Población

So far the results that we have obtained have been for bundles whose fibre Σh has been of

genus at least 3. We shall now consider the case of genus 0, where one can give a fairly precise description of the possible compact leaves of a (symplectically) flat bundle.

Examples of sphere bundles with horizontal foliations that have closed leaves of arbitrary self-intersection have been given by Mitsumatsu (cf. [Mit1]). We shall summarise his con- struction here. Let_R2 →ξk→Σg be a flat bundle of Euler classk ≤g−1 as given by [Mil].

Then the sphere bundleSk =S(ξk⊕R) is flat and has two sectionsL± corresponding to the

north and south poles of the fibre and [L±]2 _=±k.

We would of course like to have similar examples for flat bundles with symplectic holon- omy. The flat structures that one obtains via the construction of Mitsumatsu cannot have

4.2. Closed leaves of flat bundles with symplectic holonomy 77

symplectic holonomy. For if so, then one would have a vertical symplectic form ωv that is

positive on each fibre, i.e. ωv([F])6= 0, and vanishes identically on the leaves of the horizon-

tal foliation. But the set {L−, L+} generates H2(Sk,R), which is a contradiction. Thus in

order to produce horizontal foliations of sphere bundles with symplectic holonomy we will have to adapt the argument of Proposition 4.2.8. Again we letξk be a flatSL(2,R)-bundle over Σg as above. We let ai, bi ∈π1(Σg) denote the standard generators of the fundamental

group so that holonomy homomorphism for ξk is given by

ai 7→Ai

bi 7→Bi.

Then by performing the extension trick of Proposition 4.2.1 we obtain Hamiltonian diffeo- morphisms φi, ψi, which have compact support inside some disc D2− ⊂ R2. Moreover, the

product η = Qg

i=1[φi, ψi] has Cal(η) = 0 and is the identity in some neighbourhood of the

origin. So in fact η has support in some annulus A ⊂ D−2. If we consider D2− ⊂ S2 as the

southern hemisphere of the 2-sphere we see thatηmay equally well be thought of as a diffeo- morphism acting on the upper hemisphere, that is as an element inHamc_(D2

+). Then, since Cal(−η) = 0, Banyaga’s Theorem implies thatη−1 =Qg0

j=1[αj, βj], withαj, βj ∈Hamc(D2+).

We then define a flat S2_-bundleE

k over Σg+g0 with symplectic monodromies as follows: ai 7→φi, bi 7→ψi for 1≤i≤g

ag+j 7→αj, bg+j 7→βj for 1≤j ≤g0,

and by construction there is a leaf L corresponding to the south pole that has [L]2 = k. Interestingly, this can be the only compact leaf of the horizontal foliation. For ifL0 were any other leaf then {L, L0} would generate H2(Ek,R) and this would contradict the existence of

Chapter 5

Surface bundles and extended

Hamiltonian groups

Motivated by the problem of extending flat structures on the boundaries of surface bundles to their interiors, we show that an arbitrary circle bundle over a surface can be filled by a flat surface bundle after stabilisition. We may even assume that the horizontal foliation of such a bundle is symplectic if the genus of the fibre is large enough. The condition that the genus of the fibre is non-zero in the symplectic case is necessary by a result of Tsuboi, which expresses the Euler class of the boundary circle bundle in terms of the Calabi invariant of certain Hamiltonian extensions of the holonomies on the boundary to the disc.

We shall extend this result to the case of arbitrary Riemann surfaces. In the course of generalising Tsuboi’s result we are naturally lead to consider extended Hamiltonian groups as introduced by Kotschick and Morita. By considering the extended Hamiltonian groups as extensions of the mapping class group of punctured surfaces, we also obtain a formula that relates the Calabi invariant to the first MMM-class of surface bundles with boundary. Finally, we resolve a special case of a question posed in [KM1] by showing that the second MMM-class vanishes for surface bundles with holonomy in the extended Hamiltonian group.

5.1

Filling flat

S

1

-bundles

Given an arbitrary manifold M it is natural to ask what sort of manifolds bound M. If M

is an S1_{-bundle then the natural class of null-cobordisms to consider are surface bundles,}

whose fibre is a punctured surface Σ1

h. If M is in addition a flat S1-bundle, then one would

like to know when M bounds a flat surface bundle. To answer the latter question in full generality is a subtle matter. However for bundles over compact surfaces we will show that after stabilisation any flat S1_{-bundle can be filled in by a flat Σ}1

h-bundle. We first make

precise what we mean by a stabilisation in this context.

