nA2n . . .An−1,n)β−1n = (A1nA2n . . .An−1,n)

. This equality holds in the free group Un, hence the only possibility is

βn = (A1nA2n . . .An−1,n)m

for some integer m. Using this information in (2.30), we thus obtain:

β−1

n−1Ainβn−1 = (A1nA2n . . .An−1,n)mAin(A1nA2n . . .An−1,n)−m ∈ Un (2.30)

Equation (2.31) expresses the action of the element βn−1 on the free generators

A1n,A2n, . . . ,An−1,n of Un by conjugation. By Corollary 8, this action induces a faith-

ful representation of Pn−1 as a group of automorphisms of the free group Un. Now, a

calculation based on equation (2.31) shows that the elements

[A12(A13A23) . . . (A1,n−1A2,n−1 . . .An−2,n−1)]−m

has precisely the effect which our element β−1

n−1 is required to have, hence β−1

n−1 must

be precisely [A12(A13A23) . . . (A1,n−1A2,n−1 . . .An−2,n−1)]−m for some integer m. Thus

our original element β = βn−1βn can only have been:

β = [(A12)(A13A23) . . . (A1,n−1A2,n−1 . . .An−2,n−1)(A1,nA2,n . . .An−1,n)]m

(Just substitute the value of βn−1 and βn and then use property ii) to get β)

Now, It can be easily seen that β = (σ1σ2 . . . σn−1)m,∀m.

Since by property i) this element is in fact in the center for every integer m, the

proof of Corollary 9 is complete. ∎

Theorem 12. (Artin, 1925). Let Fn =< x1, . . . , xn > be a free group of rank n. Let β

be an endomorphism of Fn. Then β ∈ Bn ⊂ AutFn iff β satisfies the two conditions

(xi)β = AixµiA−1i 1 ≤ i ≤ n (2.31)

(x1x2 . . . xn)β = x1x2 . . . xn (2.32)

where (µ1, . . . , µn) is a permutation of (1, . . . , n) and Ai = Ai(x1, . . . xn) is a word in

the generators of Fn.

Proof. The necessity of condition (2.32) and (2.33) follows immediate, (by Corollary

8) and therefore we need to establish only that they are sufficient. This will be

accomplished by proving that every endomorphism of Fn which satisfies (2.32) and

(2.33) is a product of powers of σ1, . . . , σn−1, and hence is in Bn. To prove this, we

examine how cancellations occurs in the equality

A1xµ1A−11 A2xµ2A

−12 . . .AnxµnA

−1n = x1x2 . . . xn (2.33)

which results from (2.32) and (2.23). we assume that each term AixµiA−1i is freely

reduced.

We state that in order for (2.34) to hold in the free group Fn, there must exist

some µ = 1,2, . . . , n − 1 such that either

a) xµvA−1v is absorbed by Av+1

or

b) A−1v absorbs Av+1xµv+1

This assertion will imply Theorem 12 by the following reasoning: Define the

“length” of the automorphism β to be the sum of the letter lengths of the words

AixµiA−1i ,1 ≤ i ≤ n. that is,

l(β) = l(A1xµ1A−11 A2xµ2A

−12 . . .AnxµnA

−1n )

that is “length of alphabets”. We will show that if (a) is true, then σvβ has shorter

length then β, while if (b) is true σ−1v β has shorter length then β.(Note that the braid

automorphisms act on the right, hence σvβ means apply σv first, then apply β). This

implies that every automorphism β of Fn which satisfies conditions (2.32) and (2.33)

can be reduced to be the identity by repeated application of appropriate elementary

automorphisms σv or σ−1v . That is,

l(σvβ) < l(β), then

let γ1 = σvβ Ô⇒ l(σv1γ1) = l(σv1σvβ) < l(γ)

. . .

γn = σvnβ Ô⇒ l(σvnγn) = l(σ1σ2 . . . σn−1β) = l(γn)

Ô⇒ β = σε11 σε22 . . . σεn−1n−1 where εi = +1or − 1

Hence β ∈ Bn

To show that the length can always be reduced as indicated, suppose first that (a)

is true. Then the action of β on xv and xv+1 is given by:

(xv)β = AvxµvA−1v , (xv+1)β = Avx−1

µvAv+1xµv+1A−1v+1xµvA

−1v (2.34)

(where Av+1 and A−1v+1 are residue after cancellation of Av+1 and A−1

v+1 respectively)

Using the action given in Corollary 8, we now compute the product σvβ:

(xv)σvβ = (xvxv+1x−1v )β = (xv)β(xv+1)β(x−1

v )β

=(AvxµvA−1v )(Avx−1

µvAv+1xµv+1A−1v+1xµvA

−1v )(Avx−1

µ A−1v )

=AvAv+1xµv+1A−1v+1A

−1v ,we get by applying equation (2.35)

(xv+1)σvβ = (xv)β = AvxµvA−1v (2.35)

Since both β and σvβ have the same effect on xj if j ≠ v, v + 1, a comparison of

(2.35) and (2.36) shows that σvβ has shorter length than β as

l(β) = 4l(Av) + 3l(xµv) + l(xµv+1) + 2l(Av+1) + . . .

l(σvβ) = 4l(Av) + 2l(Av+1) + l(xµv+1) + l(xµv)

So, clearly, l(σvβ) < l(β).

A same kind of argument holds in case (b). Thus, Theorem 12 will be true if we

can show that our assertion about the cancellation is true.

