François Laudenbach
email: francois.laudenbach@univ-nantes.fr

Gaël Meigniez
email: gael.meigniez@univ-ubs.fr

Haefliger's codimension-one singular foliations, open books and twisted open books in dimension 3

Keywords: 2000 Mathematics Subject Classification. 57R30 Foliations, Haefliger's Γ-structures, open book

We consider singular foliations of codimension one on 3-manifolds, in the sense defined by André Haefliger as being Γ 1 -structures. We prove that under the obvious linear embedding condition, they are Γ 1 -homotopic to a regular foliation carried by an open book or a twisted open book. The latter concept is introduced for this aim. Our result holds true in every regularity C r , r ≥ 1. In particular, in dimension 3, this gives a very simple proof of Thurston's 1976 regularization theorem without using Mather's homology equivalence.

Introduction

In this paper we prove a regularization theorem for singular foliations of codimension one on closed 3-manifolds. They will even be homotopic to some specific regular models. Let us recall from [START_REF] Haefliger | Homotopy and integrability[END_REF] that, on a manifold M, a Haefliger structure ξ of codimension one, or a Γ 1 -structure, consists of:

-a line bundle ν = (E(ν) → M), that is, a real vector bundle of rank 1, called the bundle normal to ξ; -in the total space E(ν), a germ, still noted ξ, of codimension-one foliation along the zero section i of ν, transverse to the fibers. The Γ 1 -structure ξ is said to be regular when the foliation ξ is also transverse to the zero section i. Then the trace ξ ∩ i(M) is a genuine foliation of M. If not, this trace is a singular foliation.

A homotopy of ξ is defined as a Γ 1 -structure on M × [0, 1] inducing ξ on M × {0}. A regularization of ξ is a homotopy to a regular Γ 1 -structure. It does not exist in general. An obvious necessary condition is that ν must embed into the tangent bundle τ M. When ν is trivial and dim M = 3 this condition is fulfilled.

In what follows, the manifold M will be C ∞ and ξ will be a Γ r 1 -structure (1 ≤ r ≤ ∞), meaning that ξ viewed as a foliation is tangentially C ∞ and transversely C r , that is, the foliation charts are C r in the direction transverse to the leaves.

Of course, "carried by a (twisted) open book decomposition" needs explanation. But we first comment on the regularization aspect only.

Under this aspect, our result is a particular case of a general regularization theorem due to W. Thurston (see [START_REF] Thurston | Existence of codimension-one foliations[END_REF]). Thurston's proof was based on a deep result due to J. Mather [START_REF] Mather | On Haefliger's classifying space I[END_REF][START_REF] Mather | Integrability in codimension 1[END_REF], namely, the homology isomorphism between on the one hand the classifying space of the group Diff c (R) endowed with the discrete topology, and on the other hand the loop space ΩB(Γ 1 ) + .

We present here a proof of the above regularization theorem which does not depend on the latter result. After the present research was achieved, a regularization theorem in all dimensions, still avoiding any difficult result, was provided in [START_REF] Meigniez | Regularization and minimization of Γ 1 -structures[END_REF], without the open book models that we get here in dimension 3.

Our proof provides model foliations in all homotopy classes of Γ 1 -structure. In this introduction, to make short, we explain the model in the co-orientable case: that is, ν is trivial.

In this case, the models are based on the notion of open book decomposition. Recall that such a structure on M consists of a link B in M, called the binding, and a fibration p : M B → S 1 such that, for every θ ∈ S 1 , the fiber p -1 (θ) is the interior of an embedded surface, called the page P θ , whose boundary is the binding. The existence of an open book decomposition could have been proved by J. Alexander when M is orientable, as a consequence of [START_REF] Alexander | Note on Riemann spaces[END_REF] (every orientable closed 3-manifold is a branched cover of the 3-sphere) and [START_REF] Alexander | A lemma on systems of knotted curves[END_REF] (every link can be braided); but the concept was introduced by H. Winkelnkemper in 1973 [START_REF] Winkelnkemper | Manifolds as open book[END_REF]. Henceforth, we refer to the more flexible construction by E. Giroux, which includes the case when M is non-orientable (see section 3).

It is well-known that every open book B gives rise to a foliation F B , as follows. The pages endow B with a normal framing. So, a tubular neighborhood N(B) of B is trivialized: N(B) ∼ = B × D 2 . Out of N(B) the leaves of F B are the pages modified by spiraling around N(B); some neighborhood of ∂N(B) in N(B) is a union of compact leaves; and the rest of N(B) is foliated by a Reeb component.

This foliation of N(B) will be called a thick Reeb component and it is introduced for technical reasons in the homotopy argument of section 5. This technical point could be avoided by using a theorem of F. Sergeraert [START_REF] Sergeraert | Feuilletages et difféomorphismes infiniment tangents à l'identité[END_REF] on Reeb components. We call such a foliation an open book foliation.

The latter foliation can be modified by inserting a so-called suspension foliation. Precisely, let K be a compact subsurface in some leaf of F B out of N(B), and let K ×[-ε, +ε] be a foliated thickening of it: each When working with a Γ 1 -structure ξ whose normal bundle is twisted, it is necessary to introduce the notion of twisted open book, which is inspired by work of E. Giroux and will be explained in section 3. It is worth noticing that the existence of twisted open book decompositions on M with a given normal bundle ν follows from our regularization theorem applied to the so-called twisted trivial Γ 1 -structure with normal bundle ν (see definition 2.2 and theorem 1.2).

K × {t} is contained in a leaf of F B . Let ϕ : π 1 (K) → Diff c (] -ε, +ε[)
The paper is organized as follows (see the text for definitions). In section 2, the given Γ r 1 -structure ξ is modified by a C 0 -small homotopy to make it Morse (that is, with Morse singularities), and to give it a pseudo-gradient X whose dynamics is trivial. We deduce a decomposition of M of the form M = N(Σ) ∪ H, where N(Σ) is a tubular neighborhood of an embedded closed surface Σ which is transverse to X, and where H is a handlebody with one or two connected components. Moreover, Σ is Poincaré-dual to w 1 (ν), the first Stiefel-Whitney class of ν.

In section 3, we build an open book B (possibly twisted) which is somewhat adapted to Σ: the surface Σ minus a disk is contained in a page, and the bundle normal to B is isomorphic to ν. In passing, we obtain the following theorem: Theorem 1.2. Let ν be a line bundle over M which embeds into the tangent space τ M. Then there exists a twisted (or non-twisted) open book decomposition B of M whose normal bundle is isomorphic to ν. It is twisted whenever ν is so.

In section 4, we apply to ξ a new homotopy to put it into plateau form, meaning that it is trivial in H, and in the interior of N(Σ), it is transverse to X.

In section 5 a final homotopy puts ξ into the desired form F B,ϕ carried by B.

Many arguments of this paper become significatively simpler when ν is trivial. The reader who is mainly interested in this case may refer to [START_REF] Laudenbach | Regularization of Γ 1 -structures in dimension 3[END_REF]. Also, the reader interested in prescribing the homotopy class of the plane field tangent to the resulting foliation is refered to the same preprint.

We are very grateful to Vincent Colin, Étienne Ghys and Emmanuel Giroux for their comments, suggestions and explanations.

Morsification of the singularities and dynamics of a pseudo-gradient

For proving theorem 1.1, the setting is a closed 3-manifold M endowed with a Γ r 1 -structure ξ, with r ≥ 1. Nevertheless, the next proposition 2.1 holds true in any dimension.

Denote by ν the normal bundle to ξ, by E(ν) its the total space, and by i : M → E(ν) the zero section. Regard ξ as a foliation on some neighborhood of i(M) in E(ν).

A point x ∈ M is said to be a singularity of ξ when the zero section i is not transverse to ξ at x. One says that ξ is Morse when at each singularity the contact of i with ξ is quadratic non-degenerate in some foliated chart where i is C 2 . In that case, the trace ξ ∩ i(M) is a singular foliation which is locally defined by a Morse function.

For every section s of ν of class C r which is C 0 -close to i, the pullback s * ξ is a Γ r 1 -structure on M homotopic to ξ. Such a homotopy of ξ is said to be a C 0 -small homotopy.

Proposition 2.1. After a C 0 -small homotopy, ξ is Morse.

Proof. Of course, in case r ≥ 2 this proposition is obvious: any C r -generic choice of s in a small enough C 0 -neighborhood of i works. In case r = 1 one has the following method which works in any dimension of M.

At the beginning the zero section is covered by finitely many boxes bi-foliated with respect to ξ and to the R-fibers. These boxes form a C r atlas. In each chart, the zero section becomes the graph of a real function of class C r . We choose a triangulation T r of M = i(M) so fine that each simplex lies in such a box. Then, for each simplex σ in T r we choose a bi-foliated neighborhood V σ such that when τ is a face of σ we have V τ ⊂ V σ . We need the two following lemmas.

W ⊂ int V ⊂ V ⊂ U
Then there exists a real function g : D k → R such that:

g coincides with f on W and has no critical point in V ; g is C ∞ on the complement of V ; g is a Morse function (which makes sense since g has no critical points in V ).

