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

Since Mod(Sg,1) contains torsion, it follows that the short exact sequence

(6.7) does not split, and so we thus obtain a nontrivial element of H2(Mod(Sg,1); Z), called the Euler class.

We now give two different constructions of the Euler class, that is, we give two derivations of the short exact sequence (6.7) defining th e Euler class. The first comes directly from the classical Euler class.

THE EULER CLASS VIA LIFTED MAPPING CLASSES

As in Chapter 4, an element of Mod(Sg,1) gives rise to a homeomorphism of the circle at infinity in hyperbolic space. We briefly recall t he construction. Assume that g ≥ 2 and regard the puncture of Sg,1 as a marked point p. If we choose a hyperbolic metric on the closed surface Sg , its universal cover is isometric to H2. Let pe be some distinguished lift of p to H2.

We can represent any f Mod(Sg,1) by a homeomorphism φ : Sg → Sg

We thus have a well-defined map

Mod(Sg,1) ֒→ Homeo+(S1).

This map is clearly a homomorphism. It is injective because if e fixes each

∂φ

γ∞±1 ∂H2 (using the notation from Section 4.1), it follows that φ fixes each γ π1(Sg ), and then, since Sg is a K(G, 1) space, it follows that φ is homotopic to the identity. The construction of the map Mod(Sg,1) → Homeo+(S1) is due to Nielsen; he used this as a starting point for his analysis and classification of mapping classes.

]

We finally define the group Mod(Sg,1) as the pullback of Mod(Sg,1) to

^ + 1

Homeo (S ):

172 CHAPTER 6

tral in ] g,1 . As above, the central extension (6.8) has an associated

Mod(S )

cocycle, giving an element e H2(Mod(Sg,1; Z). The element e is called the euler class for Mod(Sg,1).

under the map on cohomology

Homeo (S ).

THE RESTRICTION OF THE EULER CLASS TO THE POINT PUSHING SUB-

GROUP

We will next evaluate the Euler class e H2(Mod(Sg,1); Z) on a concrete 2–cycle, namely, the one coming from the point pushing subgr oup. We will do this by constructing an easy-to-evaluate cohomology class and by proving that this class equals the Euler class.

Another way to obtain an element of H2(π1(Sg ); Z) is by considering the unit tangent bundle S1 → U T (Sg ) → Sg. Since Sg is aspherical, the long

exact sequence associated to this fiber bundle gives a short e xact sequence

1 → Z → π1(U T (Sg )) → π1(Sg ) → 1.

This is a central extension, and so it has an associated cocycle e′ H2(Sg ; Z). We claim that e′ is nontrivial. Indeed, if e were trivial, there would be a splitting π1(Sg ) → π1(U T (Sg ), and hence a section of U T (Sg ) → Sg. The latter would give a nonvanishing vector field on Sg, which is prohibited by the Poincar´e–Bendixon theorem (for g ≥ 2). We thus have that e′ is nontrivial. In fact this argument gives that e′ has infinite order in H2(Sg ; Z).

Proposition 6.10 The elements Push (e) and e′ of H2(π1(Sg ); Z) are equal.

Proposition 6.10 implies that the evaluation of the pullback via Push of the Euler class for Mod(Sg,1) on the fundamental class of π1(Sg ) is the Euler number of the unit tangent bundle, which is equal 2 − 2g (the euler number of the tangent bundle to a Riemannian manifold is always equal to the Euler characteristic of the manifold). In particular, we have that the Euler class for Mod(Sg,1) is nontrivial even when restricted to the point pushing subgroup.

Proof. By the five lemma it suffices to exhibit a homomorphism π1(U T (Sg )) →

^ that makes the following diagram commutative:

π1(Sg )

1 Z π1(U T (Sg )) π1(Sg ) 1

The key is the following claim.

Claim: The image of π1(Sg ) in Homeo+(S1) given by the composition π1(Sg ) → Mod(Sg,1) → Homeo+(S1) coincides with the image of the composition π1(Sg ) → Isom+(H2) → Homeo+(S1) obtained by identifying π1(Sg ) with the group of deck transformations of the covering H2 → Sg .

