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 −1 ˜ X such∶ that→ for each α, p 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. S S = 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 R2 1 axes of . Next, for a fixed number λ,0<λ< /2, for = 1 example λ /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- n−1 tance λ from all theThis previous covering endpoints. space The is union called of all the theseuniversal open segments cover is of X because, it is a covering a graph, with vertices the intersection points of horizontal and vertical segments, and edges the subsegmentsspace between of every adjacent other vertices. connected We label covering all the horizontal space of X.edges a, oriented to the right, and all the vertical edges b , oriented upward. Universal cover: A covering space is a universal covering space if it is simply This covering space is called the universal cover of X because, as our general theory will show, itconnected. 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- Remark: We will use covering space theory to deduce some of the interesting damental groups are free with bases represented by the loops specified by the listed e words in a and b ,properties starting at the in basepoint Infinite groupx0 indicated Theory by the like: heavilythe freeshaded group ver- on two generators can contain tex. This can be proved in each case by applying van Kampen’s theorem. One can
also interpret the listas of a subgroup words as generators a free group of the on image any subgroup finite number ofe e generators, or even on a countably p∗ π1(X,x0) = in π1(X, x0) a,infinite b . A general set of fact generators. we shall prove about covering spaces is that e e 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 group1.1.1 on any Lifting finite number Properties of generators, or even on a countably infinite set of generators as in examples (10) and (11).
e Changing the basepoint vertex changes the subgroup p∗ π (X,xe ) to a conju- A lift of a map f Y X is a map1f˜ Y0 X˜ such that pf˜ f. We will describe 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 Xe joining one basepoint to the other. For three special lifting∶ properties→ of covering∶ spaces→ and see a few= applications of these. example, the covering spaces (3) and (4) differ only in the choice of basepoints, and First we have the homotopy lifting property, or covering homotopy prop- the corresponding subgroups of π1(X, x0) differ by conjugation by b . The main classification theorem for covering spaces says that by associating the erty
as,(we will also prove it as the construction is quite important) e e e 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 Proposition 2. Given a covering space p X˜ X, a homotopy F Y I X, and conjugacy classes of subgroups of π1(X, x0). If one keeps track of the basepoint vertex xe ∈ Xe , then this is˜ a one-to-one˜ correspondence between covering spaces ˜ ˜ 0 a map f0 Y X lifting f0, then there exits∶ a→ unique homotopy F ∶ Y ×I → X of F : e e → and actual subgroups of , not just conjugacy classes. p (X,x0) (X, x0) ˜ ˜ π1(X, x0) Of course, for thesesuch statements that ∶F toY →× make{0} sensef0. one has to have a precise notion of ∶ × → when two covering spaces are the same, or ‘isomorphic.’ In the case at hand, an iso- S = −1 Proof. Pick an open cover Uα of X such that p 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( × V)t⊂. 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. SupposẽF∶ V× [ 0, t]i → ̃X has been constructed. As, F V t , t U , we may choose some set U X containing F y, t such that U is =i i+1 αi ̃ ∶ × [ i] → ̃ i (αi × homeomorphic to U under p. Shrinking V if needed, by continuity we may assume [ ]) ⊂ αi ̃ ∈ ̃ ̃( ) ̃ ˜ −1 that F V ti Uαi . We may define F on the set V ti, ti+1 to be p F , where p−1 denotes the homeomorphism p−1 U U . By pasting lemma, the resulting ̃( × { }) ⊂ ̃αi αi × [ ] ( ) function F V 0, ti+1 X is continuous.∶ → This̃ completes the induction furnishing the map F̃ ∶ V × I[ X]that→ ̃ lifts F V ×I . ̃ ∶ × → ̃ S 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 t t t 1 such that F t t , t U for ̃ ̃ ∶ { } ×0 → 1 n ̃( ) = ̃ ( ) i i+1 αi ′ some index αi. We claim= that

lie in the same open set Uαi X that is homeomorphic to Uαi under p. But as p ˜ ̃({ } × [ ]) ̃ ({ } × [ ]) ̃( ) = ̃ ( ) Uαi ′ ′ is injective and pF F ̃pF∈ ,̃ this implies F F on y ti, ti+1 completingS 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 ṼS I X , F V S I X constructed̃ in Step 1, by Step 2, F ′ ′ and F must agreẽ ∶ on×V →Ṽ. Therefore,̃ ∶ × by→ pasting̃ together lifts F V I X for̃ each̃ point y Y , one obtains∩ a well defined lift F Y I X, completing̃ ∶ × the→ proof̃ of the proposition.∈ ̃ ∶ × → ̃ ∎ Corollary 1. (The path lifting property) Let f I X be a path such that f 0 x0. −1 ˜ ˜ Given a point x˜0 p x0 , there exits a unique lift∶ →f I X of f such that f(0) = x˜0. 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, x˜0 π1 X, x0 induced by a covering space

p X, x˜0 X, x0 is injective.∶ ( Thẽ ) image→ subgroup( ) p∗ π1 X, x˜0 in π1 X, x0 consists∶ ( ̃ of) the→ ( homotopy) classes of loops in X based at x0 (whose( ̃ lifts)) to X starting( ) at x˜0 are loops. ̃

Proposition 4. The number of sheets of a covering space p X, x˜0 X, x0 with

X and X path-connected equals the index of p∗ π1 X, x˜0 in∶ (π1̃ X, x) 0→.( ) ̃ ( ( ̃ )) ( ) Proposition 5. (lifting criterion) Suppose given a covering space p X, x˜0

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,( x˜0 )of f exists iff f∗ π1 Y, y0 p∗ π1 X, x˜0 . ∶ ( ) → ( ̃ ) ( ( )) ⊂ ( ( ̃ )) 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→ all̃ 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.) 2 1 Example: Consider the subspace A R , consisting of the circles of radius n centered at the point 1 , 0 for n 1, 2,...,. This is an example of a space that is not n ⊂ 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 X X taking a basepoint x˜ p−1 x to a basepoint p−1 x iff 1 2 ∶ ̃ → 1 ∶ ̃ 0 → 2 2 0 p1∗ π1 X1, x˜1 ∶ ̃p2∗→π̃1 X2, x˜2 . ∈ ( ) ∈ ( ) ̃ ̃ (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, x˜0 X, x0 and the set of the subgroups of π1 X, x0 , obtained by associating the∶ ( ̃ subgroup) → ( p∗ π1) X, x˜0 to the covering space X,(x˜0 . 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 group̃ → G̃ X under composition. 1 Example: For the covering space p R S 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 liftsx, ˜ x˜ of x, a deck∶ ̃ → transformation takingx ˜ tox ˜ . ∈ 1 1 1 n Example: The∃ covering R S and S S (defined by z z ) 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, x˜0 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, x˜0

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 (X̃,) 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 g∈2. ∈ ( )∩ ( ) ≠ ∅ 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 g(2 ̃U) ̃for some g1, g2 G̃ ⊂X̃, then g1 x˜1 g2 x˜2 for somex ˜1⊂, x˜2 U. Since(̃) ∩x ˜1 (and̃) ≠x ˜2∅must lie in the same∈ ( ̃ set) p−1 x , which intersects U in only one point, we must havex ˜ x˜ . Then g−1g fixes ( ) = ( ) ∈ ̃ 1 2 1 2 this point, so g−1g Id and g g . ( ) 1 2 ̃ 1 2 = 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( )S ∈ 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

nontrivialCovering element Spacesg G with g y y Section. 1.3 73 An action without fixed points is called a free action. 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 withoutIt is satisfying easy to see(∗) that, for examplefreeness the implies group of covering symmetries space action. But the converse of the familiar tiling of R2 by regular hexagons. The reason why the action of this group on R2 fails to satisfyis not(∗ true,) 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 120Example: degree rotations(Covering about space these action points, doesn’t and the midpoints implies freeness) Consider the action of of edges are fixed by 180 degree rotations. An action without fixed points is called a free action. Thus for a freeZ actionon S1 ofinG whichon Y , onlya generator the identity of elementZ actsby of G rotationfixes any through an angle α that is an 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) −1 irrational= multiple of 2π.∗ is equivalent to g1 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 exerciseExample at the endLet of Ythe be section the closed is to show orientable that for surface actions of genus 11, an ‘11-hole torus’ on Hausdorff spaces, freeness plus proper discontinuity implies condition (∗). Note that proper discontinuityas is shown automatic in the for∶ actions figure.(taken by a finite from group. Hatcher’s Algebraic Topology) 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 Z ∗ the cyclic group 5 acting on Y , and the condition ( ) is C4 C 3 Z obviously satisfied. The quotient space Y/ 5 is a surface of genus 3, obtained from one of the five subsurfaces of C C ··· 5 2 Y cut off by the circles C1, ,C5 by identifying its two C1 boundary circles Ci and Ci+1 to form the circle C as shown. Thus we have a covering space M →M where 11 3 p Mg denotes the closed orientable surface of genus g .

