L’INSTITUTFOURIER DE ANNALES

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AN N A L

E S

E D L ’IN IT ST T U

F O U R IE R

ANNALES

DE

L’INSTITUT FOURIER

Tomoo YOKOYAMA & Takashi TSUBOI

Codimension one minimal foliations and the fundamental groups of leaves Tome 58, n^o2 (2008), p. 723-731.

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CODIMENSION ONE MINIMAL FOLIATIONS AND THE FUNDAMENTAL GROUPS OF LEAVES

by Tomoo YOKOYAMA & Takashi TSUBOI (*)

Abstract. — LetFbe a transversely orientable transversely real-analytic codimension one minimal foliation of a paracompact manifoldM. We show that if the fundamental group of each leaf ofF is isomorphic toZ, thenF is without holonomy. We also show that ifπ2(M)∼= 0and the fundamental group of each leaf of F is isomorphic toZ^k(k∈Z_>0), thenF is without holonomy.

Résumé. — SoitFun feuilletage minimal de codimension un transversalement orientable, transversalement analytique réel sur une variétéM paracompacte. On démontre que le feuilletageFest sans holonomie si le groupe fondamental de toute la feuille deF est isomorphe àZ. On démontre aussi que le feuilletageF est sans holonomie si le groupe d’homotopieπ2(M)∼= 0et que le groupe fondamental de toute la feuille deFest isomorphe àZ^k(k∈Z_>0).

1. Introduction

Let F be a codimension one smooth foliation of a closed manifoldM. We study the relationship between the topology of the leaves ofM and the topology ofM. If each leaf of the foliationF is simply connected, then the foliationF is without holonomy. Then the classical result of Tischler [10]

says thatM fibers over the circleS¹, all leaves ofF are diffeomorphic to a covering space of the fiber, and all leaves are dense if the leaves are not compact.

We would like to know how is the foliation if the fundamental group of each leaf is isomorphic to an elementary group. In this direction, we show the following theorems. A foliation is said minimal if all leaves are dense.

Keywords:Foliations, real-analytic, holonomy, fundamental groups of leaves.

Math. classification:57R30, 53C12.

(*) The authors are partially supported by Grants-in-Aid for Scientific research 16204004, 17104001 and 18654008, Japan Society for Promotion of Science, Japan, and by the 21st Century COE Program at Graduate School of Mathematical Sciences, the University of Tokyo.

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724 Tomoo YOKOYAMA & Takashi TSUBOI

Theorem 1.1. — LetF be a transversely orientable transversely real- analytic codimension one minimal foliation of a paracompact manifoldM. Assume that the fundamental group of each leaf ofF is isomorphic toZ.

Then the foliationF is without holonomy.

Theorem 1.2. — LetF be a transversely orientable transversely real- analytic codimension one minimal foliation of a paracompact manifoldM. Assume thatπ2(M) ∼= 0 and the fundamental group of each leaf of F is isomorphic toZ^k(k∈Z_>0). Then the foliation F is without holonomy.

Remark 1.3. — For foliations of closed manifolds, if we assume that the end set of each leaf is homeomorphic to neither the Cantor set nor the empty set, then the assumption of minimality in Theorems 1.1 and 1.2 are satisfied by a result of Duminy (see [2]). For, if the foliation is not minimal and there are no compact leaves, then there is an exceptional minimal set and the end set of a semi-proper leaf of the exceptional minimal set is the Cantor set.

Remark 1.4. — The assumption of transverse real-analyticity in Theo- rems 1.1 and 1.2 is only used to guarantee that there are no null homotopic closed transverse curves for the foliations ([5]).

Thus for example, for a codimension one real-analytic foliation F of a closed manifoldM, if all leaves ofFare homeomorphic toS¹×R^k(k>2), then the foliation is without holonomy. In this case,M is anS¹-bundle over T^k+1. For, by the classical result of Novikov ([8]), the universal covering Mfis homeomorphic toR^k+1×R, whereR^k+1 is the universal covering of S¹×R^k, and π1(M)/i∗(π1(L)) is free abelian, wherei : L−→ M is the inclusion map of a leafL. Thus M is aspherical and homotopy equivalent to anS¹-bundle over T^k+1. Then by the result of Farrell-Jones [4], M is homeomorphic to theS¹-bundle overT^k+1.

