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

Figure 5.5 The Lickorish generating set for Mod(S).

first found by Lickorish, and so we call these Dehn twists the Lickorish generators [108].

THEOREM 5.12 (Lickorish generators) Let Sg be closed surface of genus g ≥ 1. Then the Dehn twists about the isotopy classes

a1, . . . , ag , m1, . . . , mg , c1, . . . , cg−1

shown in Figure 5.5 generate Mod(Sg ).

In the proof of Theorem 5.12 we refer to the Dehn twists in the statement of the theorem as “Lickorish twists,” so as not to confuse the is sue that we will be proving that they are indeed generators for Mod(S).

Proof. We proceed by induction on g. Since the Lickorish twists for the torus T 2 ≈ S1 are the standard generators for Mod(T 2), the theorem is true for the case of g = 1, and we may assume that g ≥ 2.

We again apply Lemma 5.9 to the action of Mod(Sg ) on the 1–dimensional

simplicial complex b g from Section 5.1. By Lemma 3.16 we have

N(S )

Ta1 Tm1 Ta1 (m1) = a1. Thus by Lemma 5.9 it suffices to show that Mod(Sg , m1), the stabilizer in Mod(Sg ) of m1, lies in the group generated by Lickorish twists.

If Mod(Sg , m~1) is the subgroup of Mod(Sg ) consisting of elements that preserve the orientation of m1, then we have

1 → Mod(Sg , m~1) → Mod(Sg , m1) → Z/2Z → 1.

Since the product of Lickorish twists Ta1 Tm2 1 Ta1 reverses the orientation of m1, it suffices to show that Mod(Sg , m~1) lies in the group generated by the Lickorish twists.

By Proposition 3.14 we have the following exact sequence:

1 → hTm1 i → Mod(Sg , m~1) → PMod(Sm1 ) → 1

where Sm1 ≈ Sg−1,2 is the surface obtained by deleting a representative of m1 from Sg (this is perhaps a slight abuse of notation, since we usually write Sm1 to mean the surface obtained from a surface S by cutting along a curve m1). Since Tm1 is a Lickorish twist, it is enough to show that PMod(Sm1 ) is generated by the images of the Lickorish twists.

We apply the Birman Exact Sequence (Theorem 5.5) twice. Let Sm′ 1 denote the surface obtained from Sm1 by forgetting the first puncture m−, and let Sm′′ 1 be the surface obtained from Sm′ 1 by forgetting the second puncture m+. We then have the following maps of exact sequences, where each square commutes:

(5.3)

and

m+

αk

Figure 5.7 Standard generators for π1(Sm′′ 1 , m+)

In the discussion below, we use the notation Sm1 , Sm′ 1 , and Sm′′ 1 instead of

the simpler notations Sg−1,2, Sg−1,1, and Sg−1 in order to emphasize the point that each of these surfaces comes with fixed maps Sm1 → Sm′ 1 →

Sm′′ 1 . In particular, there is no choice for the images of the Lickorish twists in Mod(Sm′ 1 ) and Mod(Sm′′ 1 ).

We start with sequence (5.4). The goal is to show that Mod(Sm′ 1 ) is generated by the images of the Lickorish twists in Mod(Sm′ 1 ); that is, we want to show that Mod(Sm′ 1 ) is generated by the Dehn twists about the simple closed curves shown on the bottom of Figure 5.6. By induction, Mod(Sm′′ 1 ) ≈ Mod(Sg−1) is generated by the Dehn twists about the images of these curves in Sm′′ 1 ≈ Sg , and so by the exact sequence (5.4), it suffices to show that each element of Push′(π1(Sm′′ 1 )) is a product of the Dehn twists given in the bottom of Figure 5.6.

Standard generators for π1(Sm′′ 1 ) ≈ π1(Sg−1) are shown in Figure 5.7. The mapping class Push′(α1) is equal to the product Tc1 Tm−21 (refer to Figure 5.6) so this element is a product of Lickorish twists.

We now explain how to write Push′(β1) as a product of Lickorish twists.

Using Lemma 3.16 we see that

Tm2 Ta2 (α1) = β1.

Thus, by Fact 5.7, Push′(β1) is conjugate to Push′(α1) by a product of Lickorish twists, and hence itself is a product of Lickorish twists.

Repeating this conjugation trick, we see that the image of each standard generator for π1(Sm′′ 1 ) under Push′ is a product of the images of the Lickorish twists in Mod(Sm′ 1 ). The required formulas are:

(Tc−i 1Ta−i+11 )(Ta−i 1Tc−i 1)(βi−1) = βi

Ta−i+11 Tm−i1+1 (βi) = αi

We remark that the Lickorish twists seem to be exactly designed for completing this step.

m′2 m′3

Sm1

m−

...

m+

Turning to sequence (5.3), it now remains to show that Push(π1(Sm′ 1 , m−)) lies in the group generated by the Dehn twists about the simple closed curves shown on the top of Figure 5.6. The proof is essentially the same as the previous argument. To facilitate the argument, it is helpful to notice that each Tm′i is a product of Lickorish twists, where the m′2, . . . , m′g−1 are the isotopy classes shown in Figure 5.8. This follows from the chain relation

THE HUMPHRIES GENERATORS

We can now give Humphries' proof that the Humphries generators do indeed form a generating set for Mod(Sg ).