Proof of claim. For α π1(Sg , p), we have that Push(α) acts by conjugation on π1(Sg , p), and so the lift of any representative of Push(α) fixing pe sends γ · pe to (αγα−1) · pe for all

174 CHAPTER 6

γ π1(Sg , p). On the other hand, the deck transformation corresponding to α sends γ ·pe to (αγ) ·pe. We can modify this deck transformation by pushing each point (αγ) · pe along the unique lift of α−1 starting at that point. This induces an isotopy of H2 moving points a uniformly bounded amount, and hence does not change the corresponding element of Homeo+(S1). At the end of this isotopy, each point (αγ) ·p gets sent to (αγα−1) ·p. Since the lift of Push(α) and the (modified) deck transforma-

Now let αb be an element of π1(U T (Sg )). In order to construct the associated

b ^ + 1

element ψ Homeo (S ), we need two ingredients:

1.a homeomorphism ψ Homeo+(S1), and

2.a path τ in S1 from some basepoint x0 S1 to ψ(x0).

Mod(Sg , (p, v)), the group of isotopy classes of diffeomorphisms of Sg

The homeomorphism f αb is the desired element of Homeo+(S1). It remains to construct the path τ in S1 from some fixed basepoint x0 to f αb (x0).

If we forget the datum of the vector v, and only remember the point p, then fαb also represents Push(α), where α π1(Sg ) is the image of αb under the forgetful map π1(U T (Sg )) → π1(Sg ). Thus, it follows from the claim that

as an element of Homeo+(S1) the mapping class fαb agrees with the deck transformation corresponding to α.

Let (p,e ve) be a fixed lift of (p, v) to U T (H2). Let x0 be the point of ∂H2 ≈ S1 to which (p,e ve) points. Because fαb agrees with the deck transformation

, and since deck transformations are isometries, the lifted map eαb takes

α φ

(p,e ve) to an element of U T (H2) that points to f αb (x0).

Recall that φαb is a diffeomorphism obtained at the end of a smooth isotopy

e

of Sg. Thus, φαb is a diffeomorphism obtained at the end of a smooth isotopy

of H2. At each point in time during the isotopy of H2, the pair (p,e ve) has a well-defined image, which in turn points to some point on ∂H2. Thus, the isotopy of H2 coming from α determines a path ταb in ∂H2 ≈ S1. Again, at the end of the isotopy, the image of (p, v) points to the image of x0, and so

implies that fαb agrees with a deck transformation, we have in fact con-

THE EULER CLASS VIA CAPPING THE BOUNDARY

We now give a different construction of the group ] g,1 , and hence a

Mod(S )

different derivation of the Euler class for Mod(Sg,1). Let Sg1 be the genus g surface with one boundary component. Recall from Proposition 3.13 that there is a short exact sequence

1 → Z → Mod(Sg1) → Mod(Sg,1) → 1

where the kernel Z is generated by the Dehn twist about the boundary of Sg1 and is thus central. Since the extension is central, it thus gives an element e′′ H2(Mod(Sg,1); Z). Corollary 7.3 gives that Mod(Sg1) is torsion

duce that e′′ is again the Euler class.

THE BIRMAN EXACT SEQUENCE DOES NOT SPLIT

Let g ≥ 2. The Birman exact sequence (Theorem 5.5) is:

1 → π1(Sg ) → Mod(Sg,1) → Mod(Sg ) → 1.

Above, we described an embedding Mod(Sg,1) → Homeo+(S1). Since finite subgroups of Homeo+(S1) are cyclic, it follows that the same is true for Mod(Sg,1). It is easy to find finite subgroups of Mod(Sg ) that are not cyclic (for example the dihedral group of order 2g), and so we have the following.

Corollary 6.11 Let g ≥ 2. The Birman Exact Sequence

1 → π1(Sg ) → Mod(Sg,1) → Mod(Sg ) → 1

does not split.

6.6 SURFACE BUNDLES AND THE MEYER SIGNATURE COCYCLE

Our next goal is to construct an element of H2(Mod(Sg ); Z) that is not a power of the euler class e. This element, called the Meyer signature cocycle, is defined using the theory of surface bundles over surfaces.