We examine the manner in which the LHS of (2.34) reduces to RHS. Suppose first

that one of the term AvxµvA−1v is completely absorbed by the other terms during the

free cancellations which reduces the LHS of (2.34) to RHS. We ask how the letter xµ

is absorbed in these cancellations? If xµ is absorbed by a letter to the left of xµv−1 ,

then (a) is satisfied. If xµv is absorbed by a letter in A−1v−1, then (b) is satisfied. If xµv

is absorbed by a letter in Av+1, then (a) is satisfied. If xµv is absorbed by a letter to

the right of xµv+1 , then (b) is satisfied. Since xµv cannot be absorbed by either xµv−1

or xµv+1 , both of which have subscripts which are different from the subscript µv, all

possible cases have been treated.

Only, it remains to consider the case where there is no subscript v with the property

that AvxµvA−1v is completely absorbed. In that case, some residue Rv will remain for

each AvxµA−1v after all free reductions have been made. Then (2.34) implies

R1R2 . . .Rn = x1x2 . . . xn

. This implies that Ri = xi for each i = 1,2, . . . , n. Now examine the term A1xµ1A−11 .

The initial letter in this term can only be x1. We consider first the case where

A1xµ1A−11 is not identically x1, that is,

A1xµ1A−11 = x1A1xµ1A

−11 x

−11

Since by hypothesis A1xµ1A−11 x

−11 is completely absorbed, there are two possibilities:

xµ1 is absorbed by A2 (in which case (a) is satisfied) or by a letter to the right of

xµ2 (in which case (b) is satisfied). If, on the other hand, A1xµ1A−11 = x1 the entire

argument can be repeated for A2xµ2A−12 , etc. In this way we see that in every case

either (a) or (b) is true, hence Theorem 12 is completely proved. ∎

Now using Theorem 12, we will give a new geometric interpretation on the group

Bn.

2.4.1 Braids as Automorphisms of free groups or Mapping

Class groups

Let D2 be a disc, and let Qn = q1, . . . , qn be a set of fixed, distinguished points of D2.

The fundamental group π1(D2 −Qn) is free group of rank n. Let x1, . . . , xn be a basis

for π1(D2 −Qn), where xi(i = 1, . . . , n) is represented by a simple loop which encloses

the boundary points qi, but no boundary point qj for j ≠ i. See Figure below(taken

from the paper ‘Basic results of braid groups’ [22]).

J. Gonzalez-Meneses

distinct heights, that is, for distinct values of t ∈ [0, 1]. In this way, it isclear that every braid is a product of braids in which only two consecutivestrands cross. That is, if one considers for i = 1, . . . , n − 1 the braids σiand σ−1

i as in Figure 2, it is clear that they are the inverse of each other,and that σ1, . . . , σn−1 is a set of generators of Bn, called the standardgenerators, or the Artin generators of the braid group Bn.

1.6. Braids as automorphisms of the free group

We shall now give still another interpretation of braids. This is one ofthe main results in Artin’s paper [2]. There is a natural representationof braids on n strands as automorphisms of the free group Fn of rank n.Although Artin visualized braids as collections of strands, we believe thatit is more natural to define their representation into Aut(Fn) by means ofmapping classes, as was done by Magnus [57].

Figure 3. The loops x1, . . . , xn are free generators of π1(Dn).

We remark that the fundamental group of the n-times punctured discDn

is precisely the free group of rank n: π1(Dn) = Fn. If we fix a base point, sayin the boundary of Dn, one can take as free generators the loops x1, . . . , xndepicted in Figure 3. Now a braid β ∈ Bn can be seen as an automorphismof Dn up to isotopy, so β induces a well defined action on π1(Dn) = Fn,where a loop γ ∈ π1(Dn) is sent to β(γ). This action is clearly a grouphomomorphism (respects concatenation of loops), which is bijective as β−1

yields the inverse action. Hence β induces an automorphism of Fn, andthis gives a representation:

ρ : Bn −→ Aut(Fn)β 7−→ ρβ.

8

Theorem 13. Let M be the group of automorphisms of π1(Dn) (where Dn = D2−Qn)

which are indicated by the homeomorphisms of Dn which keep the boundary of D2 fixed

pointwise. Then M is precisely the group Bn.

Proof. For geometric reasons, each elements β ∈ M satisfies conditions (2.32) and

(2.33), hence M ⊆ Bn.

We fix a base point, say in the boundary of Dn, one can take as free generators the

loops x1, x2, . . . , xn depicted in figure above. Now a braid β ∈ Bn can be seen as an

automorphism of Dn upto isotopy, so β induces a well defined action on π1(Dn) = Fn,

where a loop γ ∈ π1(Dn) is sent to β(γ). It is easy to check that this action is

a group homomorphism (respects concatenation of loops), which is bijective as β−1

yields the inverse action. Hence β induces an automorphism of Fn, and this gives a

representation:

ρ ∶ Bn → Aut(Fn)

which sends β to ρβ

The automorphism ρβ can be easily described when β = σi, by giving the image of

the generators x1, . . . , xn of Fn. Namely:

ρσi(xi) = xi+1, ρσi(xi+1) = x−1i+1xixi+1, ρσi(xj) = xj (ifj ≠ i, i + 1)

BASIC RESULTS ON BRAID GROUPS

The automorphism ρβ can be easily described when β = σi, by giving theimage of the generators x1, . . . , xn of Fn (see Figure 4). Namely:

ρσi(xi) = xi+1, ρσi

(xi+1) = x−1i+1xixi+1, ρσi

(xj) = xj (if j 6= i, i+ 1).

