топология / Farb, Margalit, A primer on mapping class groups

Let (c1, . . . , c2g ) be a chain of isotopy classes of simple closed curves in Sg . As in Section 1.3, this means that i(ci, ci+1) = 1 and i(ci, cj ) = 0 otherwise. For concreteness, we take the curves shown in the left hand side of Figure 4.2. Orient each ci so that each algebraic intersection number

ˆ

i(ci, ci+1) is +1.

Recall that free homotopy classes of oriented curves in Sg correspond to conjugacy classes of elements of π1(Sg ); each ci is the conjugacy class of the element γi shown in the right hand side of Figure 4.2.

Figure 4.2 A chain on a genus 2 surface (left) and representatives in the fundamental group (right). The labels γ1, γ2, and γ3, and γ4 on the right hand side should be inferred.

Since Φ is an automorphism of π1(Sg ), it acts on the set of conjugacy classes of π1(Sg ). We claim that {Φ(ci)} is also a chain of isotopy classes of simple

closed curves, and that the algebraic intersections ˆ i i+1 are all i(Φ(c ), Φ(c ))

+1 or all −1. We prove this claim in four steps:

1.Φ(ci) is a simple closed curve for each i.

2.i(Φ(ci), Φ(cj )) = 0 for |i − j| > 1.

3.i(Φ(ci), Φ(ci+1)) = 1 for each i.

ˆ

4.i(Φ(ci), Φ(ci+1)) does not depend on i.

Each of the four steps will follow from Lemma 4.5 (or Corollary 4.6). For Step 1, recall that a conjugacy class of a primitive element of π1(Sg ) has a simple representative if and only if each pair of representatives for the class is not linked at infinity (cf. the proof of Proposition 1.6). N ow simply note that, as proved in Lemma 4.5, Φ preserves whether or not axes are linked. We could just as well note that the property of being linked at infinity is clearly preserved by any homeomorphism of ∂H2, and that Φ induces a homeomorphism Φ : ∂H2 → ∂H2, as proved in Corollary 4.6.

Similarly, for Step 2, we use the fact that two conjugacy classes have geometric intersection number 0 if and only if any pair of representatives

is unlinked at infinity, and this latter property is preserve d by Φ . For Step 3 we use the following Φ -invariant characterization of when two conjugacy classes have representatives with geometric intersection number 1 (plus Lemma 4.5):

Two conjugacy classes a and b have geometric intersection number 1 if and only if for some representative α of a that is linked at infinity with a given representative β of b, the set of representatives of a that are linked at infinity with β is precisely {βk αβ−k : k Z}, .

Step 4 is more intricate. It suffices to prove that given three conjugacy classes a, b, and c with i(a, b) = i(b, c) = 1 and i(a, c) = 0, we can

characterize the agreement of the signs of ˆ with ˆ in terms of i(a, b) i(b, c)

data we know to be preserved by Φ. Let α, β, and γ be any representatives for a, b, and c so that the axes for α and γ are disjoint, and the axis for β intersects each of the axes for α and γ once each. Now note the following.

ˆ ˆ

With the above notation, i(a, b) has the same sign as i(b, c) if and only if the axes for αβα−1 and γβγ−1 lie on different sides of the axis for β.

Replacing a, b, and c with ci, ci+1, and ci+2, we apply Lemma 4.5 and Corollary 4.7 to complete Step 4, thus proving the claim.

By the Change of Coordinates Principle, more precisely by Example 6 in Section 1.3, there is a homeomorphism φ that fixes the basepoint of π1(Sg ) and satisfies φ (ci) = Φ (ci) (with orientation) for each i. Here φ and Φ denote the induced actions on (conjugacy classes of) elements of π1(Sg ).

To complete the proof of the theorem we must now prove that the mapping class [φ], acting on π1(Sg ), induces the outer automorphism [Φ]. Since the representatives γi generate π1(Sg ), it suffices to show that there is an inner automorphism Iα of π1(Sg ) so that

Iα ◦ φ−1 ◦ Φ(γi) = γi

for each i. We will use the notation Iβ for the inner automorphism of π1(Sg ) given by γ 7→βγβ−1.

Note that it is simply not true in general that if an automorphism of a group fixes the conjugacy class of each generator, then it is an inne r automorphism. As an example, take the free group on {x, y, z} and consider the automorphism given by x 7→yxy−1, y 7→y, and z 7→z.

We will use the fact that the particular representatives γi of the ci shown in Figure 4.2 form a “chain” in the sense that the lifts of γi and γi+1 to

γi+1 are linked on the surface; more precisely, if we take a small closed neighborhood of the basepoint of π1(Sg ), γi and γi+1 are linked on the boundary of this neighborhood. Arbitrary lifts of ci and ci+1 may or may not be linked at infinity.

