Regularization of Gamma_1-structures in dimension 3
aa r X i v : . [ m a t h . G T ] S e p Regularization of Γ -stru tures in dimension 3FRANÇOIS LAUDENBACH AND GAËL MEIGNIEZAbstra t. For Γ -stru tures on 3-manifolds, we give a very simple proof of Thurston's regular-ization theorem, (cid:28)rst proved in [13℄, without using Mather's homology equivalen e. Moreover,in the o-orientable ase, the resulting foliation an be hosen of a pre ise kind, namely an(cid:16)open book foliation modi(cid:28)ed by suspension(cid:17). There is also a model in the non o-orientable ase. 1. Introdu tionA Γ -stru ture ξ , in the sense of A. Hae(cid:29)iger, on a manifold M is given by a line bundle ν = ( E → M ) , alled the normal bundle to ξ , and a germ of odimension-one foliation F alongthe zero se tion, whi h is required to be transverse to the (cid:28)bers (see [8℄). To (cid:28)x ideas, onsiderthe o-orientable ase, that is, the normal bundle is trivial: E ∼ = M × R ; for the general asesee se tion 7. The Γ -stru ture ξ is said to be regular when the foliation F is transverse to thezero se tion, in whi h ase the pullba k of F to M is a genuine foliation on M . A homotopy of ξ is de(cid:28)ned as a Γ -stru ture on M × [0 , indu ing ξ on M × { } . A regularization theoremshould laim that any Γ -stru ture is homotopi to a regular one. It is not true in general.An obvious ne essary ondition is that ν must embed into the tangent bundle τ M . When ν istrivial and dim M = 3 this ondition is ful(cid:28)lled.The C ∞ ategory is understood in the sequel, unless otherwise spe i(cid:28)ed. In parti ular M is C ∞ . One alls ξ a Γ r -stru ture ( r ≥ ) if it is tangentially C ∞ and transversely C r , that is,the foliation harts are C r in the dire tion transverse to the leaves. We will prove the followingtheorem.Theorem 1.1. If M is a losed 3-manifold and ξ a Γ r -stru ture, r ≥ , whose normal bundleis trivial, then ξ is homotopi to a regular Γ r -stru ture.Moreover, the resulting foliation of M may have its tangent plane (cid:28)eld in a pres ribedhomotopy lass (see proposition 6.1).This theorem is a parti ular ase of a general regularization theorem due to W. Thurston (see[13℄). Thurston's proof was based on the deep result due to J. Mather [9℄, [10℄: the homologyequivalen e between the lassifying spa e of the group Di(cid:27) c ( R ) endowed with the dis retetopology and the loop spa e Ω B (Γ ) + . We present a proof of this regularization theorem whi hdoes not need this result. A regularization theorem in all dimensions, still avoiding any di(cid:30) ultresult, is provided in [12℄. But there are reasons for onsidering the dimension 3 separately.2000 Mathemati s Subje t Classi(cid:28) ation. 57R30.Key words and phrases. Foliations, Hae(cid:29)iger's Γ -stru tures, open book.FL supported by ANR Floer Power. FRANÇOIS LAUDENBACH AND GAËL MEIGNIEZOur proof provides models realizing ea h homotopy lass of Γ -stru ture. The models arebased on the notion of open book de omposition. Re all that su h a stru ture on M onsistsof a link B in M , alled the binding, and a (cid:28)bration p : M r B → S su h that, for every θ ∈ S , p − ( θ ) is the interior of an embedded surfa e, alled the page P θ , whose boundary isthe binding. The existen e of open book de omposition ould be proved by J. Alexander when M is orientable, as a onsequen e of [1℄ (every orientable losed 3-manifold is a bran hed overof the 3-sphere) and [2℄ (every link an be braided); but he was ignoring this on ept whi hwas introdu ed by H. Winkelnkemper in 1973 [16℄. Hen eforth, we refer to the more (cid:29)exible onstru tion by E. Giroux, whi h in ludes the non-orientable ase (see se tion 3). An openbook gives rise to a foliation O onstru ted as follows. The pages endow B with a normalframing. So a tubular neighborhood T of B is trivialized: T ∼ = B × D . Out of T the leavesare the pages modi(cid:28)ed by spiraling around T ; the boundary of T is a union of ompa t leaves;and the interior of T is foliated by a Reeb omponent, or a generalized Reeb omponent in thesense of Wood [17℄. For te hni al reasons in the homotopy argument of se tion 4, the Reeb omponents of O , instead of being usual Reeb omponents, will be thi k Reeb omponents inwhi h a neighborhood of the boundary is foliated by tori ompa t leaves. We all su h afoliation an open book foliation.The latter an be modi(cid:28)ed by inserting a so alled suspension foliation. Pre isely, let Σ bea ompa t sub-surfa e of some leaf of O out of T and Σ × [ − , +1] be a foliated neighborhoodof it (ea h Σ × { t } being ontained in a leaf of O ). Let ϕ : π (Σ) → Diff c (] − , +1[) besome representation into the group of ompa tly supported di(cid:27)eomorphisms; ϕ is assumed tobe trivial on the peripheral elements. It allows us to onstru t a suspension foliation F ϕ on Σ × [ − , +1] , whose leaves are transverse to the