The automorphism ρσ−1

ican be easily deduced from ρσi

. For a general

braid β, written as a product of σ1, . . . , σn−1 and their inverses, the auto-morphism ρβ is just the composition of the corresponding automorphismscorresponding to each letter.

Figure 4. Action of σi on the generators xi and xi+1.

Later we will see that the braid group Bn admits the presentation

Bn =

⟨

σ1, . . . , σn−1

∣

∣

∣

∣

σiσj = σjσi, |i− j| > 1σiσjσi = σjσiσj, |i− j| = 1

⟩

. (1.1)

It is then very easy to check that ρ is well defined, as ρσiσj≡ ρσjσi

if|i − j| > 1, and ρσiσjσi

≡ ρσjσiσjif |i − j| = 1. Artin [3] showed that ρ is

faithful by topological arguments, making no use of the above presentation.Notice that for every β ∈ Bn, the automorphism ρβ sends each generator

xj to a conjugate of a generator. Notice also that for each i = 1, . . . , n−1,one has ρσi

(x1 · · · xn) = x1 · · · xn. Hence ρβ(x1 · · · xn) = x1 · · · xn for every

9

The automorphism ρσ−1i can be easily deduced from ρσi . For a general braid β,

written as a product of σ1, . . . , σn−1 and their inverses, the automorphism ρβ is just

the composition of the corresponding automorphisms corresponding to each letter.

It is easy to check that ρ is well defined, as ρσiσj = ρσjσi if ∣i − j∣ > 1, and ρσiσjσi ≡

ρσjσiσj if ∣i − j∣ = 1.

Note that for every β ∈ Bn, the automorphism ρβ sends each generator xj to

a conjugate of a generator. Notice also that for each i = 1,2, . . . , n − 1, one has

ρσi(x1x2 . . . xn) = x1x2 . . . xn. Hence ρβ(x1 . . . xn) = x1 . . . xn for every β ∈ Bn. This

is clear as x1 . . . xn corresponds to a loop that runs parallel to the boundary of Dn,

enclosing the n punctures, hence it is not deformed by any braid (upto isotopy).

Hence, Bn ≡M(see [22] for more details). ∎

Remark: Let ∆ be an autohomeomorphism of D2 −Qm which keeps boundary

of D2 fixed pointwise. Thus ∆ represents an element of M . Then ∆ has a unique

extension ∆ to D2 which permutes the points of Qm. The map ∆ is isotopic to the

identity in D2. This isotopy may be used to define an autohomeomorphism of D2 × I

which preserves D2 × 0 pointwise, preserves D2 × t, t ∈ I, setwise, and coincides

with ∆ on D2 × 1. The image of Qn × I under this extension is a geometric braid in

the sense defined earlier.

2.5 The braid group of the sphere

Theorem 14. (Fadell-Van Buskrik,1962). the braid group π1B0,nS2 of the 2-sphere

S2 admits a presentation with generators δ1, . . . , δn−1 and defining relations:

δiδj = δjδi if ∣i − j∣ ≥ 2,1 ≤ i, j ≤ n − 1 (2.36)

δiδi+1δi = δi+1δiδi+1 1 ≤ i ≤ n − 2 (2.37)

δ1 . . . δn−2δ2n−1δn−2 . . . δ1 = 1 (2.38)

Proof. The proof will only be outlined. It rests on an inductive argument which is

exactly like that used in the proof of Theorem 11. The only difficulty is that the

fundamental exact sequence (2.13), which was crucial in establishing Theorem 11, is

only valid for n ≥ 4 when M = S2. Therefore the inductive argument which was the

basis of the proof of Theorem 11 begins with n = 3 rather than 1. The following

additional facts are needed to establish Theorem 14:

(i)π1F0,2S2 = 1

(ii)π1B0,2S2 = cyclic group of order 2

(iii)π1F0,3S2 = cyclic group of order 2

(iv)π1B0,3S2 = ZS,metacyclic group of order 12

(v)π2F0,3S2 = 1

By, metacyclic group we mean an extension of a cyclic group by a cyclic group.

That is, it is a group for which a short exact sequence:

1→K → G→H → 1

where H and K are cyclic.

The proof of (i) follows easily from the fibration of Theorem 6 using the well-known

facts that π1F1,1S2 and π1F0,1S2 are both trivial groups:

⋅ ⋅ ⋅→ π2(M −Q0)→ π1F1,1S2 → π1F0,2S

2 → π1S2 → π0S

2 → . . .

To prove (ii), one need only note that π1B0,2S2 maps homeomorphically onto Σ2,

which is of order 2, and use (i). For proofs of (iii)-(v) one may refer to Fadell and

Van Buskrirk, 1967 (see [17]) ∎

Note, for completeness, that the faithful representation found for π1B0,nE2 as a

group of automorphisms of the fundamental group of the n−punctured plane (Corol-

lary 8 and Theorem 13) does not generalize to a faithful representation of π1B0,nS2

as a group of automorphisms of the fundamental group of the n−punctured sphere.

To be sure, the action given in equations (2.23) does induce an action of π1B0,nS2 on

Fn−1, where Fn−1 is the quotient group of Fn obtained by adding the single relation

x1x2 . . . xn = 1. This action is, however, not a representation of π1B0,nS2, because re-

lation (2.39) is not satisfied. Moreover, one can verify that the element (δ1δ2 . . . δn−1)n

induces the identity automorphism of Fn−1, yet the relation (δ1δ2 . . . δn)n cannot be a

consequence of relations (2.37), (2.38) and (2.39), see [17] for details. This induced

action on Fn−1 can be used to study the mapping class group of the n−punctured

sphere .