We will use some homological algebra to show that the Meyer signature cocycle gives rise to nontrivial elements of H2(Mod(Sg )), H2(Mod(Sg1)), and H2(Mod(Sg,1), and to then complete the proof of Theorem 6.8.

In order to define the Meyer signature cocycle properly, we mu st clarify the connection between the mapping class group and the theory of surface bundles, so this is where we start.

SURFACE BUNDLES

The basic problem in the theory of surface bundles is to classify, for fixed (Hausdorff, paracompact) base space B, all isomorphism classes of bundles

Sg → E → B.

Recall that a bundle isomorphism is a fiberwise homeomorphism of total spaces covering the identity map. The reduction of the Sg –bundle classification problem to a problem about Mod(Sg ), at least for g ≥ 2, begins with

the following theorem. This theorem is a special case of Theorem 1.14, proved by Hamstrom. For the statement, recall that Homeo0(Sg ) denotes the topological group of homeomorphisms of Sg that are isotopic to the identity.

THEOREM 6.12 If g ≥ 2 then Homeo0(Sg ) is contractible.

Let BHomeo+(Sg ) denote the classifying space of the topological group Homeo+(Sg ). The theory of classifying spaces gives a bijective correspondence as follows.

This bijection is realized concretely in the following way. There is a bundle ζ given by

Sg → E → BHomeo+(Sg )

with the universal property that any Sg –bundle over any space B is the pullback of ζ via a continuous map (the classifying map) f : B → BHomeo+(Sg ). Homotopic classifying maps gives isomorphic bundles. Conversely, any bundle induces such a map f . The bundle ζ is called the universal Sg – bundle. Thus BHomeo+(Sg ) plays the same role for surface bundles as the (infinite) Grassmann manifolds BSO(n) play for vector bundles.

Consider the exact sequence

Homeo0(Sg ) → Homeo+(Sg ) → Mod(Sg ).

Theorem 6.12 together with Whitehead's theorem implies that Homeo+(Sg ) is homotopy equivalent to the discrete topological group Mod(Sg ) for g ≥ 2. In other words we have the following.

Proposition 6.13 For g ≥ 2 the space BHomeo+(Sg ) is a K(Mod(Sg ), 1) space.

A continuous map f : B → K(Mod(Sg ), 1) induces a homomorphism f : π1(B) → Mod(Sg ). Basic algebraic topology gives that the map f is determined up to free homotopy by the conjugacy class of the representation f , and that every representation is induced by some continuous map. In other words there is a bijection between free homotopy classes of

maps f : B → K(Mod(Sg ), 1) and conjugacy classes of representations π1(B) → Mod(Sg ). This bijection, together with Proposition 6.13, gives the following bijective correspondence.

The simplest (but already interesting) instance of this fact is that isomorphism classes of Sg –bundles over S1 are in bijection with conjugacy classes of elements in Mod(Sg ). A more remarkable consequence is that, given any group extension

there exist topological spaces (indeed closed manifolds) E and B and a fi- bration Sg → E → B inducing the given group extension. Why is this surprising? Well, if we are given a representation ρ : π1(B) → Homeo+(Sg ), it is easy to see how to build a bundle Sg → E → B with monodromy

◦ → × e

π ρ : π1(B) Mod(Sg ): just takes the quotient of Sg B by the obvious π1(B)–action. However, the data specified by the group extension ( 6.9)

only determines a representation ρ : π1(B) → Mod(Sg ). That is, elements of the monodromy are specified only up to isotopy, so it is not a t all clear how to use this data to build a well-defined Sg -bundle. In fact Morita has constructed examples where the monodromy ρ : π1(B) → Mod(Sg ) does not lift to a representation ρe : π1(B) → Homeo+(Sg ). Yet the bijection above gives a fiber bundle Sg → E → B with B and E closed manifolds that has monodromy ρ.

The above discussion should clarify why the problem of classifying conjugacy classes of representations of various groups into Mod(Sg ) is an important problem.

Another corollary of Proposition 6.13 is that

H (BHomeo+(Sg ); Z) ≈ H (Mod(Sg ); Z).

This isomorphism is one of the main reasons that we care about the cohomology of Mod(Sg ). It is the reason we think of elements of H (Mod(Sg ); Z) as “characteristic classes of surface bundles,” as we now ex plain.

