ON THE COHOMOLOGY OF FOLIATED BUNDLES ∗

We prove a de Rham-like theorem for foliated bundles F → (M,F) π → B showing that the cohomology H∗(F) is isomorphic to the equivariant cohomology HΓ ( B̃, C∞ (F ) ) ,Γ = π1 (B) and B̃ the universal covering of B. When B is an EilenbergMac Lane space K (Γ, 1) the cohomology H∗ (F) is the cohomology of the Γ-module C∞ (F ). This gives algebraic models for H∗ (F) and geometrial models for the cohomology of the Γ-module C∞ (F ). Using this isomorphism and a theorem of J. Palis and J.C. Yoccoz on the triviality of centralizers of diffeomorphisms, [14] and [15] we show that H∗(F) is infinite dimensional for a large class of foliated bundles. AMS (MOS) Subj class: 57R30


Introduction
The cohomology of foliated manifolds appears naturally in the study of locally free actions of Lie groups and characteristic classes of foliations, [3], [17] and [18].In this article we study the cohomology of foliated bundles F → (M, F) π → B suspension of actions ϕ : Γ → Dif f (F ), where B and F are connected C ∞ manifolds and Γ = π 1 (B).We show in Theorem 2.1 that the foliated cohomology H * (F) of a foliated bundle π is isomorphic to the equivariant cohomology H * Γ ( B, C ∞ (F )) where B is the universal covering of B and C ∞ (F ) has the Γ-module structure given by the action ϕ.This isomorphism is via a natural de Rham mapping.When B is an Eilenberg-Mac Lane space K(Γ, 1), i.e.B is contractible then H * (F) is isomorphic to the cohomology H * (Γ, C ∞ (F )) of the Γ-module C ∞ (F ), the action on C ∞ (F ) being γ • h = h • ϕ γ , γ ∈ Γ and h ∈ C ∞ (F ).In this way we have both algebraic models for the foliated cohomology H * (F) and geometrical models for the cohomology H * (Γ, C ∞ (F )) of the Γmodule C ∞ (F ).Theorems 2.4 and 3.3 and J. Palis and J.C. Yoccoz results on the triviality of centralizers of diffeomorphisms, [14], [15] show that the cohomology H * (F) of a foliated bundle suspension of an action ϕ : Γ → Dif f (F ) is infinite dimensional for a large class of actions.In section 4 we discuss the case B = T p and show that the cohomology of groups gives a procedure for the computation of H * (Z p , C ∞ (F )) which can be used to give an alternative simple way for computing the cohomology of linear foliations of T n , [2].We also state R.U.Luz, [9] computation of the cohomology of the actions of Z p by affine transformations of T q .

The Cohomology of a Foliated Bundle
The foliated cohomology introduced by B.L. Reinhart in [16] appears naturally in the study of locally free actions of Lie groups and characteristic classes of foliations, [3], [17] and [18].
Let F be a p-dimensional foliation of M and Λ (M ) the graded algebra of all C ∞ differential forms on M .If I(F) ⊂ Λ (M ) is the annihilating ideal of F, then I(F) q+1 = 0, q = m − p being the codimen-sion of F. Thus Λ (F) = Λ (M ) I(F) is a graded algebra, called the algebra of differential forms along F. The elements of Λ j (F) may be thought of as sections of the j-th exterior power of the dual bundle of the tangent bundle T F of F. Since by Frobenius's theorem dI(F) ⊂ I(F) the differential d : Λ (M ) → Λ (M ) induces the foliated differential d f : Λ (F) → Λ (F).The kernel Z(F) of d f is the set of d f -closed forms and the image B(F) of d f is the set of d f -exact forms.The cohomology H * (F) of the differential complex (Λ (F) , d f ) is the cohomology of the foliated manifold (M, F).This is a natural generalization to foliations of the de Rham cohomology.Let τ : Λ (M) → Λ (F) be the canonical projection.We say that ξ and we denote by ξ the cohomology class of τ (ξ) in H j (F).
