Mapping Class Groups

University of Cambridge

Mathematics Tripos

Part III Mapping Class Groups

Michaelmas, 2019

Lectures by H. Wilton

Notes by Qiangru Kuang Contents

Contents

0 Introduction 2 0.1 Classification of surfaces ...... 2 0.2 Mapping class groups ...... 2 0.3 Context & Motivation ...... 3

1 Curves, Surfaces & Hyperbolic geometry 4 1.1 The hyperblic plane ...... 4 1.2 Hyperbolic structures ...... 5 1.3 Curves on hyperbolic surface ...... 6

2 Simple closed curves & Intersection number 8

Index 9

1 0 Introduction

0 Introduction

0.1 Classification of surfaces The subject of this course is surfaces, i.e. two-dimensional manifolds. We assume throughout the manifold 푆 is connected, smooth and oriented. More importantly, we assume it is of finite type: 푆 = 푆 − {finite set} where 푆 is a compact manifold possibly with boundary, and the finite set is contained in its interior.

Theorem 0.1 (classification of surfaces of finite. type) Every connected orientable surface of finite type is diffeomorphic to some 푆푔,푛,푏 which is a surface of genus 푔 with 푛 punctures and 푏 boundary components.

Figure 1: 푆푔,푛,푏

It has Euler characteristic 휒(푆) = 2 − 2푔 − (푛 + 푏). A closed surface is a surface with 푛 = 푏 = 0.A compact surface is a surface with 푛 = 0.

Example. Suppose 휒(푆) > 0 then 푔 = 0, 푛 + 1 = 0 or 1. Thus 푆 is either 푆2, or 푆2 with a puncture which is isomorphic to C, or 푆2 with one boundary component which is isomorphic of 퐷2, which can be thought as sitting inside C as the unit disk. Note that we can either think of a puncture as a deleted point, or as a marked point on the surface. These two views are equivalent and we will use whichever that is more convenient.

Example. Suppose 휒(푆) = 0 then either 푔 = 1, 푛 + 푏 = 0 or 푔 = 0, 푛 + 푏 = 2. 1 Thus it is either the torus, the punctured plane C∗, 푆 × 퐼 or the punctured 2 disk 퐷∗ .

0.2 Mapping class groups The natural group associated to a surface 푆 is the group of orientation-preserving homeomorphism Homeo+(푆). However this is a huge group. It might be easier to consider the maps up to homotopy. Suppose 휙0, 휙1 ∶ 푆 → 푆 are homeomor- phisms. We can define an equivalence relation 휙0 ∼ 휙1 if there exists an isotopy 휙푡 ∶ 푆 × 퐼 → 푆 from 휙0 to 휙1. Another way to think about this: give Homeo+(푆) the compact-open topology. Let Homeo0(푆) be the path component of 1푆. It is an exercise to show that Homeo0(푆) is a normal subgroup of Homeo(푆).

2 0 Introduction

Definition (mapping class group). The mapping class group of 푆 is

+ Mod(푆) = Homeo (푆, 휕푆)/ Homeo0(푆, 휕푆)

where Homeo(푆, 휕푆) is the subgroup of Homeo(푆) that fixes 휕푆 pointwise.

We might ask if we can replace homeomorphism by diffeomorphism, or we replace isotopy by homotopy. Indeed we have

Theorem 0.2 (Baer, Munkres). For any smooth surface 푆 of finite type,

+ + Mod(푆) ≅ Diff (푆, 휕푆)/ Diff0(푆, 휕푆) ≅ Homeo (푆, 휕푆)/ ∼

where the equivalence relation is homotopy.

0.3 Context & Motivation bundles Let 휙 ∈ Diff(푆). We can define 푀휙 = 푆 × [0, 1]/ ∼ where (푥, 1) ∼ 1 (휙(푥), 1). This is called a surface bundle over 푆 . Note that 푀휙 only depends on [휙] ∈ Mod(푆). More generally, if 퐵 is a space and 휌 ∶ 휋1(퐵) → Mod(푆) then we get a bundle 퐵̃ × 푆 ... this leads to an 푆-bundle over 퐵. moduli space More handwavy motivation. Let 푆 = 푆푔 = 푆푔,0,0. It turns out the moduli space ℳ푔 of geometric structures on 푆푔 is the same as the moduli space of complex structures on 푆푔. Morally (but not actually true), the ̃ universal cover ℳ푔 is 풯푔, the Teichüller space, and 휋1(ℳ푔) is Mod(푆), so we have ℳ푔 = Mod(푆푔) \ 풯푔. analogy There is an analogy between the torus and a surface 푆.