Definition 5.1.1. LetE be a flatS1-bundle over Σg with holonomy representation given by

π1(Σg) ρ

→Dif f0(S1). Let Σg+g0 → Σ_g be the map that collapses Σ_g0 in the decomposition

Σg+g0 = Σ_g#Σ_g0. Then the stabilisation ofρ is the flat bundle associated to the composition

of ρ with this collapsing map.

80 5. Surface bundles and extended Hamiltonian groups

If we allow the monodromies to be arbitrary, then it is an easy matter to show the following.

Proposition 5.1.2. Let h≥3 or h= 0 and let M be a flat S1_{-bundle. Then there is a flat}

bundle Σ1

h →E →Σg, whose boundary is a stabilisation of M. In particular, there exist flat

Σ1_h-bundles whose boundaries have non-trivial Euler class.

Proof. Let ai, bi ∈ π1(Σg) denote the standard generators of the fundamental group and

let φi = ρ(ai) and ψi = ρ(bi) be the images of these generators in Dif f0(S1) under the

monodromy homomorphismρ. Sinceφi, ψi are isotopic to the identity, we may extend them

to diffeomorphisms ¯φi,ψ¯i on a collar of the boundary [0,1]×S1 in such a way that

¯

φi(t, x) = (t, φi(x)) , ¯ψi(t, x) = (t, ψi(x)) for 0≤t <

and

¯

φi(t, x) = ¯ψi(t, x) =Id for 1− < t≤1.

We then extend by the identity to obtain ¯φi,ψ¯i ∈Dif f+(Σ1h) such thatη =

Qg

i=1[ ¯φi,ψ¯i] lies

inDif fc(Σ1_h).

In the proof of Proposition 4.1.4 we saw that for h ≥ 3 the group Dif fc(Σ1_h) is perfect and for h = 0 this is the classical result of Thurston (cf. [Th1]). Thus we may write

η−1 = Qg0

i=1[αi, βi], where αi, βi ∈ Dif fc(Σ1h). We define a flat bundle E over Σg+g0 by the

holonomy representation

ai 7→φ¯i, bi 7→ψ¯i for 1≤i≤g

ag+j 7→αj, bg+j 7→βj for 1≤j ≤g0.

The boundary of E is a flat S1-bundle and by construction it has the following holonomy representation:

ai 7→φi, bi 7→ψi for 1≤i≤g

ag+j 7→Id, bg+j 7→Id for 1≤j ≤g0

so that ∂E is a stabilisation of M as required.

The second statement follows from the existence of flatS1_{-bundles with non-trivial Euler}

classes (cf. [Mil]).

Proposition 5.1.2 implies that any flat circle bundle can be filled in by a flat disc bundle after a suitable stabilisation. On the other hand, if we require that the bundle have symplectic holonomy, then this is no longer true (cf. Theorem 5.2.1 below). However, if the fibre has genush≥3, then one can indeed find a filling by a symplectically flat bundle after a suitable stabilisation. To this end we shall need an analogue of the extension trick of Proposition 5.1.2 in the symplectic case.

Proposition 5.1.3. Let π1(Σg) ρ

→ Dif f0(S1) be a flat structure on an S1-bundle M and

let φi, ψi denote ρ(ai), ρ(bi) respectively. Then there are symplectic extensions φ˜i,ψ˜i on the

annulus A = S1 _{× [0,1] that are the identity in a neighbourhood of S}1 _{× {1} such that}

Qg

5.1. Filling flat S1-bundles 81

Proof. LetF be the horizontal foliation given by the flat structure on M and letα∈Ω1_(M)

be a defining 1-form for F. We choose a function φ on [0,1], which is equal to t on a neighbourhood of 0 and is identically zero for all t in a neighbourhood of 1. We set ω =

dt∧α+φ(t)dα on E = M ×[0,1] and let _∂θ∂ denote a vector field that is tangent to the fibres of M. Then ω(∂ ∂t, ∂ ∂θ) =α( ∂ ∂θ)6= 0,

since F is transverse to the fibres of M and, thus, ω is a nowhere vanishing 2-form on E. Furthermore, since F is a foliation we compute:

ω2 = (dt∧α+φ(t)dα)2 = 2φ(t)dt∧α∧dα = 0.

Thus Fω = Ker(ω) is a well-defined distribution that is transverse to the (annular) fibres

of E → Σg. Moreover, since ω =d(tα) in a neighbourhood of M × {0} this distribution is

integrable and transversally symplectic on this neighbourhood, and restricts toF onM×{0}. On a neighbourhood ofM×{1}the formωreduces todt∧αand again the kernel distribution is integrable and agrees with F on this neighbourhood.