Proof. Let X be a continuous vector field on U such that X • f > 0. Let α be some smooth bump function on D k which equals 1 on W and has support in V . Let c > 0 be less than the minimum of X • f on V . Let m > 0 be the maximum of |X • α| on V . We choose a C ∞ function h : D k → R whose restriction to V is very close to f in the C 1 topology. More precisely, we require that the following inequalities hold on V :

-

X • h > c -m|f -h| < c 2 .
The function h which has no critical points on V may be chosen to be a Morse function on D k . Then g := αf + (1 -α)h has the wanted properties. Indeed, at every point of V ,

X • g = αX • f + (1 -α)X • h + (X • α)(f -h)
which is larger than c/2. -g|D k = g; g coincides with f in a neighborhood of ∂D k ; g is a Morse function and its critical points are those of g.

Proof.

Near each critical point of g, the extension g is constructed by adding a non-degenerate quadratic form in the coordinates of a transversal to D k . Hence, the last condition is satisfied. Away from the critical points we have to solve the following extension problem: we are given a submanifold N and a function g : N → R without critical points. We have to extend g to a neighborhood of N, the germ of the extension being already given near ∂N. It is easy by using the projection p : V → N of some tubular neighborhood of N in M, well chosen near ∂N, and setting g = g • p.

We now return to the proof of proposition 2.1. It is done by induction on the skeleta of T r. Assume we already have a section s with the following property: for each simplex τ of dimension less than k, the restriction (s * ξ)|τ is Morse, and there is a neighborhood U of the (k -1)-skeleton such that s * ξ is non-singular on U T r [k-1] . If σ is a k-simplex, s|σ viewed in the foliated chart V σ is the graph of a function f to which it is allowable to apply successively the two preceding lemmas. It is clear from the construction that the new section is C 0 -close to the initial zero section i.

For example, the Morsification applies to the twisted trivial Γ 1 -structure in the following sense.

Definition 2.2. Let ν be a line bundle over M, whose first Stiefel-Whitney class is viewed as a representation w 1 (ν) : π 1 (M) → Z/2 ⊂ GL 1 (R). The twisted trivial Γ 1 -structure ξ 0 with normal bundle ν is the foliation of E(ν) which is the suspension of w 1 (ν).

Then, all points of M are singular. In other words, the zero section is a leaf. For every loop γ in this leaf, its holonomy is non-trivial if and only if ν is a twisted bundle over γ.

In what follows, ξ is assumed to be Morse. We are interested in the dynamics of a so-called pseudo-gradient of ξ, defined as follows. A twisted vector field X on M is a C ∞ section X of Hom(ν, τ M). Then, the sign of X • ξ is well-defined at each point of M = i(M) where X is transverse to ξ. Definition 2.3. A pseudo-gradient for ξ is a twisted vector field X on M such that the Lyapunov inequality X • ξ < 0 holds everywhere but the singularities.

As ν is possibly non-orientable, one cannot distinguish the index of a singularity from its coindex; but one can distinguish saddle points from center points thanks to their phase portraits. >From each saddle point s start two separatrices, that is, isolated orbits of X that begin at s. Proposition 2.4. After some C 0 -small homotopy, ξ is still Morse and admits a pseudo-gradient X whose dynamics has no recurrence (that is, every orbit has a finite length) and no separatrix joins two saddle points.

Proof. The existence of a smooth pseudo-gradient is easy to prove even if ξ is C 1 only. Indeed, near the singularities, there are smooth charts of the singular foliation and the usual negative gradient in Morse coordinates is convenient. Away from them, the Lyapunov inequality allows one to approximate a C 0 gradient by a smooth pseudo-gradient. Let X 0 be such a pseudogradient; its dynamics is not controlled.

Finitely many mutually disjoint open 2-disks, d 1 , . . . , d N , are chosen in regular leaves of the trace ξ ∩ i(M) such that every orbit of X 0 crosses at least one shrinked disk 1 2 d k , for some k ∈ 1, . . . , N. Following Wilson's idea [START_REF] Wilson | On the minimal sets of non-singular vector fields[END_REF], the zero section i of ξ and X 0 are changed in a neighborhood D 2 × [-1, +1] of each disk into a plug such that every orbit of the modified pseudo-gradient X is trapped by one of the plugs. Here are a few more details.

In these neighborhoods D 2 × [-1, +1], whose last coordinate is t, the foliation ξ ∩ i(M) is defined by the function t. By a C 0 -small homotopy supported in D 2 × [-1, 0], we just change the zero section i to a section s such that the singular foliation ξ ∩ s(M) is made of a cancelling pair of singularities, center-saddle. More precisely, it is defined by a Morse function f which coincides with t near the boundary and which has a pair of critical points of indices [START_REF] Alexander | A lemma on systems of knotted curves[END_REF][START_REF] Geiges | An Introduction to Contact Topology[END_REF]. In these cylinders, the new pseudo-gradient X is the gradient of f for the flat metric. By using D 2 × [0, 1] in a convenient way, it is easy to make the plug have the mirror symmetry with respect to D 2 × {0}: any orbit of X entering the cylinder at p × {-1} is trapped if |p| ≤ 3 4 or gets out the cylinder at p × {+1}. In the cylinder D 2 × [0, 1], the singular foliation ξ ∩ s(M) is defined by a Morse function with a pair of critical points of indices (0, 1).

Thanks to the mirror symmetry, very much like in Wilson's paper, the new twisted vector field X has a finiteness property: Each orbit of X has a finite length, and connects two singularities.

Let us give a proof. We consider any half orbit λ of X, starting at some point x outside the plugs. We have to prove that the ω-limit set ω(λ) is a singularity of X. Let λ 0 be the half orbit of X 0 through x, with the same germ as λ at x. If λ is trapped by some plug, that is, if it ends at some singularity in this plug, then we are done. So, we may assume that λ is not trapped by any plug. Due to the mirror symmetry, as recalled before, whenever λ enters a plug at p×{∓1}, p ∈ D 2 , it gets out at p × {±1}. In particular, outside the plugs, we have λ = λ 0 . It remains to show that ω(λ 0 ) is a singularity of X 0 . Indeed, if not, ω(λ 0 ) would contain a whole orbit of X 0 , thus would intersect some disk 1 2 d k . Thus λ would be trapped by the corresponding plug, a contradiction.

Unfortunately, the mirror symmetry in the plugs implies that X has some separatrices connecting two saddle points. Nevertheless, one can destroy these connecting separatrices by a perturbation of X. We claim that the finiteness property is preserved when the perturbation is small enough.

By contradiction, assume that X is the C 0 -limit of some sequence (X n ) of twisted vector fields such that each X n has an orbit segment λ n of length n. Then the Hausdorff accumulation set of (λ n ) in M for n → +∞ would contain either a half orbit of X of infinite length, which does not exist by the finiteness property; or a half infinite (or periodic) broken orbit Λ of X: that is, Λ is an infinite (or periodic) sequence of orbits connecting successive saddle singularities. Then, if Λ enters a plug at p × {∓1} it must get out at p × {±1}. Just as in the above proof of the finiteness property for X, we conclude that there is some half infinite (or periodic) broken orbit Λ 0 of X 0 such that Λ = Λ 0 outside the plugs.

Let λ 0 be one of the orbits of X 0 which are contained in Λ 0 . Then λ 0 meets some disk 1 2 d k . Thus, Λ is trapped by the corresponding plug and ends at the corresponding center singularity. Since Λ is contained in the accumulation set of (λ n ), infinitely many λ n 's are trapped in the same way as λ 0 and their lengths arer bounded, a contradiction. >From the pseudo-gradient X of ξ given by proposition 2.4 we deduce the following topological objects. Let G be the topological closure in M of the separatrices of all saddle points.

It is a graph whose vertices (resp. midpoints of the edges) are the center (resp. saddle) singularities of ξ. Thus G admits an arbitrarily small tubular neighborhood H whose boundary is transverse to X. Let M be the complement of int H in M. Since each orbit of X from a point of ∂H = ∂ M has a finite length, it must return to the boundary. Therefore, M is fibered over a surface Σ, ρ : M → Σ, the fibers being intervals ( ∼ = [-1, 1]) tangent to X. By taking a section we think of Σ as a closed surface embedded in M H and M becomes a tubular neighbohood N(Σ) of Σ in M. By construction of X, the normal bundle to Σ in M is ν|Σ. Proposition 2.5.

1)

The line bundle ν|G is orientable and, for a suitable choice of the orientation, X enters H along ∂H.

2) When ν is orientable, G has two connected components and Σ is two-sided.

3) When ν is non-orientable, G is connected and Σ is one-sided. 4) If ν embeds into the tangent bundle τ M, the Euler characteristic of H is even.

Proof. 1) We orient each separatrix from its saddle end point to its center end point. Since the separatrices are transverse to ξ, this is an orientation of ν|G over the complement of the singularities. It is easily checked that this orientation extends over the singularities. Thus X becomes a usual vector field near G transverse to ∂H.

2) When ν is a trivial bundle, by the isomorphism above-mentioned, ν(Σ, M) is trivial and Σ is two-sided. Thus, ∂ M has two connected components and G also does.

3) Assume Σ be two-sided. Then M ∼ = [-1, 1] × Σ and G has two connected components since each connected component of H has a connected boundary. In that case, ν is orientable since there are no arcs in H from Σ × {+1} to Σ × {-1}. 4) By assumption, the pseudo-gradient X is homotopic to a non-vanishing one. Then the number of zeroes of X is even.

As a conclusion of this section, we shall be able to continue the proof of theorem 1.1 with a ξ having a pseudo-gradient X for which the following statement holds true.

Corollary 2.6. The above decomposition M = N(Σ) ∪ H has the following properties:

1) The handlebody H has two or one components depending on that ν is trivial or not.

