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On smooth foliations with Morse singularities

Lilia Rosati

Universita di Firenze,Dipartimento di Matematica “U. Dini”,viale Morgagni 67/A, 50134 Firenze

e-mail: rosati@math.unifi.it

Abstract

Let M be a smooth manifold and letF be a codimension one,C∞ foliation onM , with isolated singularitiesof Morse type. The study and classification of pairs(M,F) is a challenging (and difficult) problem. In thissetting, a classical result due to Reeb [Reeb] states that a manifold admitting a foliation with exactly two center-type singularities is a sphere. In particular this is true ifthe foliation is given by a function. Along these linesa result due to Eells and Kuiper [Ee-Kui] classify manifoldshaving a real-valued function admitting exactlythree non-degenerate singular points. In the present paper, we prove a generalization of the above mentionedresults. To do this, we first describe the possible arrangements of pairs of singularities and the correspondingcodimension one invariant sets, and then we give an elimination procedure for suitable center-saddle and somesaddle-saddle configurations (of consecutive indices).In the second part, we investigate if other classical results, such as Haefliger and Novikov (Compact Leaf) the-orems, proved for regular foliations, still hold true in presence of singularities. At this purpose, in the singularset,Sing(F) of the foliationF , we considerweakly stablecomponents, that we define as those componentsadmitting a neighborhood where all leaves are compact. IfSing(F) admits only weakly stable components,given by smoothly embedded curves diffeomorphic toS1, we are able to extend Haefliger’s theorem. Finally,the existence of a closed curve, transverse to the foliation, leads us to state a Novikov-type result.

Acknoledgements

I am very grateful to prof. Bruno Scardua for proposing me such an interesting subject and for his valuableadvice. My hearthy good thanks to prof. Graziano Gentili forhis suggestions on the writing of this article.

1 Foliations and Morse Foliations

Definition 1.1 A codimensionk, foliated manifold(M,F) is a manifoldM with a differentiable structure,given by an atlas{(Ui, φi)}i∈I , satisfying the following properties:(1) φi(Ui) = Bn−k × Bk;(2) inUi ∩ Uj 6= ∅, we haveφj ◦ φ

−1i (x, y) = (fij(x, y), gij(y)),

where{fij} and {gij} are families of, respectively, submersions and diffeomorphisms, defined on naturaldomains. Given a local chart (foliated chart) (U, φ), ∀x ∈ Bn−k andy ∈ Bk, the setφ−1(·, y) is aplaqueandthe setφ−1(x, ·) is atransverse section.

The existence of a foliated manifold(M,F) determines a partition ofM into subsets, theleaves, definedby means of an equivalence relation, each endowed of an intrinsic manifold structure. Letx ∈ M ; we denoteby Fx or Lx the leaf ofF throughx. With the intrinsic manifold structure,Fx turns to be an immersed (butnot embedded, in general) submanifold ofM .In an equivalent way, a foliated manifold(M,F) is a manifoldM with a collection of couples{(Ui, gi)}i∈I ,

1

where{Ui}i∈I is an open covering ofM , gi : Ui → Bk is a submersion,∀i ∈ I, and thegi’s satisfy the cocyclerelations,gi = gij ◦ gj, gii = id, for suitable diffeomorphismsgij : Bk → Bk, defined whenUi ∩ Uj 6= ∅.EachUi is said afoliation box, andgi a distinguished map. The functionsγij = dgij are the transition maps[Stee] of a bundleNF ⊂ TM , normal to the foliation. More completely, there exists a G-structure onM[Law], which is a reduction of the structure groupGL(n, R) of the tangent bundle to the subgroup of the

matrices

(A B0 C

), whereA ∈ GL(n− k, R) andC ∈ GL(k, R).

A codimension one,C∞ foliation of a smooth manifoldM , with isolated singularities, is a pairF =(F∗, Sing(F)), whereSing(F) ⊂ M is a discrete subset andF∗ is a codimension one,C∞ foliation (in theordinary sense) ofM∗ = M \Sing(F). Theleavesof F are the leaves ofF∗ andSing(F) is thesingular setof F . A pointp is aMorse singularityif there is aC∞ function,fp : Up ⊂ M → R, defined in a neighborhoodUp of p, with a (single) non-degenerate critical point atp and such thatfp is a local first integral of the foliation,i.e. the leaves of the restrictionF|Up

are the connected components of the level hypersurfaces offp in Up\{p}.A Morse singularityp, of indexl, is asaddle, if 0 < l < n (wheren = dimM ), and acenter, if l = 0, n. Wesay that the foliationF has asaddle-connectionwhen there exists a leaf accumulated by at least two distinctsaddle-points. AMorse foliationis a foliation with isolated singularities, whose singularset consists of Morsesingularities, and which has no saddle-connections. In this way if a Morse foliation has a (global) first integral,it is given by a Morse function.Of course, the first basic example of a Morse foliation is indeed a foliation defined by a Morse function onM .A less evident example is given by the foliation depicted in figure 2.

In the literature, the orientability of a codimensionk (regular) foliation is determined by the orientability ofthe(n− k)-plane field tangent to the foliation,x → TxFx. Similarly transverse orientability is determined bythe orientability of a complementaryk-plane field. A singular, codimension one foliation,F , is transverselyorientable[Cam-Sc] if it is given by the natural(n − 1)-plane field associated to a one-form,ω ∈ Λ1(M),which is integrable in the sense of Frobenius. In this case, choosing a Riemannian metric onM , we may finda global vector field transverse to the foliation,X = grad(ω), ωX ≥ 0, andωxXx = 0 if and only if x is asingularity for the foliation (ω(x) = 0). A transversely orientable, singular foliationF of M is a transverselyorientable (regular) foliationF∗ of M∗ in the sense of the classical definition. Viceversa, ifF∗ is transverselyorientable, in general,F is not.

Thanks to the Morse Lemma [Mil 1], Morse foliations reduce tofew representative cases. On the otherhand, Morse foliations describe a large class among transverseley orientable foliations. To see this, letF be afoliation defined by an integrable one-form,ω ∈ Λ1(M), with isolated singularies. We proceed with a localanalysis; using a local chart around each singularity, we may supposeω ∈ Λ1( Rn), ω(0) = 0, and 0 is theonly singularity ofω. We haveω(x) =

∑i hi(x)dx

i and, in a neighborhood of0 ∈ Rn, we may writeω(x) =ω1(x) +O(|x|2), whereω1 is the linear part ofω, defined byω1(x) =

∑i,j aijx

idxj , aij = ∂hi(x)/∂xj . Werecall that the integrability ofω implies the integrability ofω1 and that the singularity0 ∈ Rn is said to be nondegenerate if and only if(aij) ∈ R(n) is non degenerate; in this latter case(aij) is symmetric: it is the hessianmatrix of some real functionf , defining the linearized foliation (ω1 = df ). We have

{transverseley orientable foliations, with Morse singularities} ={foliations, defined by non degenerate linear one-forms} ⊂

{foliations, defined by non degenerate one-forms}.

Let (σ, τ) be the spaceσ of integrable one-forms inRn, with a singularity at the origin, endowed with theC1-Whitney topology,τ . If ω, ω′ ∈ σ, we sayω equivalentω′ (ω ∼ ω′) if there exists a diffeomorphismφ : Rn → Rn, φ(0) = 0, which sends leaves ofω into leaves ofω′. Moreover, we sayω is structurally stable,if there exists a neighborhoodV of ω in (σ, τ) such thatω′ ∼ ω, ∀ω′ ∈ V .Theorem 1.2 (Wagneur)[Wag] The one-formω ∈ σ is structurally stable, if and only if the index of0 ∈Sing(ω) is neither2 nor n− 2.

Let us denote byS the space of foliations defined by non degenerate one-forms with singularities, whoseindex is neither2 norn− 2. If S1 ⊂ S is the subset of foliations defined by linear one-forms, thenwe have:Corollary 1.3 There exists a surjective map,

s : S1 → S/∼.

