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Lectures on the Mapping Class Group of a Surface

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Lectures on the Mapping Class Group of a Surface LECTURES ON THE MAPPING CLASS GROUP OF A SURFACE THOMAS KWOK-KEUNG AU, FENG LUO, AND TIAN YANG Abstract. In these lectures, we give the proofs of two basic theorems on surface topology, namely, the work of Dehn and Lickorish on generating the mapping class group of a surface by Dehn-twists; and the work of Dehn and Nielsen on relating self-homeomorphisms of a surface and automorphisms of the fundamental group of the surface. Some of the basic materials on hyper- bolic geometry and large scale geometry are introduced. Contents Introduction 1 1. Mapping Class Group 2 2. Dehn-Lickorish Theorem 13 3. Hyperbolic Plane and Hyperbolic Surfaces 22 3.1. A Crash Introduction to the Hyperbolic Plane 22 3.2. Hyperbolic Geometry on Surfaces 29 4. Quasi-Isometry and Large Scale Geometry 36 5. Dehn-Nielsen Theorem 44 5.1. Injectivity of ª 45 5.2. Surjectivity of ª 46 References 52 2010 Mathematics Subject Classi¯cation. Primary: 57N05; Secondary: 57M60, 57S05. Key words and phrases. Mapping class group, Dehn-Lickorish, Dehn-Nielsen. i ii LECTURES ON MAPPING CLASS GROUPS 1 Introduction The purpose of this paper is to give a quick introduction to the mapping class group of a surface. We will prove two main theorems in the theory, namely, the theorem of Dehn-Lickorish that the mapping class group is generated by Dehn twists and the theorem of Dehn-Nielsen that the mapping class group is equal to the outer-automorphism group of the fundamental group. We will present a proof of Dehn-Nielsen realization theorem following the argument of B. Farb and D. Margalit, [De, FM]. Along the way of the proof, we will introduce some of the basic notions and tools used in the surface theory. This paper is the result of a series of lectures given by the second author in the Center of Mathematical Sciences, Zhejiang University, China during the summer of 2008. It aims at providing a concise understanding of the surface theory, especially suitable for graduate students. The prerequisites for these lectures have been kept to a minimum. Basic knowledge of algebraic topology and Riemannian geometry is all one needs to follow the lectures. Let § be a compact oriented surface. A homeomorphism Á :§ ! § is called a Dehn twist if its support A, the closure of fx 2 § j Á(x) 6= xg, is homeomorphic to an annulus under a homeomorphism I : S1 £[0; 2] ! A and I¡1ÁI is a 2¼-twist on the annulus. A more precise de¯nition will be given later in De¯nition 1.16. The goal of these lectures is to prove the following two theorems. Dehn-Lickorish Theorem . Suppose h :§ ! § is an orientation preserving self-homeomorphism of a surface § so that hj@§ = id. Then there exists a ¯nite set of Dehn twists Á1;:::;Án of § such that h is homotopic to the composition Á1 ± ¢ ¢ ¢ ± Án. In fact, Dehn and Lickorish proved a stronger theorem that Ái's can be chosen from a ¯xed ¯nite set. 2 THOMAS AU, FENG LUO, AND TIAN YANG Dehn-Nielsen Theorem . Let § be a closed surface of nonzero genus. Then the group of self-homeomorphisms on § modulo homotopy is isomorphic to the outer automorphism group of the fundamental group of §. The theorem may be stated in a simplier form in terms of the notion of mapping class groups, which we will discuss later. In the ¯rst section, we will introduce the notions of mapping class groups and Dehn twists. Simple examples of mapping class groups and basic properties of Dehn twists will also be given. In the second section, how the mapping class group of a surface is generated by Dehn twists will be discussed. Indeed, we will prove the Dehn-Lickorish Theorem. Section three is a discussion of the hyperbolic geometry of surfaces. It consists of a quick introduction of the hyperbolic plane and the geometric interpretation of the fundamental group of a surface. In section four, quasi-isometries and large scale geometry are introduced. Our attention is on facts pertinent to our study of mapping class groups. In the last section, we will prove the Dehn-Nielsen Theorem. The authors are very grateful to the Center of Mathematical Sciences and the hosts for the wonderful and quiet environment where people could think and work well. The workshop also provided enlightenment to a number of mathematicians and graduate students. Particular thanks should be addressed to the organiz- ers especially Professor Lizhen Ji. We also thank the referee for many helpful suggestions on improving the exposition in this paper. 