II. Organización de los Elementos Estéticos y Académicos en la dimensión comunicativa
10. Análisis e interpretación de datos
10.3 Factores Institucionales en el desarrollo del Carnaval
In [KM1] it was shown that thek-th power of the first MMM-class is non-trivial in the group
H∗(Sympδ(Σh)) if h ≥ 3k (cf. [KM1], Theorem 3). On the other hand, the Bott vanishing
Theorem (cf. [Bott], p. 35) implies that the higher MMM-classesekvanish for all flat surface
bundles if k ≥ 3. Thus the question remains open as to the non-triviality of e2 for flat
surface bundles (cf. [KM1], Problem 4). Motivated by this, we will show that the higher MMM-classes as well as all the higher powers of e1 vanish on the extended Hamiltonian
group Ham](Σh) =Ker(F lux]), where F lux] is the extended Flux map for closed surfaces as
defined in [KM1].
As a first step we show that the first MMM-class itself is non-trivial on Ham](Σh) for
h≥3.
Proposition 5.4.1. The image of e1 under the map induced by the projection Ham](Σh)→
Γh is non-trivial for h≥3.
Proof. Applying the five-term exact sequence to the extension
1→Ham(Σh)→Ham](Σh)→Γh →1,
exactness and the perfectness of Ham(Σh) allow us to conclude that the map
H2(Γh)→H2(Ham]δ(Σh))
is injective. Since e1 ∈H2(Γh) is non-trivial for h≥3 (cf. [Iva1]), the result follows.
In the remainder of this section we will show that the classes ek
1, e2 ∈H∗(Ham]δ(Σh),R)
are trivial. To this end we shall need the following two facts. The first is a proposition due to Morita.
Proposition 5.4.2 ([Mor1], Prop. 3.1). Let Σh →E π
→ B be any oriented surface bundle.
Then the Serre spectral sequence for the fibration collapses on the second page for cohomology
with real or rational coefficients. In particular, the map π∗ is injective on cohomology.
Next, in accordance with [KM1] we let v ∈ H2(Symp
δ(Σh),R) denote the vertical sym-
plectic class, normalised so that v(F) = 2h−2 on the fibre. We further let e denote the vertical Euler class. With this notation we may state the following theorem.
Theorem 5.4.3 ([KM1], Th. 2). The projection ofe+v to theE1∞,1-term in the Serre spectral
sequence of the fibration Σh →ESympδ(Σh)→BSympδ(Σh) is the cohomology class of the
extended flux homomorphism
[F lux]]∈E1∞,1 ⊂H1(Sympδ(Σh), H1(Σh,R)). We are now ready to prove the main result of this section.
Theorem 5.4.4. The second MMM-class e2 vanishes inH4(Ham]δ(Σh),R)forh≥2. More-
5.4. The second MMM-class vanishes onHamg 93
Proof. Let
Σh →EHam]δ(Σh) π
→BHam]δ(Σh)
be the total space of the universal bundle over BHam]δ(Σh) and let ι denote the inclusion
]
Ham(Σh) ,→ Symp(Σh). Then by the naturality of the Serre spectral sequence combined
with Theorem 5.4.3, we see that the projection ofι∗(e+v) to theE1∞,1-term isι∗[F lux]], which is trivial by the definition of Ham](Σh) as the kernel of the extended Flux map. Thus by
Proposition 5.4.2
ι∗(e+v)∈E2∞,0 =E22,0 =π∗H2(Ham]δ(Σh),R)
and we conclude thatι∗(e+v) =π∗β for some class β ∈H2(Ham]
δ(Σh),R). We rewrite the
equation fore in H2(Ham]δ(Σh),R) given above, dropping the ι∗ for notational convenience
e=−v+π∗β. (5.8)
Since v2 = 0, we compute that
e1 =π!e2 =−2(2h−2)β. (5.9)
By Bott vanishing, v ` e2 = 0 in H∗(Symp
δ(Σh),R) (cf. [KM2], Section 7), and thus
e3 = (e+v)3. We conclude that e2 =π!e3 =π!(e+v)3 = 1 (4−4h)3π!π ∗ e31 = 0,
and, hence, e2 vanishes inH4(Ham]δ(Σh),R).
On the other hand equations (5.8) and (5.9) imply that
e3 = −3 (4−4h)2v `π ∗ e21+ 1 (4−4h)3π ∗ e31.
Thus applying the transfer map we obtain 0 = e2 =π!e3 = −3(2h−2) (4−4h)2 e 2 1 so that e2
1 = 0 in H4(Ham]δ(Σh),R). Finally, the fact that e21 vanishes implies that ek1 also
Chapter 6
Characteristic classes of symplectic
foliations
In [KM3] Kotschick and Morita defined foliated characteristic classes of transversally sym- plectic foliations in terms of factorisations of ordinary characteristic classes. Motivated by this we give a geometric construction for defining foliated characteristic classes as factori- sations of certain Pontryagin classes. In contrast to [KM3] our construction yields foliated characteristic classes for foliations that are only transversally volume preserving rather than transversally symplectic.