Denote φ−1 ◦ Φ by F . Since φ induces an automorphism of π1(Sg ), we see that F still preserves linking at infinity. Again, the goal is to find an element α so that Iα ◦ F is the identity automorphism of π1(S).

Since F (ci) = ci with orientation for all i, we have in particular F (c1) = c1, and so F (γ1) = α−1 1γ1α1 for some α1 π1(Sg ). Thus,

Iα1 ◦ F (γ1) = γ1.

We know that F (c2) = c2, that Iα1 ◦ F preserves linking at infinity, and that γ1 and γ2 are linked. It follows from the characterization of conjugacy classes with geometric intersection number 1 given above that Iα1 ◦F (γ2) = γ1−k γ2γ1k for some k Z. Therefore

Iγ1k α1 ◦ F (γ1) = Iγ1k ◦ Iα1 ◦ F (γ1) = γ1

and

Iγ1k α1 ◦ F (γ2) = Iγ1k ◦ Iα1 ◦ F (γ2) = γ2.

We can now see inductively that Iγ1k α1 ◦ F (γi) = γi for each i ≥ 3, and so is the desired inner automorphism. Indeed, since γ1 and γ2 are both

fixed by I k ◦ F , it follows that each element of {γl γ2γ−l} is fixed. But

γ1 α1 1 1

since γ3 is linked with γ2, it is characterized in π1(Sg ) by the properties that it is linked with γ2, and that it “lies between” γ2l γ1γ2−l and γ2l+1γ1γ2

for some particular l. Thus γ3 is fixed by Iγ1k α1 ◦ F and, by induction, each γi for i > 3 is also fixed (the inductive step for γi uses that both γi−1 and γi−2 are fixed). We have thus found the required inner automorphis m, and

The punctured case. There is a version of the Dehn–Nielsen–Baer Theorem for punctured surfaces, as follows. Let Out (π1(S)) be the subgroup of Out(π1(S)) consisting of elements that preserve the set of conjugacy classes of the simple closed curves surrounding individual punctures. Note that these conjugacy classes are precisely the primitive conjugacy classes that correspond to the parabolic elements of Isom(H2).

THEOREM 4.8 Let S = Sg,p be a hyperbolic surface of genus g with p punctures. Then the natural map

Mod±(S) → Out (π1(S))

is an isomorphism.

The proof of this more general theorem follows the same outline as in the proof of the closed case (Theorem 4.1). We content ourselves to point out the two main differences.

1.In the case S = Sg , we knew automatically that any automorphism of π1(Sg ) must send hyperbolic elements to hyperbolic elements since all nontrivial elements of π1(Sg ) are hyperbolic. If S is not closed

then an arbitrary automorphism of π1(S) can exchange hyperbolic elements with parabolic elements. But the fact that we consider Out (π1(S)) instead of Out(π1(S)) in the statement of Theorem 4.8 exactly accounts for this.

2.The map π1(S) → H2 given by taking the orbit in H2 of a single point is not a quasi-isometry. To remedy this, we truncate S by deleting a

small neighborhood of each puncture. We can choose the neighborhoods to be small enough so that the preimage in H2 of the truncated surface is a connected space X. If we endow X with the path metric,

then the action of π1(S) on X satisfies the conditions of Theorem 4.2 and so π1(S) is quasi-isometric to X.

The proof of Lemma 4.5 now proceeds similarly as before. Points are farther in X than they are in H2, so there is no problem in choosing N

so that the sets Oγ and OδN are far apart. Also, there is no obstruction to choosing the paths {αi} and {βi}. If {Φ(αi)} and {Φ(βi)} were to cross, we would still have a short path in X between two vertices of the paths, which would give the desired contradiction.

We already mentioned the theorem of Nielsen that Out(F2) ≈ GL(2, Z). Thus, we have

Out(F2) ≈ GL(2, Z) ≈ Mod±(S1,1).

In the language of Theorem 4.8, this means that the group Out (π1(S1,1)) is the entire group Out(π1(S1,1)). In other words, every element of the outer automorphism group of F2 = hx, yi preserves the conjugacy class [x, y].

Thus GL(n, Z), Mod±(S) and Out(Fn) can be viewed as three different generalizations of the same group.

