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

surprising, the proof is more subtle than the usual Change of Coordinates Principle.

To state the result we need the fact that a separating simple closed curve in Sg (or its isotopy class) induces a splitting of H1(Sg ; Z). By a splitting of H1(Sg ; Z) we mean a decomposition as a direct product of subgroups that are orthogonal with respect to skew-symmetric bilinear pairing given by al-

gebraic intersection number ˆ on 1 g Z . A simple closed curve that i H (S ; ) γ

separates Sg into two subsurfaces S′ and S′′ gives a splitting of H1(Sg ; Z) into the product of the two subgroups H1(S′; Z) and H1(S′′; Z), each subgroup consisting of those homology classes supported on one side of γ or the other. We say that two isotopy classes of simple closed curves are I(Sg)– equivalent if there is an element of I(Sg) taking one to the other.

The following theorem, observed by Johnson [90, §6], gives that the obvious necessary condition for two simple closed curves on Sg to be I(Sg)– equivalent is also sufficient.

Proposition 8.14 Let c and c′ be two isotopy classes of simple closed curves in Sg . If c and c′ are separating, then they are I(Sg )–equivalent if and only if they induce the same splitting of H1(Sg ; Z). If c and c′ are nonseparating, then they are I(Sg )–equivalent if and only if, up to sign, they represent the same element of H1(Sg ; Z).

Proof. For both cases, one direction is obvious, and so we only need to prove that the obvious necessary condition for I(Sg )–equivalence is suffi-

cient. Let γ and γ′ be representative curves for the isotopy classes c and c′.

Suppose that c and c′ are separating. Let S1 and S2 be the two embedded subsurfaces of Sg bounded by γ, and let S1′ and S2′ be the two embedded subsurfaces bounded by γ′. Up to renumbering, our hypothesis tells us that H1(S1; Z) and H1(S1′ ; Z) are equal as subsets of H1(Sg ; Z). Therefore, S1 and S1′ have the same genus, and hence are homeomorphic. Fix a homeomorphic identification of γ with γ′, and choose any homeomorphism φ1 : (S1, γ) → (S1′ , γ′) respecting this identification. By Theorem 8.4 and by the hypothesis, there is a homeomorphism ψ1 Homeo+(S1, γ′) so that ψ1 ◦ φ1 is the identity automorphism of H1(S1; Z) = H1(S1′ ; Z). Here we are invoking our claim that all of the results in Section 8.2 work for surfaces with one boundary component. We similarly choose ψ2 ◦ φ2 : S2 → S2′ . Together, the maps ψ1 ◦φ1 and ψ2 ◦φ2 induce a homeomorphism of Sg that takes γ to γ′ and acts trivially on H1(Sg ; Z).

Now suppose that c and c′ are nonseparating. We would like to proceed similarly to the previous case. One difficulty is that we do no t have a surjectivity statement for the action of the stabilizer of c in Mod(Sg ) on the homology of the surface obtained by cutting along c. Instead we proceed as follows.

Let β be any simple closed curve in Sg that intersects γ once. By the argument in the third proof of Theorem 8.4 there is a simple closed curve β′ that intersects γ′ once and is homologous to β. Let δ be the boundary of a regular neighborhood of β γ, and let δ′ be the boundary of a regular neighborhood of β′ γ′. Applying the present proposition to the case of separating simple closed curves, there is an element of I(Sg ) taking δ to δ′. Since I(S1,1) is trivial (Theorem 2.5), it follows that this element of I(Sg )

The statement of Proposition 8.14 can be sharpened in the case of isotopy classes of oriented simple closed curves. Two isotopy classes of oriented nonseparating simple closed curves are I(Sg)–equivalent if and only if they represent the same element of H1(Sg ; Z). Two isotopy classes of oriented separating simple closed curves are I(Sg)–equivalent if and only if they induce the same ordered splitting of H1(Sg ; Z), where the ordering of the factors comes from the fact that the curve has well-defined le ft and right sides.