2.6 Survey of 2-manifold braid groups

If M is a closed orientable 2−manifolds and either n ≥ 4 or P 2 ≠ M ≠ S2 and n ≥ 2,

then as shown in Theorem 7, there is an exact sequence

1→ π1Fn−1,1M → π1F0,nM → π1F0,n−1M → 1

This sequence was the basis for the structural analysis of π1B0,nE2 carried out in

Section 2.3. The same sort of analysis can be carried out with successively more

difficulty for other 2−manifold braid groups. The result appear in the following papers:

E2[Chow(1948): Fadell-Van Buskirk (1962)] (see [12] [17] respectively)

P 2[Van Buskirk (1966)](see [23])

S2[Fadell- Van Buskirk (1962)](see [17])

Torus[Birman(1969a)](see [18])

All closed 2−manifolds [G.P. Scott(1970)](see [24])

2.7 Some examples where braiding appears in math-

ematics, unexpectedly

Here we discuss, a variety of examples, outside of knot theory, where “braiding” is

an essential aspect of a mathematical or physical problems. This examples are taken

from a paper “Braids: A Survey” by Tara E. Brendle and Joen S. Birman (see [25]).

The most important among these is in ‘Algebraic Geometry’

2.7.1 Algebraic geometry

The configuration space of n points on the complex plane C is:

F0,n = F0,n(C) = (z1, . . . , zn) ∈ C × . . .C∣zi ≠ zj, i ≠ j

The orbit space of the action is B0,n = B0,n(C) = F0,n/Σn and the orbit space projection

is φ ∶ F0,n → B0,n.

Configuration spaces and the braid group appear in a natural way in algebraic

geometry. Consider the complex polynomial

(X − z1)(X − z2) . . . (X − zn) =Xn + a1Xn−1 + ⋅ ⋅ ⋅ + an−1X + an (2.39)

of degree n with n distinct complex roots z1, z2, . . . , zn. The coefficients a1, a2, . . . , an

are the elementary symmetric polynomials in z1, . . . , zn, and so we get a continuous

map Cn → Cn which takes roots to coefficients. Two points have the same image iff

they differ by a permutation, so we get the same identification as the quotient map

φ ∶ F0,n → B0,n, in quite a different way. Since we are requiring that our polynomial

have n distinct roots, a point a1, . . . , an is in the image of z under the root to

coefficient map iff the polynomial Xn + a1Xn−1 + ⋅ ⋅ ⋅ + an−1X + an has n distinct roots,

i.e. iff its coefficients avoid the points where the discriminent

∆ =∏i<j

(zi − zj)2, (2.40)

expressed as a polynomial in a1, . . . , an, vanishes. Thus, B0,n(C) can be interpreted

as the complement in Cn of the algebraic hypersurface(A hypersurface is a manifold

or an algebraic variety of dimension n − 1, which is embedded in an ambient space

of dimension n, generally a Euclidean space, an affine space or a projective space)

defined by the equation ∆ = 0, where ∆ is rewritten as a polynomial in coefficients

a1, . . . , an.(For example, the polynomial X2 + a1X + a2 has distinct roots precisely

when a21 − 4a2 = 0). In this setting the basepoint φ(p) is regarded as the choice of

a complex polynomial of degree n which has n distinct roots, and an element in the

braid group is a choice of a continuous deformation of that polynomial along a path

on which two roots never coincide. There is a substantial literature in this area, from

which we mention only one paper, by Gorin and Lin [8].

Here are few more examples:

∗ Braiding also appears in the theory of Operator algebra of ‘type II1 factors’, details

can be found in the papers by Vaughan Jones [26], [27]

∗ Braids have played a role in homotopy theory for many years, mostly in the work

of F. Cohen and his students see [28] for a substantial literature.

∗ There is also an application of configuration spaces to robotics. A vast literature

on this subject, which can be found in [29] by R.Ghrist.

∗ In this example braids are important for somewhat different reasons than they

were in our earlier examples. In our earlier examples the basic phenomenon

which was being investigated involved actual braiding, although sometimes in

a concealed way. Here some of the particular properties of the braid groups

Bn, n = 1,2,3, . . . , are used in some clever way to construct new methods for

encrypting data rather than the actual interweaving of braid strands. Further

literature can be found in [25].

Chapter 3

Mapping Class Groups

3.1 Definition and Examples

This chapter is based on the author’s understanding of Chapter 4 of the book [1]. Let

Tg denotes a closed orientable surface of genus g; and let z01 , z

02 , . . . , z

0n be the n fixed

but arbitrarily chosen points on Tg. Recall that like in Chapter 2 the symbol π1F0,nTg

denoted the pure braid group on Tg (with base point (z01 , . . . , z

0n)) and π1B0,nTg denoted

the full braid group on Tg (with the same base point).

Let FnTg denotes the group of all orientation preserving homeomorphisms h ∶ Tg →

Tg such that, for each i, h(z0i ) = z0

i .

BnTg denotes the group of all orientation preserving homeomorphisms h ∶ Tg → Tg

such that h(z01 , . . . , z

0n) = z0

1 , . . . , z0n.

Now we, give these two groups compact-open topology.