Once-punctured versus closed. The Dehn–Nielsen–Baer Theorem can be used to relate the group Mod(Sg ) to the group Mod(Sg,1), where Sg,1 is the genus g ≥ 2 surface with one marked point. This is done by the following isomorphism of exact sequences, where each square is a commutative diagram:

1 Inn(π1(Sg )) Aut(π1(Sg )) Out(π1(Sg )) 1

1 π1(Sg ) Mod±(Sg,1) Mod±(Sg ) 1

≈

Out (π1(Sg,1))

The first row is the usual relationship between the automorph ism group and outer automorphism group of any group. For the second row, notice that π1(Sg ) clearly lies in the kernel of the natural map Mod±(Sg,1) → Mod±(Sg ); that is, pushing the basepoint around a loop is isotopic to the identity on Sg (of course this such an isotopy will not leave fixed the marked point). We will prove in Section 5.2 that this is the entire kernel. The resulting short exact sequence, namely, the second row in the diagram, is a special case of what we will call the “Birman Exact Sequence. ”

The isomorphism Inn(π1(Sg )) ≈ π1(Sg ) is equivalent to the statement that π1(Sg ) has trivial center, and the isomorphism Out(π1(Sg )) ≈ Mod±(Sg )

is the Dehn–Nielsen–Baer Theorem. Now, there certainly is a map Mod±(Sg,1) → Aut(π1(Sg )) that makes the diagram (as described so far) commutative—

simply choose the basepoint for π1(Sg ) to be the marked point. The Five Lemma then tells us that the middle vertical map is an isomorphism from

Mod±(Sg,1) to Aut(π1(Sg )).

Finally, we examine the isomorphism Out (π1(Sg,1)) → Aut(π1(Sg )). At first it seems odd to have an outer automorphism group of a surf ace group be the same as the automorphism group of another surface group. However, given φ Out (π1(Sg,1)) we get an element of Aut(π1(Sg )) by taking the unique representative automorphism of φ that fixes the loop corresponding to the puncture (not just up to conjugacy), and this gives the desired isomorphism.

4.2 TWO OTHER VIEWPOINTS

In this section we provide two other proofs of the Dehn–Niels en–Baer Theorem; one inspired by 3–manifold theory (adapted from [79, Thm 13.6]) and one using harmonic maps. There are various other proofs, each involving a different kind of mathematics. For example in Theorem 1.8 of [39] Calegari exploits the relationship between simple closed curves on Sg and HNN extensions of π1(Sg ) to give an inductive argument for the Dehn– Nielsen–Baer Theorem. Zieschang–Vogt–Coldewey give a com binatorial- group-theoretical proof in [175, §5.6], and Seifert gives an elementary covering space argument in [151].

Let S be a surface with χ(S) < 0. Since S is a K(π1(S), 1) space, every outer automorphism of π1(S) is induced by some (unbased) map S → S. By the Whitehead theorem and the fact that πi(S) = 0 for i > 1, we have that this self-map of S is a homotopy equivalence. Thus, for the surjectivity part of the Dehn–Nielsen–Baer Theorem, it suffices to sho w that every homotopy equivalence of S is homotopic to a homeomorphism of S.

THEOREM 4.9 If g ≥ 2 then any homotopy equivalence φ : Sg → Sg is homotopic to a homeomorphism.

We give two approaches to Theorem 4.9 below, one topological and one analytical.

PANTS DECOMPOSITIONS: THE TOPOLOGICAL APPROACH

Recall that a pair of pants is a compact surface of genus 0 with three boundary components. Let S be a compact surface with χ(S) < 0. A pair of pants decomposition of S, or pants decomposition of S, is a collection of disjoint simple closed curves in S with the property that when we cut S along these curves, we obtain a disjoint union of pairs of pants. Equivalently, a pants decomposition of S is a maximal collection of disjoint, essential simple closed curves in S with the property that no two of these curves are isotopic.

We can easily prove the equivalence of the two definitions of a pants decomposition. First, suppose we have a collection of simple closed curves that cuts S into pairs of pants. We immediately see that every curve is essential since there are no disk components when we cut S. Further, since any simple closed curve on a pair of pants is either homotopic to a point or to

a boundary component, it follows that the given collection is maximal. For the other direction, suppose we have a collection of disjoint, nonisotopic essential simple closed curves in S. If the surface obtained from S by cutting along these curves is not a collection of pairs of pants, then it follows from the Classification of Surfaces and the additivity of Eul er characteristic that one component of the cut surface either has positive genus or is a sphere with more than three boundary components. On such a surface there exists an essential simple closed curve that is not homotopic to a boundary component. Thus the original collection of curves was not maximal.

A pair of pants has Euler characteristic −1. If we cut a surface along a collection of disjoint simple closed curves, the cut surface has the same Euler characteristic as the original surface. Thus, a pants decomposition of S cuts S into −χ(S) pairs of pants. It follows that, for a compact surface S of genus g with b boundary components, a pants decomposition for S has

−3χ(S) − b

2

= 3g + b − 3

curves. Indeed, each pair of pants has three boundary curves and, aside from the curves coming from ∂S, these curves match up in pairs to form curves in S. In particular, a pants decomposition of Sg for g ≥ 2 has 3g −3 curves, cutting Sg into 2g − 2 pairs of pants.

