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A Short Introduction to Mapping Class Groups

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A SHORT INTRODUCTION TO MAPPING CLASS GROUPS GWENA´ EL¨ MASSUYEAU Abstract. These informal notes accompany a talk given in Strasbourg for the Master Class on Geometry (spring 2009). We introduce the mapping class group of a surface and its enigmating subgroup, the Torelli group. One hour and seventeen pages are certainly not enough to present this beautiful and rich subject. So, we recommend for further reading Ivanov's survey of the mapping class group [16] as well as Farb and Margalit's book [9], which we used to prepare this talk. Johnson's survey [20] gives a very nice introduction to the Torelli group, while Morita's paper [30] reports on more recent developments of the subject. Contents 1. Definition and first examples 1 2. Generation 4 3. Presentation 8 4. The Dehn{Nielsen{Baer theorem 11 5. The Torelli group of a closed surface 13 References 16 1. Definition and first examples We consider a compact connected orientable surface Σ. By the classification theo- rem of surfaces, Σ is determined (up to homeomorphism) by the number of connected components of its boundary b := jπ0(@Σ)j and by its genus 1 g := · (rank H (Σ; ) − b + 1) : 2 1 Z We also fix an orientation on Σ: 1 g ··· ··· 1 b + Date: May 4, 2009. We are grateful to Sylvain Gervais for reading and commenting these notes. 1 2 In the sequel, if we wish to emphasize its topological type, then we denote the surface Σ by Σg;b. We are mainly interested in the closed surface Σg := Σg;0. Definition 1.1. The mapping class group of Σ is the group M(Σ) := Homeo+;@(Σ)= =∼ of orientation-preserving homeomorphisms Σ ! Σ whose restriction to @Σ is the identity, up to isotopy among homeomorphisms of the same kind. Other common notations for the mapping class group of Σ = Σg;b include MCG(Σ), Mod(Σ), Mg;b and Γg;b. Mapping class groups are also called \homeotopy groups" in the literature. There are variations for the definition of M(Σ), which may or may not give exactly the same group: We could fix a smooth structure on Σ and we could replace homeomorphisms by diffeomorphisms or, alternatively, we could triangulate Σ and replace homeomor- phisms by PL-homeomorphisms. However, this would not affect the definition of M(Σ) because we are in dimension two. (See Epstein's paper [8] for the PL case.) We could consider homeomorphisms up to homotopy relative to the boundary (') instead of considering them up to isotopy (=∼). Again, this would not affect the definition of M(Σ) since an old result of Baer asserts that two homeomor- phisms Σ ! Σ are homotopic relative to the boundary if and only if they are isotopic [1, 2]. We could allow homeomorphisms not to be the identity on the boundary: Let Mg(Σ) be the resulting group. Then, we have an exact sequence of groups b g (1.1) Z !M(Σ) !M (Σ) ! Sb ! 1: Here, we have numbered the boundary components of Σ from 1 to b, the map b b 1 Z !M(Σ) sends the i-th canonical vector of Z to the Dehn twist along a curve parallel to the i-th component of @Σ, the map M(Σ) !Mg(Σ) is the canonical one and the map Mg(Σ) ! Sb records how homeomorphisms permute the components of @Σ. We could allow homeomorphisms not to preserve the orientation: Let us de- note by M±(Σ) the resulting group. If the boundary of Σ is non-empty, any boundary-fixing homeomorphism must preserve the orientation: For b > 0, M±(Σ) = M(Σ): If the boundary is empty, then we have a short exact sequence of groups: ± For b = 0, 1 !M(Σ) !M (Σ) ! Z=2Z ! 1: Note that this sequence is split since there exists an involution Σg ! Σg which 3 reverses the orientation. (Indeed, one can embed Σg in R in such a way that 3 there is an affine plane H ⊂ R , such that the symmetry with respect to H leaves Σg globally invariant.) Remark 1.2. The set Homeo+;@(Σ) can be given the compact-open topology. Then, a continuous path ρ : [0; 1] ! Homeo+;@(Σ) is the same thing as an isotopy between ρ(0) +;@ and ρ(1). Therefore, we have M(Σ) = π0 Homeo (Σ) : 1See x2 below for the definition of a Dehn twist. 