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

αb

Push(α) a

Figure 5.4 The point pushing map Push from the Birman Exact Sequence.

Fact 5.7 For any h PMod(S ) and any α π1(S, x), we have

Push(h (α)) = hPush(α)h−1 .

Fact 5.7 follows immediately from the definitions.

The proof. We now give the proof of the existence of the Birman Exact Sequence.

Proof of Theorem 5.5. We will denote by PHomeo+(S) the group of orientation- preserving homeomorphisms of S that induce the trivial permutation on the

set of punctures and that restrict to the identity map on each component of ∂S. There is a fiber bundle

with total space PHomeo+(S), with base space S (i.e., the configuration space of a single point in S), and with fiber the subgroup of PHomeo+(S) consisting of elements that fix the point x. The map E is evaluation at the point x.

We now explain why E : PHomeo+(S) → S is a fiber bundle, that is, why PHomeo+(S) is locally homeomorphic to a product of an open set U of S with PHomeo+(S, x) so that the restriction of E is projection to the first factor. Let U be some open neighborhood of x in S that is homeomorphic to a disk. Given u U we can choose a φu Homeo+(U ) so that φu(x) = u and so that φu varies continuously as a function of u. We have a homeomorphism U × PHomeo+(S, x) → E−1(U ) given by

(u, ψ) 7→φu ◦ ψ.

y S, we can choose a homeomorphism ξ of S taking x to y. Then there is a homeomorphism E−1(U ) → E−1(ξ(U )) given by ψ 7→ξ ◦ ψ, and so we have verified the fiber bundle property.

The theorem now follows from the long exact sequence of homotopy groups associated to the above fiber bundle. The relevant part of the sequence is the following.

·· · → π1(PHomeo+(S)) → π1(S) → π0(PHomeo+(S, x))

→π0(PHomeo+(S)) → π0(S) → · · ·

By Theorem 1.14 the group π1(PHomeo+(S)) is trivial (note that the connected components of the identity in PHomeo+(S) and Homeo+(S) are the same), and of course π0(S) is trivial. The remaining terms are isomorphic to the terms of the Birman Exact Sequence.

Finally, the maps given by the long exact sequence of homotopy groups are

There is a version of Theorem 5.5 where one forgets multiple punctures instead of a single version; see Chapter 9. However, in most cases, one can simply apply Theorem 5.5 iteratively in order to forget one puncture at a time.

Surfaces with χ(S) ≥ 0. In the proof of Theorem 5.5 we used the assumption that χ(S) < 0 in order to say that π1(PHomeo+(S)) = 1. But we can still use the long exact sequence coming from the fiber bun dle (5.1) for other surfaces. For instance, for the torus T 2 we have π1(PHomeo+(T 2)) ≈ π1(T 2) ≈ Z2, and the relevant part of the short exact sequence becomes

This gives another proof that Mod(S1,1) ≈ Mod(T 2).

GENERATING Mod(S0,n)

Let S0,n be a sphere with n punctures. As per Section 2.2, PMod(S0,n) = 1 for n ≤ 3. To understand the situation for more punctures, we can apply the Birman Exact Sequence:

1 → π1(S0,3) → PMod(S0,4) → PMod(S0,3) → 1.

Since π1(S0,3) ≈ F2, we obtain that PMod(S0,4) ≈ F2. Moreover, the Birman Exact Sequence gives geometric meaning to this algebraic statement: elements of π1(S0,3) represented by simple loops map to Dehn twists in PMod(S0,4), and so the standard generating set for π1(S0,3) gives a generating set for PMod(S0,4) consisting of two Dehn twists about simple closed curves with geometric intersection number 2.

We can increase the number of punctures using the Birman exact sequence:

1 → π1(S0,4) → PMod(S0,5) → PMod(S0,4) → 1.

Since π1(S0,4) ≈ F3 and PMod(S0,4) ≈ F2, we obtain PMod(S0,5) ≈ F2 F3. Inductively, we see that PMod(S0,n) is an iterated extension of

free groups. Applying Fact 5.6, plus the fact that π1(S0,n) is generated by simple loops, we find the following.

THEOREM 5.8 For n ≥ 0 the group PMod(S0,n) is generated by finitely many Dehn twists.

To generate all of Mod(S0,n), we again apply the following exact sequence:

1 → PMod(S0,n) → Mod(S0,n) → Σn → 1.