We choose a base point x0 ∈ Σg and embedded representatives ai, bi for the standard

generators of π1(Σg, x0). We let ¯φi,ψ¯i be the holonomies of the curves ai, bi given by the

distribution Fω. Then on S1× {0} and near S1× {1} these diffeomorphisms are given by

φi×Idand ψi×Idrespectively, whereφi, ψi are the images of the standard basis under the

holonomy representation of M. Since φi, ψi lie in Dif f0(S1), we may alter the maps ¯φi,ψ¯i

near S1_{× {1}so that they restrict to the identity in a neighbourhood of S}1_{× {1}. We shall}

continue to denote these altered maps by ¯φi,ψ¯i.

We let Ω be the restriction ofωto the annular fibre overx0. Then the forms ¯φ∗iΩ−Ω and

¯

ψ_i∗Ω−Ω are trivial in compactly supported cohomology, since the holonomies ¯φi,ψ¯i have

support in S1×[0,1) and the distribution defining them was transversally symplectic in a neighbourhood of M × {0}. By applying a Moser isotopy, which will have support in the interior of S1 _{×[0,1], we obtain symplectomorphisms ˜φ}

i,ψ˜i that are symplectic extensions

of φi, ψi respectively, and by construction Qg_i[ ˜φi,ψ˜i] has support in the interior ofA.

Proposition 5.1.3 is the main step in extending flat structures symplectically and the following result follows from this and the perfectness of Sympc_(Σ1

h).

Theorem 5.1.4. LetM be a flatS1_{-bundle and assume thath≥3. Then some stabilisation}

of M bounds a flat Σ1

h-bundle with symplectic holonomy.

Proof. Let π1(Σg)

ρ

→ Dif f0(S1) be the holonomy representation associated to M and let

˜

φi,ψ˜i ∈ Symp(A) be the extensions given by Proposition 5.1.3. After a suitable choice of

symplectic form on Σ1_h, we may symplectically embedA=S1×[0,1] in Σ1_h so that S1× {0}

maps to ∂Σ1 h . We then consider η= Qg i[ ˜φi,ψ˜i] as an element inSymp c_(Σ1 h). This group is

perfect by Lemma 4.2.7 and, thus, we may write η−1 as a product of g0 commutators. We then define the associated flat bundle E0 over Σg+g0 as in the proof of Proposition 5.1.2, and

82 5. Surface bundles and extended Hamiltonian groups

Theorem 5.1.4 can be interpreted in terms of the five-term exact sequence of a certain extension of groups. For this we let Symp(Σ1_h) as usual denote the group of symplectomor- phisms of Σ1

h. We further letSymp(Σ1h, ∂Σ1h) denote those symplectomorphisms that restrict

trivially to the boundary. Then as a consequence of Proposition 5.1.3 the following sequence, which is given by restriction to ∂Σ1_h, is exact:

1→Symp(Σ1_h, ∂Σ1_h)→Symp(Σ1_h)→Dif f+(∂Σ1_h) =Dif f0(S1)→1.

With this notation we have the following proposition.

Proposition 5.1.5. Forh ≥3the connecting homomorphism in the five-term exact sequence in real cohomology associated to the following exact sequence is trivial:

1→Symp(Σ1_h, ∂Σ1_h)→Symp(Σ1_h)→Dif f+(∂Σ1_h) =Dif f0(S1)→1.

Proof. By the Universal Coefficient Theorem it suffices to show that the map

H2(Sympδ(Σ1h))→H2(Dif f0,δ(S1))

is surjective on integral cohomology. This follows immediately from Theorem 5.1.4, since any flat S1_{-bundle extends after stabilisation and this does not change the homology class}

represented by this bundle in H2(Dif f0,δ(S1)).

The Godbillon-Vey class of the horizontal foliation of a flat S1_{-bundle M defines an}

elementGV inH2(Dif f0,δ(S1),R), which is non-trivial by the work of Thurston (cf. [Bott]).

It is possible that the Godbillon-Vey class provides an obstruction to the existence of a flat symplectic bundle E that boundsM. However, by Proposition 5.1.5 the image of the class

GV in H2_(Symp

δ(Σ1h),R) is non-trivial. Geometrically, this means that after stabilisation

the horizontal foliation of any S1-bundle extends to a transversally symplectic foliation on some surface bundle E with fibre Σ1

h. In particular, the Godbillon-Vey class is not an

obstruction to finding a null-cobordism that extends the horizontal foliation of M to the interior of E symplectically.

In document Reflexiones en torno a la competencia de creación artística: aportes para la formación integral del sujeto, en la Escuela Superior Distrital María Montessori, Grado 4° (2014) (página 89-92)