In particular, we see that π1(M3) contains the ‘larger’ C group π1(M11) as a normal subgroup of index 5, with Z quotient 5 . 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 Mh then g = mn + 1 and h = mThis+ 1 for has some a 5-foldm and rotationaln. symmetry, generated by a rotation of angle 2π 5 Thus

As a special case of the final statement of the preceding proposition we see that we have the cyclic group Z5 acting on Y , and the condition for the covering space 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 action is satisfied(by Proposition 12 below). The quotient space Y 5 is a surface isomorphism an element g ∈ G corresponds to a loop in Y/G that is the projection of Z 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∃ gx is continuous we have for each g G, an open∩ neighborhood≠ Wg −1 of x0 such thatz→ gWg Vk if gx0 xk(one can take W∈g V∃0 g Vk). Then we have ⊂ = U V0 g∈G Wg = ∩ 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

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

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

0 0 identifying the two endpoints of each Iα with points of X . The points of X 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.

0 Since X has the quotient topology from the disjoint union X 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.∶ Sincẽ → X is a graph( by( ̃ the)) = preceding proposition,( ̃) ≈ the group G π1 X is free by Propositioñ 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

n n 0, 1 . The boundary−∂I of I 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 n n to be the set of homotopy classes of maps f I , ∂I X, x∈0 , where homotopies( ) n ft are required to satisfy ft ∂I x0 for all∶ (t. The) definition→ ( ) extends to the case 0 0 n 0 by taking I to be a point( ) and= ∂I 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 ( )

1 f 2s1, s2, . . . , sn , s1 0, f g s , s , . . . , s 2 1 2 n ⎧ ⎪ 1 ⎪g(2s1 1, s2, . . .) , sn , s1 ∈ [ 2 , 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

n πn X, x0 is a group, with identity element the constant map sending I 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. ( ) n n n n n Maps≥ I , ∂I X, x0 are the same as maps of the quotient I ∂I S to x n n taking the( basepoint) → s(0 ∂I) ∂I to x0. This means that we can view~ πn =X, x0 as n homotopy classes of maps= S ~, s0 X, x0 , where homotopies are through( maps) of n the same form S , s) X,( x0 .) In→ this( interpretation) of πn X, x0 , the sum f g is f∨g n c n n n−1 n composition S ( S →S( )X where c collapses the equator( S ) in S to a+ point Ð→ ∨ ÐÐ→ n−1 and we choose the basepoint s0 to lie in this S . 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 n n another basepoint( ) x1 ( γ 1 ), we may associate to each map∶ f → I , ∂I =X,( x1) a n n new map γf I , ∂I = ( X,) x0 by shrinking the domain of f ∶to( a smaller) → concentric( ) n cube in I , 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( : + ) ( + )

1 γ f 0 2 t s1, s2, . . . , sn s1 0, h s , s , . . . , s 2 t 1 2 n ⎧ ⎪ 1 ⎪γ(0 +g)((2 −t)s1 t 1, s2,) . . . , sn s1 ∈ [ 2 , 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 π))n, 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 π . The module structure on π is given by [ ] ∑ ∈1 ∈ n

i niγi α i ni γiα for α πn. Sometimes people say πn is a π module rather than a π module. (∑ Z) 1= ∑ ( ) ∈ − 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, x˜0 X, x0 induces isomor- phisms p∗ πn X, x˜0 πn X, x0 for all n 2. ∶ ( ̃ ) → ( ) ∶ ( ̃ ) → ( ) ≥ Proof. To prove surjectivity of p∗ we apply the lifting criterion in Propostion 5, which

n implies that every map S , s0 X, x0 lifts to X, x˜0 provided that n 2 so that

n S 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

1 n applies for example( to)S=. In general,≥ the n dimensional torus T , the product of n n n circles, has universal cover R , so πi T −0 for i 1. This is marked in contrast n to− the homology groups Hi T 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.

n n Taking Y to be S∶ and→ S 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. n−1 n n−1 To define these,( regard) I as the( face) of I with last coordinate∈ sn 0 and let J be n n−1 n the closure of ∂I I , the union of the remaining faces of I . Then= πn X, A, x0 for − ( ) n n n−1 n 1 is defined to be the set of homotopy classes of maps I , ∂I ,J X, A, x0 , with≥ homotopies through maps of the same form. Note that( πn X, x0, x)0 → (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

1 0 0 is abelian for n 3. For n ( 1 we have) I 0, 1 ,I ≥0 , and J 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. n Just as elements of πn ∈X, x0 can be regarded as homotopy classes of maps S , s0 X, x0 , there is also an alternative( ) definition of πn X, A, x0 as the set of homotopy( ) → n n−1 n−1 (classes) of maps D ,S , s0 X, A, x0 , since collapsing( )J to a point converts n n n−1 n n−1 I , ∂I ,J into( D ,S ), s→0 (. From this) viewpoint, addition is done via the map n n n n−1 n (c D D )D collapsing( D ) D 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:

n n−1 (A map )f D ,S , s0 X, A, x0 represents zero in πn X, A, x0 iff it is

n−1 ● homotopic∶ rel( S 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 n 0 n [D] =onto s . Conversely, if f 0 via a homotopy F D I X, then by restricting n n F to a familly of n disks in[ ]D= I starting with D ∶ 0× and→ ending with the disk n n−1 D 1 S I,− all the disks in× the family having the× { same} boundary, then we get n−1 a homotopy× { } ∪ from× f to a map into A, stationary on S .

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, y(0 . ) = = = ≃ ( The most) → ( useful property) of the relative groups πn X, A, x0 is that they fit into a long exact sequence ( )

i∗ j∗ ∂ πn A, x0 πn X, x0 πn X, A, x0 πn−1 A, x0 π0 X, x0

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

n n n−1 n−1 The map ∂ is just the restrictions of( the) map↪ ( I , ∂I) ,J ( X,) A,↪ x0( to I ), n n−1 n−1 or by restricting maps D ,S , s0 X, A, x0( to S . 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.

k Corollary 4. πn S 0 for n k.

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

n k points, then every basepoint-preserving map S S can be homotoped,− fixing the basepoint, to be cellular, and hence constant if n→ k. < ∎ 1.3.3 Freudenthal suspension theorem

n n+1 Theorem 4. The map πi S πi+1 S is an isomorphism for i 2n 1 and a

surjection for i 2n 1. More( ) generally→ ( this) holds for the suspension πi

(n − ) Corollary 5. πn S Z, generated by the identity map, for all n 1. In particular, n the degree map πn( S) ≈ 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. p This sort of short exact sequence F E B, called a fiber bundle, is prominent from the type A X X A in that→ it hasÐ→ more homogeneity: All the subspace −1 p 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 p algebraic homogeneity in a short exact sequence of groups 0 K G H 0

−1 where the ‘fibers’ p h are the cosets of K in G. In a few fiber→ bundles→ FÐ→ 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:

1 3 2 1 3 πi S πi S πi S πi−1 S πi−1 S ...