THEOREM 5.13 (Humphries generators) Let g ≥ 2. Then the group Mod(Sg ) is generated by the Dehn twists about the 2g + 1 isotopy classes of nonseparating simple closed curves

a1, . . . , ag , c1, . . . , cg−1, m1, m2

shown in Figure 5.5.

In Proposition 8.6 below we show that Theorem 5.13 is sharp in the sense that, for g ≥ 2, any generating set for Mod(Sg ) consisting only of Dehn twists must have at least 2g + 1 elements.

Proof of Theorem 5.13. By Theorem 5.12 it suffices to show that the Lickorish twists Tm3 , . . . , Tmg can each be written in terms of the other Lickorish twists.

For any 1 ≤ i ≤ g − 2 we will find a product h of Dehn twists about the ai, ci, and mi+1 that takes mi to mi+2. It will then follow from Fact 3.6 in §3.3 that

Tmi+2 = hiTmi h−i 1,

and the theorem will be proved.

d

Figure 5.9 Taking mi to mi+2.

The top left of Figure 5.9 shows the simple closed curves we will use. In the

top right of the figure we see mi. The bottom right shows Tmi+1 Tai+1 Tci Tai (mi), and the bottom left shows the image d of the latter under the product

Tci+1 Tai+1 Tai+2 Tci+1 .

twists about nonseparating simple closed curves if each Dehn twist about a boundary curve is a product of Dehn twists about nonseparating simple closed curves. It turns out that for g ≥ 2 this is possible. Consider the simple closed curves shown in Figure 5.11. A special case of the star relation from Section 6.2 gives that

(Tc1 Tc2 Tc3 Tb)3Td−1Td−1

1 2

is equal to the Dehn twist about the boundary curve d.

Figure 5.11 Writing the Dehn twist about the boundary in terms of Dehn twists about nonseparating curves.

We thus have the following.

Corollary 5.15 Let S be any surface of genus g ≥ 2. The group PMod(S) is generated by finitely many Dehn twists about nonseparatin g simple closed curves in S.

In particular, for any surface S with punctures and/or boundary, PMod(S) is generated by the Dehn twists about the simple closed curves shown in Figure 5.10 (in the picture, one can interpret the small circles as either boundary components or as punctures).

On the other hand, for a genus 1 surface S with more than one boundary component, Mod(S) is not generated by Dehn twists about nonseparating curves. In this case there is a generating set consisting of fi nitely many Dehn twists about nonseparating curves and b − 1 Dehn twists about boundary curves, where b is the number of boundary components. It follows from the computation of H1(Mod(S); Z) (Section 6.1 below) that all b − 1 Dehn twists are needed.

LANTERN RELATION

The lantern relation is a relation in Mod(S) between seven Dehn twists, all lying on a subsurface of S homeomorphic to S04, a sphere with 4 boundary components.

z b4

Figure 6.1 Two views of the lantern relation on S04.

Proposition 6.1 (Lantern relation) Let x, y, z, b1, b2, b3, and b4 be simple closed curves in a surface S that are arranged as the curves shown in Figure 6.1. Precisely, this means that there is an orientation-preserving embedding S04 ֒→ S and that each of the above 7 curves is the image of the curve with the same name in Figure 6.1. In Mod(S) we have the relation

TxTy Tz = Tb1 Tb2 Tb3 Tb4 .

Proof. As discussed in Section 3.1, any embedding of a compact surface S′ into a surface S induces a homomorphism Mod(S′) → Mod(S). Since relations are preserved by homomorphisms, it suffices to check that the stated relation holds in Mod(S04).

To check the relation in Mod(S04), we cut S04 into a disk using three arcs and apply the Alexander Method (actually, two arcs would suffice). The computation is carried out in Figure 6.2.

For the computation, it is important to keep track of three conventions: Dehn twists are to the left, the simple closed curves x, y, and z are configured clockwise on the surface, and the relation is written using functional nota-

Tz

Tx

Figure 6.2 Proof of the lantern relation. The simple closed curves x, y, and z are shown in Figure 6.1.

Any surface S with χ(S) ≤ −2 contains an essential subsurface S′ home-

omorphic to S04. Indeed, if x and y are any two simple closed curves in S

with and ˆ , then ′ can be taken to be any closed i(x, y) = 2 i(x, y) = 0 S

regular neighborhood of x y. To see this, one can use the fact that if α and β are any two simple closed curves in S, and N is any regular neighborhood of α β, then |χ(N )| = |α ∩ β|. As such, we see that the lantern relation occurs in any such S.

The lantern relation implies another relation that is simpler, yet still interesting, namely:

TxTyTz = Ty Tz Tx = Tz TxTy.

This relation follows easily from the lantern relation plus the relation that each Tbi commutes with each of Tx, Ty , and Tz . We can contrast this result with Theorem 3.18, which states that there are no relations between Dehn twists Ta and Tb with i(a, b) = 2. Note that TxTyTz is not equal to Tz TyTx.