3 Let us now see a few examples of mapping class groups. First, let us consider the case of the disk D2 (g = 0; b = 1). Proposition 1.3 (Alexander's trick). The space Homeo@(D2) = Homeo+;@(D2) is con- tractible. In particular, we have M(D2) = f1g. Proof. For any homeomorphism f : D2 ! D2 which is the identity on the boundary, 2 2 and for all t 2 [0; 1], we define a homeomorphism ft : D ! D by t · f(x=t) if 0 ≤ jxj ≤ t; f (x) := t x if t ≤ jxj ≤ 1: 2 Here D is seen as a subset of C and jxj denotes the modulus of x 2 C. Then, the map @ 2 @ 2 H : Homeo (D ) × [0; 1] ! Homeo (D ); (f; t) 7−! ft @ 2 is a homotopy between the retraction of Homeo (D ) to fIdD2 g and the identity of @ 2 @ 2 Homeo (D ). Thus, Homeo (D ) deformation retracts to fIdD2 g. The mapping class group of the sphere S2 (g = 0; b = 0) can be deduced from this. Corollary 1.4. We have M(S2) = f1g. Proof. Let f : S2 ! S2 be an orientation-preserving homeomorphism, and let γ be a simple closed oriented curve in S2. Since f(γ) is isotopic to γ, we can assume that f(γ) = γ. Then, Proposition 1.3 can be applied to each of the two disks into which the 2 curve γ splits S . The mapping class group of the torus S1 × S1 (g = 1; b = 0) is non-trivial. 1 1 1 Proposition 1.5. Let (a; b) be the basis of H1(S × S ; Z) defined by a := [S × 1] and b := [1 × S1]. Then, the map 1 1 M : M(S × S ) −! SL(2; Z) 1 1 1 1 which sends the isotopy class [f] to the matrix of f∗ : H1(S × S ; Z) ! H1(S × S ; Z) relative to the basis (a; b), is a group isomorphism. 1 1 Proof. The fact that we have a group homomorphism M : M(S × S ) ! GL(2; Z) is clear. To check that M takes values in SL(2; Z), we observe that f (a) • b f (b) • b 8[f] 2 M(S1 × S1);M([f]) = ∗ ∗ −f∗(a) • a −f∗(b) • a 1 1 1 1 where • : H1(S × S ; Z) × H1(S × S ; Z) ! Z denotes the intersection pairing. Since f preserves the orientation, it also leaves invariant the intersection pairing. So, we have det M([f]) = (f∗(b) • b) · (f∗(a) • a) − (f∗(b) • a) · (f∗(a) • b) = f∗(a) • f∗(b) = a • b = 1: 1 1 2 2 The surjectivity of M can be proved as follows. We realize S × S as R =Z , in such a way that the loop S1 × 1 lifts to [0; 1] × 0 and 1 × S1 lifts to 0 × [0; 1]. Any matrix T 2 2 2 2 SL(2; Z) defines a linear homeomorphism R ! R , which leaves Z globally invariant 2 2 2 2 and so induces an (orientation-preserving) homeomorphism t : R =Z ! R =Z . It is easily checked that M([t]) = T . To prove the injectivity, let us consider a homeomorphism f : S1 × S1 ! S1 × S1 1 1 such that M([f]) is trivial. Since π1(S × S ) is abelian, this implies that f acts trivally 2 2 2 at the level of the fundamental group. The canonical projection R ! R =Z gives the universal covering of S1 × S1. Thus, f can be lifted to a unique homeomorphism 4 2 2 2 fe : R ! R such that fe(0) = 0 and, by assumption on f, feis Z -equivariant. Therefore, the “affine” homotopy 2 2 H : R × [0; 1] −! R ; (x; t) 7−! t · fe(x) + (1 − t) · x between IdR2 and fe, descends to a homotopy from IdS1×S1 to f. Since homotopy 1 1 coincides with isotopy in dimension two, we deduce that [f] = 1 2 M(S × S ). The mapping class group of the annulus S1 × [0; 1] (g = 0; b = 2) can be computed by the same kind of arguments (i:e: using the universal covering). 1 1 Proposition 1.6. Let a be the generator S × 1=2 of H1(S × [0; 1]; Z), and let ρ be the 1-chain 1 × [0; 1] of S1 × [0; 1]. Then, the map 1 N : M(S × [0; 1]) −! Z which sends the isotopy class [f] to the number k such that [−ρ + f(ρ)] = k · a, is a group isomorphism. 2. Generation As we shall now see, mapping class groups are generated by \Dehn twists." Those are homeomorphisms Σ ! Σ whose support is the regular neighborhood of a simple closed curve. In the sequel, a simple closed curve on Σ is simply called a circle, and is not necessarily oriented. Given two circles α and β on Σ, we define their geometric intersection number by 0 0 0 0 0 0 i(α; β) := min jα \ β j α isotopic to α; β isotopic to β; α t β : Definition 2.1. Let α be a circle on Σ. We choose a closed regular neighborhood N of α in Σ and we identify it with S1 × [0; 1] in such a way that orientations are preserved.
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