2) The restricted bundle ν|H is trivial; equivalently, Σ is Poincaré dual to w 1 (ν). For a convenient orientation, X enters H along ∂H.

3) The fibres of N(Σ) → Σ are contained in orbits of the pseudo-gradient X.

Open books, twisted open books

E. Giroux explained to us a Morse theoretical approach to open book decomposition, which is based on [START_REF] Giroux | Convexité en topologie de contact[END_REF] and is recalled below, up to a change of terminology. An easy adaptation allows us to handle similarly the twisted open book decompositions. Definition 3.1. Let W be a compact 3-manifold and f : W → R be a Morse function which is constant and maximal on ∂W . A compact surface S properly embedded in W (that is, ∂S ⊂ ∂W ) will be called a Giroux surface for f when f |S is a Morse function having the same critical points and the same isolated local extrema as f .

The saddle points of f |S may have index 1 or 2 for f . The property for f of having a Giroux surface is kept when deforming f among the Morse functions, even when crossing the critical values; for instance, through a deformation ending at a self-indexing Morse function (the value of a critical point is its Morse index in W ). This definition is made for W ⊂ M; of course, we are mainly interested in the case when W is closed (W = M).

Theorem III.2.7 in Giroux's article [START_REF] Giroux | Convexité en topologie de contact[END_REF] states the following with different words:

Theorem 3.2.
Let M be a closed 3-manifold (orientable or not). There exist a self-indexing Morse function f : M → R and a surface S which is a 2-sided Giroux surface in M for f .

In that case, S is separating (see below) and this data immediatly gives rise to an open book decomposition B in the sense that is recalled in the introduction. Indeed, let N be the level set f -1 (3/2). The smooth curve B := N ∩S will be the binding of the open book decomposition we are looking for. It can be proved that the following holds for every regular value a, 0 < a ≤ 3/2 :

-the level set f -1 (a) is the union along their common boundaries of two surfaces, N a 1 and N a 2 , each one being diffeomorphic to the sub-level surface S a := S ∩ f -1 ([0, a]); -the sub-level set M a := f -1 ([0, a]) is divided by S a into two parts P a 1 and P a 2 which are isomorphic handlebodies (with corners); -S a is isotopic to N a i through P a i , for i = 1, 2, by an isotopy fixing its boundary curve S a ∩ f -1 (a). This claim is obvious when a is small and the property is preserved when crossing the critical level 1. In this way the handlebody (1) The union of each page with the binding is a compact (topological) surface.

H -:= f -1 ([0, 3/2]) is divided by S 3/2 into two diffeomor- phic parts P 3/2 i , i = 1,
( a regular cover. The property (3) above forces this cover to be of degree one. In the twisted case, property (3) above forces that each meridian meets each regular leaf in two points only.

So, in the twisted case, p is a singular fibration (or Seifert fibration) from M B onto [-1, +1] which has two one-sided exceptional surface fibers p -1 (±1), and which is a proper smooth submersion over the open interval. A non-exceptional page is a 2-fold covering of any exceptional page p -1 (±1); notice that this covering is trivial over a collar neighborhood of B in the considered exceptional page. The union of an exceptional page with the binding B is a smooth surface with boundary. But the union of a non-exceptional page with B is a closed surface showing (in general) an angle along B.

Notice that, since B is co-orientable, a twisted open book B gives rise to a smooth foliation F B where each component of the binding is replaced with a Reeb component, the pages spiraling around it. Remark 3.5. We may choose freely the external holonomy of this Reeb component, that is, the germ by which the pages are rolled up around the binding, among the germs of diffeomorphisms of (R, 0) which are the identity on one side and without fixed point on the other side. For the needs of the future homotopy argument at the end of our proof of theorem 1.1 (see section 5), it is essential that this germ ψ be chosen as a product of commutators. Of course, this assumption is known as being easily satisfied: for example, in Af f (R) the unit translation is a commutator; a classical conjugation yields ψ as above (see e.g. [START_REF] Meigniez | A compactly generated pseudogroup which is not realizable[END_REF], section 3). Moreover, according to Herman's theorem [START_REF] Herman | Simplicité du groupe des difféomorphismes de classe C ∞ , isotopes à l'identité, du tore de dimension n[END_REF][START_REF] Herman | Sur le groupe des difféomorphismes du tore[END_REF], it is always satisfied; but we want not to use this difficult result.

The normal bundle to B, noted ν(B), is defined as being the normal bundle ν(F B ); it is well defined up to isomorphism and is not thought of as a sub-bundle of τ M, though it embeds into the tangent bundle τ M. Here is the simplest example of a twisted open book.

Example 3.6. Here, M = S 1 × S 2 is thought of as the double of the solid torus S 1 × D 2 . The binding is the linear curve B = (2, 1) drawn on the flat separating torus T 2 . One exceptional fiber is the Möbius band M in the solid torus, and its boundary is ∂M = B. The other exceptional fiber is its mirror copy. Notice that if T 2 is cut along B, one gets an annulus A which doubly covers M when projecting along the normals to M. This covering is trivial over a collar neighborhood of ∂M. Proposition 3.7. Let f : M → R be a self-indexing (or ordered) Morse function and S be a one-sided Giroux surface for f . The following middle condition is assumed:

(MC) The curve B := f -1 ( 3 2 ) ∩ S is two-sided in its level set f -1 ( 3 2
). Then there is a twisted open book B whose binding is the curve B and such that S B is the union of the two exceptional pages.

Proof. Observe first that S 3/2 := S ∩ f -1 ([0, 3 2 ]) is one-sided. Indeed, if a 0 and a 1 are two points in a tubular neighborhood N(S 3/2 ) which do not lie on S, they are joined by an arc α in the complement of S as S is one-sided. By using the descending gradient lines of f , we can push α into N(S 3/2 ) by an isotopy fixing the end points.

Knowing that the normal bundle to S 3/2 is twisted, it is possible to construct a piece of twisted open book with binding B inside N(S 3/2 ). The gradient of f allows one to extend the open book structure on f -1 ([0, 3 2 ]) so that the complement of B in the level set f -1 ( 3 2 ) is a page. A similar construction in the upper part f -1 ([ 3 2 , 3]) ends to build B.

In general such a one-sided Giroux surface (or a twisted open book) does not exist in M; the obstruction lies in the existence of a twisted line subbundle of τ M (compare theorem 1.2). Right now, we continue the proof of theorem 1.1 starting from the setting which has been stated in corolllary 2.6.

We have a decomposition M = N(Σ) ∪ H where Σ is a closed surface, N(Σ) is a I-bundle over Σ, whose projection is ρ : N(Σ) → Σ, and H is a handlebody with one or two components. Let d 0 be a small 2-disk in Σ; set Σ ′ the closure of Σ d 0 and M ′ := ρ -1 (Σ ′ ). Consider the handlebody H ′ := H ∪ ρ -1 (d 0 ). We have a new decomposition M = M ′ ∪ H ′ . We also recall the line bundle ν which is normal to the Γ 1 -structure ξ under consideration; by assumption it embeds into τ M. Proposition 3.8. 1) When ν is trivial, there exists an open book B such that Σ ′ is contained in one page.

2) When ν is twisted, there exists a twisted open book B such that Σ ′ is contained in an exceptional page and the normal bundle ν(B) is isomorpic to ν.

The first part is due to E. Giroux [5].

Proof. For beginning with, assume that ν is twisted. The case when ν is trivial admits a similar treatment, with a few modifications which will be specified in the end. According to proposition 2.5 the handlebody H is connected and its genus g is odd; so, the genus of H ′ is g + 1 and even. A basis of compression disks in H ′ is a family of disjoint compression disks D = {D 0 , . . . , D g } such that cutting H' along D gives rise to a 3-ball. One passes from one basis to another by a sequence of elementary moves called slidings.

Given such a basis, the disk D k is said to be orientation-preserving (resp. orientationreversing) if cutting H ′ along all the disks other than D k gives rise to a solid torus (resp. a solid Klein bottle).

The following sliding property holds true: if D is changed to D ′ by sliding D k over D j and if D k is orientation-reversing, then the orientation type of D j in D ′ is reversed and the other types remain unchanged. Lemma 3. The handlebody H ′ admits a basis D = {D 0 , . . . , D g } of compression disks verifying: i) D 0 , . . . , D g are disjoint from the small disk d 0 = Σ ∩ H ′ ; ii) D 1 , . . . , D g are orientation-preserving;

iii) H ′ splits into two connected domains A 0 , A 1 whose common boundary is either

d 0 ∪ D 0 (in case H is orientable) or d 0 ∪ D 0 ∪ D g (in case H is not orientable); iv) For every even (resp. odd) 1 ≤ k ≤ g -1, the disk D k is interior to A 0 (resp. A 1 ); v) In case H is orientable, D g is interior to A 1 .
Proof of lemma 3. i), ii) and iii). First let D 1 , . . . , D g be any basis of compression disks for the handlebody H, and choose D 0 a compression disk for H ′ parallel to d 0 .

In case H is orientable, i) -iii) are immediate.

In case H is non-orientable, we shall change this basis by sliding the disks one over another. There is at least one orientation-reversing D k , k ≥ 1. Sliding if necessary D k over D 0 and D g , one makes D 0 and D g orientation-reversing. Then, sliding if necessary D g over D 1 , . . . , D g-1 , one makes D 1 , . . . , D g-1 orientation-preserving. Finally, sliding D 0 over D g , one makes D g orientation-preserving. Properties i) -iii) are verified. iv) and v). One can pass any D k , 1 ≤ k ≤ g -1, as well as D g in case H is orientable, from A 0 to A 1 , or from A 1 to A 0 , by sliding twice D 0 over D k . The orientation types are not changed.