verti al segments { x } × [ − , +1] and whoseholonomy is ϕ . The modi(cid:28) ation onsists of removing O from the interior of Σ × [ − , +1] and repla ing it by F ϕ . The new foliation, denoted O ϕ , is an open book foliation modi(cid:28)ed bysuspension. Theorem 1.1 an now be made more pre ise:Theorem 1.2. Every o-orientable Γ r -stru ture, r ≥ , is homotopi to an open book foliationmodi(cid:28)ed by suspension.The proof of this theorem is given in se tions 2 - 4 when r ≥ . In se tion 5, we explainhow to get the less regular ase ≤ r < . We have hosen to treat the ase r = 1 + bv (theholonomy lo al di(cid:27)eomorphisms are C and their (cid:28)rst derivatives have a bounded variation).Indeed, Mather observed in [11℄ that Dif f bvc ( R ) is not a perfe t group and it is often believedthat the perfe tness of Dif f rc ( R ) plays a role in the regularization theorem.In se tion 6, the homotopy lass of the tangent plane (cid:28)eld will be dis ussed. Finally the aseof non o-orientable Γ r -stru ture will be sket hed in se tion 7 where the orresponding models,based on twisted open book, will be presented.We are very grateful to Vin ent Colin, Étienne Ghys and Emmanuel Giroux for their om-ments, suggestions and explanations.2. Tsuboi's onstru tionA Γ -stru ture ξ on M is said to be trivial on a odimension 0 submanifold W when, forevery | t | small enough, W × { t } lies in a leaf of the asso iated foliation.Every losed 3-manifold M has a so- alled Heegaard de omposition M = H − ∪ Σ H + , where H ± is a possibly non-orientable handlebody (a ball with handles of index 1 atta hed) and Σ istheir ommon boundary. A thi k Heegaard de omposition is a similar de omposition where thesurfa e is thi kened: M = H ′− ∪ Σ ×{− } Σ × [ − , +1] ∪ Σ ×{ +1 } H ′ + . The following statement is due to T. Tsuboi in [14℄ where it is left to the reader as an exer ise.Proposition 2.1. Given a Γ -stru ture ξ of lass C r , r ≥ , on a losed 3-manifold M , thereexists a thi k Heegaard de omposition and a homotopy ( ξ t ) t ∈ [0 , from ξ su h that:1) ξ is trivial on H ′± ;2) ξ is regular on Σ × [ − , +1] and the indu ed foliation is a suspension.Proof. With ξ and its foliation F de(cid:28)ned on an open neighborhood of the zero se tion M × in M × R , there omes a overing of the zero se tion by boxes, open in M × R , bi-foliated withrespe t to F and the (cid:28)bers. We hoose a C -triangulation T r of M so (cid:28)ne that ea h simplexlies entirely in a box. With T r omes a ve tor (cid:28)eld X de(cid:28)ned as follows.First, on the standard k -simplex there is a smooth ve tor (cid:28)eld X ∆ k , tangent to ea h fa e,whi h is the (des ending) gradient of a Morse fun tion having one riti al point of index k atthe bary enter and one riti al point of index i at the bary enter of ea h i -fa e. When ∆ i ⊂ ∆ k is an i -fa e, X ∆ i is the restri tion of X ∆ k to ∆ i . Now, if σ is a k -simplex of T r , thought of asa C -embedding σ : ∆ k → M , we de(cid:28)ne X σ := σ ∗ ( X ∆ k ) . The union of the X σ 's is a C ve tor(cid:28)eld X whi h is uniquely integrable. After a reparametrization of ea h simplex we may assumethat the stable manifold W s ( b ( σ )) of the bary enter b ( σ ) is C .The Γ -stru ture ξ ( o-oriented by the R fa tor of M × R ) is said to be in Morse positionwith respe t to T r if:(i) it has a smooth Morse type singularity of index k at the bary enter of ea h k -simplexand it is regular elsewhere;(ii) X is (negatively) transverse to ξ out of the singularities.Lemma 2.2. Let F be the foliation asso iated to ξ . There exists a smooth se tion s su h that s ∗ F is in Morse position with respe t to T r .Note that, as s is homotopi to the zero se tion, the Γ -stru ture s ∗ F on M is homotopi to ξ .Proof. Assume that s is already built near the ( k − -skeleton. Let σ be a k -simplex. Weexplain how to extend s on a neighborhood of σ . After a (cid:28)bered isotopy of M × R over theidentity of M , we may assume that F is trivial near { b ( σ ) } × R . Now, near b ( σ ) , we ask s to oin ide with the graph of some lo al positive Morse fun tion f σ whose Hessian is negativede(cid:28)nite on T b ( σ ) σ and positive de(cid:28)nite on T b ( σ ) W s ( b ( σ )) . This fun tion is now (cid:28)xed up to apositive onstant fa tor. We will extend s as the graph of some fun tion h in the F -foliated hart over a neighborhood of σ . This fun tion is already given on a neigborhood N ( ∂σ ) of ∂σ where it is C r , the regularity of ξ , and satis(cid:28)es X.h < ex ept at the bary enter of ea h fa e.On the one hand, hoose an arbitrary extension h of h to a neighborhood of σ vanishingnear b ( σ ) . On the other hand, hoose a nonnegative fun tion g σ su h that: FRANÇOIS LAUDENBACH AND GAËL MEIGNIEZ- g σ = 0 near ∂σ ;- g σ = f σ near b ( σ ) ;- X.g σ < when g σ > ex ept at b ( σ ) .Then, if c > is a large enough onstant, h := h + cg σ has