⋅ ⋅ ⋅ → ( ) → ( ) → ( ) → 2 ( ) → 1 ( ) → The exact sequence gives an isomorphism π2 S π1 S since the two adjacent

3 3 terms π2 S and π1 S are zero by cellular( approximation(see) ≈ ( ) Theorem 3 above). ( ) ( ) 2 1 Thus we have a direct homotopy-theoretic proof that π2 S Z. Also, since πi S

0 for i 1 by Proposition 16, the exact sequence implies( that) ≈ there are isomorphisms( ) = 3 2 2 3 πi S > πi S for all i 3, so in particular π3 S π3 S , 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 mapg ˜0 X E lifting g0, so pg˜0 g0, then a homotopyg ˜t 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 formg ˜t x gt x , h×x →whereg ˜0 x g0 x , h x .

Theorem( ) = 5.( Suppose( ) ( p)) E B (has) = the( homotopy( ) ( )) lifting property with respect to

k −1 disks D for all k 0. Choose∶ → basepoints b0 B and x0 F p b0 . Then the map p∗ πn E, F, x≥0 πn B, b0 is an isomorphism∈ for all∈ n= 1.( Hence) if B is path-connected,∶ ( there) is→ a long( exact) sequence ≥

p∗ πn F, x0 πn E, x0 πn B, b0 πn−1 F, x0 π0 E, x0 0

⋅ ⋅ ⋅ → ( ) → ( ) Ð→ ( ) → ( ) → ⋅ ⋅ ⋅ → ( ) → 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 homotopyg ˜t X E starting with a given( lift) g ˜0 and extending a given∶ → liftg ˜t A E. In other words,∶ → the homotopy lifting property for X,A is the lift extension∶ property→ for X I,X 0 A I . k k k k k ( Since) the pairs D I,D 0 and (D ×I,D × {0} ∪∂D× I) are homeomorphic, k we see that the homotopy( × lifting×{ property}) ( for×D 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 liftingg ˜t one cell of X A

k at a time. Composing with the characteristic map Φ D X of a cell then gives− a k k reduction to the case X,A D , ∂D . 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

−1 a homeomorphism∶ → h p U U F making the diagram (below) commute, where the unlabeled map is∶ projection( ) → onto× the first factor. p−1 U h U F p ( ) × U

−1 Commutativity of the diagram means that h carries each fiber Fb p 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.

p Proposition 19. A fiber bundle F E B has the homotopy extension property for disks. → Ð→

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

−1 G G G that takes x, y xy ), 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. ∎

−1 −1 Proof. Note that p p U g∈GUg u∈U uF hence p 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

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

Lg L ( g) κ g V g p−1 g V gp−1 V

( ) ( ( )) = ( ) −1 So, κg x gκ g x 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∈ S ( 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 3 2 2 u 1. The cyclic group C3 1, u, u has a representation= ( ~ )µ on C given by:

= = { } 1 0 µ 1 ⎡ ⎤ ⎢0 1⎥ ⎢ ⎥ ( ) = ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣1 0⎦ µ u ⎡ ⎤ ⎢0 u⎥ ⎢ ⎥ ( ) = ⎢ ⎥ ⎢ ⎥ ⎢1 0⎥ µ u2 ⎣ ⎦ ⎡ 2⎤ ⎢0 u ⎥ ⎢ ⎥ ( ) = ⎢ ⎥ ⎢ ⎥ 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

< S = = = > 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 :

−1 Sλ,a λa λa λ Λ, a a1, a2, . . . , an

= {( )( ) S ∈ ∈ { }} 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:

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

R a1 a2 . . . ak ,  1 i j1 j2 jk i

th = ( = ± ) th Case I: Look at the i letter aji , if its index i 1 then the i letter of the

corresponding relator of H will be Sλ ,a where i ji = +

λ a1 a2 . . . ai−1 i j1 j2 ji−1 = Case II: If its index i 1, then letter of the corresponding relator of H will be S−1 where λi,aji = − λ a1 a2 . . . ai i j1 j2 ji

Then the corresponding relation of H= will be

1 2 k Ri Sλ ,a Sλ ,a ...Sλ ,a 1 1 j1 2 j2 k jk ̃ = =

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

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

2 10 establishes the relationships between> Artin’s classical braid groups on E 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

2 and defining relations for full braid group Bn of E . 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 Dn Bn.

2 Section 2.5 discusses the braid group of S , which willM( 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. n Let M be a manifold of dimension 2. let i=1 M denote the n-fold product space, and let F M denote the subspace 0,n ≥ ∏ n F0,nM z1, . . . , zn M zi zj, i j (2.1) i=1 = {( ) ∈ M S ≠ ≠ } 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′ , z′ , . . . , z′ of z′ by a permutation. Let B M de- 1 2 n ( 0,n ) 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. ∶ → 2 The classical braid group of Artin (see [4] and [13]) is the braid group π1B0,nE ,

2 2 where E denotes the Euclidean plane. Artin’s geometric definition of π1B0,nE can be recovered from the definition above as follows:

0 0 0 2 0 2 Choose a base point z z1, . . . , zn F0,nE and a pointz ˜ B0,nE such that 0 0 2 2 0 p z z˜ . Any element in π=1(B0,nE π)1 ∈B0,nE , z˜ is represented∈ by a loop

( ) = 2= ( ) 0 α I B0,nE , where α 0 α 1 z˜

, which lifts uniquely to a∶ path→ ( ) = ( ) =

2 0 α˜ I F0,nE starting atα ˜ 0 z

∶ → ( ) =

Ifα ˜ t α˜1 t , α˜2 t ,..., α˜n t , t I, then each of the coordinate functionsα ˜i

2 2 defines( ) (via= ( its( ) graph)( ) an arc ( i)) α˜∈i t , t in E I. Sinceα ˜ t F0,nE the arcs 1,..., n (as at any particular℘ time= ( (t,) any) twoα ˜×i can’t be different)( ) ∈ are disjoint. th ℘Their union℘ 1 ... n is called a geometric braid. The arcs i is called the i 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

2 ′ ′ π1B0,nE . Thus, if and are geometric braids, then (i.e. they represent

2 ′ the same element of℘π1B0,n℘E ) if the pathsα ˜ andα ˜ which℘ ∼ defines℘ these braids are 0 0 2 homotopic relative to the base point z1, . . . , zn in the space F0,nE . Thus we require

the existence of a continuous mapping( )

2 F I I F0,nE

∶ × → with

F t, 0 F1 t, 0 ,...,Fn t, 0 α˜1 t ,..., α˜n t

( ) = ( ( ) ( )) = ( ′( ) (′ )) F t, 1 F1 t, 1 ,...,Fn t, 1 α˜1 t ,..., α˜n t

( ) = ( ( ) ( )) = ( ( )0 0 ( )) F 0, s F1 0, s ,...,Fn 0, s z1, . . . , zn

F (1, s ) = (F (1, s ),...,F (1, s )) = (z0 , . . . , z0 ) 1 n µ1 µn

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

′ 2 2 if there℘ is an℘ isotopic deformation Gs of E I which℘ ℘ ⊆ is the× identity on E 0 2 and℘ ≈ ℘ on E 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

2 ● is, s meets∈ [ each] plane E t℘0 (, t)0 I℘, in precisely n points, and moreover ′ 0℘( ) , 1 × { } ∈ ℘( ) = ℘ ℘( ) = ℘ It was proved by Artin (see [13]) that ′ iff ′. Thus a braid homotopy

2 may always be “extended” to E 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

2 plane E 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

2 homotopies of the individual paths αi t , t in E 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 2 2 group π1B0,3E which are homotopic to the identity element of π1B0,3E . She goes on ˆ ˆ to define a ”homotopy braid group,” Bn, and finds a group presentation for Bn which ˆ 2 exhibits Bn as a quotient group of the group π1B0,nE . 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 α→1, 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 −1 multiply as in( the) groupoid of paths. Now define the inverse f of a motion f to be the inverse of the path f in H M , translated so that its initial point is 1M .

−1 Eventually, let motions f and( g)be equivalent if the path f g 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

−1 classes of these elementary braids will be denoted by the symbol σi and σi . In the example of Figure 1,

2 −1 Ω σ1σ2σ3

2 Intuitively, σ1, σ2, . . . , σn−1 generate= the group π1B0,nE , a fact which will be proved later.

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

σiσj σjσi if i j 2; 1 i, j n 1 (2.2)

= S − S ≥ ≤ ≤ − 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

2 π1B0,nE . 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 Qm. 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.