It follows that a generating set for Mod(S0,n) is obtained from a generating set for PMod(S0,n) by adding lifts of generators for Σn. We know that Σn is generated by transpositions. A simple lift of a transposition is a “half-twist,” defined in Chapter 9.

CAPPING THE BOUNDARY

By souping up the proof of the Birman exact sequence we can give another perspective on the boundary capping sequence (Proposition 3.13) that unifies it with the Birman exact sequence.

Let S◦ be a surface with nonempty boundary, and let S be the surface obtained from S◦ by capping some component β of ∂S◦ with a punctured disk. As in Proposition 3.13, we have a short exact sequence:

Note that hTβ i is central in Mod(S◦), since any element of Mod(S◦) has a representative that is the identity in a neighborhood of ∂S◦.

We now give our second proof of Proposition 3.13, using the notation from the sequence (5.2).

Second proof of Proposition 3.13. The proof has two steps. Step 1 is to identify Mod(S◦) with a different group and to reinterpret the capping map in the new context, and Step 2 is to apply the method or proof of the Birman exact sequence to the corresponding fiber bundle.

Step 1. Let S denote the surface obtained from S◦ by capping the boundary

b

where D is some smoothly embedded closed disk in S, and diffeomorphisms are taken to fix D pointwise. This identification can be realized

where Emb ((D, S), (p, v)) is the space of smooth, orientation-preserving

126 CHAPTER 5

To get the middle row directly, one can consider the fiber bund le

5.3 PROOF OF FINITE GENERATION OF Mod(S)

To show that is finitely generated we consider its action on b .

Mod(S) N(S)

Note that indeed acts on b since homeomorphisms take non-

Mod(S) N(S)

separating simple closed curves to nonseparating simple closed curves, and homeomorphisms preserve geometric intersection number. It is a basic principle from geometric group theory that if a group G acts on a path-connected space X, and if D is a subspace of X whose G–translates cover X, then G is generated by the set {g G : gD ∩ D 6= }. The proof of this is implicit in our proof of Theorem 4.2. The next lemma is a specialized version of this fact, designed specifically so that we can apply it to the acti on of Mod(S)

on b (S).

N

Lemma 5.9 Suppose that a group G acts by simplicial automorphisms on a connected, 1–dimensional simplicial complex X. Suppose that G acts transitively on the vertices of X, and that it also acts transitively on pairs of vertices of X that are connected by an edge. Let v and w be two vertices of X that are connected by an edge, and choose h G so that h(w) = v. Then the group G is generated by the element h together with the stabilizer of v in G.

Proof. Let g G. We would like to show that g is contained in the subgroup H < G generated by the stabilizer of v together with the element h. Since X is connected, there is a sequence of vertices

v = v0, . . . , vk = g(v)

where adjacent vertices are connected by an edge. Since G acts transitively on the vertices of X, we can choose elements gi of G so that gi(v) = vi. We take g0 to be the identity and gk to be g. We will prove by induction that gi H. The base case g0 H clearly holds. Now assume that gi H. We must prove that gi+1 H.

Applying the element gi−1 to the edge between vi = gi(v) and vi+1 =

gi+1(v), we obtain the edge between v and gi−1gi+1(v). Since G acts transitively on ordered pairs of vertices of X that are connected by an edge, there

is an element r G that takes the pair (v, gi−1gi+1(v)) to the pair (v, w). In particular, r lies in the stabilizer of v and rgi−1gi+1(v) = w. We then have that hrgi−1gi+1(v) = v, which means that hrgi−1gi+1 lies in the stabilizer of v. In particular, hrgi−1gi+1 H. Since h and r lie in H by the definition of H and since gi lies in H by induction, we have that gi+1 lies in H. In

We are now ready to prove the following theorem, which contains Theorem 5.1 as the special case n = 0.

THEOREM 5.10 Let Sg,n be a surface of genus g ≥ 1 with n ≥ 0 punctures. Then the group PMod(Sg,n) is finitely generated by Dehn twists

about nonseparating simple closed curves in Sg,n.

Recall that we already showed that PMod(S0,n) is finitely generated by Dehn twists for n ≥ 0 (Theorem 5.8).

Proof. We will use double induction on genus and the number of punctures of S, with base cases T 2 = S1,0 and S1,1.