We now start constructing a one-sided Giroux surface satisfying (MC) and hence, according to proposition 3.7, a twisted open book. The surface Σ ′ , which is one-sided, is a Giroux surface in M ′ with respect to some Morse function f ′ : M ′ → R having one minimum, (g + 1) critical points of index 1 and which is constant on ∂M ′ . Now, we follow Giroux's algorithm for completing Σ ′ to a closed Giroux surface. First, one has to attach 2-handles to ∂M ′ in such a way that:

(vi) each attaching curve intersects ∂Σ ′ in two points exactly. Thus, each 2-handle will produce simultaneously a 1-handle attached to Σ ′ , which allows one to extend both f ′ and the Giroux surface for it (cf. [START_REF] Giroux | Convexité en topologie de contact[END_REF]). The previous disks D 0 , . . . , D g are devoted to be cores of these 2-handles, after convenient isotopies: in order that their attaching curves satisfy condition (vi), pairs of intersection points with ∂Σ ′ will be created. The following process is applied in order to control the attachment of the last cell, after the (g + 1) surgeries of index 2.

At the first step, a simple arc is drawn on ∂M ′ ∩ A 1 from ∂D 1 to ∂Σ ′ , otherwise disjoint from the compression disks, and ∂D 1 is pushed by isotopy along this arc in order to create two intersection points with ∂Σ ′ . Let (M 1 , Σ 1 ) be the outcome of the handle gluing to (M ′ , Σ ′ ). The boundary of Σ 1 is made of an essential curve c 1 0 and a curve c 1 1 which bounds a disk in ∂M 1 . By an isotopy of M which preserves H ′ and leaves D 0 , D 2 . . . D g fixed, one makes c 1 0 = ∂d 0 and c

D 2 c 1 1 c 1 0 d 0 D 0 D 5 D 3 D 4 Figure 2.
The surface ∂M 1 and the isotopy prior to the gluing of D 2 , in case g = 5 and H is not orientable.

At the second step, a simple arc is drawn on ∂M 1 from ∂D 2 to c 1 1 , crossing c 1 0 once, otherwise disjoint from the compression disks, and ∂D 2 is pushed by isotopy along this arc in order to create two intersection points with c 1 0 and also to surround c 1 1 ; condition (iv) guarantees that such an arc does exist (compare figure 2).

This surrounding amounts to a handle sliding of D 2 over D 1 . According to condition (ii), this operation does not change the type of the compression disks. And so on, until the gluing of D g .

After step g, we have (M g , Σ g ) which is bounded by a torus or a Klein bottle whose complement in M is a handlebody of genus 1 for which D 0 is a compression disk. The boundary of Σ g is made of an essential curve c g 0 = ∂d 0 , and a curve c g 1 which is a union of g parallel circles bounding nested disks in ∂M g ∩ A 1 .

At the last step, a simple arc is drawn on ∂M g from ∂D 0 to c g 1 , crossing c g 0 , and ∂D 0 is pushed by isotopy along this arc in order to create two intersection points with c g 0 and also to surround c g 1 . This amounts to a handle sliding of D 0 over D 1 ∪ . . . ∪ D g . By property ii) of lemma 3, when D 0 is orientation reversing, all the other handles become orientation-reversing. After attaching this 2-handle, one gets M g+1 whose complement in M is a 3-ball β, and a Giroux surface Σ g+1 in M g+1 whose boundary is made of (g + 2) parallel circles. In order to close the Giroux surface by only one 2-handle, it is necessary to have one circle only. We now explain this last step of Giroux's algorithm. Let γ 0 , γ 1 , . . . , γ g+1 be the boundary curves of the Giroux surface that we have in ∂β, the numbering being chosen so that two consecutive circles bound an annulus avoiding the other circles. The regions of their complement in the 2-sphere are colored with two colors alternatively. A trivial 1-handle is attached to M g+1 inside β whose core is a simple unknotted arc h 1 in β having one end point in γ 0 and the other in γ 2 ; moreover, one of the attached disks is turned by half a turn in order that the coloring extends along each side of the band which is attached to Σ g+1 . Now, there is an obvious 2-handle which kills the previous 1-handle and whose core satisfies condition (vi) with respect to the previous Giroux surface. After this surgery, we have a Giroux surface in M with a 3-ball removed, whose boundary is made of g parallel circles in a 2-sphere. By repeating this operation we finally get the ideal situation where the boundary of the Giroux surface consists of one curve in the sphere which bounds the last cell of M: a 3-cell for closing M containing a 2-cell for closing the Giroux surface.

This construction yields a Giroux surface S equipped with a Morse function f whose critical values are not ordered with respect to their indices in M. We have added trivial 1-handles, h 1 , h 3 , . . . , h g at a level higher than critical points of index 2.

As in the classical Morse theory it is easy to make the reordering. But, one has to take care of the middle condition (MC) in proposition 3.7. This condition could fail only when H ′ is not orientable. We continue the proof in this case.

The reordering mainly consists of extending the 1-handles h k 's by the gradient lines of f so that they are thought of as attached to ∂Σ ′ . Call h1 , h3 , . . . , hg these extended 1-handles. One checks that a case when (MC) certainly holds true is when they are compatible with the orientation Or 0 defined near ∂Σ ′ . However, by construction, they are compatible with the orientation Or β of β. As each compression disk of H ′ , after all the slidings we made, is orientation reversing, each time one crosses one D k when traversing ∂Σ ′ , the sign Or β /Or 0 changes. Fortunately, the shortest path in ∂Σ ′ joining the feet of h1 crosses exactly two compression disks, as the feet of h 1 are in γ 0 and γ 2 respectively. And similarly for the other hk 's. So, we are done.

Finally, having a Giroux surface and an ordered Morse function at hand, we have a twisted open book structure B with Σ ′ in an exceptional page. It remains to check that its normal bundle ν(B) is isomorphic to ν.

The isomorphim class of a real line bundle is determined by its first Stiefel-Whitney class w 1 (ν), or by its Poincaré dual. By construction, Σ is a Poincaré dual of w 1 (ν) with Z/2 coefficients. On the other hand, the Poincaré dual of w 1 ν(B) is the Giroux surface S that we have built. But, d 0 and S int(Σ ′ ) are homologous in (H ′ , ∂H ′ ) since they share a common boundary in the aspherical manifold H ′ . Therefore, S and Σ are homologous and the considered bundles are isomorphic.

The case when ν is trivial is very similar. Here, Σ is two-sided and the handlebody H has two connected components, H = H 1 ⊔ H 2 , each one having the same genus g. The handlebody H ′ is made of the union of H and the 1-handle ρ -1 (d 0 ); it has genus 2g. Since the two handlebodies have diffeomorphic boundaries, if one is non-orientable, the other is neither.

There are compression disks D 1 , D 3 , . . . , D 2g-1 in H 1 and compression disks D 2 , D 4 , . . . , D 2g in H 2 . All together they are compression disks which cut H ′ into one ball exactly. So, they will become the core of 2-handles attached to M ′ after some suitable isotopy of their attaching curves. The boundary of D 1 is moved so that it intersects ∂Σ ′ in two points. Next, we move the boundary of D 2 and so on. It is easy to check that the configuration of nested disks, similar to the previous case, can be realized at each step. Corollary 3.9. Theorem 1.2 holds true.

Proof. Start with the trivial or twisted trivial Γ 1 -structure ξ 0 according to whether its normal bundle is trivial or not (compare definition 2.2). Apply the Morsification process (section 2) until the decomposition M = N(Σ) ∪ H of corollary 2.6. Thus, proposition 3.8 yields the conclusion.

3.10. Modification by suspension. We end this section by explaining how the foliation F B associated above to any twisted open book B, can be modified by suspension in a similar manner as we did in section 1 for open book foliations.

Let K be a compact subsurface in the exceptional page p -1 (-1). Consider a tubular neighborhood N(K) of K in M, indeed a [-ε, ε]-bundle over K, which is compatible with F B in the following sense: the trace of F B on N(K) is the foliation suspension of the restricted representation ϕ 0 := w 1 (ν) :

π 1 (K) → Z/2 = Aut([-ε, ε]) (compare definition 2.2). Let ϕ : π 1 (K) → Diff([-ε, ε]
) be a representation such that:

1) for each α ∈ π 1 (K) and x ∈ [-ε, ε] close to the end points, we have: After this definition, the statement of theorem 1.1 is meaningful.

ϕ(α)(x) = ϕ 0 (α)(x); 2) if α is peripheral, ϕ(α) = ϕ 0 (α).

Homotopy to the plateau form

This section is a more step toward proving the regularization theorem 1.1. We introduce the following definition. Definition 4.1. Given a Γ 1 -structure ξ on a space G, by an upper (resp. lower) completion of ξ one means a foliation F of G × (-ε, 1] (resp. G × [-1, ε)), for some positive ε, which is transverse to every fiber {x} × (-ε, 1] (resp. {x} × [-1, ε)), whose germ along G × {0} is ξ, and such that G × {t} is a leaf of F for every t close enough to +1 (resp. -1). Proposition 4.2. Every co-orientable Γ r 1 -structure ξ on a simplicial complex G of dimension 1, r ≥ 1, admits an upper (resp. lower) completion of class C r .