0 0 Proof. let us first consider, for some base point z1, . . . , zr in Fm,rM, the fibre π−1 z0, . . . , z0 : 1 r ( ) ( )

−1 0 0 0 0 0 0 π z1, . . . , zr z1, . . . , zr , yr+1, . . . , yn ,where z1, . . . , zr , yr+1, . . . , yn

are distinct points in M Qm . (2.6) ( ) = {( } ) −

0 0 define Qm+r Qm z1, . . . , zr , then

= ∪ { }

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

and there is a natural homeomorphism

−1 0 0 h:Fm+r,n−r M π z1, . . . , zr (2.8)

( ) Ð→ ( )

0 0 defined by h yr+1, . . . , yn z1, . . . , zr , 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 Q∈ m+1−and then= pick= a

homeomorphism α M M, fixed on Qm, such that α qm+1 x0. Let U denote ¯ a neighborhood of x∶0 in →M Qm which is homeomorphic( to an) open= ball, and let U ¯ ¯ denote the closure of U. Define− a map θ U U U with the following properties. Setting θz y θ z, y we require: ∶ × Ð→ ¯ ¯ ¯ i) θ U( )U= is( a homeomorphism) which fixes ∂U. ii) θ∶z 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 ∶ × →

φ −1 ( ) = ∉ U Fm+1,n−1M π U φ−1 is given by × ( )

−1 −1 φ z, z2, . . . , zn z, θz α z2 , . . . , θz α zn (2.9)

−1 −1 −1 φ (z, z2, . . . , zn) = (z, α θ(z z)2 , . . . , α θ(z z))n (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,n≥M Fm,1M( M )Q=m

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 π2→Fm+1(,n−1−M 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∎

0 0 0 0 z1, . . . , zn be the base point for π1F0,nM. Let Fn−1,1M M Qn−1 M z1, . . . , zn−1 .

Define( j to) be the inclusion map from Fn−1,1M to F0,nM= as− = −{ }

0 0 0 0 j zn z1, . . . , zn−1, zn zn M z1, . . . , zn−1 (2.12)

Theorem 7. If π(2 M) = (Qm π3 M )Qm π0∈M −Q{ m 1 for} every m 0, then the following sequence( of− groups) = and( homomorphism− ) = ( − is exact:) = ≥

0 j∗ 0 0 π∗ 0 0 1 π1 Fn−1,1M, z π1 F0,nM, z1, . . . , zn π1 F0,n−1M, z1, . . . , zn−1 1

→ ( ) Ð→ ( ( )) Ð→ ( ( )) (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,∎

2 2 to determine group presentation( ) for π1B0,nE and π1B0,nS .

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 n M induces a homomorphism ∶ → ∏

ik ∗ πkF0,nM πkM (2.14) n which is surjective if dim M (k )and∶ also injective→ M 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∎

2 2 π1B0,nE and Artin’s pure braid− group π1F0,nE 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].

0 0 2 Choose z1, . . . , zn be the base point for the group π1Fm,nE , as before, and regard E2 as an open disc in M which contains the n points z0, . . . , z0 and also the ( ) 1 n 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 2 view, the space Fm,nE may be identified with a subset( of −Fm,nM) by composing any 2 2 2 map from Pn to Fm,nE with the inclusion map E M. Let em,n Fm,nE Fm,nM

∗ 2 be the required identification. Then the induced map⊆ em,n π1Fm,n∶ E →π1Fm,nM 2 takes any n string braid on E Qm and considers it as a braid∶ in M→ Qm .

− ( − ) 2 2 (∗ − ) Theorem 9. If M is any compact surface except S or P , then ker e0,n 1 .

= { } Proof. ([18]; see also Goldberg, 1973 [20] for a different proof). The homomorphism

∗ em,n together with the exact sequences of Theorem 7, yield a commutative diagram:

2 2 2 1 π1 E Qn−1 π1F0,nE π1F0,n−1E 1

e∗ e∗ e∗ (n−1,1− ) 0,n 0,n−1 1 π1 M Qn−1 π1F0,nM π1F0,n−1M 1

∗ Note that en−1,1 (is injective− ) for each n(using Van-kampan Theorem, just think 2 2 ∗ E Qn−1 as D Qn−1). Now we can use the diagram inductively to prove that e0,n is injective as well. To be precise, e∗ is injective since π F E2 π E2 1. This − − 0,1 1 0,1 1 begins the induction. Suppose inductively that e∗ is injective. Then the strong 0,n−1 = = ∗ 5-lemma implies that e0,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  are isomorphism, then γ is an isomorphism as well.

j A i B C k D l E

α β γ δ  ′ j′ ′ ′ A′ i B′ C′ k D′ l E′

∎ 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 n M be the inclusion map. Then in the following sequence of (not necessarily abelian) groups ∶ → ∏

e∗ n 2 0,n i∗ 1 π1F0,nE π1F0,nM π1M 1 i=1 → ÐÐ→ Ð→ M → 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

2 2 From Theorem 8, 9 and 10, clearly, the groups π1B0,nE and π1F0,nE need special attention. So, in this section we will consider only the case M E2. And, we will

2 2 adopt abbreviations B0,n and F0,n for the spaces B0,nE and F0,n=E . In this section the short exact sequence of Theorem 7 will be used inductively to

2 show that π1F0,n is constructed in nice ways from the building blocks π1 E Qi , 1 i n 1(Corollary 6), finding simultaneously generators and relations for( the− groups) ≤ π1≤B0,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)

= S − S ≥ ≤ ≤ − 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 symbolsσ ˜1,..., σ˜n−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 σ and σ−1 in Figure 2.) we now give an equivalent ∶ → 1 1 definition which is more precise. Recall the covering projection φ F0,n B0,n. Choose

0 the point φ 1, 0 ,..., n, 0 z˜ as base point for the group π1B∶ 0,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( ))σ ˜i 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)

( ) = (( ) 2 ( − ) ( ) ( ) ( + 2 ) ( )) where li t i t, t t and li+1 t i 1 t, t t . That is, l t is constant

th √ st √ on all but( ) the= (i+ and− i −1 )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

0 µ˜ π1 B0,n, z˜ Σn

∶ ( ) → be defined as follows: Letα ˜ π1B0,n be represented by a loop

∈ ˜ 0 β I, 0, 1 B0,n, z˜

∶ ( { })0→ ( ) ˜ and let β β1, . . . , βn I, 0 F0,n, z be the unique lift of β. Define

= ( ) ∶ ( { }) → ( )

β1 0 , . . . , βn 0 µ˜ α˜ Σ (2.18) ⎡ ⎤ n ⎢β1 1 , . . . , βn 1 ⎥ ⎢ ( ) ( )⎥ ( ) = ⎢ ⎥ ∈ ⎢ ⎥ The kernel of these homomorphism⎢ (µ ˜)is the pure( )⎥ braid group π F . With respect ⎣ ⎦ 1 0,n to the homomorphismµ ˜ 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 Pn is an isomorphism onto π1F0,n. ∶ → S

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:)S ≤ ≤ − } µ 1 Pn Bn Σn 1

in=iSPn i 1 µ˜ 1 π1F0,n π1B0,n Σn 1

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.

= S ∎ To show that i Pn is an isomorphism onto, we next find a presentation for Pn.