In a similar way, for a codimension one real-analytic foliation F of a closed manifoldM with π2(M)∼= 0, if all leaves ofF are homeomorphic toT^k×R^` (k+`>3), then the foliation is without holonomy andM is a T^k-bundle overT^`+1.

We note that if the fundamental group of each leaf is isomorphic to a com- plicated group, it is no longer true that the foliation is without holonomy.

In fact, we give an example (Example 3.2) of foliation with nontrivial holonomy of closed 3-manifold whose leaves are diffeomorphic. Etienne Ghys told us that he also knew of this example.

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2. Proof of main theorems

To prove our main theorems, we introduce the notion of trivial fences andπ₁-carriers.

Definition 2.1 (Trivial fence). — Let F be a transversely oriented foliation of a manifoldM andT a foliation transverse toF. For a compact subsetKof a leafL₀ofFand forε >0, an embeddingF:K×[0, ε]−→ M is called a trivial positive fence ifF(K× {t})is contained in a leaf Lt of F,F|K× {0}is the inclusionK⊂L0andF({x} ×[0, ε])is an orientation preserving embedding to a leafτ_x ofT. A trivial negative fence F : K× [−ε,0] −→ M and a trivial two-sided fence F : K×[−ε, ε] −→ M are defined in a similar way.

Remark 2.2. — For an arcwise connected compact subsetKof a leafL, a trivial positive fence on K exists, if and only if the holonomy on the positive side along any loop inKis trivial. IfLis a leaf without holonomy, there is a trivial two-sided fence on any arcwise connected compact subset KofL.

Definition 2.3 (π1-carrier). — An arcwise connected subset K of a leafL ofF is called aπ₁-carrier for L ifπ₁(K)−→π₁(L)is surjective.

Now we give the proof of Theorem 1.1.

Proof of Theorem 1.1. — By a result of [3], the union of leaves without holonomy is a residual saturated subset ofM. LetLbe a leaf ofF without holonomy andK∼=S¹⊂L0be aπ1-carrier forL0. LetF :K×[0, ε]−→M be a trivial positive fence onKand put Ft(x) =F(x, t)

If the image of (F_t)_∗ : π₁(K) −→ π₁(L_t) is of finite index, then L_t is without holonomy. For, for a loopγ inLt, its powerγ^k is in(Ft)_∗(π1(K)) and the holonomy alongγ^k is trivial by the existence of the trivial fenceF. Since the holonomy group of a leaf of a codimension one foliation is torsion- free, the holonomy alongγ is trivial.

Assume that for a non empty open interval (δ0, δ1) ⊂ [0, ε] and any t∈ (δ0, δ1), (Ft)_∗(π1(K))is of finite index. Then F is without holonomy by the minimality ofF, i.e. the saturation ofF_t(K×(δ₀, δ₁))is the whole manifoldM.

Now, if (F_t)_∗(π₁(K)) is of infinite index for some t ∈ [0, ε], then that π1(Lt)∼=Zimplies that(Ft)_∗(π1(K)) = 0.

Thus ifFis with nontrivial holonomy, then{t∈[0, ε]

(F_t)_∗(π₁(K)) = 0}

is a dense subset of[0, ε].

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Assume that F is with nontrivial holonomy. Let p : Mf−→ M be the universal covering ofM. Let Fe =p^∗F be the pullback foliation ofMf. By the above argument, for a leafLwithout holonomy ofF and any liftLe of L, p|eL is a diffeomorphism and π1(eL) ∼=Z. For a leaf L with nontrivial holonomy, the real-analyticity of F implies that its lift Le is simply connected. For, otherwise,Leis a finite cover ofLandLe is still with nontrivial holonomy. However the argument of Haefliger [5] shows that transversely orientable real-analytic foliations on simply connected manifolds are without holonomy.

Thus the foliation Fe of Mf is without holonomy and Fe contains the residual subset formed by the non-simply connected lifts of leavesL of F without holonomy, and simply connected leaves formed by the lifts of leaves with nontrivial holonomy.