the required properties, ex eptsmoothness. Returning to M × R , the se tion s we have built is C r , smooth near the singular-ities, and X is transverse to s ∗ ξ ex ept at the singularities. Therefore, there exists a smooth C r -approximation of s , relative to a neighborhood of the bary enters whi h meets all the re-quired properties. (cid:3) Thus, by a deformation of the zero se tion whi h indu es a homotopy of ξ , we have put ξ in Morse position with respe t to T r . In the same way, applying lemma 2.2 to the trivial Γ -stru ture ξ , we also have a Morse fun tion f su h that X.f < ex ept at the bary enters.Let G − (resp. G + ) denote the losure of the union of the unstable (resp. stable) manifoldsof the singularities of X of index 1 (resp. 2). The following properties are lear:(a) In M , the subset G − (resp. G + ) is a C - omplex of dimension 1.(b) It admits arbitrarily small handlebody neighborhoods H ′− (resp. H ′ + ) whose boundaryis transverse to X .( ) Every orbit of X outside H ′± has one end point on ∂H ′− and the other on ∂H ′ + . This alsoholds true for any smooth C -approximation e X of X (in parti ular e X is still negativelytransverse to ξ ).Given a ( o-orientable) Γ -stru ture ξ on a spa e G , by an upper (resp. lower) ompletionof ξ one means a foliation F of G × ( − ǫ, (resp. G × [ − , ǫ ) ), for some positive ǫ , whi h istransverse to every (cid:28)ber { x } × ( − ǫ, (resp. { x } × [ − , ǫ ) ), whose germ along G × { } is ξ ,and su h that G × { t } is a leaf of F for every t lose enough to +1 (resp. − ).Lemma 2.3. Every o-orientable Γ r -stru ture on a simpli ial omplex G of dimension 1, r ≥ ,admits an upper (resp. lower) ompletion of lass C r .Proof. One redu es immediately to the ase where G is a single edge. In that ase, using apartition of unity, one builds a line (cid:28)eld whi h ful(cid:28)lls the laim. This line (cid:28)eld is integrable. (cid:3) By (a), the Γ -stru ture ξ admits an upper (resp. lower) ompletion over G + (resp. G − ),and thus also over an open neighborhood N + (resp. N − ) of G + (resp. G − ). By (b), there is ahandlebody neighborhood H ′± of G ± ontained in N ± and whose boundary is transverse to X .So we have a foliation F de(cid:28)ned on a neighborhood of ( M × { } ) ∪ ( H ′− × [ − , ∪ ( H ′ + × [0 , whi h is transverse to X on M r ( H ′− ∪ H ′ + ) and tangent to H ′± × { t } for every t lose to ± .By ( ), there is a di(cid:27)eomorphism F : M r Int ( H ′− ∪ H ′ + ) → Σ × [ − , +1] for some losedsurfa e Σ , whi h maps orbit segments of e X onto (cid:28)bers.For a small ǫ > , hoose a fun tion ψ : R → [ − , +1] whi h is smooth, odd, and su h that:- ψ ( t ) = 0 for ≤ t ≤ − ǫ and ψ (1 − ǫ ) = ǫ ;- ψ is a(cid:30)ne on the interval [1 − ǫ, − ǫ ] ;- ψ (1 − ǫ ) = 1 − ǫ and ψ ( t ) = 1 for t ≥ ;- ψ ′ > on the interval ]1 − ǫ, .Let s : M → M × R be the graph of the fun tion whose value is ± on H ′± and ψ ( t ) at thepoint F − ( x, t ) for ( x, t ) ∈ Σ × [ − , +1] . When ǫ is small enough, it is easily he ked that,for every x ∈ Σ , the path t s ◦ F − ( x, t ) is transverse to F ex ept at its end points. Then, ξ := s ∗ F is homotopi to ξ and obviously ful(cid:28)lls the onditions required in proposition 2.1. (cid:3)
3. Giroux's onstru tionWe use here theorem III.2.7 from Giroux's arti le [5℄, whi h states the following:Let M be a losed 3-manifold (orientable or not). There exist a Morse fun tion f : M → R and a o-orientable surfa e S whi h is f -essential in M .Giroux says that S is f -essential when the restri tion f | S has exa tly the same riti al pointsas f and the same lo al extrema. In the sequel, we all su h a surfa e a Giroux surfa e.Giroux explained to us [6℄ how this notion is related to open book de ompositions. In theabove statement, the fun tion f an be easily hosen self-indexing (the value of a riti al pointis its Morse index in M ). Thus, let N be the level set f − (3 / . The smooth urve B := N ∩ S will be the binding of the open book de omposition we are looking for. It an be proved thatthe following holds for every regular value a, < a ≤ / :- the level set f − ( a ) is the union along their ommon boundaries of two surfa es, N a and N a , ea h one being di(cid:27)eomorphi to the sub-level surfa e S a := S ∩ f − ([0 , a ]) ;- the sub-level M a := f − ([0 , a ]) is divided by S a into two parts P a and P a whi h areisomorphi handlebodies (with orners);- S a is isotopi to N ai through P ai , for i = 1 , , by an isotopy (cid:28)xing its boundary urve S a ∩ f − ( a ) .This laim is obvious when a is small and the property is preserved when rossing the riti allevel 1. In this way the handlebody H − := f − ([0 , / is divided by S / into two di(cid:27)eomor-phi parts P / i , i = 1 , , and we have N = N / ∪ N / . We take S / , whi h is isotopi to N / i in P / i , as a page. The (cid:28)gure is the same in H + := f − ([3 / , . The open book