Lemma 2. The groupS Pn admits a presentation with generators

2 −1 −1 −1 Aij σj−1σj−2 . . . σi+1σi σi+1 . . . σj−2σj−1 1 i j n (2.20)

= ( ≤ < ≤ ) and defining relations

Ai,j if r s i j or i r s j, ⎧ ⎪ −1 ⎪Ar,jAi,jAr,j if

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 n of all products of the form j=2 Mj,kj ; j kj 1 where Mj,i σj−1σj−2 . . . σiifj i, or 1 if j i. For example, coset representatives for P in B are the set M M {∏ ≥ ≥ } 3 3 = 22 33≠ 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{ byS the≤ η

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∗ 2 π1F∶n−1,1 π1→E Qn−1 , which is a free group of rank n 1. = It is easy= ( to see− that) the following diagram is commutative:− η 1 Un Pn Pn−1 1

inSUn in in−1

π∗ 1 π1Fn−1,1 π1F0,n π1F0,n−1 1

with exact rows. In the bottom row(using the notation used in Theorem 7) the base

0 0 0 2 point for π1F0,n is z1, . . . , zn , so that zn is the base point for π1Fn−1,1 π1 E z0, . . . , z0 . Now, from equation (2.20) and our picture definition of the geometric 1 n−1 ( ) = ( − {braidsσ ˜i })in σn one may identify the image in Ajn of the generator Ajn of Un as being represented by a loop based at z0 which encircles the point z0 once and = ( ) n ( ) j 0 0 0 0 separates it from z1, . . . , zj−1, zj+1, . . . , zn−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)S ≤ < 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 Un is an isomorphism onto. Now observe that P1 1 and π1F0,1 1. Therefore i1 isS 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 completesS 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 2 π1B0,n of the plane E . 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 U∈n and βn−1 Pn−1. By induction,

= ( > ) ∈ β β2β3 . . . β∈nπβ

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:

−1 xi xixi+1xi ⎧ ⎪ σi ξ ⎪xi+→1 xi (2.22) ⎪ ⎪ ( ) ∶ ⎨xj →xj if j i, i 1. ⎪ ⎪ The restriction of ξ to the pure⎪ braid→ group Pn maps≠ + the generator Ars of Pn to the ⎩⎪ automorphism:

xi if s i or i r, ⎧ ⎪ −1 ⎪xrxixr if

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 byx ˙ 1,..., x˙ n, we find= from~[ 2.23 ] that the automorphism of Fn induced by σi mapsx ˙ i x˙ i+1, x˙ i+1 x˙ i andx ˙ j ( x˙j)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. ~ ≃ S 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( )U=n+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 =

φ(σ)=ρσ Pn AutUn+1 ξ ≈

AutFn

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 β=n 1. Now, β commutes with each element of Un+1, hence≠ β commutes= with=A⋅i,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,...,A(j−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:

−1 βiAi,n+1βi 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= −1 πAi,n+1π An,n+1 i 1, . . . , n (2.25)

= ( = ) −1 πAk,iπ Ak,n+1 k 1, . . . , i 1 (2.26)

To establish equation (2.26), note= first that ( = − )

2 −1 −1 −1 −1 2 Ai,n+1 σn . . . σi+1σi σi+1 . . . σn σi . . . σn−1σnσn−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:

−1 2 −1 −1 −1 −1 −1 −1 πAi,n+1π σnσn−1 . . . σi σnσn−1 . . . σi+1σi σi+1 . . . σn σi σi+1 . . . σn−1σn σ σ . . . σ σ−1 . . . σ−1 σ2σ . . . σ σ−1σ−1 . . . σ−1 n n−1 = ( i i n)(−1 n n−1 i i i+1 n)( ) 2 = (σn 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:

−1 2 −1 −1 −1 −1 −1 πAk,iπ σnσn−1 . . . σi σi−1σi−2 . . . σkσk+1 . . . σi−1 σi σi+1 . . . σn Ak,n+1 by definition

= ( )( )( ) = ( ) Transforming equation (2.25) by π, and applying (2.26), we obtain:

−1 −1 −1 πβiπ An,n+1 πβi π An,n+1 (2.28)

( ) ( −1 ) = But, by equation (2.27), the elements πβiπ 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

−1 s that group. Since An,n+1 is a generator of Un+1, it then follows that πβiπ An,n+1 for some integer s. But then β π−1As π, and by equation (2.26) this is precisely i n,n+1 = As . We now have established the sought-for contradiction, for β belongs to the i,n+1 = i s free group Ui, and the only way that Ai,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

2 group of Theorem 11, and the geometric braid group π1B0,nE , 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

≥ n σ1σ2 . . . σn−1 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( A1)(nA2n ...A) n−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:

2 2 n ∆n σ1σ2 . . . σn−1 σ1σ2 . . . σn−2 ... σ1σ2 σ1 σ1σ2 . . . σn−1

= (( )( ) ( ) ) = ( ) We will verify it for n 3 and then by induction we can prove it in general,

for n 3 =

= 3 2 2 σ1σ2 σ1σ2σ1 σ2σ1σ2 σ1σ2σ1 σ1σ2σ1 σ1σ2σ1 ∆3

( ) = ( )( ) = ( )( ) = ( ) = 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-twist achieved by keeping the top of the braid fixed and turning over the row of the lower ends by an angle of π. See Figure 1.11 for a diagram of Δ5. The braid 2 θn = Δn can be obtained from the trivial braid 1n 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.Wehave

π(Δ )=(n, n 1,...,1) S . n − ∈ n

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 to bottom, one easily obtains for all i =1, 2,...,n 1, −

σi Δn = Δn σn i . (1.8) − 2 The braid θn ∆n 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:

= ∈ ∈ −1 −1 ∈ ( ) βn−1βnAinβn βn−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 − = −

−1 −1 βn A1nA2n ...An−1,n βn βn−1 A1nA2n ...An−1,n βn−1

( ) = ( ) Since βn−1 Pn−1, condition ii) above then implies:

∈ −1 βn A1nA2n ...An−1,n βn A1nA2n ...An−1,n

( ) = ( ) . This equality holds in the free group Un, hence the only possibility is

m βn A1nA2n ...An−1,n

= ( ) for some integer m. Using this information in 2.30 , we thus obtain:

−1 m ( ) −m βn−1Ainβn−1 A1nA2n ...An−1,n Ain A1nA2n ...An−1,n 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

−m A12 A13A23 ... A1,n−1A2,n−1 ...An−2,n−1

[ ( ) ( −1 )] −1 has precisely the effect which our element βn−1 is required to have, hence βn−1 must −m be precisely A12 A13A23 ... A1,n−1A2,n−1 ...An−2,n−1 for some integer m. Thus our original element[ ( β β)n−1β(n can only have been: )]

= m β A12 A13A23 ... A1,n−1A2,n−1 ...An−2,n−1 A1,nA2,n ...An−1,n

= [( )( ) ( )( )] (Just substitute the value of βn−1 and βn and then use property ii) to get β) m Now, It can be easily seen that β σ1σ2 . . . σn−1 , 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 = β satisfies the two conditions

∈ ⊂−1 xi β Aixµi Ai 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

−1 −1 −1 A1xµ1 A1 A2xµ2 A2 ...Anxµn An x1x2 . . . xn (2.33)

= −1 which results from (2.32) and (2.23). we assume that each term Aixµi Ai 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) x A−1 is absorbed by A µ=v v − v+1 or

−1 b) Av 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

−1 Aixµi Ai , 1 i n. that is,

≤ ≤ −1 −1 −1 l β l A1xµ1 A1 A2xµ2 A2 ...Anxµn An

( ) = ( ) that is “length of alphabets”. We will show that if (a) is true, then σvβ has shorter

−1 length then β, while if (b) is true σv 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

−1 automorphisms σv or σv . That is,

l σvβ l β , then

let( γ1) <σv(β ) l σv1 γ1 l σv1 σvβ l γ ... = Ô⇒ ( ) = ( ) < ( )

γn σvn β l σvn γn l σ1σ2 . . . σn−1β l γn β σ1 σ2 . . . σn−1 where  1or 1 = Ô⇒1 2 ( n−1) = ( i ) = ( ) HenceÔ⇒ 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:

−1 −1 ˜ ˜−1 −1 xv β Avxµv Av , xv+1 β Avxµv Av+1xµv+1 Av+1xµv Av (2.34)

˜ ( ) ˜−=1 ( ) = −1 (where Av+1 and Av+1 are residue after cancellation of Av+1 and Av+1 respectively)

Using the action given in Corollary 8, we now compute the product σvβ:

−1 −1 xv σvβ xvxv+1xv β xv β xv+1 β xv β

−1 ( −1) ˜ = ( ˜−1 ) −1= ( ) −(1 −1 ) ( ) =(Avxµv Av Avxµv Av+1xµv+1 Av+1xµv Av Avxµ Av

)( )( ) ˜ ˜−1 −1 =AvAv+1xµv+1 Av+1Av , we get by applying equation (2.35) −1 (xv+1 σvβ xv β Avxµv Av (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

−1 that one of the term Avxµv Av 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 , −1 then (a) is satisfied. If xµv is absorbed by a letter in Av−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

−1 that Avxµv Av is completely absorbed. In that case, some residue Rv will remain for −1 each AvxµAv after all free reductions have been made. Then (2.34) implies

R1R2 ...Rn x1x2 . . . xn

= −1 . This implies that Ri xi for each i 1, 2, . . . , n. Now examine the term A1xµ1 A1 .