For a lift Lf₀ of a leaf L₀ of F without holonomy, p|fL₀ is a diffeomorphism, and the positive trivial fenceF :K×[0, ε]−→M lifts to a unique positive trivial fenceFe:Ke ×[0, ε]−→Mfwhere Ke = (p|fL0)⁻¹(K). Hence {t∈[0, ε]

(Fet)_∗(π1(K)) = 0}e is a dense subset of [0, ε].

The following lemma 2.4 shows that in this situation, the union of simply connected leaves is a residual subset ofMfand this completes the proof of

Theorem 1.1.

Lemma 2.4. — LetFebe a foliation without holonomy of a simply con- nected manifold Mf. Let p : Mf −→ R be a continuous submersion such that a leaf ofFeis a connected component ofp⁻¹(y)(y∈R). Assume that (∗) for any compact subset K of a leaf Le and any two-sided trivial fence F : K×[−ε, ε] −→ Mf, there exists t ∈ (−ε, ε) such that (F_t)_∗(π₁(K)) = 0⊂π₁(eL_t).

Then the union of simply connected leaves is a residual subset.

Proof. — First we take an increasing sequence of compact submanifolds Mi ofMfwith several nice properties.

For each pointx∈Mf, take a closed foliated product neighbourhoodU ∼= Dⁿ⁻¹×[−ε, ε]such thatDⁿ⁻¹× {∗}is on a leaf ofFeand{∗} ×[−ε, ε]is on a leaf of the transverse foliationTe. SinceMfis paracompact, it is covered by countably many such distinguished neighbourhoodsU_i∼=Dⁿ⁻¹_i ×[−εi, ε_i] (i∈Z_>0). PutMi= [

k6i

Dⁿ⁻¹_k ×[−εk, εk]. ThenMf=[

i

Mi. By modifying Mi, we may assume thatMiis a compact submanifolds ofMfwith boundary and codimension two corners. ∂M_i is a union of compact submanifolds of leaves and compact submanifolds transverse to Fe. Moreover, we may

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assume that, for any leafLe₀ of F,e Le₀∩M_i is a compact submanifolds of Le0.

We observe that if Lf₀ contain no components of tangential boundary ofMi, then there is a two-sided trivial fenceF : (Mi∩Le0)×[−ε, ε] −→

Mf (ε > 0) such that F((Mi∩Le0)× {t}) is contained in a leaf Le and p⁻¹(p(Im(F)))∩M_i = Im(F), where Im(F) is the image of the fence F. For a component(M_i∩Le₀)_j ofM_i∩Le₀, if Le₀ contains no components of tangential boundary ofMiand the induced mapπ1((Mi∩Le0)j)−→π1(eL0) is the zero map, then for sufficiently smallt in[−ε, ε],π1((Mi∩Let)j)−→

π₁(eL_t)is the zero map. The reason is as follows. Take a finite generating set ofπ1((Mi∩Le0)j). For a generator[γ]ofπ1((Mi∩Le0)j), there is a continuous map h: D² −→ Le₀ such that ∂h = γ. Then this hextends to bh : D²× [−ε⁰, ε⁰]−→Mfsuch that bh({∗} ×[−ε⁰, ε⁰])⊂τe_h(∗)and bh(D²× {t})⊂Let. Since π₁((M_i∩Le_t)_j) is generated by the loops [F_t(γ)] and these bound bh(D²× {t}),π1((Mi∩Let)j)−→π1(eLt)is the zero map.

By the assumption (∗), for the trivial two-sided fenceF, {t∈[−ε, ε]

π₁((M_i∩Le_t)_j)−→π₁(eL_t)is the zero map}

is a dense subset of[−ε, ε], and this is an open subset by the above argument.

Sincep⁻¹(p(Im(F)))∩Mi= Im(F), the number of connected components of Im(F) is finite and the number of the connected components of the tangential boundary ofMiis finite,

G⁰_i={s∈p(M_i)

π₁((M_i∩L)e _j)−→π₁(eL) is the zero map for any connected component (Mi∩L)e j ofMi∩Le for any leafLein p⁻¹(s)}

contains an open dense subset ofp(Mi).

Put Gi = (R\p(Mi))∪G⁰_i. ThenGi contains an open dense subset of R. ThusG=

∞

\

i=1

Gi is a residual subset of R. Now, fors∈G, any leaf of p⁻¹(s) is simply connected . Hence the union of simply connected leaves containsp⁻¹(G)which is a residual subset ofMf. Thus Theorem 1.1 is shown. The proof of Theorem 1.2 also uses the homomorphism(Ft)∗:π1(K)−→π1(Lt)for the trivial fenceF.