de ompositionis now lear.Proposition 3.1. Let K ⊂ M be a ompa t onne ted o-orientable surfa e whose boundary isnot empty. Then there exists an open book de omposition whose some page ontains K in itsinterior.Proof. (Giroux) A ording to the above dis ussion it is su(cid:30) ient to (cid:28)nd a Morse fun tion f and a Giroux surfa e S (with respe t to f ) ontaining K . Let H be the quotient of K × [ − , by shrinking to a point ea h interval { x } × [ − , when x ∈ ∂K × [ − , . After smoothing,it is a handlebody whose boundary is the double of K . On H there exists a standard Morsefun tion f whi h is onstant on ∂H , having one minimum, the other riti al points being ofindex 1. The surfa e K × { } an be made f -essential. This fun tion is then extended to a FRANÇOIS LAUDENBACH AND GAËL MEIGNIEZglobal Morse fun tion ˜ f on M . At this point we have to follow the proof of Theorem III.2.7in [5℄. The fun tion ˜ f is hanged on the omplement of H , step by step when rossing its riti al level, so that K × { } extends as a Giroux surfa e in M . (cid:3) Let now ξ be a Γ -stru ture meeting the on lusion of proposition 2.1, up to res aling theinterval to [ − ε, + ε ] . Let F ϕ be the suspension foliation indu ed by ξ on Σ × [ − ε, + ε ] . Choose x ∈ Σ × { } ; the segment x × [ − ε, + ε ] is transverse to F ϕ . Let K be the surfa e obtainedfrom Σ × by removing a small open disk entered at x . The foliation F ϕ foliates K × [ − ε, + ε ] so that K × { t } lies in a leaf, when t is lose to ± ε , and ∂K × [ − ε, + ε ] is foliated by parallel ir les. We apply proposition 3.1 to this K .Corollary 3.2. There exists an open book foliation O of M indu ing the trivial foliation on K × [ − ε, + ε ] (the leaves are K × { t } , t ∈ [ − ε, + ε ] ).Therefore, we have an open book foliation modi(cid:28)ed by suspension by repla ing the abovetrivial foliation of K × [ − ε, + ε ] by e F ϕ , the tra e of F ϕ on K × [ − ε, + ε ] . Let O ϕ be the resultingfoliation of M and ξ ϕ be its regular Γ -stru ture. For proving theorem 1.2 (when r ≥ ) it issu(cid:30) ient to prove that ξ and ξ ϕ are homotopi . This is done in the next se tion.4. Homotopy of Γ -stru turesWe are going to des ribe a homotopy from ξ ϕ to ξ . Re all the tube T around the binding.For simpli ity, we assume that ea h omponent of T is foliated by a standard Reeb foliation;the same holds true if T is foliated by Wood omponents (in the sense of [17℄). Let T ′ be aslightly larger tube.Lemma 4.1. There exists a homotopy, relative to M r int ( T ′ ) , from ξ ϕ to a new Γ -stru ture ξ on M su h that:1) ξ is trivial on T ;2) ξ is regular on T ′ r int ( T ) with ompa t tori leaves near ∂T and spiraling half- ylinderleaves with boundary in ∂T ′ (as in an open book foliation).Proof. Re all from the introdu tion that we only use thi k Reeb omponents. So there is athird on entri tube T ′′ , T ⊂ int ( T ′′ ) ⊂ int ( T ′ ) , so that T ′′ r int ( T ) is foliated by tori leaves;and int ( T ) is foliated by planes.On ea h omponent of ∂T we have oodinates ( x, y ) oming from the framing of the binding,the x -axis being a parallel and the y -axis being a meridian. Let γ be a parallel in ∂T . The Γ -stru ture whi h is indu ed by ξ ϕ on γ is singular but not trivial and the germ of foliation G along the zero se tion in the normal line bundle A ∼ = γ × R is shown on (cid:28)gure 1.The annulus A is endowed with oordinates ( x, z ) ∈ γ × R . The orientation of the z -axis,whi h is also the orientation of the normal bundle to the foliation O ϕ along γ , points to theinterior of T . So the leaves of G are parallel ir les in { z ≤ } and spiraling leaves in { z > } .Take oordinates ( x, y, r ) on T where r is the distan e to the binding; say that r = 1 on ∂T ′′ and r = 1 / on ∂T . Let λ ( r ) be an even smooth fun tion with r = 0 as unique riti al point,vanishing at r = 1 / and λ (1) < < λ (0) . Consider g : T ′′ → A , g ( x, y, r ) = (cid:0) x, λ ( r ) (cid:1) . z = 0 Figure 1It is easily seen that ξ ϕ | int ( T ′′ ) ∼ = g ∗ G . Let now ¯ λ ( r ) be a new even fun tion oin iding with λ ( r ) near r = 1 , having negative values everywhere and whose riti al set is { r ∈ [0 , / } . Let ¯ g : T ′′ → A , ¯ g ( x, y, r ) = (cid:0) x, ¯ λ ( r ) (cid:1) . A bary entri ombination of λ and ¯ λ yields a homotopyfrom g to ¯ g whi h is relative to a neighborhood of ∂T ′′ . The Γ -stru ture ξ we are looking foris de(cid:28)ned by ξ | int ( T ′′ ) = ¯ g ∗ ( G ) and ξ = ξ ϕ on a neighborhood of M r int ( T ′′ ) . (cid:3) Re all the domain K × [ − ε, + ε ] from the previous se tion. After the following lemma we aredone with the homotopy problem.Lemma 4.2. There exits a homotopy from ξ to ξ relative to K × [ − ε, + ε ] .Proof. Let us denote M ′ := M r int ( K × [ − ε, + ε ]) whi h is a manifold with boundary and orners. It is equivalent to prove that the restri tions of ξ and ξ to M ′ are