DIFFERENTIAL GEOMETRY AND APPLICATIONS Satellite Conference of ICM in Berlin, Aug. 10 – 14, 1998, Brno Masaryk University in Brno (Czech Republic) 1999, 413–418

THE FLUX HOMOMORPHISM IN THE FOLIATED CASE

TOMASZ RYBICKI

Abstract. By using the cotangent bundle of a foliation and the foliated cohomology we extend some results concerning the flux homomorphism. In analogy to symplectomorphisms one defines the flux for the automorphism group of a regular Poisson manifold. Next the flux extends to the case of ‘foliated’ loops. In view of the perfectness theorem the transgression homomorphism is an obstacle to further extension.

1. We will work in the category of (regularly) foliated smooth manifolds (M, F) and leaf preserving diffeomorphisms.

Theorem 1 [8]. There is a contravariant functor from this category to the category of ℤ-graded associative algebras with continuous homomorphisms which assigns to (M, F) the graded algebra H^{∗}(F). Here H^{k}(F) denotes the k-th foliated (or tangent) cohomology group.

Let us recall briefly some facts connected with the foliated cohomology. A foliated (or tangential) k-form at x_{0} is a skew-symmetric k-form on the tangential space T_{x_{0}}F at x_{0}. If a foliated form assumes the form

ω̄ = ∑_{I} a_{I}(x_{1}, …, x_{p}, y)d̄x_{i_{1}} ∧ ⋯ ∧ d̄x_{i_{k}} ,

1991 Mathematics Subject Classification. Primary 58D05; Secondary 22E65. Key words and phrases. Foliation, diffeomorphism group, foliated cohomology, flux homomorphism. Supported by KBN 2 P03A 024 10. This paper is in final form and no version of it will be submitted for publication elsewhere.

where y = (y_{1}, …, y_{q}), I = (i_{1}, …, i_{k}), p = dim F, q = codim F, then its exterior derivative is

d̄ω̄ = ∑_{I,j} \frac{∂a_{I}}{∂x_{j}} (x_{1}, …, x_{p}, y)d̄x_{j} ∧ d̄x_{i_{1}} ∧ ⋯ ∧ d̄x_{i_{k}} .

More precisely, the complex of foliated forms can be defined as follows. We have the subcomplex Ω^{∗}(M, F) of the de Rham complex (Ω^{∗}(M), d): ω ∈ Ω^{r}(M, F) if and only if ω ∈ Ω^{r}(M) and ω(X_{1}, …, X_{r}) = 0 for all X_{1}, …, X_{r} tangent to F. We set

Ω^{∗}(F) = Ω^{∗}(M)/Ω^{∗}(M, F) ,

and by ω̄ ∈ Ω^{r}(F) we will denote the class of ω ∈ Ω^{r}(M).

It is visible that the exterior product ∧ in Ω^{∗}(M) descends to Ω^{∗}(F). Next, it is easily checked that ι(X)ω̄ = \overline{ι(X)ω} and f^{∗}ω̄ = \overline{f^{∗}ω} are correct definitions whenever X is tangent to F and f is leaf preserving. For such an X we have the Lie derivative L_{X} ω̄ = ι(X)d̄ω̄ + d̄ι(X)ω̄.

Finally, if ω̄_{1} = ω̄_{2} then \overline{d ω}_{1} = \overline{d ω}_{2}. In particular, ω̄ = d̄λ̄ whenever ω = d λ. Consequently, d̄ is a well-defined differential on Ω^{∗}(F) By H^{∗}(F) we denote the cohomology of Ω^{∗}(F), and by [ω̄] the cohomology class of ω̄. Clearly H^{r}(F) = 0 if r > p = dim F.

Likewise H_{c}^{∗}(F) will stand for the foliated cohomology with compact support, that is with Ω^{∗}(M) replaced by Ω_{c}^{∗}(M) and so on.

The foliated cohomology can be also introduced by means of R_{t}, the sheaf of germs of tangentially locally constant smooth functions. Namely, one has

0 → R_{t} → \underline{Λ}^{0}(T^{∗}F) ⟶^{d̄} \underline{Λ}^{1}(T^{∗}F) → ⋯ → \underline{Λ}^{p}(T^{∗}F) → 0 ,

the exact sequence of sheaves. Then Ω^{k}(F) is identified with the space of global sections of \underline{Λ}^{k}(T^{∗}F) and we have H^{k}(M; R_{t}) = H^{k}(F).

Each H^{k}(F) is a topological vector space, but possibly infinite dimensional (contrary to de Rham), and possibly non-Hausdorff as, in general, the image of d̄ fail to be closed. So one may consider, if necessary,

H̄^{k}(F) := ker(d̄^{k}) / \overline{im(d̄^{k−1})} ,

the Hausdorff foliated cohomology. For instance, if F is a linear foliation with a diophantine irrational slope on the 2-dimensional torus then H^{1}(F) is infinite dimensional while H̄^{1}(F) = ℝ (cf. [8]).