Proof. Let us show the lower completion. After a fine subdivision of the edges, one reduces oneself to the case where, over each edge α of G, the germ of foliation ξ is given by the level sets of a real function f (x, t), (x, t) ∈ α×] -ε, +ε[; this function is smooth in x, is C r in t, and satisfies ∂f ∂t > 0 everywhere. Moreover, we may assume that the completion is already given over the 0-skeleton of G. At this point, we argue as in proposition 2.1: keeping the germ of f fixed along α × {0} ∪ G [0] × [-1, 0] one can arrange that f is C ∞ near t = -ε. Finally, the dimension of α × [-1, 0] being 2, we are reduce to build a smooth line field which fulfills the statement; the flow lines of this line field are the leaves of the wanted foliation. This can be done by partition of unity.

N + × R M × {0} X X H + × R H -× R N -× R Figure 3
. Homotopy to the plateau form tracked by some orbit of X.

Now we come back to the proof of theorem 1.1. Recall the Morse Γ r 1 -structure obtained in proposition 2.4, with its normal bundle ν, its pseudo-gradient X, and the associated decomposition M = N(Σ) ∪ H (proposition 2.5, corollary 2.6). Definition 4.3. A Γ 1 -structure ξ plat on M with normal bundle ν is said to be in plateau form with respect to this decomposition, if:

ξ plat is trivial over H; ξ plat is regular over the interior of N(Σ) and, in this domain, it is transverse to the fibers of N(Σ) → Σ.

In other words, ξ plat is a suspension foliation in N(Σ) and trivial in H.

Proposition 4.4. The Γ r 1 -structure ξ is homotopic to one, noted ξ plat , in plateau form with respect to the decomposition M = N(Σ) ∪ H. This move is due to T. Tsuboi in [START_REF] Tsuboi | Classifying spaces for groupoid structures, Conference Foliations[END_REF], where it is given as an exercise.

Proof. The homotopy of Γ 1 -structures will actually be a homotopy of the zero section in a foliated domain of the total space E(ν). We give the proof when ν is trivial only, the other cases being similar. Then, E(ν) = M × R, and X is a genuine vector field, and the graph G formed by the separatrices of X splits into G + ⊔ G -, where G + (resp. G -) is a repeller (resp. an attractor) of X. By proposition 4.2, ξ admits an upper (resp. a lower) completion over G + (resp. G -), and thus also over an open neighborhood N + (resp. N -) of G + (resp. G -) in M. Recall that in section 2.1 H was defined as an arbitrarily small handlebody neighborhood of G whose boundary is transverse to X. Thus, we can arrange that the connected component

H + (resp. H -) of H containing G + (resp. G -) is contained in N + (resp. N -). So we have a foliation F defined on a neighborhood of (M × {0}) ∪ (H -× [-1, 0]) ∪ (H + × [0, 1]) in M × R which is transverse to X on (M int(H -∪ H + )) × {0}
and tangent to H ± × {t} for every t close to ±1. Lemma 4. There exists a homotopy, relative to M int(N 1 (B)), from F B,ϕ to a Γ 1 -structure ξ ϕ on M which is trivial on N 1 (B).

Proof. Let R be any connected component of N 1 (B). Since the Reeb components of F B,ϕ are thick, the holonomy of ∂R is trivial outside. This holonomy is generated by the germ at 0 of some self-diffeomorphism λ of the real line, whose support is contained in [0, +∞). Let F λ be the suspension of λ. It is a foliation on the annulus S 1 × R, whose closed leaves are S 1 × {t}, t ≤ 0.

Then, on some small neighborhood N(R) of R, the foliation F B,ϕ is the pullback of F λ by some smooth map F : N(R) → S 1 × R. More precisely, in cylindrical coordinates (r, θ, φ) of R ∼ = D 2 × S 1 where φ is the normal projection onto S 1 and D 2 is of radius 1, we take F (r, θ, φ) := (φ, ǫ(1 -r 2 )) with ǫ > 0 small. Write F (x) = (s(x), f (x)) and define G(x) := (s(x), g(f (x))) where g is any smooth function on the real line such that g(t) = 0 for every t ≥ 0 and that g ′ (t) > 0 for every t < 0. Then, ξ ϕ := G * F λ obviously works. The homotopy from F B,ϕ to ξ ϕ is clear.

In the second step ξ ϕ will be homotoped to ξ plat relative to ∂H ′ . This second step will be different depending on whether ν is trivial or not.

Second step, co-orientable case. Here the bundle ν is assumed to be trivial and B is a genuine open book decomposition.

After the following lemma we shall be done with the homotopy problem.

Lemma 5. There exits a homotopy from ξ ϕ to ξ plat relative to N(Σ ′ ).

Proof. Recall the decomposition M = N(Σ ′ ) ∪ H ′ . We have to prove that the restrictions of ξ It is trivial on the small disk d and regular on the annulus D int(d). In the regular part, the leaves are circles near ∂d and the other leaves are spiraling, crossing ∂D transversely. The restriction of ξ ϕ to H ′ has the form f * ξ D for some map f : H ′ → D. Namely, on N(B) ∼ = D ×S 1 we take for f the canonical projection onto D; and we extend it continuously to H ′ by sending every leaf of ξ ϕ |H ′ , which is a subset of a page of B, to a point of ∂D. In particular, f sends the 2-dimensional cylinder ρ -1 (∂d 0 ) to an interval I embedded in ∂D.

In a similar way, ξ plat |H ′ is the pullback of ξ D by the map g : H ′ → I ⊂ D which equals f on ∂H ′ and sends H to ∂I (recall that H has two connected components in the co-orientable case). Each regular leaf of ξ plat is sent to the same point of I as its trace on ρ -1 (d 0 ). Since D retracts by deformation onto I, consequently ξ ϕ is homotopic to ξ plat on H ′ relative to ∂H ′ . This finishes the proof of theorem 1.1 in the co-orientable case.

Second step, twisted case. In the twisted case, the homotopy argument will be a little more sophisticated, because the classifying space of the Seifert fibrations over the interval is infinite-dimensional, as the classifying space of the line bundles is.

In the compact unit 3-ball D 3 , let σ be the orientation-preserving involution of D 3 defined by σ(x, y, t) = (x, -y, -t). Lemma 6. There is a σ-invariant Γ 1 -structure ξ σ on D 3 , whose restriction to the disk (∂D 3 ) ∩ {t ≤ 0} is conjugate to the Γ 1 -structure ξ D represented on figure 4.

Proof. It is more convenient to regard D 3 as the solid cylinder, namely, the product D 2 × [-1, +1]. In this model one has σ(z, t) = (z, -t), where z = x + iy.

The trace of ξ σ on ∂(D 2 × [-1, +1]) will be trivial over D 2 × {±1} and regular on ∂D 2 × [-1, +1], where it will indeed be the suspension of some diffeomorphism ζ of the interval [-1, +1]. We first build this diffeomorphism.

Recall from remark 3.5 that the rolling-up germ ψ is chosen as a product of commutators. Let 0 < ǫ < 1/2. One easily makes a diffeomorphism γ of the interval [-1, +1] such that: a) One has γ(t) ≥ t with equality if and only if t iii) ζ(t) ≥ t. Indeed, for every u ∈ [-1, +1] one has γ -1 (u) ≤ u ≤ τ γ -1 τ (u). Applying this at u = γ(t) one has t ≤ γ(t) ≤ [τ, γ -1 ](t). iv) One has ζ(t) = t if and only if t = γ(t) and τ (t) = γ(τ (t)); that is, t ≤ -1 + ǫ or t ≥ 1 -ǫ. One will define ξ σ as the foliation given in D 2 × [-1, +1] by the heigth function h(z, t) := t, modified as follows. Create in the interior of D 2 × [-1, +1] a number g of pairs of singularities,

Theorem 1 . 1 .

 11 Let M be a closed 3-manifold equipped with a Haefliger structure ξ of codimension one and differentiability class C r , 1 ≤ r ≤ ∞. Assume that the bundle normal to ξ embeds into the bundle tangent to M. Then ξ is C r -homotopic to a regular Haefliger structure (that is, a foliation) carried by an open book decomposition of M, or by a twisted open book decomposition of M.

  be some representation into the group of compactly supported diffeomorphisms; ϕ is assumed to be trivial on the peripheral elements (necessarily ∂K is non-empty). It allows us to construct a suspension foliationF ϕ on K × [-ε, +ε],whose leaves are transverse to the vertical segments {x} × [-ε, +ε] and whose holonomy is ϕ. The modification consists of removing F B from the interior of K ×[-ε, +ε] and replacing it by F ϕ . The new foliation, denoted F B,ϕ , is an open book foliation modified by suspension. It is also said to be carried by the open book decomposition B.

Lemma 1 .

 1 Let D k be the standard compact k-disk and let U be an open neighborhood in D k of the boundary ∂D k . Let f : U → R be a function of class C 1 without critical points. Let V and W be compact collar neighborhoods of ∂D k in D k with:

Lemma 2 .

 2 Let D k be a compact k-disk embedded in M. Let f be a real C 1 function defined on some neighborhood U of ∂D k in M. The functions f and f |U ∩ D k are assumed to have no critical points. Let g : D k → R be a Morse function which extends f |U ∩ D k . Then, there are a neighborhood V of D k in M and a function g : V → R with the following properties:

2 , 2 .