In this article we study the cohomology of a foliated bundle.Let B and F be connected orientable smooth manifolds, p and q dimensional, respectively, and ϕ : π 1 (B) → Dif f (F ) be a left action.The foliated bundle F → (M, F) π → B suspension of ϕ is constructed as follows: let p : B → B be the universal covering of B and x o ∈ B. Associated to ϕ there is an action given by Φ γ (x, y) = (γ • x, ϕ −1 γ (y)) where γ • x denotes the image of x by the deck transformation of B corresponding to the homotopy class γ of π 1 (B, x o ).The orbit space M of Φ is a manifold and actually we have a fiber bundle F → M π → B. Every object of B × F which is invariant under Φ induces a corresponding object on M .To the natural foliation F given by the projection B × F → F , which is invariant under Φ, corresponds a foliation F on M which is transverse to the fibers of π.We think of Λ (F) as the set of differential forms ξ of Λ F which are invariant under Φ i.e.Φ * γ (ξ) = ξ.We observe that ξ ∈ Λ F invariant under the action Φ is equivalent to where ξ y = j * y (ξ), j y : B → B × F being the inclusion j y (x) = (x, y) and [γ] the automorphism of B × F given by [γ](x, y) = (γ • x, y).
The image of π * : Λ (B) → Λ (F) is called the space of basic forms.In [3] to each invariant Borel probability measure µ on F there was associated an epimorphism of differential complexes i.e.P µ is a continuous surjective linear mapping which commutes with the differentials: The split short exact sequence 0 → Ker → Λ (F) gives the split short exact cohomology sequences

The Cohomology of a Foliated Bundle as Equivariant Cohomology
Let F → (M, F) π → B be the foliated bundle suspension of an action ϕ : π 1 (B, x o ) → Dif f (F ).The fundamental group Γ = π 1 (B, x o ) acts freely on the universal covering B of B by deck transformations γ : x → γ • x.A (left) ZΓ-module, also called a Γ-module, consists of an abelian group A together with a homomorphism of the integral group ring ZΓ of Γ to the ring of endomorphisms of A, [4], [5] and [12].
Since Γ acts freely on B the simplicial complex of all C ∞ functions h : F → R also has a Γ-module structure given by the action ϕ : Γ → Dif f (F ).Γ acts on C ∞ (F ) by γ • h = h • ϕ γ .We also write C ∞ (F ) for this Γ-module.Let Hom Γ (S * ( B), C ∞ (F )) be the group of all Γ-homomorphisms of S * ( B) into for all σ in S * ( B) and γ in Γ.
In this section we prove a de Rham type theorem which says that the cohomology H * (F) of a foliated bundle π is isomorphic to the equivariant cohomology H * Γ ( B, C ∞ (F )) via the natural de Rham mapping.Let F be the foliation on B × F given by the natural projection B × F → F .We have a natural de Rham mapping for ξ in Λ j ( F), σ ∈ S j ( B) and y ∈ F .This mapping retricts to the subcomplex Λ (F) of Φ-invariant forms of Λ F as a natural de Rham mapping To see this notice that from (1.2) and (2.2) a form ξ in Λ j ( F) is Φinvariant if and only if Then the de Rham mapping (2.3) induces an isomorphism The proof follows the same basic pattern of Massey's proof, [10] of the classical de Rham's theorem, based on Milnor's proof of the Poincaré duality theorem.For Massey's proof to work we have to show the existence of Mayer-Vietoris sequences for the equivariant cohomology.Mayer-Vietoris sequences for foliated cohomology, are well known, see [6].
Let U = {U j } be an open covering of B such that π −1 (U j ) π → U j are trivial and U j are contractible.Denote by S j ( B, U) the Γ-module generated by "U-small" j-simplices i.e. simplices whose ranges lie in elements of U and let 2.2 Lemma.The restriction homomorphism j U induces an isomorphism Proof: The proof is essentially the same as for the classical isomorphism, [7] Let Sd : S j ( B) → S j ( B) be the subdivision homomorphism and be the corresponding homotopy operator i.e.
On the cohomology of foliated bundles The proof follows now as in the classical case observing that if l is a Γ-cocycle then l = l•T is a Γ-homomorphism and δl = l−l•sd.There exists an integer n > 0 such that (sd) n (σ) is a linear combination of U-small singular simplices.Thus l•(sd) n is cohomologous to l, proving the lemma.
