La memoria no está en el cuerpo, el cuerpo es memoria (Jerzy Grotowski )
ILUSTRACIÓN 2 DIMENSIONES DE UNA EXPERIENCIA CORPORAL
In Section 6.1 we defined characteristic classes of transversally volume preserving foliations in foliated cohomology. The restriction of such a class to a leaf of a foliation defines a cohomology class in ordinary cohomology and, thus, determines a characteristic class of the individual leaves. By using results on normal forms for germs of Hamiltonian diffeomor- phisms we will show that there exist symplectically foliatedR2-bundles with holonomy in the groupSympk(
R2,0) ofCk-symplectomorphisms that fix the origin, for which the Kotschick- Morita class γKM restricts non-trivially on the central leaf that corresponds to the origin.
Moreover, these bundles may be chosen to be topological trivial, which means that the same holds for γ1 by Proposition 6.2.1. These examples show that the classes γ1 and γKM carry
information that is sensitive to the geometry of the foliation and not just to the homotopy class of the underlying distribution as is the case for p1.
We letPk
ham(F) be the principalJhamk -bundle ofk-jets of localF-projections that preserve
the transverse symplectic structure. Now if L is the leaf of a foliation, then the restriction of the bundle Phamk (F) to L is a flat principalJhamk -bundle, which is then determined by its holonomy representation π1(L)→Jhamk . An analogous construction to that given in Section
6.2, associates to any class in HGF∗ (ham02n,un), a class in H∗(L). Moreover, by composing
with the natural map
HGF∗ (ham02n,sp2n)→HGF∗ (ham02n,un)
one obtains the following commutative diagram where the rightmost arrow is given by re- striction HGF∗ (ham02n,sp2n) // HF∗(M) HGF∗ (ham02n,un) //H∗(L).
We remark that these observations still hold for the cohomology of the truncated Lie algebra
gkham−1 =ham02n/hamk2−1n , in the case that the foliation F is only of class Ck.
In general, any element in the relative Lie algebra cohomology H∗(g,k) of a pair (G, K) defines a characteristic class of flat principalG-bundles via the construction described above, provided thatG/K is contractible. In the context of flatG-bundles this is often referred to as
theReinhard construction. It is clear that these classes are natural with respect to pullbacks
under both maps between manifolds and between pairs of Lie groups. Furthermore, Rein- hard’s construction may be used to define elements in the group cohomology ofGconsidered as a discrete group. For in order to define an elementφ∈Hk(G
δ,R) it suffices to define how
φ evaluates on integral homology classes. For homology classes that are representable by (singular) manifolds we may use the Reinhard construction directly. Moreover, any two sin- gular manifolds that are homologous are cobordant after possibly taking multiples. Hence, by Stokes’ Theorem the map φ is well-defined on the subgroup Hmank ⊂ Hk(Gδ) of ele-
6.3. Characteristic classes of leaves 103
Theorem that the quotient of these two groups consists of torsion and, thus, any homomor- phismHmank →Rextends uniquely to a homomorphismHk(Gδ)→Ryielding a well-defined
element inHk(G δ,R).
We will restrict ourselves to the special case G = Jk
ham. In particular, we consider
the class γ = γKM in HGF2 (ham
0
2n,sp2) that was defined in [KM3] via the explicit cocycle
representative
(X3∧Y3−3X2Y ∧XY2)∗.
This cocycle is the pullback of the corresponding elementγjet in the Lie algebra chain group
C2(g2ham,sp2). Via the Reinhard construction the class γjet defines a cohomology class in
H2((J2
ham)δ,R), which will also be denoted by γjet, and we claim that γjet is non-trivial. In
fact, the restriction of γjet to the subgroup Jham,Id2 ⊂ Jham2 is non-trivial, where Jham,Id2 is
the subgroup of 2-jets with linear part equal to the identity. To see this we take the trivial bundle
E =Jham,Id2 ×T2 →T2
and let ˜E →R2 be its universal cover. SinceJ2
ham,Idis a simply connected abelian Lie group,
it is isomorphic to its Lie algebra Sym3H2
R. We let x1, ..., x4 be coordinates on the fibre
with respect to the standard basis ofSym3H2
Rand choose coordinates y1, y2 onR
2. We then
consider the horizontal foliation on ˜E given by the projection (x1, ..., x4, y1, y2)→(x1−y1, x2, x3, x4−y2).
This foliation descends to E and by inspecting the definition of γjet, or more precisely its
restriction to Sym3H2
R, it is clear that this class evaluates non-trivially on the (foliated)
principal bundle E.