 22 and we have N = N In H + := f -1 ([3/2, 3]), we do the same construction with the function 3 -f . The open book decomposition is now clear. We now generalize the notion of open book decomposition. Definition 3.3. A generalized open book decomposition of the closed connected 3-manifold M is a pair B = (B, P) where B is a co-orientable link, the binding, and P is a codimension-one foliation of M B, whose leaves are called the pages, satisfying the following properties:

) 3 )Figure 1 .

 31 Figure 1. Twisted open book.

  These two conditions allow us to remove F B from the interior of N(K) and replace it with the suspension of ϕ. The modified foliation of M is denoted by F B,ϕ . Definition 3.11. Any foliation of this form is said to be carried by the twisted open book B.

Figure 4 .

 4 Figure 4. The Γ 1 -structure ξ D on the 2-disk.

≥ 1 -

 1 2ǫ or t ≤ -1 + ǫ; b) The germ of γ at -1 + ǫ is conjugate to ψ; c) γ is a product of commutators: γ = [α 1 , β 1 ] . . . [α g , β g ]. Here each α j , each β j is a diffeomorphism of the interval [-1, +1] with support in (-1, +1); and [α, β] denotes αβα -1 β -1 . In order to construct such a γ one starts from a factorization of ψ into g commutators and extends each entry as a diffeomorphism of [-1, +1]. Then, set ζ := [τ, γ -1 ] where τ (t) := -t. One has: i) The germ of ζ at -1 + ǫ is conjugate to ψ; ii) τ ζτ = ζ -1 . In other words, the suspension of ζ is σ-invariant in ∂D 2 × [-1, +1];

  1 1 ⊂ A 1 .

			00 00 00 00 00 00 11 11 11 11 11 11
	00000000000 00000000000 00000000000 00000000000 11111111111 11111111111 11111111111 11111111111 0 0 1 1	0000000000 0000000000 0000000000 0000000000 1111111111 1111111111 1111111111 1111111111 00 00 11 11 00 00 00 11 11 11	0000000000 0000000000 0000000000 0000000000 1111111111 1111111111 1111111111 1111111111
	0000 0000 0000 0000 0000 0000 0000 1111 1111 1111 1111 1111 1111 1111		000 000 000 000 000 111 111 111 111 111

Recall (section 2.1) that there is a diffeomorphism

, +1] which maps orbit segments of X onto fibers.

For a small ε > 0, choose a function ψ : R → [-1, +1] which is smooth, odd, and such that:

ψ(t) = 0 for 0 ≤ t ≤ 1 -3ε and ψ(1 -2ε) = ε; ψ is affine on the interval [1 -2ε, 1 -ε]; ψ(1 -ε) = 1 -ε and ψ(t) = 1 for t ≥ 1; ψ ′ > 0 on the interval ]1 -3ε, 1[. Let s : M → M × R be the graph of the function whose value is ±1 on H ± and ψ(t) at the point F -1 (x, t) for (x, t) ∈ Σ × [-1, +1]. When ε is small enough, it is easily checked that, for every x ∈ Σ, the path t → s • F -1 (x, t) is transverse to F except at its end points. Then, ξ 1 := s * F is homotopic to ξ and obviously fulfills the conditions required in proposition 4.4. Indeed, at each point, < s * X, ξ > is a non-negative linear combination of < X, ξ > and < -∂ ∂t , ξ >, hence, it is non-vanishing. (cf. figure 3)

Homotopy of Γ 1 -structures

Now we complete the proof of theorem 1.1.

On the one hand, to the given Γ 1 -structure ξ was associated a decomposition M = H ∪N(Σ), and according to proposition 4.4, ξ was homotoped to ξ plat in plateau form with respect to this decomposition. So, ξ plat is trivial on the handlebody H, while on N(Σ), it is the suspension foliation of some representation ϕ : π 1 (Σ) → Diff([-1, +1]). The gluing with the trivial Γ 1structure on H implies that, for each α ∈ π 1 (Σ) and x ∈ [-1, +1] near the end points, we have:

where ϕ 0 is obtained by restricting the first Stiefel-Whitney class of ν (compare condition 1) in 3.10). Recall that d 0 is a small disk in Σ, and Σ ′ = Σ int(d 0 ).

On the other hand, according to proposition 3.8, Σ ′ is contained in a page of some open book B; when ν is twisted, B is twisted and Σ ′ lies in an exceptional page. Applying 3.10, we modify F B by the suspension of the above-mentioned representation ϕ and get a foliation F B,ϕ . By construction, this foliation coincides with

Recall that H ′ is the handlebody which is the union of H and ρ -1 (d 0 ). To establish theorem 1.1, it remains to prove that the plateau Γ-structure ξ plat and the Γ-structure F B,ϕ carried by B are homotopic as Γ 1 -structures on H ′ relative to ∂H ′ . Notice that this statement is in fact independent of the representation ϕ since the modification of F B into F B,ϕ did not modify the foliation in H ′ .

The homotopy will be done in two steps, starting from F B,ϕ .

First step. First, one flattens its Reeb components. Set N 1 (B) the tubular neighborhood of B which is the union of the Reeb components of F B,ϕ . As we are speaking here of non-thickened Reeb components we have N 1 (B) ⊂ N(B); the collar between both tubes is foliated by tori.

the j-th pair (j = 1 . . . , g) consisting of two singularities s j 1 , s j 2 of respective indices 1, 2, in cancellation position. Let f be the resulting Morse function. One chooses its singular values so that f (s j 1 ) < -1 + ǫ < 1 -ǫ < f (s j 2 ) (j = 1, . . . , g). So, the intermediate level sets have got some genus: for u ∈ [-1 + ǫ, 1 -ǫ], the level set f -1 (u) is a compact surface of genus g bounded by one circle. Its fundamental group F 2g being non-abelian free on 2g generators, the α j 's and β j 's define a representation λ :

one changes the level surfaces of f to the suspension of the representation λ. The resulting Γ 1 -structure on D 2 × [-1, +1] induces the suspension of γ on (∂D 2 ) × [-1, +1]. One pushes this structure by some convenient isotopy to make it coincide with the heigth function in the half cylinder {y ≥ 0} × [-1, +1]. Next, in the half cylinder {y ≤ 0} × [-1, +1], one performs the modification which is σ-symmetric to the preceding one. Finally, we obtain a σ-invariant Γ 1 -structure ξ σ in the solid cylinder.

Obviously ξ σ is trivial over D 2 × ±1, while its trace on (∂D 2 ) × [-1, +1] is the suspension of the diffeomorphism [τ, γ -1 ]. Thus, thanks to property i) above, the trace of ξ σ on the disk

The involution σ is suspended to get a D 3 -bundle over RP ∞ . Let E be its total space. By lemma 6, there exists a Γ 1 -structure ξ E on E whose restriction to each fiber is ξ σ . Its normal bundle is the unique twisted bundle of rank one over E. Lemma 7. There are two continuous maps f, g :

Proof. Consider the circle S 1 := {t = 0} in S 2 in the coordinates considered above. As it is σ-invariant, it defines a circle subbundle E ′ ⊂ E over RP ∞ . Every circle fiber is transverse to ξ E . Actually ξ E |E ′ coincides with the (infinite-dimensional) horizontal foliation F E ′ on E ′ which is the suspension over RP ∞ of the orientation-reversing involution of the circle, σ|S 1 . The holonomy covering space of each leaf is S ∞ , thus contractible. So, (E ′ , F E ′ ) is the Haefliger classifying space for the groupoid generated by this involution ( [START_REF] Haefliger | Groupoïdes d'holonomie et classifiants[END_REF]). In other words, (E ′ , F E ′ ) is the classifying space for foliations of codimension one which are wandering, that is, every leaf meets every transverse interval in a finite set. Any such foliation F on any manifold V is the pullback of F E ′ by some classifying map V → E ′ . One also has the relative version: if X ⊂ V is a submanifold transverse to F then every classifying map for F |X extends to some classifying map for F .

Just as in the orientable case above, ξ ϕ |N(B) is the pullback of ξ D through the canonical projection N(B) ∼ = D 2 × S 1 → D 2 . On the other hand, in the complement H ′ \ int(N(B)) the Γ 1 -structure ξ ϕ is a wandering foliation. Embed D 2 into E as one half of the boundary of some 3-ball fibre, so that ξ D is the restriction of ξ E to D 2 (lemma 6). Thanks to the relative classifying property of (E ′ , F E ′ ) , this projection extends to some map f :

Recall that ξ ϕ and ξ plat coincide on ∂H ′ . Thus, again by the relative classifying property, ξ plat |H ′ is the pullback of F E ′ by some map g : M → E ′ equal to f on ∂H ′ . If one likes better, rather than invoking Haefliger's classifying property [START_REF] Haefliger | Groupoïdes d'holonomie et classifiants[END_REF], one can in this case easily build f and g by hands.

Finally, H ′ being a handlebody, and π 2 (E), π 3 (E) being both trivial, necessarily f and g are homotopic rel. ∂H ′ . So ξ ϕ and ξ plat are homotopic in H ′ rel. ∂H ′ . This completes the homotopy argument in the twisted case, and the proof of theorem 1.1.


Fernand Meyer
email: fernand.meyer@mines-paristech.fr

Jean Stawiaski
email: jean.stawiaski@philips.com

A stochastic evaluation of the contour strength

If one considers only local neighborhoods for segmenting an image, one gets contours whose strength is often poorly estimated. A method for reevaluating the contour strength by taking into account non local features is presented: one generates a fixed number of random germs which serve as markers for the watershed segmentation. For each new population of markers, another set of contours is generated. "Important" contours are selected more often. The present paper shows that the probability that a contour is selected can be estimated without performing the effective simulations.