1. Mapping Class Group In this section, we will introduce basic notions about mapping class groups and elementary techniques in handling Dehn twists. Let § = §g;r be the compact oriented surface of genus g with r boundary com- ponents, r ¸ 0. LECTURES ON MAPPING CLASS GROUPS 3 De¯nition 1.1. The mapping class group of § is given by def ¡(§) :== Homeo(§)/ ' ; where Homeo(§) is the set of homeomorphisms from § to § and the relation ' is homotopy of maps. Remark . The relationship between homotopic and isotopic homeomorphisms on surfaces was established by Baer. Baer's theorem says that they are the same in the following sense. Theorem 1.2. (Baer) Suppose f and g are two homotopic homeomorphisms of a compact oriented surface X so that f = g on the boundary of X. Then f is isotopic to g by an isotopy leaving each point in the boundary of X ¯xed. One may consult [St2, ch. 6] or [Ep] for a proof, which includes a discussion on isotopic curve systems on surfaces. For this reason, we will not address the isotopy issues in this paper. The theory of mapping class groups plays an important role in the study of low-dimensional topology. Example 1.3. Let Hg be the genus g handlebody, that is, the regular neigh- borhood of a wedge of g circles in the 3-space. The boundary of Hg is §g;0. It is well-known that for each closed orientable 3-manifold M, there is a positive number g and an h 2 ¡(§g;0) such that M = Hg [h Hg : Such a decomposition of a 3-manifold is called a Heegaard Splitting. In other words, M is obtained by gluing up two copies of Hg by a homeomorphism h of their boundary surfaces §g;0. The resulting manifold M depends only on the homotopy class of h. 4 THOMAS AU, FENG LUO, AND TIAN YANG Example 1.4. Thurston's Virtual Fibre Conjecture states that any closed hy- perbolic 3-manifold M has a ¯nite cover N such that N = §g;0 £ [0; 1]/ f (x; 0) » (h(x); 1) : x 2 §g;0 g ; where h is a homeomorphism on §g;0. In short, N is a surface bundle over the circle. This is one of the main conjectures in 3-manifold theory after the resolution of the Geometrization Conjecture. Example 1.5. Mapping class groups are needed in several other important areas. For example, in the theory of Lefschetz ¯brations of 4-manifolds, [Au, Don, Gom]; in the TeichmÄullertheory of surfaces in which the mapping class group acts on the Teichm Ämullerspace, [BH, Pa]; and in the theory of the Moduli space of algebraic curves and Riemann surfaces, [HL]. We will assume basic facts from topology. For example, the Euler Characteristic Â(§g;r) of a surface satis¯es Â(§g;r) = 2 ¡ 2g ¡ r : 2 2 Surfaces § = §g;r with Â(§) > 0 are the sphere S and the disk D ; for Â(§) = 0, we have the torus T2 and the annulus S1 £ [0; 1]; all others are of Â(§) < 0. As a beginning, the reader is encouraged to work out the following. Example 1.6. ¡(S2) =» Z=(2Z) and ¡(D2) = f id g. The following was proved by J. Nielsen in his Ph.D. thesis in 1913. Theorem 1.7. (Nielsen) ¡(T2) =» GL(2; Z). ± Proof. We will represent T2 as the quotient space R2 Z2. There is a natural homomorphism 2 2 ½: ¡(T ) ! Aut(H1(T ; Z)) = GL(2; Z) that takes a homeomorphism class [h] to the induced map h¤ on the ¯rst homology group. LECTURES ON MAPPING CLASS GROUPS 5 First, we will show that ½ is surjective, i.e., Image(½) = GL(2; Z). Let A 2 GL(2; Z). The integral matrix A can be seen as the linear map (by abuse of language) A: R2 ! R2; x 7! Ax for x 2 R2 ; which preserves the integer lattice, i.e., A(x + n) = A(x) + A(n) with A(n) 2 Z2 for every x 2 R2 and n 2 Z2. This clearly induces a homeomorphism on the quotient A~: T2 ! T2; [x] 7! [Ax] : It is easy to verify that ½(A~) = (A~)¤ = A with a suitable identi¯cation of n and A(n) 2 Z2 with the standard basis of R2. Second, to prove that ½ is injective, we will show ker(½) = f id g. 2 2 2 Let [h] 2 ¡(T ) where h 2 Homeo(T ) and h¤ = id. Represent T as the quotient space of [0; 1] £ [0; 1] by gluing [0; 1] £ f 0 g to [0; 1] £ f 1 g and f 0 g £ [0; 1] to f 1 g £ [0; 1]. Let a; b denote the curves in T2 corresponding to quotient classes of [0; 1] £ f 0; 1 g and f 0; 1 g £ [0; 1], respectively. Let P be the point of intersection of a and b. a b b P a Figure 1.1 2 » 2 » 2 2 Then H1(T ; Z) = ¼1(T ;P ) = ha; bi = Z .
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