2

PSfrag replacementsF1F2

L

L

L0 L1

L2

L2

Figure 1: F1,F2 foliations on RP 2:Hol(L,F1) = {e}, Hol(L0,F1) = {e, g0},g20 = e, Hol(L1,F2) = {e, g1}, g1orientation reversing diffeomorphism,Hol(L2,F2) = {e, g2}, g2 generator ofunilateral holonomy.

PSfrag replacements

p2

p3

p1

q

a

a

Figure 2: A singular foliation of the sphereS2,which does not admit a first integral. With thesame spirit, a singular foliation onS3 may begiven.

2 Holonomy and Reeb Stability Theorems

It is well known that a basic tool in the study of foliations isthe holonomy of a leaf (in the sense of Ehresmann).If L is a leaf of a codimensionk foliation (M,F), the holonomyHol(L,F) = Φ(π1(L)), is the image of arepresentation,Φ : π1(L) → Germ( Rk, 0), of the fundamental group ofL into the germs of diffeomorphismsof Rk, fixing the origin. Letx ∈ L andΣx be a section transverse toL at x; with abuse of notation, we willwrite that a diffeomorphismg : Dom(g) ⊂ Σx → Σx, fixing the origin, is an element of the holonomy group.For codimension one foliations (k = 1), we may have:(i) Hol(L,F) = {e}, (ii) Hol(L,F) = {e, g}, withg2 = e, g 6= e, (iii) Hol(L,F) = {e, g}, wheregn 6= e, ∀n, andg is a (orientation preserving or reversing)diffeomorphism. In particular, among orientation preserving diffeomorphisms, we might find ag : Σx → Σx,such thatg is the identity on one component ofΣx \ {x} and it is not the identity on the other; in this case, wesay thatL hasunilateral holonomy(see figure 1 for some examples). We recall Reeb Stability Theorems (cfr.,for example, [Cam-LN] or [Mor-Sc]).Theorem 2.1 (Reeb Local Stability)LetF be aC1, codimensionk foliation of a manifoldM andF a compactleaf with finite holonomy group. There exists a neighborhoodU ofF , saturated inF (also calledinvariant), inwhich all the leaves are compact with finite holonomy groups.Further, we can define a retractionπ : U → Fsuch that, for every leafF ′ ⊂ U , π|F ′ : F ′ → F is a covering with a finite number of sheets and, for eachy ∈ F , π−1(y) is homeomorphic to a disk of dimensionk and is transverse toF . The neighborhoodU can betaken to be arbitrarily small.

The last statement means in particular that, in a neighborhood of the point corresponding to a compact leafwith finite holonomy, the space of leaves is Hausdorff.

Under certain conditions the Reeb Local Stability Theorem may replace the Poincare Bendixon Theorem[Pal-deM] in higher dimensions. This is the case of codimension one, singular foliations(Mn,F), with n ≥ 3,and some center-type singularity inSing(F).Theorem 2.2 (Reeb Global Stability)LetF be aC1, codimension one foliation of a closed manifold,M . IfF contains a compact leafF with finite fundamental group, then all the leaves ofF are compact, with finitefundamental group. IfF is transversely orientable, then every leaf ofF is diffeomorphic toF ; M is the totalspace of a fibrationf : M → S1 overS1, with fibreF , andF is the fibre foliation,{f−1(θ)|θ ∈ S1}.

This theorem holds true even whenF is a foliation of a manifold with boundary, which is, a priori, tangenton certain components of the boundary and transverse on other components [God]. In this setting, letH l ={(x1, . . . , xl) ∈ Rl|xl ≥ 0}. Taking into account definition 1.1, we say that a foliation of a manifold withboundary istangent, respectivelytransverse to the boundary, if there exists a differentiable atlas{(Ui, φi)}i∈I ,such that property (1) of the above mentioned definition holds for domainsUi such thatUi ∩ ∂M = ∅, whileφi(Ui) = Bn−k × H k, respectively,φi(Ui) = H n−k × Bk for domains such thatUi ∩ ∂M 6= ∅. Moreover,we ask that the change of coordinates has still the form described in property (2). Recall thatF|∂M is a regularcodimensionk − 1 (respectively,k) foliation of the(n− 1)-dimensional boundary. After this, it is immediateto write the definition for foliations which are tangent on certain components of the boundary and transverse

3

PSfrag replacements b

b

a

a

Figure 3:n = 2: a singular foliation with center-type singularities, having no first integral.

PSfrag replacements

R1

R2

R3

Ws(q))Wu(q)

p

q

ULF

Figure 4: A trivial couple center-saddle(p, q)(Theorem 3.5, case(i)).

on others.Observe that, for foliations tangent to the boundary, we have to replaceS1 with [0, 1] in the second statementof the Reeb Theorem 2.2 (see Lemma 5.6).

We say that a component ofSing(F) is weakly stableif it admits a neighborhood,U , such thatF|U is afoliation with all leaves compact. The problem of global stability for a foliation with weakly stable singularcomponents may be reduced to the case of foliations of manifolds with boundary, tangent to the boundary. It isenough to cut off an invariant neighborhood of each singularcomponent.

Holonomy is related to transverse orientability by the following:Proposition 2.3LetL be a leaf of a codimension one (Morse) foliation(M,F). If Hol(L,F) = {e, g}, whereg2 = e, g 6= e, thenF is non-transversely orientable. Moreover, ifπ : M → M/F is the projection onto thespace of leaves, then∂(M/F) 6= ∅ andπ(L) ∈ ∂(M/F).Proof. We choosex ∈ L and a segmentΣx, transverse to the foliation atx. Theng : Σx → Σx turns out tobeg(y) = −y. Let y → Ny a 1-plane field complementary to the tangent plane fieldy → TyFy. Supposewe may choose a vector fieldy → X(y) such thatNy = span{X(y)}. Then it shoud beX(x) = −X(x) =(dg)x(X(x)), a contraddiction. Consider the space of leaves nearL; this space is the quotient ofΣx withrespect to the equivalence relation∼ which identifies points onΣx of the same leaf. ThenΣx/∼ is a segmentof type(z, x] or [x, z), whereπ−1(x) = L.

At last we recall a classical result due to Reeb.Theorem 2.4 (Reeb Sphere Theorem) [Reeb]A transversely orientable Morse foliation on a closed manifold,M , of dimensionn ≥ 3, having only centers as singularities, is homeomorphic to then-sphere.This result is proved by showing that the foliation considered must be given by a Morse function with only twosingular points, and therefore thesis follows by Morse theory. Notice that the theorem still holds true forn = 2,with a different proof. In particular, the foliation need not to be given by a function (see figure 3).

3 Arrangements of singularities

In section 4 we will study the elimination of singularities for Morse foliations. To this aim we will describe herehow to identify special “couples” of singularities and we will study the topology of the neighbouring leaves.Definition 3.1 Let n = dimM,n ≥ 2. We define the setC(F) ⊂ M as the union of center-type singularitiesand leaves diffeomorphic toSn−1 (with trivial holonomy ifn = 2) and for a center singularity,p, we denoteby Cp(F) the connected component ofC(F) that containsp.Proposition 3.2LetF be a Morse foliation on a manifoldM . We have:(1) C(F) and Cp(F) are open inM .(2) Cp(F) ∩ Cq(F) 6= ∅ if and only if Cp(F) = Cq(F). Cp(F) = M if and only if∂ Cp(F) = ∅. In thiscase the singularities ofF are centers and the leaves are all diffeomorphic toSn−1.(3) If q ∈ Sing(F) ∩ ∂ Cp(F), thenq must be a saddle; in this case∂ Cp(F) ∩ Sing(F) = {q}. Moreover,for n ≥ 3 andF transversely orientable,∂ Cp(F) 6= ∅ if and only if ∂ Cp(F) ∩ Sing(F) 6= ∅. In thesehypotheses,∂ Cp(F) contains at least one separatrix of the saddleq.(4) ∂ Cp(F) \ {q} is closed inM \ {q}.

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q

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Figure 5: A saddleq of index1 (n− 1), accumu-lating one centerp (Theorem 3.5, case(ii) ).