Remark 6.3.1. We note that γjet and the pullback of the Euler class under the mapJham2 →
Jham1 =SL(2,R) are linearly independent, since the restriction of the Euler class to Jham,Id2
is trivial.
We will further show that the image of γjet inH2((Jhamm )δ,R) is non-trivial for arbitrary finite m, which is an immediate consequence of the following lemma.
Proposition 6.3.2. The natural map Jm
ham → Jham2 induces an injection on second real
cohomology for all m≥2.
Proof. We first note that the map factors as
Jhamm →Jhamm−1 →...→Jham2 .
Thus it suffices to show that the mapJhamm−1 →Jhamm−2 induces an injection on cohomology for all m ≥ 4. We note that the kernel of this map is a simply connected abelian Lie group of rank m+ 1, which we denote by Km−1. As such it is isomorphic to its Lie algebra, which
in this case is SymmH2
R with the trivial Lie bracket and an isomorphism is induced by the
exponential map. Restricted to the subalgebra SymmH2
R the exponential map is given by
considering the (m−1)-jet at the origin of the Hamiltonian flow generated by an element in
SymmH2
104 6. Characteristic classes of symplectic foliations
By the Universal Coefficient Theorem, injectivity in real cohomology, will follow from the surjectivity of the induced map in integralhomology. By considering the five-term exact sequence associated to the extension
1→Km−1 →Jhamm−1 →Jhamm−2 →1,
it will suffice to show that the group of coinvariants H1((Km−1)δ)Jm−2
ham is trivial.
First recall that if Φt
H is the Hamiltonian flow generated by H, then for any symplecto-
morphism ψ the flow ψΦtHψ−1 is the Hamiltonian flow generated by (ψ−1)∗H. Thus under our identification of Km−1 with its Lie algebra, conjugation by an element ψ ∈Jhamm−1 corre-
sponds to precomposition in the Lie algebra ofKm−1 byψ−1, whereKm−1 has been identified
with the additive group of homogeneous polynomials of degree m. Now if we let
u(λ) = λ 0 0 λ−1 ,
then for any monomial p(X, Y) =XkYm−k, we compute
p−p◦u−1(λ) = (1−λm−2k)XkYm−k.
If m−2k6= 0, we may choose λ >0 such that 1−λm−2k = 1
2.
Thus we conclude that 12p(X, Y) is trivial in H1((Km−1)δ)Jm−2
ham and, hence, the same holds
for p(X, Y). If 2k=m, let ρ(θ) = cos(θ) sin(θ) −sin(θ) cos(θ)
denote the rotation by θ. The coefficient of the XmYm-term in p◦ρ(θ) is
cm,m(θ) = X l+q=m (−1)m−l m l m q
cosl(θ)sinm−l(θ)sinq(θ)cosm−q(θ)
= X l+q=m (−1)q m l m q cos2l(θ)(1−cos2(θ))q.
This is a polynomial of degree 2m incos(θ), whose leading term has coefficient
X l+q=m (−1)2q m l m q = X l+q=m m q 2 6 = 0.
Since cm,m(0) = 1, there is some θ0 so that cm,m(θ0) = 1 ± for any sufficiently small .
Hence, the coefficient of the XmYm-term in p−p◦ρ(θ
0) is ±. Combining this with the
case m 6= 2k considered above, we conclude that ±XmYm is trivial in H1((Km−1)δ)Jhamm−2.
Since the group generated by such elements includes all monomials aXmYm, the group
H1((Km−1)δ)Jm−2
ham vanishes and the desired claim follows by the exactness of the five-term
6.3. Characteristic classes of leaves 105
In order to extend our results from jets to actual germs we shall need a normal form theorem for Hamiltonian germs. This is provided by a result of Banyaga, de la Llave and Wayne in [BLW]. In order to state their result we need to define several constants.
Let M be a real linear map and assume that there are constants 0< λ−1− < λ+ <1< µ−1− < µ+
so that the (complex) eigenvalues λ of M satisfy
λ−1− ≤ |λ| ≤λ+ or µ−1− ≤ |λ| ≤µ+.
We shall call such a a linear map hyperbolic. For any such constants we define
A = |lnλ+| (lnµ++|lnλ+|) |lnµ−| (lnλ−+|lnµ−|) and B = 1−2A.
With these definitions we may now state the following theorem that gives hypotheses under which a Hamiltonian diffeomorphism germ of classCr can be linearised.