First proof of Theorem 4.9. We modify φ in steps by homotopies until it is a homeomorphism; at each stage, the resulting map will be called φ. Choose some pants decomposition P of Sg consisting of smooth simple closed curves. We first approximate φ by a smooth map that is transverse to P. By choosing a close–enough approximation we can assume tha t the approximation is homotopic to φ. By transversality we have that φ−1(P) is a collection of simple closed curves. If any component of φ−1(P) is inessential we can homotope φ to remove that component since such a curve bounds a disk, and we can use that disk to define the homotopy.

Since φ induces an automorphism on π1(Sg ) it takes primitive conjugacy classes in π1(Sg ) to primitive conjugacy classes in π1(Sg ). Thus the restriction of φ to any particular component of φ−1(P) has degree ±1 as a map S1 → S1. We can therefore homotope φ so that it restricts to a homeomorphism on each component of φ−1(P).

Since φ is a homotopy equivalence it has degree ±1, and so φ is surjective. It follows that φ−1(P) has at least 3g − 3 components. If it had more, then two such components would necessarily be isotopic, and the annulus

between them would give rise to a homotopy of φ reducing the number of components of φ−1(P).

At this point φ is a homeomorphism on each component of φ−1(P), and φ maps each component of Sg − φ−1(P) to a single component of Sg − P. It therefore suffices to show that if R and R′ are pairs of pants, and if φ : R → R′ is a continuous map such that φ|∂R is a homeomorphism, then there is a homotopy of φ to a homeomorphism R → R′, so that the homotopy restricts to the identity map on ∂R.

Let X be the union of three disjoint arcs in R′, one connecting each pair of boundary components. Note that it must be that R′ − (∂R′ X) is homeomorphic to a disjoint union of two open disks. Again, we may assume that φ is smooth, and so φ−1(X) is a properly embedded 1–manifold with boundary lying in ∂R. If any component of φ−1(X) is closed, then it is necessarily nullhomotopic (since all nonperipheral simple closed curves on a pair of pants are nullhomotopic), and we may modify φ by homotopy to remove this component.

Since φ|∂R is assumed to be a homeomorphism, and so it takes distinct boundary components to distinct boundary components, φ−1(X) consists of exactly three arcs, one for each pair of boundary components of R. We can modify φ so that it restricts to a homeomorphism on each component of

HARMONIC MAPS: THE ANALYTIC APPROACH

We now give an analytic proof of Theorem 4.9. While this proof relies on the machinery of harmonic maps, it is conceptually straightforward.

A harmonic map between Riemannian manifolds is one that minimizes the

energy functional

Z

E(f ) = kDf k2.

µ

Second proof of Theorem 4.9. If we endow S with a hyperbolic metric, then it is a theorem of Eells–Sampson and Shibata that, with respe ct to this metric, there is a harmonic map h in the homotopy class of φ [49, 155].

Since h is a homotopy equivalence, we must have that the degree of h is ±1. We now apply the (highly nontrivial) theorem that any harmonic map of de-

gree 1 is a diffeomorphism (Shibata proved the harmonic map is a homeomorphism [155], which is all we need, and it follows from work of Lewy [107] and Heinz [78] that the harmonic map is in fact a diffeomorphism). 2

Chapter Five

Generating the mapping class group

Is there a way to generate all (homotopy classes of) homeomorphisms of a surface by compositions of simple-to-understand homeomorphisms? We have already seen that Mod(T 2) is generated by the Dehn twists about the latitude and longitude curves. Our next main goal will be to prove the following result.

5.1 (Dehn–Lickorish Theorem) For g ≥ 0 the group Mod(Sg ) is generated by finitely many Dehn twists about nonseparatin g simple closed curves.

Theorem 5.1 can be likened to the theorem that for each n ≥ 2 the group SL(n, Z) of n × n integer matrices of determinant 1 can be generated by finitely many elementary matrices. As with the linear case, T heorem 5.1 is fundamental to our understanding of Mod(Sg ).

In the 1920's Dehn proved that Mod(Sg ) is generated by 2g(g − 1) Dehn twists [46]. Mumford, building on Dehn's work, showed in 1967 that only Dehn twists about nonseparating curves were needed [131]. In 1964 Lickorish, apparently unaware of Dehn's work, gave an independent proof that Mod(Sg ) is generated by the Dehn twists about the 3g − 1 nonseparating curves shown in Figure 5.5 below [108].

In 1979 Humphries [82] proved the surprising theorem that the twists about the 2g + 1 curves in Figure 5.1 suffice to generate Mod(Sg ). These generators are often called the Humphries generators. Humphries further showed that any set of Dehn twist generators for Mod(Sg ) must have at least 2g + 1 elements; see Section 8.2 for a proof of this fact.

Punctures and pure mapping class groups. Theorem 5.1 is simply not true for surfaces with multiple punctures, since no composition of Dehn