Definition: π1(FnTg, Id) denotes the group of path components of FnTg and is

called the pure mapping class group of Tg.

Definition: π0(BnTg, Id) denotes the group of path components of BnTg and is

called the full mapping class group of Tg. We use M(g, n) for this group.

Note that F0Tg = B0Tg and F1Tg = B1Tg. Hence M(g,0) = π0(F0Tg) = π0(B0Tg)

and M(g,1) = π0(F1Tg) = π0(B1Tg).

The terminology M(g, n) is meant to stand for ”modular group”. Fricke called

63

the mapping class group the “automorphic modular group”. It can also be viewed as

a generalization of the classical modular group SL(2,Z)

Elements of M(g, n) are called the mapping classes.

Example: The homomorphism

σ ∶M(2,0)→ SL(2,Z)

given by the action on H1(T ;Z) ≈ Z2 is an isomorphism. (More tools are required to

prove it, see [30])

3.2 The natural homomorphism from M(g,n) to

M(g,0)

Definition: Recall that z01 , . . . , z

0n are a set of n distinguished points on Tg, and that

if h ∈ FnTg, then h(z0i ) = z0

i for each i = 1, . . . , n. Recall also that, in chapter 2,

(z01 , . . . , z

0n) was chosen to be the base point for the space F0,nTg. Using these points,

we now define (for each pair of integers n, g ≥ 0) an evaluation map as follows:

εgn ∶ F0Tg → F0,nTg

by εgn(h) = (h(z01), . . . , h(z0

n)) where F0,nTg = (p1, . . . , pn)∣pi ∈ Tg;pi ≠ pj if i ≠ j.

Observe that εgn is continuous with the given topologies on F0Tg (equipped with

compact open topology) and F0,nTg (equipped with subspace topology for F0,nTg ⊂

Tg × Tg × ⋅ ⋅ ⋅ × Tg).

Theorem 15 (Birman, 1969b). The evaluation map εgn ∶ F0Tg → F0,nTg is a locally

trivial fibering with fibre FnTg.

Proof. Note that FnTg is a closed subgroup of the topological group F0Tg (as com-

plement of a point in D2 is closed, and finite intersection of closed sets is closed)

and that two elements h and g of F0Tg have the same image under εgn iff they are

in the same left coset of FnTg in F0Tg(this can be easily verified). This observation

results in a natural identification of F0,nTg with the quotient space F0Tg/FnTg, and

this identification turns out to be a homeomorphism which turns εgn into a projection

map and show that F0,nTg is a homogeneous space.(see section 1.4 for definition of

homogeneous space)

Proposition 22, in section 1.4, Theorem 15 will follow immediately once we show

that, relative to εgn, ∃ a local cross section of the homogeneous space F0,nTg in

F0Tg at the single point z0 = εgn(FnTg) = (z01 , . . . , z

0n) ∈ F0,nTg (that is, there is a

neighborhood Uz0 of z0 in F0,nTg and a map κ ∶ Uz0 → F0Tg such that εgnκ= Id).

Choose pairwise disjoint euclidean neighborhoods Uz01 , . . . , Uz0n(such as choice is pos-

sible as Tg is a manifold and hence Hausdorff) of z01 , . . . , z

0n respectively, on Tg. Then

U(z0) = (u1, . . . , un∣ui ∈ U(z0i ) is a neighborhood of z0 in F0,nTg. Construct a

family of homeomorphisms lu ∈ F0Tg ∣u ∈ U(z0), depending continuously on u, such

that, for each u ∈ U(z0), lu(z0i ) = ui and lu(Tg − ∪ni=1U(z0

i )) = Id.(such a construction

is always possible because we know that ∃ a homeomorphism from a n-dimensional

closed unit to itself taking a point ’a’ to a point ’b’ within the interior of the ball and

fixing the boundary) Define κ(u) = lu and it is then easy to see that it is a local cross

section. ∎

Corollary 10. There is an exact sequence of homotopy groups

→ π1F0Tgεgn∗ÐÐ→ π1F0,nTg

dgn∗ÐÐ→ π0FnTgign∗ÐÐ→ π0F0Tg → π0F0,nTg = 1 (3.1)

Proof. The homomorphism εgn∗ is induced by the evaluation map εgn and the homo-

morphism ign∗ by the inclusion FnTg → F0Tg. The exact sequence is simply the exact

homotopy sequence of the fibering εgn ∶ F0Tg → F0,nTg(see section 1.3.4 for details

about fibration) ∎

Remark: The surjection i∗ = ign∗ is the analogue for the pure mapping class

groups of the homomorphism j∗ = jgn∗ ∶M(g, n) →M(g,0) of the full mapping class

groups named in the title of this paragraph and about which we will study below in

Theorem 17.

Here onwards, for simplicity we will replace the symbols ign∗ , εgn∗ , dgn∗ by i∗, ε∗, d∗

respectively. Corollary 10 becomes effective only when i∗ has been structurally deter-

mined by a careful examination of im d∗=ker i∗ and ker d∗:

Theorem 16. (Birman, 1969b). For each pair of integers g, n ≥ 0 let i∗ ∶ π0FnTg →

π0F0Tg be the homomorphism induced by the inclusion. Then ker i∗=image d∗ ≈

π0F0,nTg if g ≥ 2. If g = 1, n ≥ 2 or g = 0, n ≥ 3 then ker i∗ ≈ π1F0,nTg ≈ π1F0,nTg/center.

Proof. Let us consider the final segment of the long exact sequence of Corollary 10.