homotopi relativelyto ∂M ′ . Consider the standard losed 2-disk D = D endowed with the Γ -stru ture ξ D whi his shown on (cid:28)gure 2. d Im ( i ) Figure 2 FRANÇOIS LAUDENBACH AND GAËL MEIGNIEZIt is trivial on the small disk d and regular on the annulus D r int ( d ) . In the regular part,the leaves are ir les near ∂d and the other leaves are spiraling, rossing ∂D transversely. One he ks that the restrition of ξ to M ′ has the form f ∗ ξ D from some map f : M ′ → D . Wetake f | T : T → d to be the open book trivialization of T (re all the binding has a anoni alframing); f | ∂K × [ − ε, + ε ] to be the proje tion pr onto [ − ε, + ε ] omposed with an embedding i : [ − ε, + ε ] → ∂D and f maps ea h leaf of the regular part of ξ to a leaf of the regular part of ξ D . As K does not approa h T , we an take f ( ∂M ′ ) = i ([ − ε, + ε ]) ; a tually, ex ept near T , f is given by the (cid:28)bration over S = ∂D of the open book de omposition.On e ξ has Tsuboi's form (a ording to proposition 2.1), the restrition of ξ to M ′ has asimilar form: ξ = k ∗ ξ D for some map k : M ′ → D . Re all that M ′ is the union of twohandlebodies and a solid ylinder D × [ − ε, + ε ] . Take k to be i ◦ pr on the ylinder and k to be onstant on ea h handlebody. Observe that f and k oin ide on ∂M ′ . As D retra tsby deformation onto the image of i , one dedu es that f and k are homotopi relatively to ∂M ′ . (cid:3) This (cid:28)nishes the proof of theorem 1.2 when r ≥ .5. The ase C bv A o-oriented Γ r -stru ture ξ on M an be realized by a foliation F de(cid:28)ned on a neighborhoodof the 0-se tion in M × R ; it is made of bi-foliated harts whi h are C ∞ in the dire tion ofthe leaves and C r in the dire tion of the (cid:28)bers. Consider su h a box U over an open disk D entered at x ∈ M ; its tra e on the x -(cid:28)ber is an interval I . Ea h leaf of U reads z = f ( x, t ) , x ∈ D , for some t ∈ I . Here f is a fun tion whi h is smooth in x and C r in t , with f ( x , t ) = t ;the foliating property is equivalent to ∂f∂t > . When r = 1 + bv , there is a positive measure µ ( x, t ) on I , without atoms and depending smoothly on x , su h that: ( ∗ ) ∂∂t f ( x, t ) − ∂∂t f ( x, t ) = Z tt µ ( x, t ) . Proposition 5.1. Theorem 1.2 holds true for any lass of regularity r ≥ in luding the lass r = 1 + bv .Proof. The only part of the proof whi h requires some are of regularity is se tion 2, espe iallythe proof of lemma 2.3. Indeed, we have to avoid integrating C ve tor (cid:28)elds. For provinglemma 2.3 with weak regularity we use the lemmas below whi h we shall prove in the ase r = 1 + bv only.Lemma 5.2. Let f : D × I → R be a C r -fun tion as above. Assume I =] − ε, + ε [ and f ( x, − ε ) > − for every x ∈ D . Then there exists a fun tion F : D × [ − , → R of lass C r su h that:1) F ( x, t ) = f ( x, t ) when t ∈ [ − ε, ,2) F ( x, t ) = t when t is lose to − ,3) ∂F∂t > .Proof. Let µ ( x, t ) be the positive measure whose support is [ − ε, su h that formula ( ∗ ) holdsfor every ( x, t ) ∈ D × [ − ε, and t = 0 . There exists another positive measure ν ( x, t ) , smoothin x and whose support is ontained in ] − , − ε ] , su h that ( ∗∗ ) f ( x, − ε ) = − Z − ε − (cid:18)Z t − ν ( x, τ ) (cid:19) dt. Then a solution is F ( x, t ) = − Z t − (cid:18)Z s − (cid:0) µ ( x, τ ) + ν ( x, τ ) (cid:1)(cid:19) ds. (cid:3) Lemma 5.3. Let A and A two disjoint losed sub-disks of D . Let F and F be two solutionsof lemma 5.2. Then there exists a third solution whi h equals F when x ∈ A and F when x ∈ A .Proof. Both solutions F and F di(cid:27)er by the hoi e of the measure ν ( x, t ) in formula ( ∗∗ ) ,whi h is ν i for F i . Choose a partition of unity λ ( x ) + λ ( x ) with λ i = 1 on A i . Then ν ( x, t ) = λ ( x ) ν ( x, t ) + λ ( x ) ν ( x, t ) yields the desired solution. (cid:3) The proof of proposition 5.1 is now easy. As already said, it is su(cid:30) ient to prove lemma 2.3in lass C r , r ≥ . It is an extension problem of a foliation given near the 1-skeleton T r [1] × { } to T r [1] × [ − , . One overs T r [1] by (cid:28)nitely many n -disks D j . The problem is solved in ea h D j × [ − , by applying lemma 5.2. By applying lemma 5.3 one makes the di(cid:27)erent extensionsmat h together. (cid:3)