The initial letter in this= term can only= be x1. We consider first the case where −1 A1xµ1 A1 is not identically x1, that is,

−1 ˜ ˜−1 −1 A1xµ1 A1 x1A1xµ1 A1 x1

˜ ˜−1 −1 = Since by hypothesis A1xµ1 A1 x1 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 −1 xµ2 (in which case (b) is satisfied). If, on the other hand, A1xµ1 A1 x1 the entire

= −1 argument can be repeated for A2xµ2 A2 , etc. In this way we see that in every case either (a) or (b) is true, hence Theorem 12 is completely proved.

J. Gonzalez-Meneses´ Now using Theorem 12, we will give a new geometric interpretation on the group∎

Bn. distinct heights, that is, for distinct values of t ∈ [0, 1]. In this way, it is clear that every braid is a product of braids in which only two consecutive strands cross. That is, if one considers for i = 1,...,n − 1 the braids σi −1 2.4.1and σ Braidsi as in Figure as Automorphisms 2, it is clear that they are ofthe inverse free of groups each other, or Mapping and that {σ1,...,σn−1} is a set of generators of Bn, called the standard generatorsClass, or groups the Artin generators of the braid group Bn.

21.6. 2 Let D beBraids a disc, and as automorphisms let Qn q1, . . . , qofn thebe afree set group of fixed, distinguished points of D . We shall now give still another2 interpretation of braids. This is one of The fundamental group π1 D= { Qn is free} group of rank n. Let x1, . . . , xn be a basis the main results in Artin’s paper [2]. There is a natural representation for π 2 Q , where x i 1, . . . , n is represented by a simple loop which encloses 1ofD braidsn on n strandsi as( automorphisms− ) of the free group Fn of rank n. Although Artin visualized braids as collections of strands, we believe that the boundary points qi, but no boundary point qj for j i. See Figure below(taken (it is more− ) natural to define( = their representation) into Aut(Fn) by means of mapping classes, as was done by Magnus [57]. from the paper ‘Basic results of braid groups’ [22]). ≠

Figure 3. The loops x1,...,xn are free generators of π1(Dn).

We remark that the fundamental group of the n-times punctured disc Dn 2 Theorem 13. Let M be the group of automorphisms of π1 Dn (where Dn D Qn) is precisely the free group of rank n: π1(Dn)= Fn. If we fix a base point, say in the boundary of D , one can take as free generators the loops x ,...,x 2 which are indicated by then homeomorphisms of Dn which keep( the1) boundaryn of= D −fixed depicted in Figure 3. Now a braid β ∈ Bn can be seen as an automorphism pointwise.of Dn Thenup to isotopy,M is precisely so β induces the groupa well definedBn. action on π1(Dn) = Fn, where a loop γ ∈ π1(Dn) is sent to β(γ). This action is clearly a group Proof.homomorphismFor geometric (respects reasons, concatenation each elements of loops),β whichM issatisfies bijective conditions as β−1 (2.32) and yields the inverse action. Hence β induces an automorphism of Fn, and (2.33),this hence givesM a representation:Bn. ∈ We fix a base⊆ point, sayρ in: theBn boundary−→ Aut( ofFnD) n, one can take as free generators the β −→ ρβ. loops x1, x2, . . . , xn depicted in figure above. Now a braid β Bn can be seen as an 8 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 −1 a group homomorphism∈ ( ) (respects concatenation( ) of loops), which is bijective as β yields the inverse action. Hence β induces an automorphism of Fn, and this gives a representation:

ρ Bn Aut Fn

∶ → ( ) BASIC RESULTS ON BRAID GROUPS

which sends β to ρβ The automorphism ρβ can be easily described when β = σi, by giving the image of the generators x ,...,x of F (see Figure 4). Namely: The automorphism ρβ can be1 easilyn describedn when β σi, by giving the image of −1 ρσi (xi)= xi+1, ρσi (xi+1)= xi+1xixi+1, ρσi (xj)= xj (if j = i, i + 1). the generators x1, . . . , xn of Fn. Namely: −1 = The automorphism ρ can be easily deduced from ρσi . For a general σi braid β, written as a product of σ1,...,σn−1 and their inverses, the auto- morphism ρ is just the composition−1 of the corresponding automorphisms ρσi xi xβi+1, ρσi xi+1 xi+1xixi+1, ρσi xj xj ifj i, i 1 corresponding 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 The automorphism ρ −1 can be easily deduced from ρσ . For a general braid β, σi σiσj = σjσi, |i − j| >i 1 Bn = σ1,...,σn−1 . (1.1)  σiσjσi = σjσiσj, |i − j| = 1  written as a product of σ , . . . , σ and their inverses, the automorphism ρ is just 1 n−1 β It is then very easy to check that ρ is well defined, as ρσiσj ≡ ρσj σi if the composition|i − j| > 1, of and theρ correspondingσiσj σi ≡ ρσj σiσj automorphismsif |i − j| = 1. Artin corresponding [3] showed that to eachρ is letter. faithful by topological arguments, making no use of the above presentation.

It is easyNotice to that check for that everyρ βis∈ wellBn, defined, the automorphism as ρσiσj ρρβσjsendsσi if eachi j generator1, and ρσiσj σi xj to a conjugate of a generator. Notice also that for each i = 1,...,n − 1, ρσ σ σ if i j 1. j i j one has ρσi (x1 xn)= x1 xn. Hence ρβ(x1 =xn)= x1 S − xnS for> every ≡ NoteS that− S for= every β Bn, the automorphism9 ρβ sends each generator xj to a conjugate of a generator.∈ Notice also that for each i 1, 2, . . . , n 1, one has ρ x x . . . x x x . . . x . Hence ρ x . . . x x . . . x for every β B . This σi 1 2 n 1 2 n β 1 n 1 n= − n 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).

≡ 2 ∎ Remark: Let ∆ be an autohomeomorphism of D Qm which keeps boundary

2 of D fixed pointwise. Thus ∆ represents an element of− M. Then ∆ has a unique 2 extension ∆ to D 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

2 2 which preserves D 0 pointwise, preserves D t , t I, setwise, and coincides× × { } × { } ∈ 2 with ∆ on D 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

2 Theorem 14. (Fadell-Van Buskrik,1962). the braid group π1B0,nS of the 2-sphere

2 S admits a presentation with generators δ1, . . . , δn−1 and defining relations:

δiδj δjδi if i j 2, 1 i, j n 1 (2.36)

= S − S ≥ ≤ ≤ − δiδi+1δi δi+1δiδi+1 1 i n 2 (2.37)

= 2 ≤ ≤ − δ1 . . . δn−2δn−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:= 2 (i)π1F0,2S 1

2 (ii)π1B0,2S = cyclic group of order 2 2 (iii)π1F0,3S = cyclic group of order 2 2 (iv)π1B0,3S = ZS, metacyclic group of order 12 2 (v)π2F0,3S =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

2 2 facts that π1F1,1S and π1F0,1S are both trivial groups:

2 2 2 2 π2 M Q0 π1F1,1S π1F0,2S π1S π0S ...

⋅ ⋅ ⋅ → ( − ) → → → → → 2 To prove (ii), one need only note that π1B0,2S 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])

∎ 2 Note, for completeness, that the faithful representation found for π1B0,nE as a group of automorphisms of the fundamental group of the n punctured plane (Corol-

2 lary 8 and Theorem 13) does not generalize to a faithful representation− of π1B0,nS as a group of automorphisms of the fundamental group of the n punctured sphere.