Except for the case g = 1, whose proof will be omitted, the theorem follows im-

mediately from Lemma 3 below (which shows that Ker d∗ ⊂ centerπ1F0,nTg) and

Lemma 4-6 (which identify Z(π1F0,nTg) explicitly). Before these lemmas are stated

and proved, however, the construction of d∗ and the proof of the relationship Ker

i∗=im d∗ will be recalled.

Construction of d∗: Suppose β ∈ π1F0,nTg, with β represented by a loop

(β1, . . . , βn) ∶ In → F0,nTg

. Then it is an easy matter to construct an isotopy ht ∶ Tg → Tg(0 ≤ t ≤ 1) such that

h0 = Id, ht(xi) = βi(t)(such a isotopy exists if we embed each such strings β1 into a unit

disk and then we know that there exists a homeomorphism taking a point to any other

point fixing the boundary), and hence h1 ∈ FnTg. Indeed, the construction is obvious

for generators Aij ∣1 ≤ i < j ≤ n of π1F0,nTg (see chapter 2) and by composition of

isotopies can be extended to all elements of π1F0,nTg. Then, [h1] = d∗β

Now, Ker i∗=Im d∗. This follows immediately from the fact that the sequence (3.1)

is exact. It will be of interest to obtain image d∗ explicitly. Suppose that h ∈ FnTgwith [h] ∈ ker i∗. Since h ∈ FnTg, h fixes z0

1 , z02 , . . . , z

0n pointwise. Since [h] ∈ Ker i∗, h

is isotopic to the identity map on Tg, say by an isotopy ht(0 ≤ t ≤ 1), h0 = Id, h1 = h.

Then (ht(z01), . . . , ht(z0

n)) represents an element β of π1F0,nTg and d∗β = [h] ∎

Lemma 3. Ker d∗ ⊂ Z(π1F0,nTg)

Proof. Suppose α ∈ Ker d∗ = Im ε∗, and let H ∈ π1F0Tg be such that ε∗H = α. The

element H is represented by a loop h = ht∣0 ≤ t ≤ 1 in F0Tg, where each ht is in

F0Tg and h0 = h1 = Id. Then ε(ht) = (ht(x1), . . . , ht(xn)) (0 ≤ t ≤ 1) represents

α. Let β ∈ π1F0,nTg, with β represented by (β1(s), . . . , βn(s)) (0 ≤ s ≤ 1). Define

G ∶ I × I → F0.nTg by G(t, s) = (htβ1(s), . . . , htβn(s)) ((t, s) ∈ I × I). Then G is

continuous and G∣∂(I × I) represents the homotopy class αβα−1β−1. Since β was an

arbitrary element of π1F0,nTg, we may conclude that α ∈ Z(π1F0,nTg). ∎

Lemma 4. If g ≥ 2, then center π1F0,nTg = 1.

Proof. Recall Theorem 7, the exact sequence (2.13)

1→ π1Fn−1,1Tgδ∗Ð→ π1F0,nTg

π∗Ð→ π1F0,n−1Tg → 1

of the pure braid groups. If n = 1, π1F0,1Tg = π1Tg which is centerless.(assuming it to

be true here, the proof involves concepts from Hyperbolic Geometry and Riemannian

Geometry) Assuming by induction that π1F0,n−1Tg is centerless. Since π∗ is surjective,

π∗(center π1F0,nTg) ⊂ center π1F0,n−1Tg = 1. Hence center π1F0,nTg lies in the group

Im j∗=ker π∗. But π1Fn−1,1Tg ≈ Im j∗ is a free group of rank > 1, hence centerless.

Thus center π1F0,nTg = 1. ∎

The next two lemmas treat the case g = 0, that is Tg = S2

Lemma 5. (Gillette and Van Buskrik,1968). Let δ1, . . . , δn−1 be the standard gen-

erators of π1B0,nS2, n ≥ 3. Then the center of π1B0,nS2 is the subgroup of order 2

generated by (δ1δ2 . . . δn−1)n.

Proof. We assume this lemma without proving it, whose prove can be obtained in

[35]. ∎

Lemma 6. If n ≥ 3, then

Z(π1B0,nS2) ⊂ Z(π1F0,nS

2) ⊂ Ker d∗

Proof. By Lemma 5,

center π1B0,nS2 ⊂ π1F0,nS

2

is the cyclic subgroup of order 2 generated by (δ1 . . . δn−1)n, hence Lemma 6 will be

true if we can prove that d∗(δ1 . . . δn−1)n = 1. In the standard geometric model of

π1F0,nE2, (σ1 . . . σn−1)n can be pictured as in figure a and b below(drawn for the case

n = 4). Then as a motion of points on S2, (δ1 . . . δn−1)n can be pictured as in figure c.

(figure taken from Birman’s book [1])

Recall now the construction of d∗. Without loss of generality, z01 , . . . , z

0n are spaced

at equal distances on a longitude joining z01 at the equator with z0

n at the north pole

(as in Figure 13c), and the motion of points pictured in Figure 13c can be realized

by a rotation ht ∶ S2 → S2 for (0 ≤ t ≤ 1) about the axis joining the north and south

pole, h0 = h1 = Id. Then d∗(δ1 . . . δn−1)n = [h1] = 1 ∎

Lemmas 3-6 complete the proof of Theorem 16 except for the case g = 1. The proof

for g = 1 will not be included here. One proceeds as in the other cases to the identify

center π1B0,nT1(see [18]). The center is a free abelian group of rank 2 (isomorphic to

π1T1).