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q

γ

Figure 6: A saddleq of index1 (n− 1), accumu-lating one centerp (Theorem 3.5, case(iii) ).

Figure 7: Two saddles in trivial coupling for thefoliation defined by the functionfǫ = −x2

2 + y3

3 −

ǫy + z2

2 , (ǫ > 0).

PSfrag replacements

L1L2p q

S1

S2

Σ

Σ

no intersection

legenda

Figure 8: A dead branch of a trivial couple of sad-dles for a foliated manifold(Mn,F), n ≥ 3.

Proof. (1) C(F) is open by the Reeb Local Stability Theorem 2.1. (3) If non-empty, ∂ Cp(F) ∩ Sing(F)consists of a single saddleq, as there are no saddle connections. The second part followsby the Reeb GlobalStability Theorem for manifolds with boundary and the thirdby the Morse Lemma. (4) By the TransverseUniformity Theorem (see, for example, [Cam-LN]), it follows that the intrinsic topology of∂ Cp(F) \ {q}coincides with its natural topology, as induced byM \ {q}.

We recall the following (cfr., for example [Mor-Sc]):Lemma 3.3 (Holonomy Lemma)LetF be a codimension one, transversely orientable foliation onM , letAbe a leaf ofF andK be a compact and path-connected set. Ifg : K → A is aC1 map homotopic to a constantin A, theng has anormal extensioni.e. there existǫ > 0 and aC1 mapG : K × [0, ǫ] → M such thatGt(x) = Gx(t) = G(x, t) has the following properties:(i) G0(K) = g, (ii) Gt(K) ⊂ A(t) for some leafA(t)of F with A(0) = A, (iii) ∀x ∈ K the curveGx([0, ǫ]) is normal toF .

For the case of center-saddle pairings we prove the following descriptions of the separatrix:Theorem 3.4Let F be aC∞, codimension one, transversely orientable, Morse foliation of a compactn-manifold,M , n ≥ 3. Letq be a saddle of indexl /∈ {1, n− 1}, accumulating to one centerp. LetL ⊂ Cp(F)be a spherical leaf intersecting a neighborhoodU of q, defined by the Morse Lemma. Then∂ Cp(F) \ {q}has a single connected component (see figure 13) and is homeomorphic toSn−1/Sl−1. If F is a leaf such thatF ∩

(U \ Cp(F)

)6= ∅, thenF is homeomorphic to Bl×Sn−l−1∪φ Bl×Sn−l−1, whereφ is a diffeomorphism

of the boundary (for example, we may haveF ≃ Sl × Sn−l−1, but alsoF ≃ Sn−1, for l = n/2).Proof. Let ω ∈ Λ1(M) be a one-form defining the transversely orientable foliation. We choose a riemannianmetric onM and we consider the transverse vector fieldXx = grad(ω)x. We suppose||X || = 1. In U , wehaveX = h · grad(f) for some real functionh > 0 defined onU . Further, we may suppose that∂U followsthe orbits ofX in a neighborhood of∂ Cp(F).The Morse Lemma gives a local description of the foliation near its singularities; in particular the local topologyof a leaf near a saddle of indexl is given by the connected components of the level sets of the functionf(x) =−x2

1 − · · · − x2l + x2

l+1 + · · · + x2n. If, for c ≥ 0, we writef−1(c) = {(x1, . . . , xn) ∈ Rn|x2

1 + · · · +

5

x2l + c = x2

l+1 + · · · + x2n}, it is easy to see thatf−1(0) is homeomorphic to a cone overSl−1 × Sn−l−1

andf−1(c) ≃ Bl × Sn−l−1 (c > 0). Similarly, we obtainf−1(c) ≃ Bn−l × Sl−1 for c < 0. Therefore,by our hypothesis onl, the level sets are connected; in particular the separatrixS ⊃ f−1(0) is unique and∂ Cp(F) = S ∪ {q}; moreoverU is splitted byf−1(0) in two different components. A priori, a leaf mayintersect more than one component. AsF is transversely orientable, the holonomy is an orientationpreservingdiffeomorphism, and then a leaf may intersect only non adiacent components; then this is not the case, in ourhypotheses.Let L be a spherical leaf⊂ Cp(F) enough nearq. ThenL ∩ U 6= ∅ and it is not restrictive to suppose itis given byf−1(c) for somec < 0. We define the compact setK = Sn−1 \ Bn−l × Sl−1 ≃ L \ U . As

n ≥ 3, the compositionK≃

// L \ U �

� ı// L is homotopic to a constant in its leaf. By

the proof of the Holonomy Lemma 3.3,L \ U projects diffeomorphically ontoA(ǫ) = ∂ Cp(F), by meansof the constant-speed vector field,X . Together with the Morse Lemma, this gives a piecewise description of∂ Cp(F), which is obtained by piecing pieces toghether. It comes out∂ Cp(F) ≃ Sn−1/Sl−1, a set with thehomotopy type ofSn−1 ∨ Sl (where∨ is the wedge sum), simply connected in our hypotheses. Consequently,the mapK×{ǫ} → ∂ Cp(F), obtained with the extension, admits, on turn, a normal extension. This completesthe piecewise description ofF .

In case of presence of a saddle of index 1 orn− 1, we have:Theorem 3.5Let F be aC∞, codimension one, transversely orientable, Morse foliation of a compactn-manifold,M , n ≥ 3. Letq be a saddle of index1 or n − 1 accumulating to one centerp. LetL ⊂ Cp(F) bea spherical leaf intersecting a neighborhoodU of q, defined by the Morse Lemma. We may have:(i) ∂ Cp(F)contains a single separatrix of the saddle (see figure 4) and is homeomorphic toSn−1; (ii) ∂ Cp(F) containsboth separatricesS1 andS2 of the saddle (see figure 5) and is homeomorphic toSn−1/Sn−2 ≃ Sn−1 ∨Sn−1.If this is the case, there exist two leavesFi (i = 1, 2), such thatFi andL intersect different components ofU \ Si and we have thatFi is homeomorphic toSn−1 (i = 1, 2); (iii) q is a self-connected saddle (see figure

6) and∂ Cp(F) is homeomorphic toSn−1/S0. In this case we will refer to the couple(Cp(F),F|

Cp(F )

)as

a singular Reeb component. Moreover,U \ ∂ Cp(F) has three connected components andL intersects two ofthem. IfF is a leaf intersecting the third component ofU \ ∂ Cp(F), thenF is homeomorphic toS1 × Sn−2,or to R× Sn−2.Proof. The proof is quite similar to the proof of the previous theorem. Nevertheless we give a brief sketchhere. The three cases arise from the fact thatq has two local separatrices,S1 andS2, but not necessarily∂ Cp(F) contains both of them. When this is the case, we may have thatS1 andS2 belong to distinct leaves,or to the same leaf (in this case all spherical leaves contained in Cp(F) intersect two different components ofU \ (S1 ∪ S2) ). Using the Morse lemma, we construct the setK for the application of the Holonomy Lemma

3.3. We have, respectively:K = Bn−1, K = K1 ⊔K2 = S0 ×Bn−1 (we apply twice the Holonomy Lemma),K = B1 × Sn−2. In the first two cases, asK is simply connected, the mapK → L, to be extended, is clearlyhomotopic to a constant in its leaf. ThenL \U projects onto∂Cp(F) and on neighbour leaves. This completesthe piecewise description in case(i) and(ii) .In the third case, piecing pieces together after a first application of the Holonomy Lemma, we obtain∂ Cp(F) ≃Sn−1/S0 and∂ Cp(F) \ {q} ≃ B1 × Sn−2, simply connected forn 6= 3. With a second application of theHolonomy Lemma (n 6= 3), K projects diffeomorphically onto any neighbour leaf,F . The same also happensfor n = 3, because a curveγ : S1 → ∂ Cp(F), as the one depicted in figure 6, is never a generator of theholonomy, which is locally trivial (a consequence of the Morse lemma). Nevertheless, there are essentially twoways to piece pieces together. We may haveF ≃ S1 × Sn−2 orF ≃ R× Sn−2.