The statement of Proposition 8.14 (and its proof) apply to the cases of surfaces with either one boundary or one puncture.

GENERATORS FOR I(Sg )

Birman and Powell proved that I(Sg) is generated by the infinite collection of all Dehn twists about separating simple closed curves and all bounding pair maps [19, 144]. The general method they used is as follows.

From relations to generators. Let

ρ

1 → K → E → Q → 1

be a short exact sequence of groups. Suppose that E is generated by {e1, . . . , ek }, and that Q has a presentation with generators ρ(e1), . . . , ρ(ek ) and relations

{ri = 1}, where each ri is a word in the {ρ(ei)}. For each i let rei be the corresponding word in the ei. As elements of E each rei lies in K. It is easy

to check that the {rei} is a normal generating set for K, that is, the set of all conjugates of all ri by elements of E generate K.

An infinite generating set for I(Sg). Birman's idea was to apply the above general fact to the short exact sequence

1 → I(Sg ) → Mod(Sg ) → Sp(2g, Z) → 1.

Birman determined a finite presentation for Sp(2g, Z) and made the remark that the relators for Sp(2g; Z) give rise to generators for I(Sg ). Then her student Powell interpreted each of these generators as products of Dehn twists about separating curves and bounding pair maps, thus proving that I(Sg ) is generated by (infinitely many) such maps.

Putman has recently shown that the same generating set for I(Sg ) can be derived from methods similar to the ones that we used to show that Mod(Sg ) is generated by Dehn twists; see [146].

Whittling down the infinite generating set. Johnson showed that, for g ≥ 3, the Dehn twists about separating simple closed curves are not needed to generate I(Sg). In other words he proved that any such Dehn twist is a product of bounding pair maps. This can be deduced from the lantern relation, as shown in Figure 8.4. In the figure the pairs of sim ple closed curves (x, b3), (y, b1), and (z, b4) are all bounding pairs and so, using that fact that the Tbi commute with the Dehn twists about all seven simple closed curves in the picture, the lantern relation TxTy Tz = Tb1 Tb2 Tb3 Tb4 can be written as the desired relation in I(Sg ):

The genus of a bounding pair map TaTb−1 is the minimum of the genera of the two components of Sg − {a b}. A similar argument to the one just given, also due to Johnson, shows that I(Sg) is generated by genus 1 bounding pair maps. This implies, by the Change of Coordinates Principle, that I(Sg ) is normally generated in Mod(Sg ) by a single genus 1 bounding pair map.

Finite generation. In his clever and beautiful paper [91] Johnson proved the following.

THEOREM 8.15 For g ≥ 3 the Torelli group I(Sg ) is generated by a finite number of bounding pair maps.

b2

z

x

b4

Figure 8.4 A lantern showing how to write the twist about the separating simple closed curve a as a product of bounding pair maps.

While Mod(Sg ) can be generated by 2g + 1 Dehn twists that can easily displayed in one figure, we will see below that any generating set for I(Sg ) must have at least O(g3) generators (Theorem 8.19). Thus any such generating set for I(Sg) is not so easy to write in a single figure (consider the g = 20 case). This indicates the combinatorial complexity needed to prove Theorem 8.15. What is particularly remarkable is that for g > 2 Johnson finds a generating set for I(Sg ) with O(2g ) elements; even naming that many elements in a coherent way is not so trivial! Another point to make is that the natural strategy of trying to prove Theorem 8.15 by induction cannot work since the statement of the theorem is false for g = 2 (see below).

Johnson's strategy for Theorem 8.15 is as follows. He first pr oduces an explicit list of bounding pair maps in I(Sg), some of which are genus one, and shows that the group generated by these is normal in Mod(Sg ). To check normality it suffices to check that the conjugate of eac h bounding pair map on the list by each Humphries generator for Mod(Sg ) is a product of bounding pair maps on the list. Since any single genus one bounding pair map normally generates I(Sg ) (in Mod(Sg )), this proves the theorem.