Proof of Theorem 1.2. — LetL₀ be a leaf of F without holonomy. Let Kbe aπ1-carrier forL0.K can be taken as a bouquet ofkcircles. LetF : K×[−ε, ε]−→M be a two-sided trivial fence. As in the proof of Theorem 1.1, if F is with nontrivial holonomy, then there are no non-empty open

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intervals(δ₀, δ₁)⊂[−ε, ε]such that the image of(F_t)_∗:π₁(K)−→π₁(L_t) (t∈(δ0, δ1)) is of finite index.

Thus there are s₀andu₀ such that−ε < s₀<0< u₀< ε, (F_s₀)_∗(π₁(K))⊂π₁(L_s₀)∼=Z^k

and

(Fu₀)_∗(π1(K))⊂π1(Lu₀)∼=Z^k

are of infinite index. ThusN_s⁰₀ = Ker((Fs₀)∗)⊂π1(L0)∼=Z^k andN_u⁰₀ = Ker((F_u₀)_∗)⊂π₁(L₀)∼=Z^k are nontrivial.

First we show thatN_s⁰₀∩N_u⁰₀ = 0.

Lemma 2.5. — N_s⁰

0∩N_u⁰

0 = 0.

Proof. — Assume that N_s⁰₀ ∩N_u⁰₀ is nontrivial. Let γ : S¹ −→ L0 be a loop representing a nontrivial element ofN_u⁰₀∩N_s⁰₀. Let (M ,f F)e be the pull-back foliation of the universal coveringMfofM. Since(F_u₀)_∗(γ) = 0,γ lifts to a mapγe:S¹−→Lf0⊂Mfto the universal cover. Sinceπ1(fL0)is free abelian and[eγ]∈π1(fL0)is nontrivial,[eγ]∈H1(fL0;Z)is nontrivial. Since F is transversely real-analytic, Fe is defined by a continuous submersion Mf−→RandMf\Lf0always have two connected components,Mf⁺andMf⁻. Since[γ]∈N_u⁰

0, there is a continuous maph₊:D²−→F(γ×[0, u0])∪L_u₀ such thatγ=∂h₊. h₊ lifts to a continuous maphc₊ :D²−→ Cl(fM⁺) = Mf⁺∪Lf0 such thateγ=∂hc+.

In a similar way, there are a continuous maph₋:D²−→F(γ×[s0,0])∪ L_s₀ such that γ=∂h₋ and its lift hc₋ :D²−→Cl(fM⁻) =Mf⁻∪Lf₀ such thateγ=∂hc₋.

By the Mayer-Vietoris sequence for Mf = Cl(Mf⁻)∪Cl(fM⁺), we see thatγeinduces a nontrivial element of H2(Mf;Z)and this contradicts the assumption that0 =π₂(M)∼=π₂(Mf)∼=H₂(Mf;Z). Thus N_s⁰

0∩N_u⁰

0 = 0.

We continue the proof of Theorem 1.2.

Put t0= 0, K^t⁰ =K andF_t^t⁰ =Ft:K −→Lt. We have, s0 < t0 < u0

andN_s^t₀⁰∩N_u^t⁰₀ = 0. Note that for0 =t0< t6u0,(F_t^t⁰)_∗|N_s^t⁰₀ is injective.

For otherwise,N_s^t⁰₀∩N_t^t⁰= 0, contradicting Lemma 2.5

LetLt₁ (t0< t1< u0) be a leaf without holonomy. Take aπ1-carrierK^t¹ forLt1 containingF_t^t₁⁰(K^t⁰). Then we have a two-sided trivial fence F^t¹ : K^t¹×[t1−ε1, t1+ε1]−→M. For this fence, by the same argument as before, we can finds₁andu₁such thatt₀< s₁< t₁< u₁< u₀andN_s^t¹

1∩N_u^t¹

1= 0, where F_t^t¹ : K^t¹ −→ L_t and N_t^t¹ = Ker((F_t^t¹)_∗ : π₁(K^t¹) −→ π₁(L_t)) (t=s1, u1) are nontrivial.