The last result gives the motivation for a new concept.Definition 3.6 In a codimension one singular foliationF it may happen that, for some leafL andq ∈ Sing(F),the setL ∪ {q} is arcwise connected. LetC = {q ∈ Sing(F)|L ∪ {q} is arcwise connected}. If for someleafL the setC 6= ∅, we define the correspondingsingular leaf[Wag] S(L) = L ∪ C. In particular, ifF is atransversely orientable Morse foliation, each singular leaf is given byS(L) = L∪{q}, for a single saddle-typesingularityq, either selfconnected or not.

In the case of a transversely orientable Morse foliationF onM (n = dimM ≥ 3), given a saddleq anda separatrixL of q, we may define a sort of holonomy map of the singular leafS(L). This is done in thefollowing way.As the foliation is Morse, in a neighborhoodU ⊂ M of q there exists a (Morse) local first integralf : U → R,

6

with f(q) = 0. Keeping into account the structure of the level sets of the Morse functionf (see Theorem 3.4and Theorem 3.5) we observe that there are at most three connected components inU \ S(L) = U \ {f−1(0)}(notice that the number of components depends on the Morse index ofq).Let γ : [0, 1] → S(L) be aC1 path through the singularityq. At first, we consider the caseγ([0, 1]) ⊂ U ,q = γ(t) for some0 < t < 1. For a pointx ∈ M \ Sing(F), let Σx be a transverse section atx. ThesetΣx \ {x} is the union of two connected components,Σ+

x andΣ−x that we will denote bysemi-transverse

sections atx. Forx = γ(0) ∈ S(L) we havef(x) = 0 and we can choose semi-transverse sections atx in away thatf(Σ+

x ) > 0 andf(Σ−x ) < 0. We repeat the construction fory = γ(1), obtaining four semi-transverse

sections, which are contained in (at most) three connected components ofU \S(L). As a consequence, at leasttwo of them are in the same component. By our choices, this happens forΣ−

x andΣ−y (but we cannot exclude it

happens also forΣ+x andΣ+

y ). We define thesemi-holonomy maph− : Σ−γ(0)∪γ(0) → Σ−

γ(1)∪γ(1) by setting

h−(γ(0)) = γ(1) andh−(z) = h(z) for z ∈ Σ−γ(0), whereh : Σ−

γ(0) → Σ−γ(1) is a classic holonomy map (i.e.

such that for a leafF , it is h(F ∩ Σ−γ(0)) = F ∩ Σ−

γ(1)). In the same way, if it is the case, we defineh+.Consider now any curveγ : [0, 1] → S(L). As F is transversely orientable, the choice of a semi-transversesection for the curveγ([0, 1]) ∩ U , may be extended continuously on the rest of the curve,γ([0, 1]) \ U ; withthis remark, we use classic holonomy outsideU . To complete the definition, it is enough to say what a semi-transverse section at the saddleq is. In this way we allowq ∈ γ(∂[0, 1]). To this aim, we use the orbits ofthe transverse vector field,grad(f). By the property of gradient vector fields, there exist points t, v such thatα(t) = ω(v) = q. Let Σ+

q (Σ−q ) be the negative (positive) semi-orbit throught (v). Each ofΣ+

q andΣ−q ,

transverse to the foliation and such thatΣ+q ∩ Σ−

q = {q}, is asemi-transverse sectionat the saddleq.In this way, thesemi-holonomy of a singular leafHol+(S(L),F) is a representation of the fundamental

groupπ1(S(L)) into the germs of diffeomorphisms ofR≥0 fixing the origin,Germ( R≥0, 0).Now we consider the (most interesting) case of a selfconnected separatrixS(L) = ∂ Cp(F), with ∂ Cp(F)

satisfying the description of Theorem 3.5, case(iii) . The singular leaf∂ Cp(F), homeomorphic toSn−1/S0,has the homotopy type ofSn−1 ∨ S1. We haveHol+(∂ Cp(F),F) = {e, h−

γ }, whereγ is the non trivialgenerator of the homotopy, andh−

γ is a map with domain contained in the complement∁ Cp(F). The twooptionsh−

γ = e, h−γ 6= e give an explanation of the two possible results about the topology of the leaves near

the selfconnected separatrix.

4 Realization and elimination of pairings of singularities

Let us describe one of the key points in our work, i.e. the elimination procedure, which allows us to deletepairs of singularities in certain configurations, and, thisway, to lead us back to simple situations as in the ReebSphere Theorem (2.4). We need the following notion [Cam-Sc]:Definition 4.1 Let F be a codimension one foliation with isolated singularitieson a manifoldMn. By adead branchof F we mean a regionR ⊂ M diffeomorphic to the product Bn−1 × B1, whose boundary,∂R ≈ Bn−1×S0∪Sn−2×B1, is the union of two invariant components (pieces of leaves of F , not necessarilydistinct leaves inF ) and, respectively, of transverse sections,Σ ≈ {t} × B1, t ∈ Sn−2.Let Σi, i = 1, 2 be two transverse sections. Observe that the holonomy fromΣ1 → Σ2 is always trivial, in thesense of the Transverse Uniformity Theorem [Cam-LN], even if Σi ∩ S(L) 6= ∅ for some singular leafS(L).In this case we refer to the holonomy of the singular leaf, in the sense above.

A first result includes known situations.Proposition 4.2Given a foliated manifold(Mn,F), withF Morse and transversely orientable, withSing(F) ∋p, q, wherep is a center andq ∈ ∂ Cp(F) is a saddle of index 1 orn− 1, there exists a new foliated manifold(M, F), such that:(i) F andF agree outside a suitable regionR of M , which contains the singularitiesp, q;(ii) F is nonsingular in a neighborhood ofR.Proof. We are in the situations described by Theorem 3.5. If we are incase(i), the couple(p, q) may beeliminated with the technique of the dead branch, as illustrated in [Cam-Sc]. If we are in case(ii) , we observethat the two leavesFi, i = 1, 2 bound a region,A, homeomorphic to an anulus,Sn−1 × [0, 1]. We may nowreplace the singular foliationF|A with the trivial foliationF|A, given bySn−1 × {t}, t ∈ [0, 1]. If we are incase(iii) , we may replace the singular Reeb component with a regular one, in the spirit of [Cam-Sc]. Even inthis case, we may think the replacing takes place with the aidof a new sort of dead branch, thedead branch

7

PSfrag replacements

p

pq

qq

r

r

r

s

s s

ss

V

Figure 9: On the left: the height function on theplane V defines a foliation of the torus; on theright: a possible description of the foliation.

PSfrag replacementsP1

P2

p

q

∂ Cp(F)

Figure 10: On the left, a dead branch for the self-connected saddleq of figure 9; on the right, thefoliation obtained after the elimination of the twocouples of singularities.

of the selfconnected saddle, that we describe with the picture of figure 10, for the case ofthe foliation of thetorus of figure 9, defined by the height Morse function [Mil 1].Observe that the couples(p, q) and(r, s) of thisfoliation may be also seen as an example of the coupling described in Theorem 3.5, case(ii) . In this case theelimination technique and the results are completely different (see figure 11).Definition 4.3 If the couple(p, q) satisfies the description of Theorem 3.5, case(i) (and therefore may be elim-inated with the technique of the dead branch), we will say that (p, q) is a trivial couple.

A new result is the construction of saddle-saddle situations:Proposition 4.4 Given a foliationF on an n-manifoldMn, there exists a new foliationF on M , withSing(F) = Sing(F) ∪ {p, q}, wherep and q are a couple of saddles of consecutive indices,connectingtransversely(i.e. such that the stable manifold ofp, Ws(p), intersects transversely the unstable manifold ofq,Wu(q)).Proof. We choose the domain of (any) foliated chart,(U, φ). Observe thatR′ = U (≃ φ(U)) is a dead branchfor a foliationF ǫ′ , given (up to diffeomorphisms) by the submersionfǫ = −x2

1/2− · · · − x2k−1/2 + (x3

k/3−ǫxk) + xk+1/2+ · · ·+ x2

n/2, for someǫ = ǫ′ < 0. We considerF ǫ′′ , given by takingǫ = ǫ′′ > 0 in fǫ, whichpresents a couple of saddles of consecutive indices, and we choose a dead branchR′′ around them. We alsochoose a homeomorphism betweenR′ andR′′ which sends invariant sets ofF ǫ′ into invariant sets ofF ǫ′′ in aneighborhood of the boundary. With a surgery, we may replaceF ǫ′ with F ǫ′′ .