Theorem 6.3.3 ([BLW], Th. 1.1). Let φ be the germ of a Cr-Hamiltonian diffeomor-
phism fixing the origin, whose (k −2)-jet is linear and hyperbolic. Then there exists a
Cl-Hamiltonian germψ with D
0ψ =Id so that
ψ−1φψ=D0φ,
provided that 1≤l≤kA−B for a suitable choice of λ±, µ± and r >2k+ 4.
This is the main tool to show that the classγinduces a non-trivial class in the cohomology of Hamiltonian germs of class Ck at the origin.
Theorem 6.3.4. The class γ is non-trivial in H2(Gk
ham) for all 2≤k <∞.
Proof. By Proposition 6.3.2 the image of γjet in H2((Jhamm )δ,R) is non-trivial for allm. We
let σ be an integral homology class in H2((Jhamm )δ) that pairs non-trivially with γjet. Such
a class is equivalent to a representation of some surface group π1(Σg)→Jhamm . We let ai, bi
be standard generators for π1(Σg) and let αi, βi denote their images in Jhamm . Then if we
consider the m-jets αi, βi as (smooth) germs in the natural way, we see that them-jet of
φ=
g
Y
i=1
[αi, βi]
is the identity. Thus after multiplying φ with a hyperbolic elementM ∈SL(2,R), Theorem 6.3.3 implies that we may linearise the resulting germ by a Hamiltonian diffeomorphism germ
ψ of class Cl. Since the value of l grows linearly with m, after taking m large enough we may assume that l=k and, hence, the following holds inGk
ham ψ g Y i=1 [αi, βi]M ψ−1 =M.
106 6. Characteristic classes of symplectic foliations In other words g Y i=1 [ψαiψ−1, ψβiψ−1][ψ, M] = 1.
The image of the class inH2((Jhamk )δ) that is associated to the representation of the surface
group π1(Σg+1) given by the above relation decomposes as σ+τψ,M, where σ is our original
class and τψ,M is the class associated to the representation of the fundamental group of the
2-torus defined by the k-jets of the elements ψ and M. We note that γjet depends only on
2-jets and we have the following split exact sequence given by taking a 2-jet to its linear part 1→Jham,Id2 →Jham2 →SL(2,R)→1.
Thus any element ψ ∈ Jham2 can be written uniquely as ψ = ψId◦D0ψ, where the linear
part ofψId is the identity. As in the proof of Proposition 6.3.2, we may identifyJham,Id2 with
Sym3H2
R via the exponential map.
Now after a linear change of coordinates, we may assume that the hyperbolic element
M is diagonal and thus M commutes with an element ψ if and only if it commutes with
ψId and D0ψ. Since Jham,Id2 is just the group of homogeneous polynomials of degree 3, the
calculation of Proposition 6.3.2 shows that if ψ and M commute, then ψId must be trivial.
Furthermore, since the linear part of ψ can be chosen to be the identity, Proposition 6.3.3 implies that the 2-jet of ψ is trivial and, hence, that the homology class τψ,M is trivial in
H2((Jham2 )δ). Thus we have
γ(σ0) = γjet(σ+τψ,M) = γjet(σ)6= 0
and the claim follows.
The final step is to extend classes given by germs to classes defined by actual Ck- symplectomorphisms. This extension will rely on the following lemma, which mimics the original proof of Mather (cf. [Math1], [LeR]) that the group of compactly support homeo- morphisms is perfect.
Lemma 6.3.5. Let φ be an element in the group Hamkc(R2n\ {0}) that consists of Hamil-
tonian diffeomorphisms of class Ck with compact support. Then φ can be written as the
product of commutators of elements in Sympk(
R2n\ {0}), whose supports are disjoint from
0.
Proof. First of all since φ can be connected to the identity by a path of Hamiltonian diffeo-
morphisms with compact support, the standard fragmentation argument given in the smooth case applies and we may assume that φ is a product of elements with support in a ball (cf. [Ban], p. 110). Thus it suffices to consider the case where φ has support in a ball B. Now we let Bi be disjoint translates of B and let Ci be larger balls with the property that Ci
contains both Bi and Bi+1 and Ci is disjoint fromCi+2. We further assume that the Ci are
chosen so that there is a symplectomorphism hi with support inCi that interchangesBi and
6.3. Characteristic classes of leaves 107
We then inductively define g1 =φ andgi+1 =higih−1i . The elementsgi commute as they
have disjoint supports and the same holds forhi, hj and gi, hj, provided that|i−j| ≥2. We
may then define the following infinite products
A=g2g4g6..., B =h2h4h6..., C =g1g3g5..., D=h1h3h5...
and one computes that
φ = [A, B][C, D].