Of course, the hard part is coming up with the explicit list. The proof of Theorem 8.15 would take us too far afield, but we encourage the reader to read the proof in [91].

While Theorem 8.15 settles the question of finite generation of I(Sg ), we do not have an analogue of the Humphries generating set. In fact, Johnson has conjectured that I(Sg ) has a generating set with O(g3) elements, as the rank of H1(I(Sg ); Q) has this order. If this conjecture is true, Johnson's generating set with O(2g ) elements is far from minimal. In the case g = 3 Johnson was able to whittle down the cardinality of his generating set for I(S3) to 35, which is exactly the rank of H1(I(S3); Q). Johnson conjectures that this should persist in higher genus. However, it is still an open question even to find a generating set for I(Sg ) whose number of elements is polynomial in g.

Two related open questions are: Is I(Sg ) finitely presented for g ≥ 3? Is K(Sg ) finitely generated for g ≥ 3?

Genus two. In genus two the story is quite different. McCullough–Mille r showed that I(S2) is not finitely generated [117]. Mess sharpened this result by showing that I(S2) is an infinitely generated free group, with one Dehn twist generator for each orbit of the action of I(S2) on the set of separating simple closed curves in S2 [125]. Note that there are no bounding pairs in S2, and so I(S2) is generated by Dehn twists, that is, I(S2) = K(S2).

Non-closed surfaces. For the surfaces Sg,1 and Sg1, it follows from Theorem 8.15 and the Birman exact sequences for I(Sg) that I(Sg,1) is generated by finitely many bounding pair maps, and that I(Sg1) is generated by finitely many bounding pair maps together with the Dehn twist about the boundary curve of Sg1.

THE JOHNSON HOMOMORPHISM

We now describe another of Johnson's fundamental contributions to our understanding of the Torelli group, the so-called “Johnson ho momorphism” [89]. This is a surjective homomorphism

τ : I(Sg ) → 3H1(Sg ; Z),

where 3H1(Sg ; Z) is the third exterior power of H1(Sg ; Z). The map τ exactly captures the torsion free part of H1(I(Sg); Z) (Theorem 8.19). It is a useful invariant of elements of I(Sg ), as we shall see.

2.Hom(H, 2H) is canonically isomorphic to H 2H. Using the

nondegenerate symplectic form given by algebraic intersection number, we can canonically identify H with its dual H . This gives a canonical isomorphism Hom(H, 2H) ≈ H 2H.

3.There is a natural inclusion of 3H into H 2H given by

a b c 7→a (b c) + b (c a) + c (a b)

and we will show below that the image of τ is exactly 3H.

Naturality. The action of Mod(Sg1) on H = H1(Sg1; Z) induces an action of Mod(Sg1) on 3H. A crucial and easy to verify property of τ is the following naturality property: for any f I(Sg1) and g Mod(Sg1), we have

Computing the image of a Dehn twist. We now explain how to explicitly calculate τ on certain elements of I(Sg). Let c be the “standard” separating simple closed curve shown in Figure 8.5. We claim that τ (Tc) = 0.

To prove this claim we begin by taking the generators {αi, βi} for π1(Sg1) shown in Figure 8.5. Let k be the genus of the subsurface of Sg1 cut off by c and not containing ∂Sg1; in Figure 8.5 this is the surface to the left of c. We see that Tc fixes αi and βi for k + 1 ≤ i ≤ g. Let γ be the element of π1(Sg1) shown in the bottom right of Figure 8.5. For x {α1, β1, . . . , αk , βk }, we find that

Tc(x) = γxγ−1

and so

Tc(x)x−1 = [γ, x].

But γ is a separating simple closed curve, and so γ ′, so that [γ, x] [ , ′]. Thus, we have [γ, x] = 0 2H, and so τ (Tc) = 0.