Throughout we will assume that dim M = n, dim F = p.

2. The concept of the flux homomorphism was initiated by E. Calabi [2] and developed by A. Banyaga and others. It is a basic invariant of diffeomorphism groups, cf. [1], [6].

For any connected and locally connected by arcs topological group G its universal cover group G̃ is the set of all pairs (g, {g_{t}}), where {g_{t}} is the homotopy rel. endpoints class of the path g_{t} in G such that g_{0} = e and g_{1} = g, endowed with the pointwise multiplication.

By G_{0} we denote the component of e in G. Clearly G̃_{0}/π(G_{0}) = G_{0}, where π is the first homotopy group.

Given a foliation F we will consider the group of all leaf preserving diffeomorphisms with compact support, Diff_{c}(M, F). Let ϕ_{t} be an isotopy in Diff_{c}(M, F) (ϕ_{0} = id). Then the associated time-dependent vector field ϕ̇_{t} := \frac{d}{dt}ϕ_{t} ∘ ϕ_{t}^{−1} is tangent to F. Let ᾱ ∈ Ω^{k+1}(F) be any d̄-closed k + 1-form. As in [1] we put

I_{ϕ_{t}}(ᾱ) := ∫_{0}^{1} ϕ_{t}^{∗}ι(ϕ̇_{t})ᾱdt .

Under the assumption Lϕ̇_{t} ᾱ = 0, d̄I_{ϕ_{t}}(ᾱ) = 0 and we get the foliated cohomology class

[I_{ϕ_{t}}(ᾱ)] = [ ∫_{0}^{1} ι(ϕ̇_{t})ᾱdt] ∈ H_{c}^{k}(F) .

Theorem 2. Let G(M, F, ᾱ) := {f ∈ Diff_{c}(M, F) : f^{∗}ᾱ = ᾱ}. There is a continuous homomorphism

S̃ᾱ : { \atop G(M, F, ᾱ)}_{0} ∋ \{ϕ_{t}\} ↦ [I_{ϕ_{t}}(ᾱ)] ∈ H_{c}^{k}(F)

called the foliated flux homomorphism associated with ᾱ.

Specifically, if ᾱ is a 2-form (i.e. k = 1) and is ‘symplectic’ (cf. [3]) then we get the flux in the (regular) Poisson case. The Weinstein chart for G(M, F, ᾱ)_{0} was constructed in [12]. Proof. For an isotopy ϕ = (ϕ_{t}) in Diff_{c}(M, F)_{0} we have

ϕ̇_{t} = δ^{r}ϕ(∂_{t}|_{t})

where δ^{r} is the right logarithmic derivative, cf. [6]. Therefore for a smooth homotopy (s, t) ↦ ψ(s, t) with ψ(i, t) = ϕ_{t}^{i}, i = 0, 1, it follows from the left Maurer-Cartan equality d δ^{r}ψ − \frac{1}{2}[δ^{r}ψ, δ^{r}ψ] = 0 that

∂_{s}δ^{r}ψ(∂_{t}) = ∂_{t}δ^{r}ψ(∂_{s}) + [δ^{r}ψ(∂_{s}), δ^{r}ψ(∂_{t})] .

Hence by a standard argument

∫_{0}^{1} ι(ϕ̇_{t}^{1})ᾱdt − ∫_{0}^{1} ι(ϕ̇_{t}^{0})ᾱdt
= ∫_{0}^{1} ι(δ^{r}ψ(∂_{t}|_{(1,t)}))ᾱdt − ∫_{0}^{1} ι(δ^{r}ψ(∂_{t}|_{(0,t)}))ᾱdt
= d̄ ∫_{0}^{1} ∫_{0}^{1} ι(δ^{r}ψ(∂_{s}))(ι(δ^{r}ψ(∂_{t}))ᾱ)ds dt .

This means that S̃ᾱ is well-defined. That S̃ᾱ is a homomorphism follows from the fact that for any topological group G the pointwise multiplication of paths and the juxtaposition of paths coincide on the level of homotopy preserving endpoints.

Questions. Denote Γᾱ := im(π(G(M, F, ᾱ)_{0})) where π is the first homotopy group. Our first question is whether Γᾱ is countable. In view of some results from R. Palais [10] and by possible extensions of some difficult methods from H. Omori [9] we conjecture the affirmative answer. The second question is whether Γᾱ is discrete. Observe that it is still an open problem in the symplectic case, cf. [7].

Another question is when S̃ᾱ admits global sections. This is the case of compact symplectic manifolds by an argument involving the Hodge theory ([6]).