The converse of the above poposition is preceded by the followingRemark 4.5Given a foliationF onMn with two complementary saddle singularitiesp, q ∈ Sing(F), havinga strong stable connectionγ, there exist a neighborhoodU of p, q andγ in Mn, a δ ∈ R+ and a coordinatesystemφ : U → Rn taking p onto (0, . . . , φk = −δ, . . . , 0), q onto (0, . . . , φk = δ, . . . , 0), γ onto thexk-axis, {xl = 0}l 6=k, and such that:(i) the stable manifold ofp is tangent toφ−1({xl = 0}l>k) at p, (ii)the unstable manifold ofq is tangent toφ−1({xl = 0}l<k) at q (we are led to the situation considered in[Mil 2], A first cancelation theorem). So using the chartφ : U → Rn we may assume that we are on adead branch ofRn and the foliationF|U is defined byfǫ, for ǫ = δ2. In this way the vector fieldgrad(fǫ)defines a transverse orientation inU . For a suitableµ > 0, the pointsr1 = (0, . . . , φk = −δ − µ, . . . , 0)andr2 = (0, . . . , φk = δ + µ, . . . , 0) are such that the modification takes place in a region ofU delimited byLri , i = 1, 2.Proposition 4.6Given a foliationF on Mn with a couple of saddlesp, q of complementary indices, havinga strong stable connection, there exists a dead branch of thecouple of saddles,R ⊂ M and we can obtaina foliation F onM such that:(i) F andF agree onM \ R; (ii) F is nonsingular in a neighborhood ofR;indeedF |R is conjugated to a trivial fibration;(iii) the holonomy ofF is conjugate to the holonomy ofF inthe following sense: given any leafL of F such thatL ∩ (M \R) 6= ∅, then the corresponding leafL of F issuch thatHol(L, F) is conjugate toHol(L,F).Example 4.7 (Trivial Coupling of Saddles)Let M = Sn, n ≥ 3. For l = 1, . . . , n − 2 we may find a

Morse foliation ofM = Sn, invariant for the splittingSn = Bn−l × Sl ∪φ Sn−l−1 × Bl+1, whereφ is a

diffeomorphism of the boundary. In fact, by theorem 3.4 or 3.5, case(iii) , Bn−l × Sl admits a foliation withone center and one saddle of indexl. Similarly, Sn−l−1 × Bl+1 admits a foliation with a saddle of indexn− l− 1, actually a saddle of indexl+ 1, after the attachment. We may eliminate the trivial couple of saddlesand we are led to the well-known foliation ofSn, with a couple of centers and spherical leaves.

8

Remark 4.8The elimination of saddles of consecutive indices is actually a generalization of the elimination ofcouples center-saddle,(p, q) with q ∈ ∂ Cp(F). Indeed, we may eliminate(p, q) only when the saddleq hasindex1 orn−1. This means the singularities of the couple must have consecutive indices and, asq ∈ ∂ Cp(F),there exists an orbit of the transverse vector field havingp asα-limit (backward) andq asω-limit (forward), orviceversa. Such an orbit is a strong stable connection.

5 Reeb-type theorems

We shall now describe how to apply our techniques to obtain some generalizations of the Reeb Sphere Theorem(2.4) for the case of Morse foliations admitting both centers and saddles.A first generalization is based on the following notion:Definition 5.1 We say that an isolated singularity,p, of aC∞, codimension one foliationF onM is astablesingularity, if there exists a neighborhoodU of p in M and aC∞ function,f : U → R, defining the foliationin U , such thatf(p) = 0 andf−1(a) is compact, for|a| small. The following characterization of stablesingularities can be found in [Cam-Sc].Lemma 5.2An isolated singularityp of a functionf : U ⊂ Rn → R defines a stable singularity for df ,if and only if there exists a neighborhoodV ⊂ U of p, such that,∀x ∈ V , we have eitherω(x) = {p} orα(x) = {p}, whereω(x) (respectivelyα(x)) is theω-limit (respectivelyα-limit) of the orbit of the vector fieldgrad(f) through the pointx.

In particular it follows the well-known:Lemma 5.3If a functionf : U ⊂ Rn → R has an isolated local maximum or minimum atp ∈ U thenp is astable singularity fordf .

The converse is also true:Lemma 5.4If p is a stable singularity, defined by the functionf , thenp is a point of local maximum or minimumfor f .Proof. It follows immediately by Lemma 5.2 and by the fact thatf is monotonous, strictly increasing, alongthe orbits ofgrad(f).

With this notion, we obtainLemma 5.5LetF be a codimension one, singular foliation on a manifoldMn. In a neighborhood of a stablesingularity, the leaves ofF are diffeomorphic to spheres.Proof. Let p ∈ Sing(F) be a stable singularity. By Lemma 5.4, we may supposep is a minimum (otherwisewe use−f ). Using a local chart aroundp, we may suppose we are onRn and we may write the Taylor-Lagrange expansion aroundp for an approximation of the functionf : U → R at the second order. Wehavef(p + h) = f(p) + 1/2〈h,H(p + θh)h〉, whereH is the Hessian off and0 < θ < 1. It follows〈h,H(p+ θh)h〉 ≥ 0 in U . Thenf is convex and hence the sublevels,f−1(c), are also convex.We consider the flowφ : D(φ) ⊂ R × U → U of the vector fieldgrad(f). By the properties of gradientvector fields, in our hypothesis,D(φ) ⊃ (−∞, 0] × U and∀x ∈ U there exists theα-limit, α(x) = p. Foranyx ∈ f−1(c), the tangent space,Txf

−1(c), to the sublevels off does not contain the radial direction,−→px.This is obvious otherwise, for the convexity off−1(c), the singularityp should lie on the sublevelf−1(c), acontraddiction because, in this case,p should be a saddle. Equivalently, the orbits of the vector field grad(f)are transverse to spheres centered atp. An application of the implicit function theorem shows the existenceof a smooth functionx → tx, that assigns to each pointx ∈ f−1(c) the (negative) time at whichφ(t, x)intersectsSn−1(p, ǫ), whereǫ is small enough to have Bn(p, ǫ) ( R(f−1(c)), the compact region bounded byf−1(c) . The diffeomorphism between the leaff−1(c) and the sphereSn−1(p, ǫ) is given by the compositionx → φ(tx, x). The lemma is proved.Lemma 5.6 Let F be a codimension one, transversely orientable foliation ofM , with all leaves closed,π : M → M/F the projection onto the space of leaves. Then we may choose a foliated atlas onM anda differentiable structure onM/F , such thatM/F is a codimension one compact manifold, locally diffeomor-phic to the space of plaques, andπ is aC∞ map.Proof. At first we notice that the space of leavesM/F (with the quotient topology) is a one-dimensional Hau-sorff topological space, as a consequence of the Reeb Local Stability Theorem 2.1. As all leaves are closed andwith no holonomy, we may choose a foliated atlas{(Ui, φi)} such that, for each leafL ∈ F , L∩Ui consists, atmost, of a single plaque. Letπ : M → M/F be the projection onto the space of leaves andπi : Ui → R theprojection onto the space of plaques. With abuse of notation, we may writeπi = p2 ◦ φi, wherep2 is the pro-

9

jection on the second component. As there is a 1-1 correspondence between the quotient spacesπ|Ui(Ui) and

πi(Ui), then, are homeomorphic. LetV ⊂ M/F be open. The setπ−1(V ) is an invariant open set. We mayfind a local chart(Ui, φi) such thatπ(Ui) = V . We say that(V, πi ◦ (π|Ui

)−1) is a chart for the differentiableatlas with the required property. To see this, it is enough toprove that, if(V, πj ◦(π|Uj