Proof of Theorem 2.1 We first notice that the theorem is true in degree zero since both H o (F) and , then the restriction of to S j ( Ũ ) is a Z-cocycle and as Ũ is contractible then there exists a Z-homomorphism ˜ : S j+1 ( Ũ ) → C ∞ (F ) such that δ ˜ = .We also denote by ˜ the extension of ˜ to a Γ-homomorphism ˜ : is an isomorphism.
Case 2. B is the union of two open sets U and V and de Rham's theorem is assumed to hold for U , V and U ∩V .We show the theorem also holds for U ∪ V .To prove the theorem in this case we use Mayer-Vietoris sequences.We denote by F U , F V and F U ∩V the restrictions of the foliation ) be the inclusions.We consider the mappings given by a(w) = (k * w, * w) We have then the short exact sequence of differential complexes, [6] 0 Now we consider Mayer-Vietoris sequence for equivariant cohomology.Consider the cochain mappings and where U = {U, V } and 0 ≤ r ≤ p.These mappings are defined in the usual way, [9].Clearly a is a monomorphism.It follows from the Mayer-Vietoris sequence for singular cohomology that β is an epimorphism.For if ˜ : given by (σ) = ˜ (σ) for every σ in the ZΓ-basis B fixed before and σ ∈ S r (p −1 (U ∩ V )).Thus there exist Z-homomorphisms where we also denote by i and j the inclusions of U ∩ V in U and V , respectively.We define Γ-homomorphisms ) and analogously for ˜ 2 .Clearly β ( ˜ 1 , ˜ 2 ) = ˜ .Thus we have the short exact sequence of cochain complexes Finally, we put the sequences (2.7) and (2.8) into a commutative diagram where the first vertical mapping is the composite of de Rham mapping k and the restriction homomorphism j U and the last two mappings are de Rham mappings.Now taking the cohomology sequences associated to (2.7) and (2.8) the theorem is proved in this case using Lemma 2.2 and the five lemma, [12].
nested sequence of open sets with compact closures.It is assumed that de Rham's theorem holds for each U i ; we will show that it also holds for B. To carry out the proof in this case, we need to use inverse limits as in Massey's proof, [10] and [11].The inclusions U i ⊂ B induce cochain mappings Λ * (F) → Λ * (F i ), F i being the restriction of F to π −1 (U i ).The inverse sequence Λ * (F i ) satisfies the Mittag-Leffler condition, thus lim 1 Λ j (F i ) = 0 for all j and there is the natural short exact sequence Similarly for the cochain complexes Hom Γ (S * (U i ), C ∞ (F )) i.e. we also have the short exact sequences Now apply the de Rham homomorphism from sequence (2.2) to the sequence (2.3) and one easily prove the theorem in this case.To complete the proof in this case one passes to the limit as n → ∞ using Case 3.
Case 5. B is paracompact.In this case B is a countable union of open sets U i diffeomorphic to R p and with compact closures.Let V n = n i=1 U i .We can prove by induction on n using Cases 2 and 4 that de Rham's theorem holds true for each V n and each V n is compact.Then it follows from Case 3 that de Rham's theorem holds for B.
Let as before ϕ : Γ → Dif f (F ) be an action, Γ = π 1 (B) and B is an Eilenberg -Mac Lane space K (Γ, 1), then the augmented cellular chain complex is a free resolution of Z over ZΓ, [5].In this case the groups H j Γ B, C ∞ (F ) are called the cohomology groups of the Γ-module C ∞ (F ) and denoted by H j (Γ, C ∞ (F )) , 0 ≤ j ≤ p. Theorem 2.1 gives a geometrical interpretation for the cohomology of the Γ-module C ∞ (F ) and an alternative way of understanding the cohomology of a foliated bundle.