For the special case of codimension 2 foliations, the factorisation of the first Pontryagin class gives a foliated characteristic class γ1. A similar class, that we denote by γKM, was
defined in [KM3] and we show that the classes γ1 and γKM coincide under the assumption
that the normal bundle to the foliation is trivial.
In general, any foliated cohomology class defines a genuine characteristic class by re- stricting to a leaf. We construct transversally symplectic foliations with closed leavesLsuch that the restriction of γKM to L is non-trivial. Moreover, these foliations have trivial nor-
mal bundles so we deduce that the classes γ1 and γKM carry information that is not purely
topological.
6.1
Factorisation of Pontryagin classes
In this section we describe factorisations of certain polynomials in the Pontryagin classes of the normal bundles of transversally volume preserving foliations, which are analogous to factorisations obtained in [KM3]. We shall use Chern-Weil theory, rather than Gelfand- Fuks cohomology as in [KM3], to obtain factorisations of polynomials of total degree 4q in the Pontryagin classes of the normal bundle of any codimension 2q, transversally volume preserving foliationF. IfP denotes a polynomial of the correct degree and ω is a transverse volume form for F, then these factorisations are of the form
P(Ω) =ω∧γP.
The main benefit of our approach is that the classes γP are canonically defined in terms of
the foliation F and the formω, whereas those given in Theorem 4 of [KM3] are not. 95
96 6. Characteristic classes of symplectic foliations
In order to obtain the factorisations mentioned above, we shall need an explicit description of integration along the fibre for the case of M ×[0,1]→π M. We let α ∈Ωk(M ×[0,1]) be a k-form, which may be uniquely written as
α=ρ+σ∧dt,
where ρ has nodt component. Then integration along the fibre
π!: Ωk(M ×[0,1])→Ωk−1(M) is given explicitly via the following formula
π!α= Z 1
0 σ dt.
Furthermore, we have the following lemma.
Lemma 6.1.1. Let α∈Ωk(M×[0,1]) be ak-form and let ι
0, ι1 be the inclusions ofM× {0}
resp. M × {1} in M ×[0,1]. Then the following relation holds
π!dα−d π!α=ι∗1α−ι ∗ 0α.
Proof. We write
α=ρ+σ∧dt.
If dM denotes the exterior derivative onM, then we have
dMπ!α=dM Z 1 0 σ dt= Z 1 0 dMσ dt. Moreover dα =dρ+d(σ∧dt) =dMρ+ ∂ρ ∂t ∧dt+dMσ∧dt and thus π!dα−d π!α= Z 1 0 (∂ρ ∂t +dMσ)dt− Z 1 0 dMσ dt = Z 1 0 ∂ρ ∂tdt =ι∗1α−ι∗0α,
where we have used the Fundamental Theorem of Calculus to obtain the final equality. We shall next recall the definition of foliated cohomology. To this end we let I∗(F) denote the ideal of forms that vanish on F. The Frobenius Theorem implies that I∗(F) is closed under the exterior differential and, thus, d descends to a differential dF on the quotient complex Ω∗(M)/I∗(F). We define the foliated cohomology as the cohomology of this quotient complex:
HF∗(M) =H∗(Ω∗(M)/I∗(F), dF).
6.1. Factorisation of Pontryagin classes 97
Definition 6.1.2. LetF be a foliation on a manifoldM. ABott connection on the normal bundle νF =T M/TF is a connection ∇such that for X ∈Γ(TF) and Y ∈Γ(νF)
∇XY = [X,Y˜],
where ˜Y denotes any lift ofY to T M.
The most important properties of Bott connections are that they are flat when restricted to the leaves of F (cf. [Bott]) and that they are canonically defined along any leaf by the formula in Definition 6.1.2. Conversely, to define a Bott connection for a given foliation one chooses a splitting
T M ∼=TF ⊕νF.
IfX =XF+Xν is the decomposition of a vectorX with respect to this splitting, then after
choosing a connection ∇ onνF, one may define a Bott connection∇ as follows:
∇XY = [XF,Y˜] +∇XνY.
There is an alternate formulation of the Bott condition in terms of connection matrices. For this we let S1, ..., Sq be a local frame for νF and choose lifts ˜S1, ...,S˜q toT M. We let θij
denote the connection matrix of ∇with respect to the frame S1, ..., Sq so that
∇Si = q
X
j=1
θij ⊗Sj.
We further let θ1, ..., θq be a dual basis for S1, ..., Sq, which means that θi vanishes on TF
and
θi( ˜Sj) =δij.
To check the Bott condition it suffices to verify that the following holds for all X ∈TF and allSi:
∇X Si = [X,S˜i],
or equivalently that
θj(∇XSi) =θij(X) = θj([X,S˜i])
for all 1≤i, j ≤q. We compute that
dθj(X,S˜i) = LXθj( ˜Si)−LS˜iθj(X)−θj([X,S˜i])
=LXδji−0−θj([X,S˜i]) =−θj([X,S˜i])
and conclude that∇ is a Bott connection if and only the connection 1-forms satisfy
dθj(X,S˜i) =−θij(X).