)−1) is another chart withthe same domain,V , there exists a diffeomorphism between the two images ofV , i.e. betweenπi◦(π|Ui

)−1(V )andπj ◦ (π|Uj

)−1(V ). This is not obvious whenUi ∩ Uj = ∅. Indeed, the searched diffeomorphism exists,and it is given by the Transverse Uniformity Theorem [Cam-LN]. Observe that, in coordinates,π coincideswith the projection on the second factor.Lemma 5.7Letn ≥ 2. A weakly stable singularity for a foliation(Mn,F) is a stable singularity.Proof. Let p be a weakly stable singularity,U a neighborhood ofp with all leaves compact. We need alocal first integral nearp. As a consequence of the Reeb Local Stability Theorem 2.1, wecan find an (invari-ant) open neighborhoodV ⊂ U of p, whose leaves have all trivial holonomy. The setV \ {p} is open inM∗ = M \ Sing(F). LetF∗ = F \ Sing(F); the projectionπ∗ : M∗ → M∗/F∗ is an open map (see, forexample [Cam-LN]). As a consequence of Lemma 5.6, the connected (asn ≥ 2) and open setπ∗(V \ {p})is a 1-dimensional manifold with boundary, i.e. it turns out to bean interval, for example(0, 1). Now, weextend smoothlyπ∗ to a mapπ on U . In particular, letW ( V be a neighborhood ofp. If (for example)π∗(W \ {p}) = (0, b) for someb < 1, we setπ(p) = 0. Thesis follows by lemma 5.3.Theorem 5.8LetMn be a closedn-dimensional manifold,n ≥ 3. Suppose thatM supports aC∞, codimen-sion one, transversely orientable foliation,F , with non-empty singular set, whose elements are, all, weaklystable singularities. ThenM is homeomorphic to the sphere,Sn.Proof. By hypothesis,∀p ∈ Sing(F), p is a weakly stable singularity. Then it is a stable singularity. By lemma5.5, in an invariant neighborhoodUp of p, the leaves are diffeomorphic to spheres. Now we can proceedas inthe proof of the Reeb Sphere Theorem 2.4.Theorem 5.9 (Classification of codimension one foliations with all leaves compact)Let F be a (possiblysingular, with isolated singularities) codimension one foliation of M , with all leaves compact. Then all pos-sible singularities are stable. IfF is (non) transversely orientable, the space of leaves is (homeomorphic to[0, 1]) diffeomorphic to[0, 1] or S1. In particular, this latter case ocurs if and only if∂M,Sing(F) = ∅. Inall the other cases, denoting byπ : M → [0, 1] the projection onto the space of leaves, it isHol(π−1(x),F) ={e}, ∀x ∈ (0, 1). Moreover, ifx = 0, 1, we may have:(i) π−1(x) ⊂ ∂M 6= ∅ andHol(π−1(x),F) = {e};(ii) π−1(x) is a (stable) singularity;(iii) Hol(π−1(x),F) = {e, g}, g 6= e, g2 = e (in this case,∀y ∈ (0, 1),the leafπ−1(y) is a two-sheeted covering ofπ−1(x).Proof. If F is transversely orientable, by the Reeb Global Stability Theorem 2.2 and Lemma 5.6, the space ofleaves is either diffeomorphic toS1 or to [0, 1]. In particular,M/F ≈ S1 if and only ifM is closed andF nonsingular. When this is not the case,M/F ≈ [0, 1], and there are exactly two points (∂[0, 1]) which come froma singular point and/or from a leaf of the boundary.If F is non transversely orientable, there is at least one leaf with (finite) non trivial holonomy, which corre-sponds a boundary point inM/F to (by Proposition 2.3). By the proof of Lemma 5.6, the projection is notdifferentiable and the space of leavesM/F , a Hausdorff topological1-dimensional space, turns out to be anorbifold (see [Thu]). We pass to the transversely orientable double covering,p : (M, F) → (M,F). The fo-liation F , pull-back ofF , has all leaves compact, and singular set empty or with stable components; thereforewe apply the first part of the classification toM/F . Both if M/F is diffeomorphic toS1 or to [0, 1], M/F ishomeomorphic to[0, 1], but (clearly) with different orbifold structures.

Before going on with our main generalization of the Reeb Sphere Theorem 2.4, which extends a similarresult of Camacho and Scardua [Cam-Sc] concerning the casen = 3, we need to recall another result, that weare going to generalize.As we know, the Reeb Sphere Theorem, in its original statement, consideres the effects (on the topology of amanifoldM ) determined by the existence, onM , of a real valued function with exactly two non-degeneratesingular points. A very similar problem was studied by Eellsand Kuiper [Ee-Kui]. They considered manifoldsadmitting a real valued function with exactly three non-degenerate singular points.They obtained very interest-ing results. Among other things, it sticks out the obstruction they found about the dimension ofM , which mustbe even and assume one of the valuesn = 2m = 2, 4, 8, 16. Moreover, the homotopy type of the manifoldturns out to vary among a finite number of cases, including (orreducing to, ifn = 2, 4) the homotopy tupe ofthe projective plane over the real, complex, quaternion or Cayley numbers.Definition 5.10In view of the results of Eells and Kuiper [Ee-Kui], if a manifoldM admits a real-valued func-

10

Figure 11: Elimination technique applied in case(ii) (Theorem 3.5) for the foliation of figure 9.

PSfrag replacements q

p1

p1

p2

p2

Figure 12: A foliation ofRP2 with three singularpoints.

tion with exactly three non-degenerate singular points, wewill say thatM is anEells-Kuipermanifold.We have (see [Cam-Sc] for the casen = 3):Theorem 5.11 (Center-Saddle Theorem)Let Mn be ann-dimensional manifold, withn ≥ 2 such that(M,F) is a foliated manifold, by means of a transversely orientable, codimension-one, Morse,C∞ folia-tion F . MoreoverF is assumed to be without holonomy ifn = 2. LetSing(F) be the singular set ofF , with#Sing(F) = k + l, wherek, l are the numbers of, respectively, centers and saddles. If wehavek ≥ l + 1,then there are two possibilities:(1) k = l + 2 andM is homeomorphic to ann-dimensional sphere;(2) k = l + 1 andM is an Eells-Kuiper manifold.Proof. If l = 0, assertion is proved by the Reeb Sphere Theorem 2.4. Letl ≥ 1; we prove our thesis byinduction on the numberl of saddles. We setF l = F .So let l = 1 andF1 = F . By hypothesis, in the setSing(F) there exist at least two centers,p1, p2, withp1 6= p2, and one saddleq. We have necessarilyq ∈ ∂ Cp1

(F) ∩ ∂ Cp2(F). In fact, if this is not the case and,

for exampleq /∈ ∂ Cp1(F), then (keeping into account that forn = 2, the foliationF is assumed to be without

holonomy)∂ Cp1= ∅ andM = Cp1

(F). A contraddiction. Leti(q) the Morse index of the saddleq.For n ≥ 3 we apply the results of Theorems 3.4 and 3.5 to the couples(p1, q) and(p2, q). In particular, byTheorem 3.5,(iii) , it follows that the saddleq cannot be selfconnected. We now have the following two possi-bilities:(a) i(q) = 1, n− 1 and(p1, q) or (and)(p2, q) is a trivial couple,(b) i(q) 6= 1, n− 1 and there are no trivial couples.Forn = 2, we have necessarilyi(q) = 1 and, in our hypotheses,q is always selfconnected. With few changes,we adapt Theorem 3.5, to this case, obtaining∂ Cp(F) ≃ S1 or ∂ Cp(F) ≃ S1 ∨ S1; in this latter case we willsay that the saddleq is selfconnected with respect top. We obtain:(a’) (p1, q) or (and)(p2, q) is a trivial couple;(b’) q is selfconnected both with respect top1 and top2.In cases(a) and(a’) we proceed with the elimination of a trivial couple, as stated in Proposition 4.2, and thenwe obtain the foliated manifold(M,F0), with no saddle-type and some center-type singularities. We apply theReeb Sphere Theorem 2.4 and obtain#Sing(F) = 2 andM ≃ Sn.In case(b) (n ≥ 3), as a consequence of Theorem 3.4, we necessarily havei(q) = n/2 (and thereforen mustbe even!). MoreoverCp1

(F) ≈ Cp2(F) andM = Cp1

(F) ∪φ Cp2(F) may be thought as two copies of the

same (singular) manifold glued together along the boundary, by means of the diffeomorphismφ.In case(b’) (n = 2), we obtain the same result as above, i.e.Cp1

(F) ≈ Cp2(F) andM = Cp1

(F)∪φ Cp2(F).