Associated to the action ϕ we also have a short exact sequence The mapping ρ is defined as follows: let Σ j be the submodule of S j ( B) generated by the elements σ − γ • σ for σ ∈ S j ( B) and The mapping ρ is defined by ρ( ) = ˆ .Clearly the kernel of ρ is Hom Γ (S * ( B), H).To show ρ is an epimorphism, choose a ZΓ-basis B for S j ( B).If ˆ : S j (B) → C ∞ (F ) Γ is a homomorphism, choose a function h σ in each coset ˆ (p * σ), σ ∈ B and let (σ) = h σ .This defines a Γ-homomorphism : S j ( B) → C ∞ (F ) and clearly ρ( ) = ˆ .We notice that the sequence (2.12) is in general not split.Associated to (2.12) we have the long exact cohomology sequence Let G be a connected Lie group and Γ be a discrete cocompact subgroup of G i.e. the right coset space B = G Γ is a compact manifold.
Each right translation R g : G → G induces a diffeomorphism R o g on B and the right action induces a right action of Gon B. Let G be the Lie algebra of G.If E ∈ G is a left invariant vector field on G and a : R × G → G its flow, then a t (g) = R a t (e) (g) i.e. the restriction of the right action G × G R → G to the 1-parameter subgroup a t (e)is the flow of E. Thus the restriction of R o to a t (e)gives a flow E o on B and we have an injective homomorphism of G into the Lie algebra χ(B) of all C ∞ vector fields on B. Let G o be the image of this homomorphism.Associated to a left action ϕ of Γ on a q-manifold F there is a left action Φ : Γ → Dif f (G × F ) given by Φ γ (g, y) = (γg, γ −1 (y)).The orbit space M = G × F Γ is a manifold and we have the foliated bundle We notice that the mappings R g × id on G × F induce automorphisms H g of the foliated bundle π i.e.H g preserves the leaves of F and π Thus we have an action of G as automorphisms of π, called the canonical action of π.The restriction of H to Γ gives an action H * of Γ as automomorphisms of the cohomology H * (F), [17].Also the right translations 2.3 Proposition.The action R * of Γ as automorphisms of Choose any smooth path c : I → G, c(0) = e and c(1) = γ.We have a homotopy h : G × I → G, h(x, t) = x c(t)and let h * : S * (G × I) → S * (G)be the induced chain mapping.Consider the prism operators, [7] On the cohomology of foliated bundles 187 where j t (x) = (x, t), t = 0, 1, the homotopy h and (2.8) give Since is a Γ-cocycle we get where ˆ = • h * P .To show that ˆ is a Γ-homomorphism we show that h * P : S j (G) → S j (G) is a Γ-homomorphism.In fact, for each γ ∈ Γ we have, [7] (L γ × id) we see that (L γ ) * commutes with h * P , i.e. h * P is a Γ-homomorphism.

Theorem.
Let G be a connected Lie group, Γ a discrete cocompact subgroup of G and F → (M, F) π → B be the suspension foliated bundle of an action ϕ of Γ on a q-manifold F .Assume .. is generated by the iterates of a diffeomorphism ϕ γ where γ ∈ Γ is in the center of G.If the only ϕ-invariant functions are the constants, then there is a natural isomorphism Proof: The Theorem follows from the cohomology sequence (2.6) if we show that H j Γ (G, H) = 0 for 0 ≤ j ≤ p.To show that H is acyclic choose a probability measure µ on F which is invariant under ϕ γ and let C ∞ o (F ) be the subspace of functions h in C ∞ (F ) with µ-measure zero i.e.F h dµ = 0. Thus H ⊂C ∞ o (F ) and each function h in H can be written uniquely as 3.2 Proposition.Let ϕ : Γ → Dif f (F ) be an action of a countable generated group Γ on a closed manifold F .If ϕ has subexponential growth and infinitely many minimal sets, then C ∞ (F ) Γ is an infinite dimensional vector space.