In particular, if θ1, ..., θq is any local basis for I1(F), then by the Frobenius Theorem there
exist 1-forms θij such that
dθj = q
X
i=1
θi∧θij. (6.1)
The matrix of 1-forms (θij) then defines a local Bott connection. This description of Bott
connections will be useful in our discussion of Gelfand-Fuks cohomology in Section 6.2 below. We now come the main result of this section.
98 6. Characteristic classes of symplectic foliations
Proposition 6.1.3. Let F be a transversally volume preserving foliation of codimension 2q
on a manifold M with defining form ω and let P be any polynomial of total degree 4q in the
Pontryagin classes of the normal bundle. Then there is a factorisation
P(Ω) =ω∧γP
for a well-defined foliated class γP ∈HF2q(M).
Proof. Let∇ be a Bott connection on the normal bundle νF of F. Since a Bott connection
is flat along leaves, the components Ωij of the curvature matrix vanish on F. We choose a
local basis θ1, ..., θ2q of I1(F) such that
θ1∧θ2...∧θ2q =ω.
With respect to this basis the curvature forms Ωij ∈I2(F) can locally be expressed as
Ωij =
2q
X
k=1
θk∧αk.
Since the Chern-Weil representative for P is given by a symmetric polynomial of degree 2q
in the entries of Ω, the following holds locally:
P(Ω) =θ1∧θ2...∧θ2q∧γP =ω∧γP.
Let Ann(ω) be the subbundle of 2q-forms annihilated by ω and let Ann(ω)⊥ ⊂ Λ2q(M) be
a complement to Ann(ω). Then on the level of forms the equation
P(Ω) =ω∧γP
has a unique global solutionγP ∈Γ(Ann(ω)⊥). The formγP is well-defined modulo elements
in Γ(Ann(ω)) =I2q(F), so we obtain a class [γ
P]∈Ω2q(M)/I2q(F). Next, sinceω andP(Ω)
are closed we compute
0 =d(ω∧γ) =ω∧dγP
and dγP ∈I∗(F). Thus we have a well-defined class [γP]∈HF2q(M).
We finally need to show that the class we obtain in foliated cohomology does not depend on the choice of Bott connection. Let∇0,∇1 be two Bott connections onν
F and letπdenote
the projection M×[0,1]→M. We then define a connection on π∗νF by setting
∇=t π∗∇1+ (1−t)π∗∇0.
This connection is then a Bott connection for the foliation π∗F that is obtained as the preimage of F under the projection π. Now π∗ω is a defining form for π∗F and, as above, after the choice of a splitting Λ2q(M ×[0,1]) ∼= Ann(π∗ω)⊕Ann(π∗ω)⊥, there is a unique
formγP ∈Γ(Ann(π∗ω)⊥) so that
6.1. Factorisation of Pontryagin classes 99
Since the form P(Ω) is closed, Lemma 6.1.1 yields
−d π!P(Ω) =ι∗1P(Ω)−ι∗0P(Ω) =P(Ω1)−P(Ω0).
Hence, one has
ω∧(−d π!γP) =ω∧γP1 −ω∧γ 0 P or equivalently γP1 −γP0 ∼=−d(π!γP) mod I∗(F). Thus [γP1] = [γ0P] as classes in HF2q(M).
If one assumes further that the foliation F is transversally symplectic with defining form
ω, then one obtains a similar factorisation for any polynomial of the form ωkP(Ω), where
P is a polynomial in the Pontryagin classes of total degree 4q −2k. We note this in the following proposition, whose proof is almost identical to that of Proposition 6.1.3.
Proposition 6.1.4. Let F be a transversally symplectic foliation of codimension 2q on a
manifold M with defining form ω and let P be any polynomial of total degree 4q−2k in the
Pontryagin classes of the normal bundle. Then there is a factorisation
ωkP(Ω) =ω2q∧γP
for a well-defined foliated class γP ∈HF2q(M).
For certain polynomials it is easy to show that the class P(Ω) =ω2q∧γ
P is non-trivial
and, hence, that the classγP is non-trivial in foliated cohomology. This was shown in [KM3]
for polynomials of the form pq1. A similar argument covers the other extreme case when
P =pq is theq-th Pontryagin class. For simplicity we writeγqfor the corresponding foliated
cohomology class.
Proposition 6.1.5. There exist foliations Fq for which the classes γq ∈ HF2q(M) do not
vanish.
Proof. We let F be a codimension 2, transversally symplectic foliation on a 4-manifold such
that p1(νF) is non-trivial (cf. [KM1]). We define Fq as the product foliation on the q-fold
product Mq induced by F and by π
i the i-th projection. These foliations are transversally
volume preserving, they are even transversally symplectic. Finally, using the Whitney sum formula and the naturality of Pontryagin classes we obtain:
pq(νFq) = pq( q M i=1 πi∗νF) = q Y i=1 π∗ip1(νF)6= 0.
Although Proposition 6.1.5 shows that γq is non-trivial, we cannot conclude that the
100 6. Characteristic classes of symplectic foliations