푆 푇 푛 휋1(푆) Z Mod(푆) SL푛(Z) closed curves (up to isotopy) vectors

Table 1: Comparison of 푆 with 푇 푛

3 1 Curves, Surfaces & Hyperbolic geometry

1 Curves, Surfaces & Hyperbolic geometry

1.1 The hyperblic plane There are two models of hyperbolic geometry.

The upper half-plane model It is the half-plane H2 = {푥+푖푦 ∶ 푦 > 0} with metric 푑푥2 + 푑푦2 푑푠2 = . 푦2 The geodesics are either vertical lines or semicircles on the 푥-axis. Then both meet the 푥-axis at right angle. The orientation preserving isometries are

+ 2 Isom (H ) = 푃 푆퐿2(R) where an element in 푃 푆퐿2(R) acts by Möbius transformation: 푎 푏 푎푧 + 푏 ( ) 푧 = 푐 푑 푐푧 + 푑 where 푎, 푏, 푐, 푑 ∈ R and 푎푑 − 푏푐 > 0 (by rescaling). As the entries are real, it preserves the 푥-axis and the point at infinity.

푧−푖 The Poincaré disc model Conjugating the isometry group by 푧 ↦ 푧+푖 , we can map H2 to the interior unit disc. The metric is 푑푥2 + 푑푦2 푑푠2 = 4 . (1 − 푟2)2

Note that it is radically symmetric. As Möbius transformations are conformal, the geodesics are diameters and arcs meeting the boundary of the disc at right angle. The (Gromov) boundary (at infinity) 휕H2 is the unit circle bounding the 2 disc. We write H = H2 ∪ 휕H2.

Note. Each isometry 푓 of H2 extend uniquely to Möbius transformation 푓 of 휕H2. + 2 Let 푓 ∈ Isom H = PSL2(R) and let 푛 be the number of fixed points of 푓. Brouwer fixed point theorem says 푛 ≥ 1. On the other hand, 푛 ≤ 2 unless 푓 = id.

2 푛 = 2 Let be {휉+, 휉−} = Fix(푓) ⊆ H . If one of them is in the interior then there is a geodesic between them lying in H2. But then 푓 fixes every point on 2 the geodesic, absurd. Thus 휉+, 휉− ∈ 휕H and there is a unique geodesic between them. There exists 푔 ∈ PSL2(R) such that 푔(휉−) = 0, 푔(휉+) = ∞ in the upper half-plane model, and Fix(푔푓푔−1) = 푔 Fix(푓). Thus after conjugation, wlog 휉− = 0, 휉+ = ∞ so 푓(푧) = 휆푧 = 푒휏(푓)푧

4 1 Curves, Surfaces & Hyperbolic geometry where 휆 > 0. So 푓 now acts as translation on the 푦-axis by 휏(푓). In general, 푓 preserves a geodesic line called Axis(푓), acting by translation. Such an 푓 is called hyperbolic or loxodromic. fact: If 푥 ∉ Axis(푓) then 푑(푥, 푓(푥)) > 휏(푓).

푛 = 1 Let Fix(푓) = {휉}. If 휉 ∈ H2 wlog 휉 = 0 in the disc model. Then must have 푓(푧) = 푒푖휃푧. Such an 푓 is called elliptic. If 휉 ∈ 휕H2 wlog 휉 = ∞ in the upper half-plane model. It follows that 푓(푧) = 푧 ± 1. Such an 푓 is called parabolic.

Remark. This classification is invariant under conjugacy.

1.2 Hyperbolic structures A hyperbolic structure on 푆 is a complete, finite-area Riemannian metric of constant curvature 휅 = +1, 0 or − 1, in which every boundary components are geodesic. What kind of hyperbolic structure can we put on a surface? Recall Gauss- Bonnet which says that if 푆 has finite area then

∫휅푑퐴 = 2휋휒(푆) 푆 so the sign of 휅 is the same as the sign of 휒(푆). In particular

1. if 휒(푆) > 0 then 휅 = 1 so 푆 is locally 푆2.