Suppose one wants to associate to every Sg –bundle a (say integral) cohomology class on the base of that bundle, so that this association is natural,

that is, it is preserved under pullbacks. By applying this to the universal Sg – bundle ζ, we see that each such cohomology class must be the pullback of some element of H (BHomeo+(Sg , Z)) ≈ H (Mod(Sg ); Z). In this sense the classes in H (Mod(Sg ); Z) are universal. This is why they are called “characteristic classes” of surface bundles.

We have already seen that H1(Mod(Sg ); Z) = 0 if g ≥ 3 (Theorem 6.2). It follows from the Universal Coefficients Theorem that H1(Mod(Sg ); Z) = 0. Thus there are no natural 1–dimensional cohomology invariants for these Sg –bundles. In Section 6.4 we proved for g ≥ 4 that H2(Mod(Sg ); Z) is cyclic, so that there is at most one natural 2–dimensional in variant. This is the Meyer signature cocycle, constructed below.

Remark on the smooth case. Every aspect of the discussion above holds with the smooth category replacing the topological category. Here we replace BHomeo+(Sg ) with BDi +(Sg ), etc. The key fact is the theorem of Earle–Eells [48] (see also [63]) that the topological group Di 0(Sg ) is contractible for g ≥ 2. Following the exact lines of the discussion above, This gives a bijective correspondence between isomorphism classes of smooth Sg –bundles over a fixed base space B and conjugacy classes of representations ρ : π1(B) → Mod(Sg ).

DEFINITION OF THE MEYER SIGNATURE COCYCLE

We are now ready to describe the construction of a nonzero element σ H2(Mod(Sg ); Z): the Meyer signature cocycle. Below we will prove that σ

pulls back to a nontrivial class both in H2(Mod(Sg,1); Z) and in H2(Mod(Sg1); Z).

For any closed 4–manifold M there is a skew-symmetric pairing

given by taking the cup product of two classes and evaluating the result on the fundamental class of M . The signature of the resulting quadratic form is called the signature of M , denoted by sig(M ).

We can use signature to give a 2–cochain

σ C2(BHomeo+(Sg ); Z) ≈ Hom(C2(BHomeo+(Sg ); Z), Z)

as follows. Suppose we are given a chain c C2(BHomeo+(Sg ); Z). It follows from general facts about 2–chains in topological sp aces that c can

key ingredient in this is the fact that the signature of a fiber bundle depends only on the action of the fundamental group of the base on the homology of the fiber; another is the so-called “Novikov additivity” of s ignature.

It is not easy to prove that σ is a nonzero element of H2(BHomeo+(S); Z). The hard part is finding a good way to compute signature in term s of the monodromy data. Kodaira, and later Atiyah (see [6]), found a surface bundle over a surface with nonzero signature. This construction can be used to give such a bundle with fiber Sg for any g ≥ 4. It follows that the signature cocycle σ H2(BHomeo+(S); Z) ≈ H2(Mod(Sg ); Z) is nonzero. Indeed, this kind of argument can be used to prove that σ has infinite order in H2(Mod(Sg ); Z).

MATCHING UPPER AND LOWER BOUNDS ON H2(Mod(S); Z)

In Section 6.4 we used Hopf's formula to give an upper bound on the number of generators of the group H2(Mod(S); Z), where S is either Sg , Sg,1 or Sg1 and where g ≥ 4. So far in this section we have constructed two nontrivial elements of H2(Mod(S); Z), the Euler class and the Meyer signature cocycle. We will now use homological algebra to compute H2(Mod(S); Z) on the nose.

The universal coefficients theorem and H2(Mod(S); Z). Let S be a surface of genus at least 3. In what follows we assume that all homology and cohomology groups have Z coefficients. The Universal Coefficients Theorem gives the following short exact sequence:

1 → Ext(H1(Mod(S)), Z) → H2(Mod(S)) → Hom(H2(Mod(S)), Z) → 1.

(6.10) Since H1(Mod(S); Z) = 0 (Theorem 6.2), the Ext term in (6.10) is trivial. Thus

H2(Mod(S); Z) ≈ Hom(H2(Mod(S); Z), Z).