We start with the inductive step on the number of punctures. Let g ≥ 1 and let n ≥ 0. Assuming that PMod(Sg,n) is generated by finitely many Dehn twists about nonseparating simple closed curves {αi} in Sg,n, we will show that PMod(S ) is generated by finitely many Dehn twists about nonseparating curves in S . We may assume that (g, n) 6= (1, 0) since we know that Mod(S1,1) ≈ Mod(T 2) is generated by Dehn twists about nonseparating simple closed curves.

We have the Birman Exact Sequence

1 → π1(Sg,n) → PMod(Sg,n+1) → PMod(Sg,n) → 1.

Since g ≥ 1, we have that π1(Sg,n) is generated by the classes of finitely many simple nonseparating loops. By Fact 5.6, the image of each of these loops is a product of two Dehn twists about nonseparating simple closed curves. We begin building a generating set for PMod(S ) by taking each of these Dehn twists individually. In order to complete the generating set it remains to choose a lift to PMod(S ) of each Dehn twist generator Tαi of PMod(Sg,n). But given the nonseparating simple curve αi in Sg,n there exists a nonseparating curve in Sg,n+1 that maps to αi under the forgetful map Sg,n+1 → Sg,n. Thus the Dehn twist Tαi in PMod(Sg,n)

ple closed curve, it follows that Mod(Sg , ~a) is generated by finitely many

5.4 EXPLICIT SETS OF GENERATORS

The goal of this section is to find an explicit finite set of Dehn twist generators for Mod(S). Our strategy for accomplishing this is to sharpen our proof that Mod(S) is generated by finitely many Dehn twists. More specifically, we choose a candidate set of generators, and check that each step of the proof of finite generation can be achieved by using our can didate set.

THE CHAIN RELATION

In the very last step of our proof of Theorem 5.12 below, we will require the following relation between Dehn twists. Recall that a sequence of isotopy

classes c1, . . . , ck in a surface S is called a chain if i(ci, ci+1) = 1 for all i and i(ci, cj ) = 0 for |i − j| > 1.

Proposition 5.11 (Chain relation) Let k ≥ 0 and let c1, · · · , ck be a chain of curves in a surface S. If we take representatives for the ci that are in minimal position, and then take a closed regular neighborhood of their union, then the boundary of this neighborhood consists of one or two simple closed curves, depending on whether k is even or odd. Denote the isotopy classes of these boundary curves by d in the even case and by d1 and d2 in the odd case. Then the following relations hold in Mod(S):

In each case the relation in Proposition 5.11 is called a chain relation, or a k– chain relation. The chain relation can be proved via the Alexander Method. In Chapter 9 we will derive the chain relations as consequences of relations in the braid group.

The 2–chain relation is a well-known example of the chain relation. In this case, the relation says that if i(a, b) = 1, then

(TaTb)6 = Td

where d is the boundary of a regular neighborhood of a b. If a and b lie in T 2 or S1,1, then Td is trivial, and we have the relation (TaTb)6 = 1. Via the isomorphism of Theorem 2.5, this is simply the relation

in SL(2, Z).

There is another version of the chain relation that is sometimes useful. In the above notation, this other version reads:

(Tc21 Tc2 · · · Tck )2k = Td and (Tc21 Tc2 · · · Tck )k = Td1 Td2 ,

for k even and odd, respectively.

Dehn twists have roots. A surprising consequence of the last relation is that the Dehn twist about a nonseparating simple closed curve has a nontrivial root in Mod(Sg ) when g ≥ 2. If we consider a chain of simple closed curves c1, . . . , c2g−1 in Sg , then the two boundary components of a regular neighborhood of ci are nonseparating simple closed curves in the same isotopy class d, so we have

(Tc21 Tc2 · · · Tc2g−1 )2g−1 = Td2.

Thus, since Td commutes with each Tci , we have

[(Tc21 Tc2 · · · Tc2g−1 )1−g Td]2g−1 = Td.

McCullough–Rajeevsarathy proved that 2g − 1 is actually the largest order of a root of Td for any g ≥ 2 [118]. It is not difficult to see that Dehn twists about separating simple closed curves have roots: for example, if we imagine fixing the subsurface of Sg to once side of a separating curve d and twisting the other side by an angle π, then we get a square root of Td. A more formal way to do this is to use the first chain relation wit h a chain of even length.

THE LICKORISH GENERATORS

Our eventual goal is to show that the Humphries generating set (see the beginning of the chapter) is indeed a generating set for Mod(Sg ). As a first step we show that the Dehn twists about the 3g − 1 simple closed curves indicated in Figure 5.5 generate Mod(Sg ). This specific generating set was