As in the transitive case we have

Proposition 1. If ᾱ is exact, i.e. d̄β̄ = ᾱ, then S̃ᾱ({ϕ_{t}}) = [ϕ_{1}^{∗}β̄−β̄]. In particular, Γᾱ = 0.

We see also that the flux is an obstacle to the fragmentation of isotopies. Namely, if an isotopy ϕ_{t} in G(M, F, ᾱ) can be fragmented into isotopies ϕ_{t}^{i} supported in contractible U_{i} then {ϕ_{t}} is in the kernel of S̃ᾱ.

3. Now our purpose is to extend the flux for the group of leaf preserving isotopies (i.e. to get rid of ᾱ). It will occur that this may be accomplished for foliated loops only.

Let π : E → M be a vector bundle. By D̂iff(E) we denote the group of all fiber preserving diffemorphisms f of E such that \underline{f}, the induced diffeomorphism of M, is compactly supported. For any global section s ∈ Γ(E) we define ϕ_{s} ∈ D̂iff(E) by

ϕ_{s}(e) := (x, v − s(x)) for e = (x, v) ∈ E ,

and we have the monomorphism Φ : Γ(E) ∋ s ↦ ϕ_{s} ∈ D̂iff(E).

Now let (M, F) be a foliated manifold. We set

E_{F}^{k} := ⋃_{x∈M} Λ^{k}(T_{x}^{∗}L_{x})

where L_{x} is the leaf through x. Clearly Γ(E_{F}^{k}) = Ω^{k}(F). The space E_{F}^{k} possesses a canonical foliated k-form λ̄_{F}^{k} with respect to the foliation τ^{∗}F where τ : E_{F}^{k} → M is the canonical projection. Namely, for U ∈ (E_{F}^{k})_{x} = Λ^{k}(T_{x}^{∗}L_{x}), V_{1}, …, V_{k} ∈ T_{U}(τ^{∗}F) we put

λ̄_{F}^{k}(V_{1}, …, V_{k}) := U(τ_{L∗}V_{1}, …, τ_{L∗}V_{k}) ,

τ_{L_{x}} : T^{∗}L_{x} → L_{x} being the restricted projection. Let θ̄ ∈ Ω^{k+1}(F) be any d̄-closed form. We define

ωk+1 ; θ̄ := τ^{∗}θ̄ + d̄λ̄_{F}^{k} ∈ Z^{k+1}(τ^{∗}F).

In particular, ω̄^{k+1} = ω̄_{0}^{k+1} = d̄λ̄_{F}^{k} is said to be a canonical (k + 1)-form. Next we set

G(E_{F}^{k}) = G(E_{F}^{k}, ωk+1 ; θ̄) = \{f ∈ D̂iff(E_{F}^{k}) : f^{∗}ωk+1 ; θ̄ = ωk+1 ; θ̄, \underline{f} ∈ Diff_{c}(M, F)\} .

Proposition 2. Let ᾱ ∈ Ω^{k}(F) = Γ(E_{F}^{k}). Then d̄ᾱ = 0 iff ϕᾱ ∈ G(E_{F}^{k}). Moreover, the sequence

(∗)  0 → Z_{c}^{k}(F) ⟶^{Φ} G(E_{F}^{k})_{0} ⟶^{τ_{−}} Diff_{c}(M, F)_{0} → 1 ,

where τ_{−} is induced by τ, is exact.

Indeed, this follows from the equalities (cf. [1])

ϕ∗ ; ᾱλ̄_{F}^{k} = λ̄_{F}^{k} − τ^{∗}ᾱ

and

ϕ∗ ; ᾱωk+1 ; θ̄ = ωk+1 ; θ̄ − τ^{∗}d̄ᾱ

which are still fulfilled in the foliated case. Therefore, as in the transitive case we get

Corollary 1. π(G(E_{F}^{k})_{0}) = π(Diff_{c}(M, F)_{0}).

Now we are in a position to define the flux homomorphism on loops. For any θ̄ ∈ Z_{c}^{k+1}(F) we have already the homomorphism

S̃_{ωk+1 ; θ̄} : { \atop G(E_{F}^{k})} ∋ \{ϕ_{t}\} ↦ [I_{ϕ_{t}}(ωk+1 ; θ̄)] ∈ H_{c}^{k}(τ^{∗}F) .

(Here H_{c}^{k}(τ^{∗}F) denotes the foliated cohomology group compactly supported in the horizontal direction.) By using Corollary 1 and taking the restriction of S̃_{ωk+1 ; θ̄} we get the flux homomorphism on foliated loops

S̃θ̄ : π(Diff_{c}(M, F)_{0}) → H_{c}^{k}(F) .