2. if 휒(푆) = 0 then 휅 = 0 so 푆 is locally R2. 3. if 휒(푆) < 0 then 휅 = −1 so 푆 is locally H2. Example. Recall that 휒(푆) > 0 implies 푆 is 푆2, 퐷2 or C. 푆2 will just have the geometric structure of the sphere. For C, there is no complete finite-area metric. For 퐷2, recall that we require the boundary component to be a geodesic so the geometric structure on 퐷2 will just be the semisphere. 2 1 2 2 If 휒(푆) = 0 then 푆 is 푇 , 퐴 = 푆 × [0, 1], C∗ or 퐷∗ . Again C∗ and 퐷∗ do not admit finite-area complete metric. 푇 will then be the flat torus, i.e. Z2 \ R2. 퐴 has the geometric structure of a cylinder.

Theorem 1.1. If 푆 is connected, oriented and of finite type and 휒(푆) < 0 then there is a convex subspace 푆̃ ⊆ H2 with geodesic boundary and an action ̃ 휋1푆 on 푆 by isometry such that ̃ 푆 ≅ 휋1푆 \ 푆

has finite area. In particular, 푆 has curvature −1 everywhere.

Sketch proof when 푆 = 푆푔,0,0. Cut along 2푔 loops to get a 4푔-gon with sides identified appropriately. Then it suffices tofinda 4푔-gon in the hyperbolic disk whose internal angles sum up to 2휋. A regular 4푔-gon with vertices on 휕H has total interior angle 0, while as we shrink the polygon it resembles more and

5 1 Curves, Surfaces & Hyperbolic geometry more like a Euclidean polygon so the total interior angle approachs (4푔 − 2)휋. As 푔 > 1, by intermediate value theorem we can find a regular 4푔-gon whose total interior angle is 2휋. The metric on 푆 then has constant curvature 휅 = −1. Then by Jacobi theorem 푆̃ ≅ H2 (homeo?). The statement about fundamental group follows from algebraic topology (note that 휋1 does act by isometry since the upstairs metric is lifted from the downstairs’). Such a surface 푆 is called hyperbolic.

Remark. When 푆 has no boundary components 푆̃ = H2.

1.3 Curves on hyperbolic surface A closed curve on 푆 is a smooth map 훼 ∶ 푆1 → 푆. It gives a conjugacy class 2 [휎] ∈ 휋1푆. This leads to an isometry of H (up to isometry) when 푆 is hyperbolic. Thus we could talk about conjugacy-invariant properties of 훼.

Definition. We say 훼 is inessential if 훼 is homotopic to a point or a puncture. Otherwise 훼 is essential.

Picture of an (embedded) hyperbolic surface. Note that a puncture is a cusp because of finite-area and completeness.

Lemma 1.2. 1. If 훼 is elliptic then it is homotopic to a point. 2. If 훼 is parabolic then it is homotopic to a puncture. 3. If 훼 is hyperbolic then it is essential.

Proof.

2 1. 훼 is elliptic implies that 훼 fixes a point of H . But the action of 휋1 is free so 훼 = id as isometry. so 훼 is homotopic to a point.

2. If 훼 is parabolic then wlog 훼 ∶ 푧 ↦ 푧 + 1. Take 훼(0) = 푥0 ∈ 푆 to be a 2 basepoint and choose 0̃푥 a lift in H . Let ̃훼 be the lift of 훼 at 0̃푥 . Let 푠̃훼 (푡) = ̃훼(푡)+ 푖푠 for 푠 ∈ [0, ∞). For all 푠,

푠̃훼 (1) =푠 ̃훼 (0) + 1

so 푠̃훼 descends to a loop 훼푠 in 푆 and 훼푠 tends to a puncture of 푠 by 2 compactness of H . 3. It’s enough to prove that 훼 homotopic to a puncture then it is parabolic. By shrinking 훼 we get a sequence of annuli. Since the hyperbolic structure is complete and of finite area, there exists 훼푛 ∼ 훼 such that ℓ(훼푛) → 0 as 푛 → ∞. Lift 훼 to a path ̃훼∶ [0, 1] → H2 and moreover we get well-defined

6 1 Curves, Surfaces & Hyperbolic geometry

lifts 푛̃훼 of 훼푛. Let 푛̃푥 =푛 ̃훼 (0). Note 푛̃훼 (1) = 훼 ⋅푛 ̃푥 . Then the translation distance is

휏(훼) ≤ 푑(푛 ̃푥 , 훼 ⋅푛 ̃푥 )

= 푑(푛 ̃훼 (0),푛 ̃훼 (1))

≤ ℓ(푛 ̃훼 )

= ℓ(훼푛) → 0

as 푛 → ∞ so 훼 is not hyperbolic.