6. Homotopy lass of plane fieldsIt is possible to enhan e theorem 1.1 by pres ribing the homotopy lass of the underlying o-oriented plane (cid:28)eld (see proposition 6.1 below). The question of doing the same with respe tto theorem 1.2 is more subtil (see proposition 6.3).Proposition 6.1. Given a o-oriented Γ -stru ture ξ on the losed 3-manifold M and a ho-motopy lass [ ν ] of o-oriented plane (cid:28)eld in the tangent spa e τ M , there exists a regular Γ -stru ture ξ reg homotopi to ξ whose underlying foliation F reg ∩ M has a tangent o-orientedplane (cid:28)eld in the lass [ ν ] .Before proving it we (cid:28)rst re all some well-known fa ts on o-oriented plane (cid:28)elds (see [4℄).Given a base plane (cid:28)eld ν , a suitable Thom-Pontryagin onstru tion yields a natural bije tionbetween the set of homotopy lasses of plane (cid:28)elds on M and Ω ν ( M ) , the group of ( o)bordism lasses of ν -framed and oriented losed (maybe non- onne ted) urves in M . A ν -framing ofthe urve γ is an isomorphism of (cid:28)ber bundles ε : ν ( γ, M ) → ν | γ , whose sour e is the normalbundle to γ in M . We denote γ ε the urve endowed with this framing. Moreover, given γ ε , if γ ′ is homologous to γ in M there exists a ν -framing ε ′ su h that ( γ ′ ) ε ′ is obordant to γ ε .0 FRANÇOIS LAUDENBACH AND GAËL MEIGNIEZProof of 6.1. We an start with an open book foliation O ϕ yielded by theorem 1.2. Let ν beits tangent plane (cid:28)eld. Near the binding, the meridian loops (out of T ) are transverse to O ϕ and homotopi to zero in M . As a onsequen e, ea h 1-homology lass may be represented aswell by a (multi)- urve in a page or by a onne ted urve out of T positively transverse to allpages. We do the se ond hoi e for γ ε , the ν -framed urve whose obordism lass en odes [ ν ] with respe t to ν .Hen e we are allowed to turbulize O ϕ along γ . In a small tube T ( γ ) about γ , we put aWood omponent. Outside, the leaves are spiraling around ∂T ( γ ) . Let O turbϕ be the resultingfoliation. Whatever the hosen type of Wood omponent is, the Γ -stru tures of O turbϕ and O ϕ are homotopi by arguing as in se tion 4. But the framing ε tells us whi h sort of Wood omponent will be onvenient for getting the desired lass [ ν ] (see lemma 6.1 in [17℄). (cid:3) In the previous statement, we have lost the ni e model we found in theorem 1.2. A tually,thanks to a lemma of Vin ent Colin [3℄, it is possible to re over our model, at least when M isorientable (see below proposition 6.3).Lemma 6.2. (Colin) Let ( B, p ) be an open book de omposition of M and γ be a simple on-ne ted urve in some page P . Assume γ is orientation preserving. Then there exist a positivestabilization ( B ′ , p ′ ) of ( B, p ) and a urve γ ′ in B ′ whi h is isotopi to γ in M . When γ is amulti- urve, the same holds true after a sequen e of stabilizations.The positive Hopf open book de omposition of the 3-sphere is the one whose binding is madeof two unknots with linking number +1; a page is an annulus foliated by (cid:28)bers of the Hopf(cid:28)bration S → S . A positive stabilization is a (cid:16) onne ted sum(cid:17) with this open book. The newpage P ′ is obtained from P by plumbing an annulus A whose ore bounds a disk in M (see [7℄for more details and other referen es).Proof. If γ is onne ted, only one stabilization is needed. We are going to explain this aseonly. A tubular neighborhood of γ in P is an annulus.Choose a simple ar α in P joining γ to some omponent β of B without rossing γ again.Let ˜ γ be a simple ar from β to itself whi h follows α − ∗ γ ∗ α . The orientation assumptionimplies that the surgery of β by ˜ γ in P provides a urve with two onne ted omponents, oneof them being isotopi to γ in P . Let P π be the page opposite to P and R : P → P π the time π of a (cid:29)ow transverse to the pages (and stationary on B ). The ore urve C of the annulus A that we use for the plumbing is the union ˜ γ ∪ R (˜ γ ) . And A is (+1) -twisted around C (withrespe t to its unknot framing) as in the Hopf open book. Let H be the 1-handle whi h is the losure of A r P . Surgering B by H provides the new binding. By onstru tion, one of its omponents is isotopi to γ . (cid:3) Proposition 6.3. Let O ϕ be an open book foliation modi(cid:28)ed by suspension, whose its underlyingopen book is denoted ( B, p ) . Let ν be its tangent o-oriented plane (cid:28)eld. Let γ ε be a ν -framed urve in M and [ ν ] be its asso iated lass of plane (cid:28)eld. Assume γ is orientation preserving.Then there exists an open book foliation O ′ ϕ with the following properties:1) its tangent plane (cid:28)eld is in the lass [ ν ] ;2) the suspension modi(cid:28) ation is the same for O ′ ϕ as for O ϕ and is supported in K × [ − ε, + ε ] ;13) as Γ -stru tures, O ϕ and O ′ ϕ are homotopi .Proof. As said in the proof of 6.1, up to framed obordism, γ ε may be hosen as a simple(multi)- urve in one page P of ( B, p ) . Applying Colin's lemma provides a stabilization ( B ′ , p ′ ) su h that, up to isotopy, γ lies in the new binding. Observe that, if K is in P , K is still in thenew page P ′ ; hen e 2) holds for any open book foliation arried by ( B ′ , p ′ ) . On e γ ε is in thebinding, for a suitable Wood omponent foliating a tube about γ ε , item 1) is ful(cid:28)lled. Finallyitem 3) follows from item 2) and the proofs in se tion 4. (cid:3)