We notice that case(b’) occurs when the setCpi(F) ≃ B2/S0 is obtained by identifying two points of the

boundary in a way that reverses the orientation.In cases(b) and(b’), it turns out that#Sing(F1) = 3. Moreover,F1 has a first integral, which is given bythe projection ofM onto the space of (possibly singular) leaves. In fact, by Lemma 5.6, the space of leaves isdiffeomorphic to a closed interval ofR. In this wayM turns out to be an Eells-Kuiper manifold. This ends thecasel = 1.Let l > 1 (and#Sing(F) > 3). As above, inSing(F) there exist at least one saddleq and two (distinct) cen-ters,p1, p2 such thatq ∈ ∂ Cp1

(F)∩ ∂ Cp2(F); we are led to the same possibilities(a), (b) for n ≥ 3 and(a)’,

(b)’ for n = 2. Anyway (b) and(b’) cannot occur, otherwiseM = Cp1(F) ∪φ Cp2

(F) and#Sing(F) = 3,

11

a contraddiction. Then we may proceed with the elimination of a trivial couple. In this way we obtain thefoliated manifold(M,F l−1), which we apply the inductive hypothesis to. The theorem is proved, observingthat, a posteriori, case(1) holds ifk = l + 2 and case(2) if k = l + 1.

6 Haefliger-type theorems

In this paragraph, we investigate the existence of leaves ofsingular foliations with unilateral holonomy. Keep-ing into account the results of the previous paragraph, for Morse foliations, we may state or exclude such anoccurrence, according to the following theorem:Theorem 6.1LetF be aC∞, codimension one, Morse foliation on a compact manifoldMn, n ≥ 3, assumedto be transversely orientable, but not necessarily closed.Let k be the number of centers andl the number ofsaddles. We have the following possibilities:(i) if k ≥ l + 1, then all leaves are closed inM \ Sing(F); inparticular, if ∂M 6= ∅ or k ≥ l + 2 each regular (singular) leaf ofF , is diffeomorphic (homeomorphic) to asphere (in the second option, it is diffeomorphic to a spherewith a pinch at one point);(ii) if k = l there aretwo possibilities: all leaves are closed inM \Sing(F), or there exists some compact (regular or singular) leafwith unilateral holonomy.Example 6.2The foliation of example 4.7 is an occurrence of theorem 6.1,case(ii) with all leaves closed. TheReeb foliation ofS3 and each foliation we may obtain from it, with the introduction of l = k trivial couplescenter-saddle, are examples of theorem 6.1, case(ii) , with a leaf with unilateral holonomy.

Now we consider other possibilities forSing(F).Definition 6.3 Let F be aC∞, codimension one foliation on a compact manifoldMn, n ≥ 3, with singularsetSing(F) 6= ∅. We say thatSing(F) is regular if its connected components are either isolated points orsmoothly embedded curves, diffeomorphic toS1. We extend the definition of stability to regular components,by saying that a connected componentΓ ⊂ Sing(F) is (weakly) stable, if there exists a neighborhood ofΓ,where the foliation has all leaves compact (notice that we can repeat the proof of Lemma 5.7 and obtain that aweakly stable component is a stable component).

In the caseSing(F) is regular, with stable isolated singularities, whenn ≥ 3 we may exclude a Haefliger-type result, as a consequence of Lemma 5.5 and the Reeb GlobalStability Theorem for manifolds with bound-ary. Then we study the caseSing(F) regular, with stable components, all diffeomorphic toS1. Let J be a setsuch that for allj ∈ J , the curveγj : S1 → M , is a smooth embedding andΓj := γj(S

1) ⊂ Sing(F) isstable. ThenJ is a finite set. This is obvious, otherwise∀j ∈ J , we may select a pointxj ∈ Γj and obtainthat the set{xj}j∈J has an accumulation point. But this is not possible because the singular components areseparated. We may regard a singular componentΓj , as adegenerate leaf, in the sense that we may associate toit, a single point of the space of leaves.

We need the following definitionDefinition 6.4 Let F be aC∞, codimension one foliation on a compact manifoldM . Let D2 be the closed2-disc andg : D2 → M be aC∞ map. We say thatp ∈ D2 is a tangency point ofg with F if (dg)p( R

2) ⊂Tg(p)Fg(p).

We recall a proposition which Haefliger’s theorem (cfr. the book [Cam-LN]) is based upon.Proposition 6.5 Let A : D2 → M be aC∞ map, such that the restrictionA|∂D2 is transverse toF , i.e.∀x ∈ ∂D2, (dA)x(Tx(∂D

2)) + TA(x)FA(x) = TA(x)M . Then, for everyǫ > 0 and every integerr ≥ 2,

there exists aC∞ map,g : D2 → M , ǫ-nearA in theCr-topology, satisfying the following properties:(i)g|∂D2 is transverse toF . (ii) For every pointp ∈ D2 of tangency ofg with F , there exists a foliation boxU of F with g(p) ∈ U and a distinguished mapπ : U → R such thatp is a non-degenerate singularity ofπ ◦ g : g−1(U) → R. In particular there are only a finite number of tangency points ofg with F , since theyare isolated, and they are contained in the open discD2 = {z ∈ R2 : ||z|| < 1}. (iii) If T = {p1, . . . , pt}is the set of tangency points ofg with F , theng(pi) andg(pj) are contained in distinct leaves ofF , for everyi 6= j. In particular, the singular foliationg∗(F) has no saddle connections.

We are now able to prove a similar result, in the case of existence of singular components.Proposition 6.6LetF be a codimension one,C∞ foliation on a compact manifoldMn, n ≥ 3, with regularsingular set,Sing(F) = ∪j∈JΓj 6= ∅, whereΓj are all stable components diffeomorphic toS1 andJ is finite.LetA : D2 → M be aC∞ map, such that the restrictionA|∂D2 is transverse toF . Then, for everyǫ > 0 and

12

every integerr ≥ 2, there exists aC∞ map,g : D2 → M , ǫ-nearA in theCr-topology, satisfying properties(i) and(iii) of proposition 6.5, while(ii) is changed in:(ii’) for every pointp ∈ D2 of tangency ofg withF , wehave two cases: (1) ifLg(p) is a regular leaf ofF , there exists a foliation box,U of F , with g(p) ∈ U , and adistinguished map,π : U → R, satisfying properties as in(ii) of Proposition 6.5; (2) ifLg(p) is a degenerateleaf ofF , there exists a neighborhood,U of p, and a singular submersion,π : U → R, satisfying propertiesas in(ii) Proposition 6.5.Proof. We start by recalling the idea of the classical proof.We choose a finite covering ofA(D2) by foliation boxes{Qi}

ri=1. In eachQi the foliation is defined by

a distinguished map, the submersionπi : Qi → R. We choose an atlas,{(Qi, φi)}ri=1, such that the last

component ofφi : Qi → Rn is πi, i.e. φi = (φ1i , φ

2i , . . . , φ

n−1i , πi). We construct the finite cover ofD2,

{Wi = A−1(Qi)}ri=1; the expression ofA in coordinates isA|Wi

= (A1i , . . . , A

n−1i , πi ◦A). We may choose

covers ofD2, {Ui}ri=1, {Vi}

ri=1, such thatUi ⊂ Vi ⊂ Vi ⊂ Wi, i = 1, . . . , r; then we proceed by induction

on the numberi. Starting withi = 1 and settingg0 = A, we apply a result ([Cam-LN], Cap. VI,§2, Lemma1, pag. 120) and we modifygi−1 in a new functiongi, in a way thatgi(Wi) ⊂ Qi andπi ◦ gi : Wi → R isMorse on the subsetUi ⊂ Wi. At last we setg = gr.