Finally, by choosing the ballsBi appropriately one may assume that the supports ofA, B, C
and D are disjoint from 0.
We will also need an analogue of Theorem 4.1.14, which in essence gives an explicit description of the Euler class in H2(G
p). First of all we note that since the inclusion
Sympkc(R2\{0}) inDif fck(R2\{0}) is a weak homotopy equivalence, the group of compactly supported symplectomorphisms of R2 \ {0} modulo isotopy is isomorphic to the compactly
supported mapping class M CGc(
R2\ {0}), which is in turn Z. With the aid of this obser- vation we prove the following proposition.
Proposition 6.3.6. Let α be the natural homomorphism from Sympkc(R2\ {0}) to the com-
pactly supported mapping class group of R2 \ {0} and consider the following extension of
groups
1→Sympkc(R2\ {0})→Sympkc(R2,0)→ Ghamk →1.
Then the Euler class in H2(Gk
ham) is, up to sign, the image of α in the five-term exact
sequence associated to the above extension.
Proof. We consider the following commutative diagram of group extensions
1 //Dif f1 c(R2\ {0}) //Dif fc1(R2,0) //GL+(2,R) //1 1 //Sympk c(R2\ {0}) // O O Sympk c(R2,0) // O O Gk ham // O O 1,
where the map on the right is given by taking a germ to its linear part. We note that the Euler classeinH2(Gk
ham) is just the pullback of the Euler class inH2(GL
+
δ(2,R)) under this
map. Now the image of the Euler class in H2(Dif fc,δ1 (R2,0)) is torsion since a flat bundle with holonomy in the groupDif f1
c(R2,0) has a section and is thus topologically trivial.
Since the group Dif f1
c,0(R2\ {0}) is perfect by the classic result of Mather (cf. [Math2]),
we have that H1(Dif fc,δ1 (R2 \ {0})) = Z and an isomorphism is given via the map α
to M CGc(
R2 \ {0}). Moreover, this map is equivariant with respect to conjugation by
GL+(2,
R), thus the invariant part H1(Dif fc,δ1 (R2\ {0}))GL
+(2,
R) is isomorphic to
Z. Simi- larly the group of coinvariants H1(Dif fc,δ1 (R
2\ {0}))
GL+(2,
R) is also isomorphic to Z. Thus,
in particular, H1(Dif fc,δ1 (R2,0)) is torsion free. The Universal Coefficient Theorem then
implies that H2(Dif f1
c,δ(R2,0)) is torsion free and we conclude that the image of the Euler
class is in fact trivial.
Now since the Euler class e is a primitive class whose image in H2(Dif f1
c,δ(R2,0)) is
trivial andH1(Dif f1
c,δ(R2\ {0}))GL
+(2,
R)=
108 6. Characteristic classes of symplectic foliations
thate =±δα. By the naturality of the five-term exact sequence the same then holds for the extension involving Sympk.
We may now prove the main result of this section that the class γ is non-trivial in
H2(Sympk
δ(R2,0)).
Theorem 6.3.7. For any 2 ≤ k < ∞ there exist foliated R2-bundles over a surface with
holonomy in Sympk(R2,0) for which the characteristic class γ is non-trivial. Moreover, we
may assume that such a bundle is topologically trivial.
Proof. By Theorem 6.3.4 there exists a representation of some surface groupπ1(Σg)→ Ghamk
for whichγ is non-trivial. As usual we let ai, bi be the standard basis ofπ1(Σg) and letαi, βi
denote the images of this basis in Gk
ham. We further let ˜αi,β˜i be representatives of these
germs that have compact support in R2. Then by construction
φ=
g
Y
i=1
[ ˜αi,β˜i]
is an element inSympkc(R2\{0}). By Proposition 6.3.6 and the explicit form of the connecting homomorphism in the five-term exact sequence given by Lemma A.1, the Euler class of the bundle given by the original representation into Gk
ham is given (up to sign) by φ considered
as an element in M CGc(R2\ {0}) (see also the proof of Theorem 5.2.5). As the Euler class can be chosen to be trivial (see Remark 6.3.1), we may assume that φ lies in the identity component.
Moreover, φ lies inHamk
c(R2\ {0}) since
F luxR2\{0}(φ) = F luxR2(φ) = F luxR2(