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Now note thatN_s^t¹

1∩(F_t^t⁰

1)_∗(N_s^t⁰

0) = 0. For otherwise, there is a loopγin K^t⁰ such thatF_t^t⁰

1 ◦γ represents a nontrivial element inN_s^t¹

1∩(F_t^t⁰

1)_∗(N_s^t⁰

0) and[γ]∈N_s^t₀⁰. Since fors16t6u1,F_t^t⁰(K^t⁰)⊂F_t^t¹(K^t¹),F_s^t₁¹◦F_t^t₁⁰◦γ= F_s^t⁰

1◦γand this represents0. This contradicts the injectivity of(F_s^t⁰

1)_∗|N_s^t₀⁰. PutA^t⁰ =N_s^t⁰

0 andA^t¹=N_s^t¹

1⊕(F_t^t₁⁰)_∗(A^t⁰). Thenrk(A^t¹)>rk(A^t⁰)>0.

Note that for[γ]∈A^t¹, we have a description of the homotopy (F_s^t⁰

0)_∗((F_s^t⁰

1)_∗|N_s^t⁰

0)⁻¹(F_s^t¹

1)_∗[γ] = 0.

Note also that fort₁ < t6u₁, (F_t^t¹)_∗|A^t¹ is injective and A^t¹ ∩N_u^t¹

1 = 0.

For otherwise, the argument of Lemma 2.5 gives rise to a nontrivial element inπ2(M).

For s1 < t1 < u1, we take Lt2 (t1 < t2 < u1) without holonomy. We take a π1-carrier K^t² for Lt₂ containing F_t^t₂¹(K^t¹). Then, using a two- sided trivial fence F_t^t², we find s2 and u2 (t1 < s2 < t2 < u2 < u1) such that N_s^t²

2 ∩N_u^t²

2 = 0, where N_t^t² = ker(F_t^t²)_∗ : π₁(K^t²)−→ π₁(L_t) (t =s2, u2) are nontrivial. Since for s2 6 t 6 u2, F_t^t¹(K^t¹) ⊂F_t^t²(K^t²), N_s^t²

2 ∩(F_t^t¹

2)_∗(A^t¹) = 0. Put A^t² = N_s^t²

2 ⊕(F_t^t¹

2)_∗(A^t¹). Then, rk(A^t²) >

rk(A^t¹). Fort2< t 6u2, (F_t^t²)_∗|A^t² is injective andA^t²∩N_u^t²₂ = 0. Note that, for[γ]∈A^t²,

(F_s^t⁰

0)_∗((F_s^t⁰

1)_∗|A^t⁰)⁻¹(F_s^t¹

1)_∗((F_s^t¹

2)_∗|A^t¹)⁻¹(F_s^t²

2)_∗[γ] = 0.

Note also that fort₂ < t6u₂, (F_t^t²)_∗|A^t² is injective and A^t² ∩N_u^t²

2 = 0.

For otherwise, the argument of Lemma 2.5 again gives rise to a nontrivial element inπ2(M).

We repeat this construction and we obtain π1-carriers K^tⁱ ⊂ Lti and two-sided trivial fencesF_t^tⁱ, nontrivial subgroupsN_s^tⁱ_i,N_u^tⁱ_i ofπ1(K^tⁱ), and subgroupsA^tⁱ⊂π1(Lt_i)(i= 1,2, . . . ). HereN_t^tⁱ = ker(F_t^tⁱ)∗:π1(K^tⁱ)−→

π₁(L_t). They satisfy

s0< t0< s1< t1< s2< t2<· · ·< si< ti

< ui< u_i−1<· · ·< u2< u1< u0

and

A^tⁱ =N_s^t_iⁱ⊕(F_t^t_iⁱ⁻¹)∗(A^tⁱ⁻¹).

SincerkA^tⁱ >rkA^tⁱ⁻¹ >· · ·>rkA^t⁰ >0,rkA^tⁱ >i.