Proof: We show that the space M ϕ of invariant measures is infinite dimensional.In fact, given any positive integer n choose n distinct minimal sets M 1 , . . ., M n .Since ϕ has subexponential growth then by [13] there exist invariant probability measures µ 1 , . . ., µ n such that sup pµ j ⊂ M j , 1 ≤ j ≤ n.These measures are linearly independent.For, let f 1 , . . ., f n be smooth functions with disjoint support such that 2 The centralizer group of a diffeomorphism ϕ : F → F is the set of elements in Dif f (F ) which commute with ϕ and is denoted by C(ϕ).We denote by Z(ϕ) the cyclic group generated by ϕ.The group Dif f (F ) of all C ∞ diffeomorphisms of F is endowed with the C ∞ topology.We say that ϕ has trivial centralizer if Z(ϕ) = C(ϕ).A question posed by S. Smale, [19] is whether there exists an open dense set of diffeomorphisms in Dif f (F ) having trivial centralizer.The question was answered affirmatively in the case of the circle by N. Kopell in [8].J. Palis and J.C. Yoccoz gave an affirmative answer for a large set in Dif f (F ), [14].Smale's question can be related with the question whether for a large set of foliated bundles the foliated cohomology is infinite dimensional.In fact, it is believed that the set of foliations with finite dimensional cohomology is a very small set.
Suppose the group Γ is generated by γ 1 , . . ., γ p .We say that an action ϕ : Γ → Dif f (F ) is generated by the iterates of a diffeomor- In this case every function h in H can be written as χ − χ • ϕ = h for χ ∈ C ∞ (F ).We remark that if Γ has a non-trivial center then Palis-Yoccoz results on the centralizer of diffeomorphisms say that the actions of Γ on F which are generated by the iterates of some diffeomorphism ψ ∈ Dif f (F ) form a large class.Next we extend Theorem 2.1 of [1] to these actions.
The orbit of an action ϕ through a point x ∈ F is the set O(x) of all points ϕ γ (x), γ ∈ Γ.The closure O (x) of O(x) will be refferred to as an orbit closure of ϕ.
3.3 Theorem.Let ϕ be an action of a finitely generated group Γ on a closed manifold F .Suppose ϕ is generated by the iterates of a diffeomorphism ψ with infinitely many orbit closures.Then the space of co-invariants C ∞ (F ) Γ of ϕ is infinite dimensional.
Proof: If ψ has infinitely many minimals, the result follows from Proposition 3.2.Assume ψ has a finite number of minimals.Let C = {O (x i ), i = 0, 1, 2, . ..} be a countable family of distinct orbit closures.By Lemma 2.2 of [1] we may assume that A = ∞ j=0 a(x i ) and W = ∞ j=0 w(x i ) are both non void (here a(x i ) and ω(x i ) denote the a-limit and ω-limit sets of x i , respectively).The inclusion gives a partial ordering on C.So there are two possibilities.Case 1. C has an infinite totally ordered subset C .Case 2. Any totally ordered subset C of C is finite.
Case 1.Let n > 0 be any integer.By assumption there exist n + 1 distinct orbit closures in C , say h(ϕ j (x o )) = 0 for each integer k.Thus χ is constant on O (x 0 ) and we may as well assume that χ vanishes on On the cohomology of foliated bundles 191 O (x 0 ).Thus χ vanishes on A and W . Now for each integer k > 0. Since χ vanishes on A and W , (3.2) gives for some subsequences (k j ) and (l j ).Since the function f i are non negative and take the value 1 on x i , then the above limits are positive.Thus (3.3) gives c 1 = 0. Similarly we show that c 2 = ...
Case 2. In this case given any positive integer n there exist n Now as in Case 1 we show that these functions are linearly independent in C ∞ (F ) Γ , proving the theorem.

Foliated Bundles over the Torus T p
Since the torus T P is a K (Z p , 1) space then the cohomology of a foliated bundle F → (M, F) π → T p is, by Theorem 2.1, isomorphic to the cohomology of the Z p -module C ∞ (F )In this section we give a procedure for the computation of this cohomology.
Let Λ(R p ) * be the exterior algebra over Z generated by the canonical 1-forms dx 1 , . . ., dx p with trivial Z p -action.Consider the graded We think of the elements of Λ j (R p ) * ⊗ C ∞ (F ) as "differential forms" Recall that H o (Γ, A) = A Γ is group of invariants for any Γ-module A.