7. Case of a Γ -stru ture with a twisted normal bundleWhat happens when the bundle ν normal to ξ is twisted? It is known that a ne essary ondition to regularization is the existen e of a (cid:28)bered embedding i : ν → τ M into the tangent(cid:28)ber bundle to M . Conversely, assuming that this ondition is ful(cid:28)lled, we are going to statea normal form theorem analogous to theorem 1.2. Sin e no step of the previous proof an beimmediately adapted to this situation, we believe that it deserves a sket h of proof.7.1. In the (cid:28)rst step (Tsuboi's onstru tion), we do not have (cid:16)Morse position(cid:17) with respe t toa triangulation, sin e index and o-index of a singularity annot be distinguished. Instead oflemma 2.2, we have the following statement.After some homotopy, ξ has Morse singularities and admits a pseudo-gradient whosedynami s has no re urren e (that is, every orbit has a (cid:28)nite length).Here, by a pseudo-gradient, it is meant a smooth se tion X of Hom ( ν, τ M ) , a twisted ve tor(cid:28)eld indeed, su h that X · ξ < ex ept at the singularities (this sign is well-de(cid:28)ned whatevera lo al orientation of ν , or o-orientation of ξ , is hosen); su h a pseudo-gradient always existsby using an auxiliary Riemannian metri .Sket h of proof. Generi ally ξ has Morse singularities. Let X be a (cid:28)rst pseudo-gradient,whi h is required to be the usual negative gradient in Morse oordinates near ea h singular-ity. Finitely many mutually disjoint 2-disks of M are hosen in regular leaves of ξ su h thatevery orbit of X rosses one of them. Following Wilson's idea [15℄, ξ and X are hanged ina neighborhood D × [ − , +1] of ea h disk into a plug su h that every orbit of the modi(cid:28)edpseudo-gradient X is trapped by one of the plugs. The plug has the mirror symmetry withrespe t to D × { } . In D × [0 , we just modify ξ by introdu ing a an elling pair of singu-larities, enter-saddle. (cid:3) Let G be the losure of the one-dimensional invariant manifold of all saddles. It is a graph.We laim: ν | G is orientable. Indeed, we orient ea h edge from its saddle end point to its enterend point. This is an orientation of ν | G over the omplement of the verti es. It is easily he kedthat this orientation extends over the verti es. Thus X be omes a usual ve tor (cid:28)eld near G and we have an arbitrarily small tubular neighborhood H of G whose boundary is transverseto X , and X enters H . Now, the negative ompletion of ξ | H an be performed as in lemma2.3.The omplement ˆ M of int H in M is (cid:28)bered over a surfa e Σ , the (cid:28)bers being intervals( ∼ = [ − , ) tangent to X . By taking a se tion we think of Σ as a surfa e in M r H . Sin e2 FRANÇOIS LAUDENBACH AND GAËL MEIGNIEZ ξ is not o-orientable, Σ is one-sided and G is onne ted. Arguing as in se tion 2, after somehomotopy, ξ be omes trivial on H and transverse to X on ˆ M , hen e a suspension foliation orresponding to a representation ϕ : π (Σ) → Dif f c (] − , .7.2. In the se ond step (Giroux's onstru tion), we have to leave the open books and we needa twisted open book. It is made of the following:- a binding B whi h is a 1-dimensional losed o-orientable submanifold in M ;- a Seifert (cid:28)bration p : M r B → [ − , +1] whi h has two one-sided ex eptional surfa e(cid:28)bers p − ( ± and whi h is a proper smooth submersion over the open interval;- when t goes to ± , p − ( t ) tends to a 2-fold overing of p − ( ± ;- near B the foliation looks like an open book.The ex eptional (cid:28)bers are ompa ti(cid:28)ed by B as smooth surfa es with boundary. But, for t ∈ ] − , +1[ , p − ( t ) is ompa ti(cid:28)ed by B as a losed surfa e showing (in general) an angle along B . Noti e that, sin e B is o-orientable, a twisted open book gives rise to a smooth foliationwhere ea h omponent of the binding is repla ed by a Reeb omponent, the pages spiralingaround it.Su h an open book is generated by a one-sided Giroux surfa e, whi h is the union of the ompa ti(cid:28)ed ex eptional (cid:28)bers. Abstra tly, a one-sided Giroux surfa e in M with respe t toa Morse fun tion f : M → R is a one-sided surfa e S su h that f | S has the same riti alpoints and the same extrema as f and ful(cid:28)lls the extra ondition: for every regular value t ∈ R , f − ( t ) ∩ S is a two-sided urve in the level set f − ( t ) . Starting with ( S, f ) where f isa self-indexing Morse fun tion, a twisted open book is easily onstru ted. Its binding is the o-orientable urve f − (3 / ∩ S . In general su h a one-sided Giroux surfa e (or twisted openbook) does not exist on M ; the obstru tion lies in the existen e of a twisted line subbundle of τ M . Nevertheless, with a suitable assumption, we have an analogue of proposition 3.1:Let i : ν → τ M be an embedding of a twisted line bundle. Let K ⊂ M be a ompa t onne ted one-sided surfa e whose boundary is not empty. Assume the following: ν | K is twisted, ν | ∂K is trivial and the normal bundle ν ( K, M ) is homotopi to i ( ν ) | K . Thenthere exists a twisted open book with one ex eptional