Introduction

Image segmentation aims at extracting from an image the contours of the objects of interest. This task is extremely difficult and problem depending. We propose here a stepwise approach, in which we first extract from the image all contours which may be pertinent. These contours are the frontiers of a fine partition of the image. Each tile represents a region which is homogeneous for some criterion. In general these contours are far too numerous and the image is oversegmented. In a second step one orders the contours according to their importance in the image. A dissimilarity between adjacent tiles is estimated. Coarser partitions are then produced by merging all regions whose dissimilarity is below a given threshold ;every time two regions are merged, one reevaluates their dissimilarity with their neighborhood, and the process is stopped as soon some additional criterion is satisfied. Each new fusion produces a finer partition ; these partitions are nested and may be ranked according their coarseness. The weight of a contour is then simply the first level of coarseness where this contour is no more present. Such nested partitions are called hierarchies. There a two main classes of methods for extracting the contours of interest from the hierarchy, in order to get the final segmentation. In the first a functional is defined and the set of contours maximizing this functional is extracted from the hierarchy [START_REF] Guigues | Modèles multi-echelles pour la segmentation d'images[END_REF][START_REF] Guigues | Scale-sets image analysis[END_REF]. Another approach to segmentation uses seeds or markers for the regions of interest (including the background). Only the strongest contours separating each pair of markers are retained. For the resulting segmentation to be satisfactory, two conditions are to be met : a) the finest partition should contain all contours of interest ; b) the strength of the contours has to be correctly estimated, otherwise, the wrong contours separating markers will be selected.

Unfortunately, some contours of objects are intrinsically weak since they correspond to transitions between similar color shades for instance, or objects having almost the same luminance as their background. When such objects are large enough and appear on a clean background, they are nevertheless easily recognized by the human eye, but not easily segmented by the computer. Local methods will estimate these contours as very weak as noticed in [START_REF] Najman | Geodesic saliency of watershed contours and hierarchical segmentation[END_REF]. We present in this paper a set of new methods able to produce useful and reliable estimates of the contour strength.

Authors of [START_REF] Angulo | Stochastic watershed segmentation[END_REF] have described such a method in the context of the watershed segmentation of gradient images. Random germs are spread all over the image and server as markers for the watershed segmentation of a gradient image. Large regions, separated by low contrast gradient from neighboring regions will be sampled more frequently than smaller regions and will be selected more often. On the other hand, high gradient watershed lines will often be selected by the watershed construction, as there are many possible positions of markers which will select them. So the probability of selecting a contour will offer a nice balance between strength of the contours and size of the adjacent regions. The method produces good results for multimedia types of images as well as for 3D granular material but suffers from a major drawback : in order to obtain a robust estimation of the contour strength a relatively large number of simulations has to be made, typically on the order of 50 simulations, implying the construction of 50 watershed. This paper shows how to obtain estimates of the contour strength without simulations. Analyzing under which conditions a given piece of contour will be selected by the watershed makes it possible to compute the contour strength without the need of any simulation. More precisely, one has to determine for each contour the two zones, separated by this contour, where a marker has to be present in order for the contour to be selected.

The paper is organized as follows. In a first section we explain how to produce a hierarchy to start with, obtained with the watershed transform ; we explain why in this case, the strength of the contour is often badly estimated. We then describe an efficient representation of a hierarchy, as a weighted tree. The last part of the paper presents how to implement the stochastic watershed on this tree, in order to estimate a contour strength compatible with the features one desires to stress.

Producing and representing hierarchies

Producing a hierarchy with the watershed transform

The watershed associates to each regional minimum of a topographic surface its catchment basin. Morphological segmentation applies the watershed transform to the gradient of the image to segment. Like that, one obtains a fine partition where each region is the catchment basin associated to a minimum of the gradient image.

Suppose now that this same topographic surface is flooded up to some level λ. A number of lakes are created, some of them are still regional minima of the topographic surface, covering one or more minima of the initial surface ; others are "full", in the sense that the level of the lake reaches the lowest pass point leading to another minimum: such lakes are not regional minima anymore. The catchment basins of this flooded relief form a partition which is coarser than the partition associated to the unflooded surface ; a number of adjacent regions of the fine partition will have merged to form larger regions.

For a level µ > λ, the same process can be repeated, leading to a yet coarser partition. Hence, the partitions formed by the catchment basins of a topographic surface, for increasing levels of flooding of this surface, form a hierarchy. This hierarchy can be represented by the fine partition formed by the catchment basins of the unflooded surface plus a dissimilarity measure between adjacent regions, equal to the flooding level for which these regions merge.

Dissimilarity associated hierarchies

More generally, we may associate a hierarchy to each partition for which a dissimilarity between adjacent regions has been defined. Merging all regions with a dissimilarity between some level λ produces a coarser partition. For increasing levels λ the partitions become coarser and coarser, producing again a hierarchy.

Representing a hierarchy as a tree

In order to give a visual support to the algorithms presented below, we associate to each hierarchy a topographic surface defined as follows. This surface is completely flat, except along the edges of the tile of the finest partition Π in the hierarchy. A 0 thickness wall is erected along each piece of contour, with a height equal to the dissimilarity between both adjacent regions. Fig. 1 illustrates this construction process. The bottom row shows the successive levels of a hierarchy, where from level to level the most similar regions have merged ; the left most image represents the finest partition. The central row of fig. 1 presents the construction of a tree representing the hierarchy. In the left most image, this tree is reduced to isolated nodes, representing each a tile of the partition Π. Each region ρ i is represented by a node ν i of the tree, weighted by the area of the region.

We then flood this topographic surface in order to construct a tree representing the hierarchy. The upper row shows the topographic surface during flooding. The left most level shows the walls separating the tiles ; the color of each tile being the color of the source which will flood this tile. As the level of the flood increases, it first reaches the level of the lowest wall : the two adjacent lakes are merged and get a uniform color. As these lakes merge for the first time, an edge is created between the corresponding nodes, with a weight equal to the dissimilarity between both regions ; we write e ij for the edge linking the nodes ν i and ν j . At level 2 in fig. 1, again two lakes meet for the first time along a wall which separated both lakes and a new edge is created. The same happens at level 3, for which the surface is entirely flooded and the tree which has been constructed spans all nodes. However there exists pieces of walls still emerging from the lakes. As the flood further increases and reaches the level of such a wall, there are not two lakes which meet for the first time, as it is the same lake which is present on both sides of the wall ; in this case no edge is created.

In summary, as the flood reaches the level of a wall separating two regions ρ i and ρ j of Π, there are two possibilities : the lakes on each side of the wall are distinct lakes which merge and an edge (e ij ) is created between the corresponding nodes ν i and ν j ; or both lakes form in reality one and the same lake, created by mergings of lakes at a lower level, in this case, no edge is created between them. This shows that regions may merge at a lower level than the dissimilarity between them : it is sufficient that they belong to a chain of lakes which have merged at lower flooding levels. Based on this remark, we may replace the initial dissimilarity between adjacent regions by the flooding dissimilarity, that is the level for which both regions belong to the same lake during flooding, which is lower (in mathematical terms, this new dissimilarity is a distance, called subdominant ultrametric distance associated to the initial dissimilarity). In Fig. 2 we see on the left the initial image and in the center its gradient. The catchment basins of the watershed constitute the finest partition ; the dissimilarity between tiles being the lowest pass point on the gradient image between adjacent regions. The right image shows as dissimilarity the flooding level for which the tiles merge for the first time. The grey tone values on the right are well below the values of the gradient image, specially for large regions, as it is the case for the coat of the cameraman. On the contrary some small contrasted regions are surrounded by extremely bright pieces of contour. This is due to the fact that for larger regions, there are often some parts of the contour which are weaker, leading to early mergings with neighboring regions.

Remark 1. The tree T defined above is the minimum spanning tree of the region neighboring graph defined as follows : each region is represented by a node ; adjacent nodes are linked by an edge with a weight equal to the dissimilarity between both regions. Edges weights along paths of the minimum spanning tree are minimizing the maximum dissimilarity between each pair of nodes [START_REF] Meyer | Minimal spanning forests for morphological segmentation[END_REF]. The construction of the flooding tree corresponds also to the single-linkage hierarchy in the context of data clustering.

Marker based watershed segmentation

The flooding tree also plays an important role in the marker based watershed construction. Some regions, called markers, are selected and play the role of sources from which the topographic surface will be flooded. Their flow is such that they create lakes with a uniform and growing altitude. Consider now the evolution of a particular lake Λ i when the flooding level increases and reaches the lowest pass point leading to another region. If there is a lake on the other side, there are two possibilities : this lake is a distinct lake Λ j , produced by another source and these lakes meet for the first time. The pass point where they meet corresponds to an edge of the flooding tree. This edge will be assigned a dissimilarity 1. If on the contrary the lake on the other side is the same lake Λ i , originating from the same source, the corresponding edge does not belong to the flooding tree. If the region on the other side is still dry, it will be flooded by the overflood of our lake and they will form a unique lake ; in this case, the corresponding edge belongs to the flooding tree, but does not correspond to a contour of the marker based segmentation : it is assigned a dissimilarity 0. It is noteworthy that all significant events such as mergings of lakes and overflood of a lake into its neighbor take place along the tree constructed above.