By the Change of Coordinates Principle and the naturality property of τ (formula (8.1)) it follows that τ (Tc′ ) = 0 for any separating simple closed curve c′ in Sg1. We thus have

K(Sg1) ≤ ker(τ ).

238 CHAPTER 8

Figure 8.5 The simple closed curves c and d, and the elements of π1(Sg1) used to compute

τ (Tc) and τ (TdTe−1).

Computing the image of a bounding pair map. As in the case of Dehn twists, in order to understand the image of an arbitrary bounding pair map it suffices to compute τ (TdTe−1) for th “standard” bounding pair {d, e} shown in Figure 8.5.

is, the subsurface of Sg cut off by d e and not containing ∂Sg has genus k. It is straightforward to compute directly he induced action of f on π1(Sg1)

where δ and ǫ are the elements of π1(Sg1) shown in the bottom left of Figure 8.5. From here we can write down the product f (x)x−1 for each

i=1

and the fact that the homology classes [δ] and [βk+1] are equal. Denoting

by ↔ the correspondence between elements of N and 2H via the isomorphism described above, we have:

i=1

Xk

=([αi] ([βi] [βk+1]) + [βi] ([βk+1] [αi]) + [βk+1] ([αi] [βi]))

i=1

In summary we have shown that

Xk

τ (TdTe−1) = xi yi z

i=1

where z H is the homology class [d] = [e] and x1, y1, . . . , xk , yk, z form a (degenerate) symplectic basis for the homology of the component of Sg1 − (d e) not containing the basepoint of π1(Sg1).

The image of τ . Choosing k = 1 in the above computation gives that the wedge product

[α1] [β1] [β2]

property (8.1) together with the fact that Mod(Sg1) surjects onto Sp(2g, Z) to show that τ surjects onto 3H.

Assume that g ≥ 3. By Theorem 8.4, there is some f Mod(Sg1) so that f maps the pair ([α1], [α3]) to the pair ([α1] + [β1] − [β3], [α3] − [β1] + [β3]) and fixes all other basis elements of H. Then

f ([α1] [β1] [β2]) = [α1] [β1] [β2] − [β1] [β2] [β3].

Since we have already shown that [α1] [β1] [β2] lies in the image of τ , it follows from the naturality property of τ that [β1] [β2] [β3] does as well. Applying factor swaps and factor rotations gives that every wedge product x y z is in the image of τ , where x, y, z {[αi], [βi]}. Since such elements span 3H, this completes the proof that τ is surjective when g ≥ 3. We leave the case of g = 2 as an exercise.

We have therefore proved the following result of Johnson.

Proposition 8.16 If g ≥ 2 then τ (I(Sg1)) = 3H.

There is another way to prove the slightly weaker fact that τ (I(Sg1)) Q =3H Q. Let HQ = H Q. Then the vector space 3HQ decomposes as a direct sum of irreducible Sp(2g, Q)-modules as follows:

3HQ = 3HQ/HQ HQ.

Since these two summands are irreducible, and since τ satisfies the naturality property (8.1), we could prove that τ is a surjection onto 3H (after tensoring with Q) by finding one element with τ –image in the first summand and one element with τ –image in the second summand, and then to apply Schur's Lemma. Note that 3H is a small, non-obviously embedded subspace of 2H H. How did Johnson know to prove that the image of τ is contained in this subspace? Well, he knew that the image of τ has to be a direct sum of Sp–invariant subspaces, so after computing a few elements in the image he might have guessed which subspaces would be needed.

The kernel of τ . As K(Sg1) is contained in the kernel of τ , and since the image of τ is infinite, and indeed the image of any bounding pair map is infinite, we immediately deduce the following.

Corollary 8.17 If g ≥ 3, then K(Sg1) has infinite index in I(Sg1). In fact, no bounding pair map or any of its nontrivial powers lies in K(Sg1).