Theorem 17. For each pair of intergers g, n ≥ 0, let j∗ = jgn∗ ∶ M(g, n) → M(g,0)

be the homomorphism induced by the inclusion j ∶ BnTg ⊂ B0Tg. Then Ker j∗ is

isomorphic to π1B0,nTg for g ≥ 2. If g = 1, n ≥ 2 or g = 0, n ≥ 3, then ker j∗ is

isomorphic to π1B0,nTg/center.

Proof. The proof is essentially a repetition of the arguments used to prove Theorem

15 and 16. As in Theorem 15, we may establish that there is an evaluation map

ε′gn ∶ B0Tg → B0,nTg, which is a locally trivial fibering with fiber BnTg. As in Corollary

10, this fibering defines an exact sequence

→ π1B0Tgε′gn∗ÐÐ→ π1B0,nTg

d′gn∗ÐÐ→M(g, n)jgn∗ÐÐ→M(g,0)→ π0B0,nTg = 1 (3.2)

As in the proof of Theorem 16, we may establish that Ker jgn∗ is naturally isomorphic

to π1B0,nTg if g ≥ 2, n ≥ 2, or to π1B0,nTg/center if g = 1, n ≥ 2 or g = 0, n ≥ 3 ∎

Now, we close this section by making explicit the situation described by Theorem

16 and 17. First consider the case n = 1, g ≥ 2. In this case the homomorphism ig1∗

and jg1∗ of Theorem 16 and 17 coincide(because FnTg = BnTg if n = 0 or n = 1), and

moreover π1F0,1Tg = π1B0,1Tg ≈ π1Tg, so that our theorems imply that ker ig1∗ =Ker jj1∗

is isomorphic to π1Tg. We will now describe an explicit fashion how to find generators

for the subgroup ker jg1∗ of the mapping class group M(g,1) = π0F1Tg = π0B1Tg.

Let z01 be the base point for π1Tg, and suppose that c is any simple closed curve on

π1Tg which contains the base point z01 . Let N be a cylindrical neighborhood of c on

Tg, parametrized by (y, θ), with −1 ≤ y ≤ +1,0 ≤ θ ≤ 2π, where the curve c is described

by y = 0, and the base point z01 by (0,0). We now define a map hcz1 ∶ Tg → Tg by the

rule that if a point is in N , then its image is given by:

hcz1 ∶ (y, θ)→

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

(y, θ + 2πy) if 0 ≤ y ≤ 1

(y, θ − 2πy) if − 1 ≤ y ≤ 0,

(3.3)

while all points of Tg −N are left fixed. We call such a map a spin of z01 about c.

(see figure below). Note that hcz1 ∈M(g,1)(the proof of this fact can be found in the

Birman’s book on “Braid, Links and Mapping Class Groups”).

Now, let a1, . . . , ag, b1, . . . , bg be 2g simple closed curves on Tg, meeting in the

base point z01 but otherwise disjoint, and having the property that their homotopy

classes generate π1Tg, and also that the homotopy class of a1b1a−11 b

−11 . . . agbga−1

g b−1g is

trivial. Then the isotopy classes of the spin maps ha1z1 , . . . , hagz1 , hb1z1 , . . . , hbgz1 on

Tg,1 generate Ker jg,1∗ , and the isotopy class of the product

ha1z1hb1z1h−1a1z1h

−1b1z1

. . . hagz1hbgz1h−1agz1h

−1bgz1

is trivial.

In the case where n ia arbitrary, the single point z01 which serves as base point for

π1Tg = π1F0,1Tg = π1B0,nTg(regarded as a subgroup of π0F1Tg = π0B1Tg) is replaced by

an array (z01 , . . . , z

0n) which determines a base point for π1F0,nTg,and also for π1Bo,nTg.

Let aij, bij; 1 ≤ i ≤ g,1 ≤ j ≤ n be 2gn simple closed curves on Tg, where ai1 = ai and

bi1 = bi are as previously defined, and where each curve aij(respectively bij) is freely

homotopic to ai1(respectively bi1) on Tg, and each curve aij (respectively bij) contains

the base point z0j , but no other point z0

k if k ≠ j. Then the isotopy classes of the spin

maps

haij ,zj , hbijzj ; 1 ≤ i ≤ g,1 ≤ j ≤ n

generates Ker ign∗ . To obtain a set of generators for Ker jgn∗ one adds th this set any

set of maps on the surface Tg which generete the full group of permutations of the

points (z01 , . . . , z

0n); for example, enclose each pair (z0

j , z0j+1) in a disc Dj which avoids

all points z0k(k ≠ j) and map Tg to itself by a map hj which fixes Tg −Dj pointwise,

and interchanges z0j and z0

j+1 ”nicely.”

The case g = 1 is similar to that described above. The case g = 0 will be treated

separately in next section below.

Remark: The exact sequences (3.1) and (3.6) offer information also about the

higher homotopy groups of FnTg,F0Tg;BnTg and B0Tg. Information about these higher

homotopy groups appears in Quintas 1968, [34].

3.3 The mapping class group of the n-punctured

sphere

Theorem 18. Every orientation-preserving self-homeomorphism of a 2-sphere, or of

a 2-sphere with one point removed, is isotopic to the identity map. Thus M(0,0) =

M(0,1) = 1.