(i) A 1-cochain ξ is determined by its value on the generators dx 1 , dx 2 and dx 3 of Λ 1 (R 3 ) * .So a 1-cochain is the same as a "1-form" From this we have the equations i.e. if the coefficients satisfy the equation Notice that a 2-cocycle is a coboundary if there exists a 1-cochain λ = h 1 dx 1 + h 2 dx 2 + h 3 dx 3 such that ξ = dλ and from this we get the equations f ij = ∂ i h j − ∂ j h i for 1 ≤ i < j ≤ 3. Thus we have the system of equations

The Cohomolgy of Actions of Z p on the Affine Group of T q
Let Affin(T q ) be the group of affine transformations of the torus T q and ϕ : Z p → Affin(T q ) be an action (homomorphism).Let ϕ * be the induced action of Z p on the ring By the cohomology of the action ϕ we mean the cohomology of the Z p -module C ∞ ϕ (T q ).This cohomology was computed by J. L. Arraut and N. M. dos Santos for actions for Z p by translations of T q in [2] and, more generally, for actions of Z p by affine transformations of T q by R. U. Luz in [9].
We now discuss briefly these results.The derivative A = Dϕ gives an action of Z p on S (q, Z).The cohomology of the action ϕ depends on both the arithmetic nature of ϕ and the algebraic properties of A. The set σ(A) of all eigenvalues of all A( ), ∈ Z p is referred to as the spectrum of A. If ϕ is minimal (i.e.every orbit is dense) then σ(A) = {1}, [9].The isotropy groups I(k) of the action t A : Z p × Z q → Z q , t A( , k) = t A( )k play an important role on the cohomology of ϕ.Let G be a non-trivial isotropy group of t A and { 1 , . . ., r } be a basis of G. Consider the matrix M whose columns are ϕ( j )(0), 1 ≤ j ≤ r.We say that ϕ satisfies a Diophantine condition for G if there exist β, c > 0 such that kM ≥ c |k| 1+β for all k ∈ Z q − {0}, I(k) = G (5.1) where x = inf{|x − | , ∈ Z r }.
Definition.An action ϕ : Z p → Affin(T q ) is Diophantine if 1. t A has only non-trivial isotropy groups and there exist only a finite number of them.2. ϕ satisfies a Diophantine condition for each isotropy group of the action t A. The cohomoly of Diophantine actions is given by 5.2 Theorem.(R.U.Luz, [9]).Let ϕ : Z p → Affin(T q ) be a Diophantine action.Then H j (Z p , C ∞ ϕ (T q )) = H j DR (T q ), 0 ≤ j ≤ p.
If ϕ acts by translations, we have 5.3 Theorem (J.L. Arraut and N.M. dos Santos, [2]).Let ϕ : Z p → Trans(T q ) be an action of Z p on the group Trans(T q ) of translations of T q .Then H j (Z p , C ∞ ϕ (T q )) = H j DR (T q ), 0 ≤ j ≤ p if and only if ϕ is Diophantine.where αis a Diophantine number and β / ∈ Q.Then (i) If β is Diophantine then ϕ is Diophantine and by Theorem 5.2.
On the cohomology of foliated bundles 195 ii) If β is Liouville then is a non-Hausdorff infinite dimensional space.We finish the section with the following oustanding problem.
5.5 Problem.Compute the cohomology group H 1 (Z, C ∞ ϕ (S 1 )) where the action of Z is generated by a diffeomorphism ϕ: S 1 → S 1 with irrational rotation number.
If ϕ is C ∞ -conjugate to a rotation, then the problem reduces to actions of Z → T ransl(S 1 ), where the answer is known.

2. 1
Theorem (de Rham's Theorem for foliated bundles).Let B and F be connected paracompact C ∞ manifolds and F → (M, F) π → B be the suspension foliated bundle of an action ϕ ϕ

Case 4 .
B is an open set of R p .Thus B is a countable union of open sets U i as in Case 1 with compact closures.By Cases 1 and 2, de Rham's theorem holds for finite unions n i=1 U i , by induction on n.
For a proof of this fact, see[12, Chap.VI.6].4.1 Example.We describe the cocycles and coboundaries in the particular case p = 3.Notice that dimΛ j (R p ) * = p j = p! j!(p − j)! .