2 To be sure, the action given in equations (2.23) does induce an action− of π1B0,nS on

Fn−1, where Fn−1 is the quotient group of Fn obtained by adding the single relation

2 x̃1x2 . . . xn 1.̃ This action is, however, not a representation of π1B0,nS , because re- n 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 ( 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

→ → → → 2 This sequence was the basis for the structural analysis of π1B0,nE 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

= ( ) = {( ) ∈ × S ≠ ≠ } 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

n n−1 X z1 X z2 ... X zn X a1X 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

n n map C C 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 n n−1 coefficient map iff the polynomial{ X a1X} an−1X an ⃗has n distinct roots, i.e. iff its coefficients avoid the points+ where the+ ⋅discriminent ⋅ ⋅ + +

2 ∆ zi zj , (2.40) i

2 a1, . . . , an.(For example, the= polynomial X a1X a2 has distinct roots precisely when a2 4a 0). In this setting the basepoint φ p is regarded as the choice of 1 2 + + 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

0 0 0 Tg denotes a closed orientable surface of genus g; and let z1, z2, . . . , zn be the n fixed

but arbitrarily chosen points on Tg. Recall that like in Chapter 2 the symbol π1F0,nTg

0 0 denoted the pure braid group on Tg (with base point (z1, . . . , zn)) and π1B0,nTg denoted

the full braid group on Tg (with the same base point).

Let nTg denotes the group of all orientation preserving homeomorphisms h Tg T such that, for each i, h z0 z0. g F i i ∶ → nTg denotes the group( of) = all orientation preserving homeomorphisms h Tg Tg such that h z0, . . . , z0 z0, . . . , z0 . B 1 n 1 n ∶ → Now we,({ give these}) two= { groups compact-open} topology.

Definition: π1 nTg, Id denotes the group of path components of nTg and is called the pure mapping(F class) group of Tg. F

Definition: π0 nTg, Id denotes the group of path components of nTg and is called the full mapping(B class) group of Tg. We use M g, n for this group.B ( )

Note that 0Tg 0Tg and 1Tg 1Tg. Hence M g, 0 π0 0Tg π0 0Tg

and M g, 1 Fπ0 1=TgB π0 1TFg . = B ( ) = (F ) = (B ) The( terminology) = (F M) =g, n(Bis 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

∶ 2( ) → ( ) given by the action on H1 T ; Z Z 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)

0 0 Definition: Recall that z1, . . . , zn are a set of n distinguished points on Tg, and that 0 0 if h nTg, then h zi zi for each i 1, . . . , n. Recall also that, in chapter 2, (z0, . . . , z0) was chosen to be the base point for the space F T . Using these points, 1 ∈ F n ( ) = = 0,n g we now define (for each pair of integers n, g 0) an evaluation map as follows:

≥ gn 0Tg F0,nTg

0 0 ∶ F → by gn h h z1 , . . . , h zn where F0,nTg p1, . . . , pn pi Tg; pi pj if i j .

Observe( ) that= ( (gn )is continuous( )) with the given= {( topologies)S on ∈ 0Tg (equipped≠ ≠ with} compact open topology) and F0,nTg (equipped with subspace topologyF for F0,nTg Tg Tg Tg). ⊂

Theorem× × ⋅ ⋅ ⋅ 15 × (Birman, 1969b). The evaluation map gn 0Tg F0,nTg is a locally

trivial fibering with fibre nTg. ∶ F →

Proof. Note that nTg isF a closed subgroup of the topological group 0Tg (as com-

2 plement of a pointF in D is closed, and finite intersection of closedF sets is closed)

and that two elements h and g of 0Tg have the same image under gn iff they are

in the same left coset of nTg in F0Tg(this can be easily verified). This observation results in a natural identificationF F of F0,nTg with the quotient space 0Tg nTg, and this identification turns out to be a homeomorphism which turns gn intoF a~F 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 gn, a local cross section of the homogeneous space F0,nTg in T at the single point z0  T z0, . . . , z0 F T (that is, there is a 0 g ∃ gn n g 1 n 0,n g 0 Fneighborhood Uz0 of z in F=0,nTg (andF a) map= ( κ Uz0 ) ∈ 0Tg such that gnκ= Id).

Choose pairwise disjoint euclidean neighborhoods Uz0 ,...,Uz0 (such as choice is pos- ∶ 1 → F n 0 0 sible as Tg is a manifold and hence Hausdorff) of z1, . . . , zn respectively, on Tg. Then 0 0 0 U z u1, . . . , un ui U zi is a neighborhood of z in F0,nTg. Construct a 0 family( ) = of{( homeomorphisms}S ∈ l(u )} 0Tg u U z , depending continuously on u, such that, for each u U z0 , l z0 u and l T n U z0 Id.(such a construction u {i ∈ Fi S ∈u (g ) i=1 i 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 ∎

gn∗ dgn∗ ign∗ π1 0Tg π1F0,nTg π0 nTg π0 0Tg π0F0,nTg 1 (3.1)

→ F ÐÐ→ ÐÐ→ F ÐÐ→ F → = Proof. The homomorphism gn∗ is induced by the evaluation map gn and the homo- morphism ign∗ by the inclusion nTg 0Tg. The exact sequence is simply the exact

homotopy sequence of the fiberingF →gn F 0Tg F0,nTg(see section 1.3.4 for details about fibration) ∶ F → ∎ Remark: The surjection i∗ ign∗ is the analogue for the pure mapping class groups of the homomorphism j j M g, n M g, 0 of the full mapping class ∗ = gn∗ 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∗ , gn∗ , dgn∗ by i∗, ∗, 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∗ π0 nTg

π0 0Tg be the homomorphism induced by the inclusion. Then≥ ker i∗=image∶ F 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

∈ n β1, . . . , βn I 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 nTg. Indeed, the construction is obvious for generators Aij 1 i j n of π1F0,n∈ TFg (see chapter 2) and by composition of isotopies can be{ extendedS ≤ < 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 nTg with h ker i . Since h T , h fixes z0, z0, . . . , z0 pointwise. Since h Ker i , h ∗ n g 1 2 n ∈ F∗ is isotopic[ ] ∈ to the identity∈ mapF on Tg, say by an isotopy ht 0 t 1 , h[0 ] ∈Id, h1 h. Then h z0 , . . . , h z0 represents an element β of π F T and d β h t 1 t n 1 0,n( g≤ ≤ )∗ = = ( ( ) ( )) = [ ] ∎ Lemma 3. Ker d∗ Z π1F0,nTg

⊂ ( ) Proof. Suppose α Ker d∗ Im ∗, and let H π1 0Tg be such that ∗H α. The element H is represented∈ by= a loop h ht 0 ∈ t F 1 in 0Tg, where each= ht is in 0Tg and h0 h1 Id. Then  ht = {ht Sx1≤, . .≤ . , ht xnF (0 t 1) represents Fα. Let β π1=F0,nT=g, with β represented( ) = ( by( β)1 s , . . .( , βn))s (0≤ ≤s 1). Define G I I ∈ F0.nTg by G t, s htβ1 s , . . .( , htβ(n)s ( t,( s)) I ≤ I).≤ Then G is −1 −1 continuous∶ × → and G ∂ I I ( represents) = ( the( ) homotopy( class)) (αβα) ∈β ×. Since β was an arbitrary elementS of(π1×F0,n) Tg, 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)

δ∗ π∗ 1 π1Fn−1,1Tg π1F0,nTg π1F0,n−1Tg 1

→ Ð→ Ð→ → 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. ≈ > = ∎

2 The next two lemmas treat the case g 0, that is Tg S

= =

Lemma 5. (Gillette and Van Buskrik,1968). Let δ1, . . . , δn−1 be the standard gen-

2 2 erators of π1B0,nS , n 3. Then the center of π1B0,nS is the subgroup of order 2

n generated by δ1δ2 . . . δn−≥1 . ( )

Proof. We assume this lemma without proving it, whose prove can be obtained in [35].