The new binary distribution of weights on the flooding tree represents the marker based segmentation. Cutting all edges with weight 1 produces a forest. Each tree of this forest has its root in a distinct marker. The union of all regions belonging to a same tree of the forest constitutes a region of the marker based segmentation. In mathematical terms, marker based segmentation results in a minimum spanning forest of the neighborhood graph, in which each tree is rooted in a marker [START_REF] Meyer | Minimal spanning forests for morphological segmentation[END_REF].

3 Stochastic evaluation of the strength of the contours

Stochastic marker based watershed

The stochastic evaluation of the contour strength as defined by Angulo et al. [START_REF] Angulo | Stochastic watershed segmentation[END_REF] assigns to each piece of contour the probability to appear as contour in a watershed segmentation, with random markers. As we have seen, marker based segmentation results in a forest derived from the flooding tree by cutting some edges of this tree. In order to compute the probability that such an edge becomes a contour we have to understand for which marker distribution this happens. We have defined markers as particular regions of the hierarchy which act as sources. One way to select such regions as markers is by drawing random punctual germs onto the surface. If at least one germ falls in a given region, this region is selected as markers ; a region without a germ is not a marker and will be flooded through sources placed in other regions.

Let us first consider two markers m 1 and m 2 . Both markers become sources which pour water. Their flow is such that they create lakes with a uniform and growing altitude. These two lakes keep separated until they finally meet on the lowest edge, at an altitude λ separating them, which becomes a contour edge. Each lake will have covered a number of catchment basins separated by walls with a lower altitude than λ. Both lakes L 1 and L 2 are separated from the rest of the topographic surface by edges higher than λ.

If one of the regions, for instance L 1 is without a marker, it can only be flooded through a neighboring region. The lowest connecting edge to a neighboring region is the edge e 12 of altitude λ. So if the region L 2 has a marker, it will flood the region L 1 through e 12 , which is not selected as a contour edge.

If both regions L 1 and L 2 are without markers, then the union of both regions L 1 ∪L 2 will be flooded from outside, through a pass point of altitude greater than λ and again the edge e 12 between both regions will not be selected as contour edge.

The same analysis may now be reformulated using the flooding tree. Consider and edge e 12 with an altitude λ of the tree, joining two nodes ν 1 and ν 2 . Cutting all edges of the tree with an altitude higher than or equal to λ produces a forest. Let us call T 1 and T 2 the trees of the forest containing respectively the nodes ν 1 and ν 2 . The edge e 12 will be a contour edge in a marker based segmentation, if and only if each of the trees T 1 and T 2 has at least one node which is a marker.

The next section will propose several modes for promoting randomly a node to a marker in the tree. For each mode, we then compute the probability of the edges of the tree to become contour edges.

Contour strength for various random distribution of markers

Uniform Poisson distribution of germs on the surface : the surfacic stochastic watershed. We first use points as germs. The simplest distribution of germs is a Poisson distribution. For a fixed number N of germs this reduces to a uniform distribution of N germs on the topographic surface. As soon one or more germs fall into a region R 1 of the fine partition Π, the corresponding node is selected as marker. The probability that a germ falls into a region A of area A obeys a binomial distribution of probabilities p = A S and 1 -p, where S is the area of the domain occupied by the function f . So the probability that out of N germs no one falls into A is (1

-A S ) N .
Using the same notations as above, we consider an edge e 12 of altitude λ. We cut all edges higher or equal than λ and consider the trees T 1 and T 2 of the forest adjacent to e 12 . The edge e 12 is a contour edge in a marker based segmentation, if and only if each of the trees T 1 and T 2 adjacent to e 12 has at least one node which is a marker. This means that at least one germ has fallen in each of the regions L 1 and L 2 spanned by both trees. We have to compute the probability of the event {there is at least one germ in L 1 } and {there is at least one germ in L 2 }. The opposite event is the union of two non exclusive events {there is no germ in L 1 } or {there is no germ in L 2 }. Its probability is p{there is no germ in L 1 } + p{there is no germ in L 2 } -p{there is no germ in L 1 ∪ L 2 }, which is:

P e1,2 = 1 -1 - L 1 S N -1 - L 2 S N + 1 - L 1 + L 2 S N . (1)
As we have assigned to each node ν i of the tree T the area of the underlying region, the area L 1 is simply the sum of the weights of the nodes of the tree T 1 in the forest obtained after cutting the edges above λ. We write meas(T 1 ) = L 1 .

The volumic stochastic watershed. In order to give more importance to the contrast an alternative measure may be used: we suppose that the marker is not thrown on the surface but within the volume of the lakes. Considering again the edge e ij of altitude λ, the adjacent regions spanned by the subtrees T 1 and T 2 may be flooded up to level λ and will then be covered by two lakes L 1 and L 2 with volumes (V 1 = λ × meas(T 1 )) and (V 2 = λ × meas(T 2 )). If Λ is the highest dissimilarity between two regions of the hierarchy, the volume occupied by a lake covering the total surface is (V = Λ × meas(T )). The probability that e ij is selected as an edge is then:

P e1,2 = 1 -1 - V 1 V N -1 - V 2 V N + 1 -λ V 1 + V 2 V N . (2)
The surfacic stochastic watershed with non punctual seeds. In what precedes we have imagined punctual seeds, able to fall at any point of both regions L i and L j adjacent to the edge e ij . If the seeds are not points, but a set B, the probabilities will be changed, as the probability that the set B falls within a region X will be proportional, not to the area of X, but to the area X B of the set X eroded by the structuring element B. This value does depend upon the shape of X and we have to measure it on the initial image. Mutatis mutandis, the area stochastic watershed will weight the edges with the probability:

P (e 1,2 ) = 1-1 - L 1 B S N -1 - L 2 B S N + 1 - L 1 B + L 2 B S N .
(3) As previously we observe that this probability becomes 0 if a regions becomes too small for containing the set B.

Likewise, the volumic stochastic watershed will weight the edges with the probability:

P (e 1,2 ) = 1 -1 - V 1 V N -1 - V 2 V N + 1 - V 1 + V 2 V N . (4)
where

V 1 = w(e) × L 1 B and V 2 = w(e) × L 2 B.
Results. Fig. 3 illustrates the results one obtains for the different methods. Fig. 3(c) and Fig. 3(e) compares the stochastic watershed where points are used as markers with the result obtained if one uses disks as seeds. Notice that in this last case, many small regions have vanished, but not all ; some small regions are created as they lie between the contours of larger regions.

Furthermore, the methods presented so far start all with a hierarchy defined by a set of weights on the minimum spanning tree and produce a new set of weights which express a preference for a type of regions compared to others (large or contrasted regions for instance). This homogeneity in the representation of the hierarchies permits to chain the process. Fig. 3(f) has been obtained by computing first a surfacic stochastic watershed and on the results compute a stochastic watershed obtained with disks as markers. One sees some difference with Fig. 3(e) where only the last stochastic watershed has been used. Figure 4 illustrates some hierarchies of contours obtained on images of the Berkeley Segmentation Dataset [START_REF] Fowlkes | The berkeley segmentation dataset and benchmark (bsdb)[END_REF]. Segmentation results were obtained by thresholding the contour strength map obtained with volumic stochastic watershed with point seeds. The Volumic stochastic measures permits to obtain a good trade-off between size and contrast of the object to be detected.

Conclusion

The stochastic framework is excessively rich as it permits infinite variations : other probability laws for the distribution of seeds, random choice of seeds belonging to a family of shapes, regionalised distribution of seeds according some properties of the domain they will fall [START_REF] Noyel | Classification-driven stochastic watershed. Application to multispectral segmentation[END_REF], etc. An additional nice feature, is that the result always is a probability distribution. That is, whatever the rules which are adopted, the edges get weights between 0 and 1, which makes it easy to combine hierarchies. Given two hierarchies H 1 and H 2 with distinct dissimilarities, it is possible to combine them, by taking for each edge the minimum (resp. maximum) of its dissimilarity in H 1 and H 2 .

Furthermore, these reevaluation methods of the contours may be chained: use a first distribution of random germs and on the resulting hierarchy, apply a second and then a third, before, ultimately use a marker based segmentation with markers introduced either in an interactive way or through an automated analysis of the scene.

Fig. 1 .

 1 Fig. 1. Construction of the minimum spanning tree by flooding a topographic surface.

Fig. 2 .

 2 Fig. 2. (a) Initial image. (b) Gradient value on the contour lines of a segmented image. (c) Contours weighted by the level of flooding for which they disappear.

Fig. 3 (

 3 a) and (b) show the initial image and the level for which the contours vanish during uniform flooding. Fig.3(c) shows the result of the surfacic stochastic watershed where one uses random points spread over the surface as seeds. If the random points are spread within the volume of the lakes, one obtains Fig.3(d), with large regions and small ones if they are contrasted enough. For balls of a given size taken as random seed, all regions where the ball cannot enter vanish yielding the segmentation of Fig.3(e).

Fig. 3 .

 3 Fig. 3. (a) Initial image. (b) Flooding level for which the contours disappear. (c) Surfacic stochastic watershed with point seeds. (d) Volumic stochastic watershed with point seeds. (e) Surfacic stochastic watershed with disks as seeds. (f) Stochastic watershed with point seeds followed by a stochastic watershed with disk seeds.

Fig. 4 .

 4 Fig. 4. Volumic stochastic watershed with point seeds. Segmentation is obtained by thresholding the contour strength.