Remark: While Theorem 18, is a well-known folk theorem, Joan S. Birman was

surprised to find that no published proof seemed to exist. Therefore she fill the gap

by including a pleasant proof, due to J.H. Roberts, who has kindly located it after a

gap of 40 years, communicated it, and allowed us to use it!

Proof. We first establish that M(0,0) is isomorphic to M(0,1). This follows from the

exact sequence (3.1) and (3.6), which coincide when n = 1:

→ π1B0,1S2 →M(0,1)→M(0,0)→ π0B0,1S

2

. The space B0,1S2 = S2 − pt is path connected, hence π0B0,1S2 = 1. Also, π1B0,1S2 =

π1S2 = 1. Hence M(0,1) ≈M(0,0).

The proof that M(0,1) = 1, that is, every orientation-preserving homeomorphism

of S2 which fixes a point p ∈ S2 is isotopic to the identity map via an isotopy which

keeps p fixed at each stage, will depend on Lemmas 7-8 below,

Lemma 7. (Alexander,1923b) If g ∶ D2 → D2 is a homeomorphism from D2 to itself

which fixes ∂D2 = S1 pointwise, then g is isotopic to the identity under an isotopy

which fixes S1 pointwise. If g(0) = 0, then the isotopy may be chosen to fix 0.

Proof. Identify D2 with the closed unit disk in R2. Let φ ∶D2 →D2 be a homeomor-

phism with φ∣∂D2 equal to the identity. We define

F (x, t) =

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

(1 − t)φ( x1−t) 0 ≤ ∣x∣ < 1 − t

x 1 − t ≤ ∣x∣ ≤ 1,

(3.4)

for 0 ≤ t < 1, and we define F (x,1) to be the identity map of D2. The result is an

isotopy F from φ to the identity. ∎

In fact, the above result can be proved in general for a n-unit ball Dn.

Lemma 8. (Schoenflies Theorem) If I1 and I2 are simple closed curves in the plane

E2 and h ∶ I1 → I2 is any homeomorphism, then there is an extension h∗ of h which

takes I1 ∪ Int I1 homeomorphically onto I2 ∪ Int I2.

Proof. This lemma will be assumed without proof. The proof of this Theorem can be

found in the following very nice elementary texts [32], [33]. ∎

Now, suppose that p ∈ S2 and that ρ ∶ S2 → S2 is an orientation preserving

homeomorphism which fixes p. We must show that there is an isotopy between ρ and

the identity which fixes p. We must show that there is an isotopy between ρ and the

identity which fixes p at each stage.

If I is a simple closed curve in S2 −p, let Int I denotes the component of S2 −I

which contains p, Ext I the other component. Choose simple closed curves I1 and I2

in S2 − p such that I1 ∪ ρ(I1) ⊂ IntI2.

Let ℘1,℘2 and ℘3 be disjoint arcs with (℘1 ∪ ℘2 ∪ ℘3) ⊂ [IntI2 − (I1 ∪ IntI1)],

each of the arcs joining the simple closed curves I1 and I2. Let xi = ℘i ∩ I1 and

yi = ℘i ∩ I2(i = 1,2,3). Since ρ is orientation preserving fixes p, there are disjoint

arcs K1,K2, and K joining ρ(x1) and y1, ρ(x2) and y2, and ρ(x3) and y3, respectively

such that Ki ⊂ IntI2 − [ρ(I1) ∪ Intρ(I1)]. (see figure below which is taken from the

Birman’s book [1])

Let Rij(i ≠ j) be the component of IntI2−℘1∪℘2∪℘3∪I1∪ IntI1 which contains

℘i∪℘j in its boundary. LetRij be the component of IntI2−[K1∪K2∪3∪ρ(I1)∪Intρ(I1)]

which contains Ki ∪Kj in its boundary.

It is easy to construct a homeomorphism g ∶ S2 → S2 which has the following

properties:

(i)g∣I1 ∪ IntI1 = ρ∣I1 ∪ IntI1

(ii)g∣I2 ∪ExtI2 = Identify∣I2 ∪ExtI2

(iii)g(Rij) =R′

ij

. Indeed, (i) and (ii) define g∣Bd℘i(i = 1,2,3). This partial map can be extended

to take ℘i to Ki homeomorphically. This defines g∣BdRij for each i ≠ j. And by

Lemma 8, g∣BdRij ∶ BdRij → BdR′

ij can be extended to a homeomorphism g∣Rij ∶

Rij → R′

ij(where R denotes the closure of the region R).

By, (i) and Lemma 7, the homeomorphisms ρ and g are isotopic under an isotopy

which moves points only in Ext I1. By (ii) and Lemma 7, the homeomorphisms g

and identity are isotopic under an isotopy which moves points only in Int I2. Since

g(p) = p, it follows from the last sentence of Lemma 7 that the latter isotopy may

be chosen to fix p. The composition of the two isotopies gives the desired isotopy

between ρ and the identity map. This completes the proof of Theorem 18 ∎

Theorem 19. If n ≥ 2, then M(0, n) = π0BnS2 admits a presentation with generators

ω1, . . . , ωn−1 and defining relations:

ωiωj = ωjωi ∣i − j∣ ≥ 2

ωiωi+1ωi = ωi+1ωiωi+1

ω1 . . . ωn−2ω2n−1ωn−2 . . . ω1 = 1

(ω1ω2 . . . ωn−1)n = 1

If n = 0 or 1, then M(0, n) = 1.

Proof. The proof can obtained in page 164, Theorem 19, from the book “Braids, Links

and Mapping Class Groups”, by Joen S. Birman, [1]. ∎

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