page ontaining K in its interior.Sket h of proof. We give the proof only in the setting of 7.1 by taking K to be the losureof Σ r d where d is a small losed 2-disk in Σ . Re all the (cid:28)bration ρ : ˆ M → Σ . Let H ′ bethe handlebody made of H to whi h is glued the 1-handle ρ − ( d ) . Let M ′ be the omplementof int H ′ . Consider a minimal system of mutually disjoint ompression disks d , ..., d g of H , sothat utting H along them yields a ball; g is the genus of H . Thus d , d , ..., d g is a minimalsystem of mutually disjoint ompression disks of H ′ .We laim: g + 1 is even. Indeed, by assumption there exists a non vanishing se tion of thebundle Hom ( ν, τ M ) . Thus, the number of zeroes of the pseudo-gradient X is even and theEuler hara teristi χ ( G ) of the graph G is even. As G is onne ted, the genus g is odd, whi hproves the laim.Now, one follows Giroux's algorithm for ompleting K to a losed Giroux's surfa e. On thesurfa e ∂M ′ the union of the atta hing urves ∂d , ∂d , ..., ∂d g is not separating. Then, aftersome isotopy, for ea h i = 0 , ..., g , ∂d i rosses ∂K in exa tly two points a i , b i linked by an ar α i α ′ i ) in ∂d i (resp. ∂K ), so that α i ∪ α ′ i bounds a disk in ∂M ′ . Moreover, one an arrangethat all the ar s α , ..., α g are parallel. Also we link a i , b i by a simple ar in d i . Now, ea h ompression disk de(cid:28)nes, simultaneously, a 1-handle whi h is glued to K and a 2-handle whi his glued to M ′ , yielding a proper surfa e K in some 3-submanifold M ′′ of M , whose omple-ment is a ball. The boundary of K is made of g + 2 parallel urves in the sphere ∂M ′′ . Asthis number is odd, Giroux des ribed a pro ess of adding an elling pairs of 1- and 2- handleswhose e(cid:27)e t is to hange K into K ⊂ M ′′ su h that ∂K is made of one urve only ([5℄, p.676-677). Hen e, K an be losed into a Giroux's surfa e. (cid:3) Theorem 7.3. Let ξ be a non o-orientable Γ -stru ture on M whose normal bundle ν embedsinto τ M . Then ξ is homotopi to a twisted open book foliation modi(cid:28)ed by suspension.Sket h of proof. Let K = Σ r int D be the surfa e with a hole, where Σ was built in the(cid:28)rst step 7.1; it meets the required assumptions for building a twisted open book.The twisted open book built in the se ond step gives rise to a foliation O . Indeed, as thebinding B is o-orientable, it is allowed to spiral the pages around a tubular neighborhood of B . The tube itself is foliated by thi k Reeb omponents. As in the o-orientable ase, we anmodify the open book foliation in a neighborhood of K using the representation ϕ , yieldingthe foliation O ϕ and its asso iated regular Γ -stru ture ξ ϕ . We have to prove that ξ and ξ ϕ arehomotopi . We may suppose that ξ is in Tsuboi form (7.1).We observe that the total spa e of ν has a foliation F (unique up to isomorphism) trans-verse to the (cid:28)bers, having the zero se tion as a leaf and whose all non trivial holonomy elementshave order 2. It de(cid:28)nes the trivial Γ -stru ture ξ in the twisted sense. Using notation of 7.2,one an prove that ξ | H ′ and ξ ϕ | H ′ are both homotopi to ξ | H ′ . Moreover, both homotopies oin ide on the boundary ∂H ′ (on H ′ , it is su(cid:30) ient to think of the ase when ϕ is the trivialrepresentation. Thus ξ | H ′ and ξ ϕ | H ′ are homotopi relative to the boundary. (cid:3) O ϕ , it it possible to have the normal (cid:28)eld inany homotopy lass of embeddings ν → τ M .Indeed, a urve in M is homotopi to a urve transverse to O ϕ if and only if it does not twist ν . But these homology lasses are exa tly those whi h appear as a (cid:28)rst di(cid:27)eren e homology lass when omparing two embeddings j , j : ν → τ M , sin e a losed urve whi h twits ν isnot a y le in H ( M, Z or ( ν ) ) ∼ = H ( M, Z or ( ν ∗⊗ τM ) ) .Referen es[1℄ J.W. Alexander, Note on Riemann spa es, Bull. Amer. Math. So . 26 (1920), 370-372.[2℄ J.W. Alexander, A lemma on systems of knotted urves, Pro . Nat. A ad. S i. U.S.A. 9 (1923), 93-95.[3℄ V. Colin, private ommuni ation.[4℄ H. Geiges, An Introdu tion to Conta t Topology, Cambridge Univ. Press, 2008.[5℄ E. Giroux, Convexité en topologie de onta t, Comment. Math. Helv. 66 (1991), 637-677.[6℄ E. Giroux, private ommuni ation.[7℄ E. Giroux, N. Goodman, On the stable equivalen e of open books in three-manifolds, Geometry & Topology10 (2006), 97-114.4 FRANÇOIS LAUDENBACH AND GAËL MEIGNIEZ[8℄ A. Hae(cid:29)iger, Homotopy and integrability, 133-175 in: Manifolds-Amsterdam 1970, L.N.M. 197, Springer,1971.[9℄ J. N. Mather, On Hae(cid:29)iger's lassifying spa e I, Bull. Amer. Math. So . 77 (1971), 1111-1115.[10℄ J. N. Mather, Integrability in odimension 1, Comment. Math. Helv. 48 (1973), 195-233.[11℄ J. N. Mather, Commutator of di(cid:27)eomorphisms, III: a group whi h is not perfe t, Comment. Math. Helv.60 (1985), 122-124.[12℄ G. Meigniez, Regularization and minimization of Γ1