∎

Lemma 6. If n 3, then

≥ 2 2 Z π1B0,nS Z π1F0,nS Ker d∗

( ) ⊂ ( ) ⊂

Proof. By Lemma 5,

2 2 center π1B0,nS π1F0,nS

⊂ n is the cyclic subgroup of order 2 generated by δ1 . . . δn−1 , hence Lemma 6 will be

n true if we can prove that d∗ δ1 . . . δn−1 1.( In the standard) geometric model of 2 n π1F0,nE , σ1 . . . σn−1 can be( pictured) as= in figure a and b below(drawn for the case 2 n n 4). Then( as a motion) of points on S , δ1 . . . δn−1 can be pictured as in figure c. (figure= taken from Birman’s book [1]) ( ) 0 0 Recall now the construction of d∗. Without loss of generality, z1, . . . , zn are spaced 0 0 at equal distances on a longitude joining z1 at the equator with zn at the north pole (as in Figure 13c), and the motion of points pictured in Figure 13c can be realized

2 2 by a rotation ht S S for 0 t 1 about the axis joining the north and south

n pole, h0 h1 Id∶. Then→ d∗ δ1 .( . .≤ δn−1≤ ) 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 nTg= 0T∶ g.( Then) → Ker (j∗ is) isomorphic to π1B0,nTg for g 2. If g 1, n 2∶ orB g ⊂0B, 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

′ gn 0Tg B0,nTg, which is a locally trivial fibering with fiber nTg. As in Corollary

10,∶ thisB fibering→ defines an exact sequence B

′ ′ gn∗ dgn∗ jgn∗ π1 0Tg π1B0,nTg M g, n M g, 0 π0B0,nTg 1 (3.2)

→ B ÐÐ→ ÐÐ→ ( ) ÐÐ→ ( ) → =

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= ≥ nTg nTg if n 0 or n 1), and moreover π F T π B T π T , so that our theorems imply that ker i Ker j 1 0,1 g 1 0,1 g 1 g F = B = g1∗ = j1∗ is isomorphic to π1=Tg. We will≈ now describe an explicit fashion how to find generators=

for the subgroup ker jg1∗ of the mapping class group M g, 1 π0 1Tg π0 1Tg.

0 Let z1 be the base point for π1Tg, and suppose that c (is any) = simpleF closed= B curve on 0 π1Tg which contains the base point z1. 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 z0 by 0, 0 . We now define a map h T T by the ( ) 1 − ≤ ≤ + ≤ ≤ cz1 g g rule that= if a point is in N, then its( image) is given by: ∶ →

y, θ 2πy if 0 y 1 hcz y, θ (3.3) 1 ⎪⎧ ⎪(y, θ + 2πy) if ≤1 ≤y 0, ∶ ( ) → ⎨ ⎪ ⎩⎪( − ) − ≤ ≤ 0 while all points of Tg N are left fixed. We call such a map a spin of z1 about c. (see figure below). Note that h M g, 1 (the proof of this fact can be found in the − cz1 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 0 base point z1 but otherwise disjoint, and having the property that their homotopy −1 −1 −1 −1 classes generate π1Tg, and also that the homotopy class of a1b1a1 b1 . . . agbgag bg 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

h h h−1 h−1 . . . h h h−1 h−1 a1z1 b1z1 a1z1 b1z1 agz1 bgz1 agz1 bgz1 is trivial.

0 In the case where n ia arbitrary, the single point z1 which serves as base point for

π1Tg π1F0,1Tg π1B0,nTg(regarded as a subgroup of π0 1Tg π0 1Tg) is replaced by an array z0, . . . , z0 which determines a base point for π F T ,and also for π B T . = 1 = n F1 0,n =g B 1 o,n g Let aij, b(ij; 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 0 0 the base point zj , but no other point zk if k j. Then the isotopy classes of the spin

maps ≠ h , h ; 1 i g, 1 j n aij ,zj bijzj

generates Ker ign∗ . To obtain{ a set of generators≤ ≤ for≤ Ker≤ j}gn∗ one adds th this set any

set of maps on the surface Tg which generete the full group of permutations of the

0 0 0 0 points z1, . . . , zn ; for example, enclose each pair zj , zj+1 in a disc Dj which avoids all points z0 k j and map T to itself by a map h which fixes T D pointwise, ( k ) g ( j ) g j and interchanges z0 and z0 ”nicely.” ( ≠ )j j+1 − 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 nTg, 0T(g; n)Tg and( 0T) g. Information about these higher homotopy groups appears inF QuintasF 1968,B [34].B

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: )

( ) ( )2 = 2 π1B0,1S M 0, 1 M 0, 0 π0B0,1S

2 → 2 → ( ) → ( ) → 2 2 . The space B0,1S S pt is path connected, hence π0B0,1S 1. Also, π1B0,1S

2 π1S 1. Hence M =0, 1 − M 0, 0 . = = = ( ) ≈ ( ) The proof that M 0, 1 1, that is, every orientation-preserving homeomorphism

2 2 of S which fixes a point( )p = S 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

2 1 which fixes ∂D S pointwise, then∶ g →is isotopic to the identity under an isotopy 1 which fixes S 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 ∶ →

S x 1 t φ − 0 x 1 t F x, t 1 t (3.4) ⎪⎧ ⎪(x − ) ( ) 1 ≤ tS S

isotopy≤ F

In fact, the above result can be proved in general for a n-unit ball Dn. ∎

Lemma 8. (Schoenflies Theorem) If 1 and 2 are simple closed curves in the plane

2 E and h 1 2 is any homeomorphism,I thenI there is an extension h∗ of h which takes 1 ∶IntI →1 Ihomeomorphically onto 2 Int 2.

Proof.IThis∪ lemmaI will be assumed withoutI ∪ proof.I 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 is a simple closed curve in S2 p , let Int denotes the component of S2

whichI contains p, Ext the other component.− { } ChooseI simple closed curves 1 and− I2 2 in S p such that I1 ρ I1 IntI2. I I Let− { 1}, 2 and 3Ibe∪ disjoint( ) ⊂ arcs with 1 2 3 IntI2 I1 IntI1 , each of℘ the℘ arcs joining℘ the simple closed curves(℘ ∪I ℘1 and∪ ℘I)2.⊂ Let[ xi − ( i∪ I1 and)] yi i I2 i 1, 2, 3 . Since ρ is orientation preserving fixes p,{ there} = are℘ ∩ disjoint {arcs} = 1℘, ∩2, and( = joining) ρ x1 and y1, ρ x2 and y2, and ρ x3 and y3, respectively K K K ( ) ( ) ( ) such that i IntI2 ρ I1 Intρ I1 . (see figure below which is taken from the

Birman’s bookK ⊂ [1]) − [ ( ) ∪ ( )]

Let ij i j be the component of IntI2 1 2 3 I1 IntI1 which contains

i j inR its( boundary.≠ ) Let ij be the component−{℘ of∪℘ Int∪℘I2 ∪ 1 ∪ 2 3 }ρ I1 Intρ I1 ℘which∪℘ contains Ki Kj in itsR boundary. −[K ∪K ∪ ∪ ( )∪ ( )] 2 2 It is easy to construct∪ a homeomorphism g S S which has the following properties: ∶ →

i g I1 IntI1 ρ I1 IntI1

( ) S ∪ = S ∪ ii g I2 ExtI2 Identify I2 ExtI2

′ ( ) S ∪ iii g = ij ijS ∪

. Indeed, i and ii define g Bd( i )i (R1, 2),=3 R. This partial map can be extended to take i(to) i homeomorphically.( ) S ℘ ( This= defines) g Bd ij for each i j. And by Lemma 8, g Bd Bd Bd ′ can be extended to a homeomorphism g ℘ K ij ij ij S R ≠ ij ′ Rij Rij(whereS RR denotes∶ R the→ closureR of the region ). SR ∶ By,→ i and Lemma 7, the homeomorphisms ρ andRg are isotopic under an isotopy which moves( ) points only in Ext 1. By ii and Lemma 7, the homeomorphisms g and identity are isotopic under anI 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

2 ∎ Theorem 19. If n 2, then M 0, n π0 nS admits a presentation with generators

ω1, . . . , ωn−1 and defining≥ relations:( ) = B

ωiωj ωjωi i j 2

= S − S ≥ ωiωi+1ωi ωi+1ωiωi+1

2 ω1 . . . ωn−2ωn=−1ωn−2 . . . ω1 1

n ω1ω2 . . . ωn−1 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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