Here if A^tⁱ ∩N_u^tⁱ

i 6= 0, we find a nontrivial element in π₂(M). For, let γ∈A^tⁱ∩N_u^tⁱ_i be a loop representing nontrivial element. Since [γ]∈N_u^tⁱ_i, γ lifts to the universal covering, eγ : S¹ −→ Le_t_i ⊂Mf. By the argument similar to that in Lemma 2.5,γebounds a disk inMf_t⁺_i∪Let_i, whereMf_t⁺_i and

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Mf_t⁻

i are the components ofMf\Le_t_i. Since[γ]∈A^tⁱ, (F_s^t⁰₀)_∗((F_s^t₁⁰)_∗|A^t⁰)⁻¹(F_s^t₁¹)_∗((F_s^t₂¹)_∗|A^t¹)⁻¹· · ·(F_s^t_i−2ⁱ⁻²)_∗

((F_s^t_i−1ⁱ⁻²)∗|A^tⁱ⁻²)⁻¹(F_s^t_i−1ⁱ⁻¹)∗((F_s^tⁱ⁻¹_i )∗|A^tⁱ⁻¹)⁻¹(F_s^t_iⁱ)∗[γ] = 0.

Henceeγbounds a disk inMf_t⁻_i∪Let_i. Thus as before we see thatH2(Mf;Z)6= 0 and we find a nontrivial element inπ2(M). This contradicts the assumption that0 =π₂(M)∼=π₂(Mf)∼=H₂(Mf;Z).

Now fori > k, the fact this contradicts the fact thatπ1(L)∼=Z^k.

3. Examples

The first example shows that the assumption on the minimality of foliations in Theorems 1.1 and 1.2 are necessary.

Example 3.1. — Consider the compact manifold M =S¹×(S¹×S^m\IntD^1+m),

where m > 1 and D^1+m is an embedded disk in S¹×S^m. It is easy to construct a foliationF ofM tangent to the boundaryS¹×S^mand whose interior leaves are homeomorphic toS¹×S^m\ {∗}. Consider the double (DM, DF) which can be constructed as a real-analytic foliation and the compact leaf has nontrivial holonomy. It is easy to see that the fundamental group of each leaf is isomorphic to Z. By taking the direct product with T^k−1 we obtain an example with the the fundamental group of each leaf being isomorphic toZ^k.

Now we construct a codimension one transversely orientable minimal foliation with nontrivial holonomy on a closed manifold whose leaves are diffeomorphic. The construction is a modification of Hirsch’s example [6].

Example 3.2. — Let Σ be the surface obtained from the 2-torus by removing 3 disks. Let∂₁, ∂₂, ∂₃ be the boundary components of Σ. Let F be the product foliation ofΣ×[0,1](i.e.,F ={Σ× {x}}_x∈[0,1]). Let f : Σ−→Σbe a diffeomorphism such thatf(∂1) =∂1, f(∂2) =∂3, f(∂3) =∂2

and letM be the manifold obtained from Σ×[0,1]by identifying (x,1) with(f(x),0). Then the boundary ofM is diffeomorphic to a disjoint union of two tori T₁² (containing ∂₁ × {0}) and T₂² (containing ∂₂× {0} and

∂3 × {0}). Using the natural projection to [0,1]/0 ∼ 1 = R/Z, T₁² is identified with∂₁×R/Z and T₂² is identified with ∂₂×R/2Z. We have the induced foliation F⁰ of M and we see thatF⁰|T₁² ={∂1× {t}}_t∈R/Z

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and F⁰|T₂² = {∂2× {t}}_t∈R/2Z. Let g : T₁² −→ T₂² be a diffeomorphism sending∂1× {t} to ∂2× {2t}. LetMg be the manifold obtained from M by the identification byg. We obtain the induced real-analytic foliationF⁰⁰ of Mg which is minimal. It is easy to see that there are countably many leaves with nontrivial holonomy. Each leaf ofF⁰⁰ is an orientable surface of infinite genus and its end set is the Cantor set consisting of non planar ends. By the classifying theorem ([7], [9]) for non-compact surfaces (see also [1]), all leaves are diffeomorphic.

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[10] D. Tischler, “On fibering certain foliated manifolds overS¹”,Topology 9(1970), p. 153-154.

Manuscrit reçu le 17 janvier 2007, accepté le 10 mai 2007.

Tomoo YOKOYAMA & Takashi TSUBOI The University of Tokyo

Graduate School of Mathematical Sciences Komaba Meguro, Tokyo 153-8914, Japan [email protected]

[email protected]