Lemma 1.3. Let 푆 be a hyperbolic surface and 훼 an essential closed curve on 푆. Then there is a unique geodesic representative in the homotopy class of 훼.

Note that for Euclidean space, such as a torus, such a representative exists but is not unique.

Proof. The universal cover of 푆1 is R and the universal cover of 푆̃ ⊆ H2 of 푆. Then each 훼 ∶ 푆1 → 푆 lifts to ̃훼∶ R → 푆.̃ Note that ̃훼 is Z-equivariant 1 by considering the action of 휋1푆 . From the lemma above we know 훼 has an axis Axis(훼). Let 휋 ∶ H2 → Axis(훼) be the orthogonal projection (equivalently, it projects a point 푥 to 휋(푥) on Axis(훼) such that the length of the geodesic 2 between 푥 and 휋(푥) is the shortest so. See example sheet 1). Let 푡̃훾 ∶ [0, 1] → H be the unique constant speed geodesic from ̃훼(푡) to 휋 ∘ ̃훼(푡). Since ⟨훼⟩ acts on both ̃훼 and Axis(훼) and the 푡̃훾 are determined canonically, taking the quotient by Z = ⟨훼⟩ defines a homotopy from 훼 to some closed curve 훽 on 푆 in the image of Axis(훼). The image of 훽 is a local geodesic and after reparameterisation, 훽 is a constant-speed geodesic. Unqiueness: suppose 훼 ≃ 훽 are both geodesics on 푆. Lift 훼, 훽 to ̃훼,̃∶R → H2 geodesics in H2 that are contained in a bounded neighbourhood of each ̃ other (since sup푡∈푆1 푑(훼(푡), 훽(푡)) < ∞ by compactness). It follows that ̃훼, 훽 are geodesics in H2 with the same endpoints on 휕H2. It follws that ̃훼= 훽 ̃ so 훼 = 훽.

7 2 Simple closed curves & Intersection number

2 Simple closed curves & Intersection number

A closed curve 훼 ∶ 푆1 → 푆 is simple if it is injective. The idea is that simple closed curves are like basis of a vector space and we can understand mapping class groups by understanding its action on simplex closed curves.

Definition ((ambient) isotopy of simple closed curves). A homotopy 훼• between simple closed curves 훼0 to 훼1 is an isotopy if each 훼푡 is simple. If 휙|푏푢푙푙푒푡 ∶ 푆 → 푆 is an isotopy such that 휙0 = id푆 and 휙1 ∘ 훼0 = 훼1 then we say 훼0, 훼1 are ambient isotopic.

Lemma 2.1. Two essential simple closed curves on an orientable surface 푆 are homotopic relative to 휕푆 if and only if they are ambient isotopic.

We’ll prove this later.

푛 Definition. An element ℎ ∈ 휋1(푆) is primitive if ℎ ≠ 푔 for some 푛 > 1.

Lemma 2.2. Let 푇 2 be the torus. The homotopy class of essential simple 2 2 2 closed curves on 푇 correspond to primitive elements of 휋1푇 = Z .

Proof. Example sheet 1, question 8.

Lemma 2.3. If 훼 is an essential simple closed curve on a hyperbolic surface 푆 then 훼 ∈ 휋1푆 is primitive. In fact the it has centraliser 퐶(훼) = ⟨훼⟩.

Proof. wlog 훼 is a geodesic and we may consider Axis(훼) ⊆ H2. Let 푔 ∈ 퐶(훼), 푥 ∈ Axis(훼). Then

푑(푔푥, 훼푔푥) = 푑(푔푥, 푔훼푥) = 푑(푥, 훼푥) = 휏(푥) as 푥 is on the axis. Therefore 푔 preserves Axis(훼) so 퐶(훼 acts on Axis(훼). We have ⟨훼⟩ ⊆ 퐶(훼). 훼 is injective then deg 푝 = 1 = |퐶(훼) ∶ ⟨푎⟩| so the result follows.

8 Index ambient isotopy, 8 mapping class group, 3