Specifically for θ̄ = 0 this homomorphism is defined canonically.

4. For any topological group G we have its central extension (cf. [5])

0 → π(G) ⟶^{i} G̃ ⟶^{p} G → 1 .

Let U_{G} ∈ H^{2}(G, π(G)) be its characteristic class. Let A be an abelian group on which G acts trivially. Recall that

t : Hom(π(G), A) → H^{2}(G, A), where t(ψ) = ψ ∘ U_{G} ,

is called the transgression homomorphism. The following fact is well-known

Theorem 3 [5]. The sequence

0 → Hom(G, A) ⟶^{p^{∗}} Hom(G̃, A) ⟶^{i^{∗}} Hom(π(G), A) ⟶^{t} H^{2}(G, A) ⟶^{p^{∗}} H^{2}(G̃, A)

is exact and natural.

Corollary 2. If G̃ is perfect then t is bijective, and t is an obstruction for extending Hom(π(G), A) to Hom(G̃, A).

Now we use an analogue of a difficult theorem of Thurston on the simplicity of the diffeomorphism group.

Theorem 4 [11]. Diff_{c}(M, F)_{0} and its universal covering are perfect.

Observe that Theorem 4 holds as well for generalized foliations with no leaves of dimension 0, cf. [4]. In this case Theorems 1 and 2 still holds unlike to the rest of considerations.

Corollary 3. The transgression homomorphism

t : Hom(π(Diff_{c}(M, F)_{0}), H_{c}^{k}(F)) → H^{2}(Diff_{c}(M, F)_{0}, H_{c}^{k}(F))

is bijective, and there is in general no extension of S̃θ̄ to { \atop Diff_{c}(M, F)}_{0}.

For any θ̄ ∈ Z_{c}^{k+1}(F) we have constructed the central extension (∗). Let us denote by Ûθ̄ ∈ H^{2}(Diff_{c}(M, F)_{0}, H_{c}^{k}(F)) the image of its characteristic class under the projection Z_{c}^{k}(F) → H_{c}^{k}(F). Simultaneously we have the k-flux on loops S̃θ̄ : π(Diff_{c}(M, F)_{0}) → H_{c}^{k}(F). The following theorem relates these concepts similarly as in the transitive case, cf. [1].

Theorem 5. Under the above notation and assumptions one has t(S̃θ̄) = Ûθ̄, i.e. the transgression of the foliated flux is the image of the characteristic class of (∗). Moreover, S̃θ̄ depends only on [θ̄] and we have a group homomorphism

H_{c}^{k+1}(F) ∋ [θ̄] ↦ t(S̃θ̄) ∈ H^{2}(Diff_{c}(M, F)_{0}, H_{c}^{k}(F)) .

Acknowledgements. The author expresses his gratitude to the referee for helpful remarks.

References

[1] Banyaga, A., The Structure of Classical Diffeomorphism Groups, Kluwer Academic Publishers, Dordrecht 1997.

[2] Calabi, E., On the group of automorphisms of a symplectic manifold, Problems in Analysis (symposium in honour of S.Bochner), Princeton University Press (1970), 1–26.

[3] Dazord, P., Hector, G., Intégration symplectiques des variétés de Poisson totalement asphériques, Symplectic Geometry, Groupoids and Integrable Systems, Séminaire Sud Rhodanien de Géométrie à Berkeley (1989) (P. Dazord and A. Weinstein, eds.), MSRI Publ. 20, Springer- -Verlag, 1991, 37–72.

[4] Haller, S., Rybicki, T., On the perfectness of nontransitive groups of diffeomorphisms, preprint ESI (Vienna) n. 584.

[5] Hilton, P., Stamback, U., Homological Algebra, Graduate Text in Math., Springer-Verlag.

[6] Kriegl, A., Michor, P. W., The Convenient Setting of Global Analysis, American Mathematical Society, 1997.

[7] McDuff, D., Fibrations in symplectic topology, I.C.M. Berlin, 1998.

[8] Moore, C., Schochet, C., Global Analysis on Foliated Spaces, Springer-Verlag 1988.

[9] Omori, H., Infinite-Dimensional Lie Groups, Transl. of Math. Monographs, A.M.S., 1997.

[10] Palais, R., Homotopy theory of infinite dimensional manifolds, Topology, 5 (1966), 1–16.

[11] Rybicki, T., The identity component of the leaf preserving diffeomorphism group is perfect, Mh. Math. 120 (1995), 189–205.

[12] Rybicki, T., On the flux homomorphism for regular Poisson manifolds, Supl. Rend. Circ. Mat. Palermo (1998).

Department of Appl. Mathematics AGH, al. Mickiewicza 30 30-059 Kraków, POLAND E-mail: rybicki@im.uj.edu.pl