In the present case, essentially, it is enough to choose a setof couples,{(Uk, πk)}k∈K , where{Uk}k∈K isan open covering ofM , πk : Uk → R, for k ∈ K, is a (possibly singular) submersion and, ifUk ∩ Ul 6= ∅

for a couple of indicesk, l ∈ K, there exists a diffeomorphismplk : πk(Uk ∩ Ul) → πl(Uk ∩ Ul), suchthatπl = plk ◦ πk. By hypothesis, there exists the set of couples{(Ui, πi)}i∈I , where{Ui}i∈I , is an opencovering ofM \ Sing(F), and, fori ∈ I, the mapπi : Ui → R, is a distinguished map, defining the foliatedmanifold (M \ Sing(F),F∗). Let y ∈ Sing(F), theny ∈ Γj , for somej ∈ J . As y ∈ M , there existsa neighborhoodC ∋ y, homeomorphic to ann-ball. Let h : C → Bn be such a homeomorphism. As themapγj : S1 → Γj is a smooth embedding, we may suppose that, locally,Γj is sent in a diameter of theball Bn, i.e. h(C ∩ Γj) = {x2 = · · · = xn = 0}. For each singular pointz = h−1(b, 0, . . . , 0), the setD = h−1(b, x2, . . . , xn), homeomorphic to a small(n−1)-ball, is transverse to the foliation atz. Moreover, ifz1 6= z2, thenD1 ∩D2 = ∅. The restrictionF|D is a singular foliation with an isolated stable singularityatz.By lemma 5.5, the leaves ofF|D are diffeomorphic to(n− 2)-spheres. It turns out thaty has a neighborhoodhomeomorphic to the product(−1, 1)×Bn−1, where the foliation is the image of the singular trivial foliation of(−1, 1)×Bn−1, given by(−1, 1)×Sn−2×{t}, t ∈ (0, 1), with singular set(−1, 1)×{0}. Letπy : Uy → [0, 1)be the projection. If, for a couple of singular pointsy, w ∈ Sing(F), we haveUy ∩Uw 6= ∅, we may supposethey belong to the same connected component,Γj . We haveπw ◦ π−1

y (0) = 0 and, as a consequence of lemma5.6, there exists a diffeomorphism betweenπy(Uy ∩ Uw \ Γj) andπw(Uy ∩ Uw \ Γj). The same happens ifUy ∩Ui 6= ∅ for someUi ⊂ M \Sing(F). It comes out thatπy is singular onUy ∩Sing(F) and non-singularonUy \ Sing(F), i.e. (dπy)z = 0 ⇔ z ∈ Uy ∩ Sing(F). At the end, we setK = I ∪ Sing(F).Let g : D2 → M be a map. Theng defines the foliationg∗(F), pull-back ofF , on D2. Observe that ifSing(F) = ∅, thenSing(g∗(F)) = {tangency points ofg with F}, but in the present case, asSing(F) 6= ∅,we haveSing(g∗(F)) = {tangency points ofg with F} ∪ g∗(Sing(F)). Either if p is a point of tangency ofg with F or if p ∈ g∗(Sing(F)), we haved(πk)p = 0. With this remark, we may follow the classical proof.

As a consequence of proposition 6.6, we have:Theorem 6.7 (Haefliger’s theorem for singular foliations)LetF be a codimension one,C2, possibly singularfoliation of ann-manifoldM , withSing(F), (empty or) regular and with stable components diffeomorphic toS1. Suppose there exists a closed curve transverse toF , homotopic to a point. Then there exists a leaf withunilateral holonomy.

7 Novikov-type theorems

We end this article with a result based on the original Novikov’s Compact Leaf Theorem and on the notion ofstable singular set. To this aim, we premise the following remark. Novikov’s statement establishes the existenceof a compact leaf for foliations on 3-manifolds with finite fundamental group. This result actually proves theexistence of an invariant submanifold, sayN ⊂ M , with boundary, such thatF|N contains open leaves whoseuniversal covering is the plane. Moreover these leaves accumulate to the compact leaf of the boundary. In whatfollows, a submanifold with the above properties will be called aNovikov component. In particular a Novikovcomponent may be a Reeb component, i.e. a solid torus endowedwith its Reeb foliation. We recall that two

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PSfrag replacements

p

q

L

FU

∂ Cp(F)

Figure 13: p − q is not a trivial coupling when1 < l < n− 1, wherel is the index of the saddleq.

PSfrag replacements ST1

ST2

γ

Figure 14: A singular foliation ofS3, with no van-ishing cycles.

Reeb components, glued together along the boundary by meansof a diffeomorphism which sends meridians inparallels and viceversa, give the classical example of the Reeb foliation ofS3.If F is a Morse foliation of a 3-manifold, as all saddles have index 1 or 2, we are always in conditions ofproposition 4.2 and then we are reduced to consider just two (opposite) cases:(i) all singularities are centers,(ii) all singularities are saddles. In case(i), by the proof of the Reeb Sphere Theorem 2.4, we know that allleaves are compact; in case(ii) , all leaves may be open and dense, as it is shown by an example of a foliationof S3 with Morse singularities and no compact leaves [Ros-Rou].As in the previous paragraph, we study the case in whichSing(F) is regular with stable components,Γj , j ∈ J ,whereJ is a finite set. We have:Theorem 7.1Let F be aC∞, codimension one foliation on a closed3-manifoldM3. SupposeSing(F) is(empty or) regular, with stable components. Then we have twopossibilities: (i) all leaves ofF are compact;(ii) F has a Novikov component.Proof. If Sing(F) = ∅, thesis (case(ii) ) follows by Novikov theorem. LetSing(F) 6= ∅. We may supposethatF is transversely orientable (otherwise we pass to the transversely orientable double covering). IfSing(F)contains an isolated singularity, as we know, we are in case(i). Then we supposeSing(F) contains no isolatedsingularity, i.e.Sing(F) =

⋃j∈J Γj . SetD(F) = {Γj, j ∈ J} ∪ { compact leaves with trivial holonomy}.

By the Reeb Local Stability Theorem 2.1,D(F) is open. We may have∂D(F) = ∅, and then we are incase(i), or ∂D(F) 6= ∅, and in this case it contains a leaf with unilateral holonomy, F . It is clear thatFbounds a Novikov component, and then we are in case(ii) ; in fact, from one side,F is accumulated by openleaves. IfF ′ is one accumulating leaf, then its universal covering isp : R2 → F ′. Suppose, by contraddiction,that the universal covering ofF ′ is p : S2 → F ′. By the Reeb Global Stability Theorem for manifolds withboundary, all leaves are compact, diffeomorphic top(S2). This concludes the proof sinceF must have infinitefundamental group.

The last result may be reread in terms of the existence of closed curves, transverse to the foliation. We have:Lemma 7.2LetF be a codimension one,C∞ foliation on a closed3-manifoldM , with singular set,Sing(F) 6=∅, regular, with stable components. ThenF is a foliation with all leaves compact if and only if there exist noclosed transversals.Proof. (Sufficiency) If the foliation admits an open (inM \ Sing(F)) leaf,L, it is well known that we mayfind a closed curve, intersectingL, transverse to the foliation. Viceversa (necessity), letF be a foliation withall leaves compact. If necessary, we pass to the transversely orientable double coveringp : (M, F) → (M,F).In this way, we apply Lemma 5.6 and obtain, asSing(F) 6= ∅, that the projection onto the space of leaves isa (global)C∞ first integral ofF , f : M → [0, 1] ⊂ R. Suppose, by contraddiction, that there exists aC1

closed transversal to the foliationF , the curveγ : S1 → M . The lifting of γ2 is a closed curve,Γ : S1 → M ,transverse toF . The setf(Γ(S1)) is compact and then has maximum and minimum,m1,m2 ∈ R. A contrad-diction, becauseΓ cannot be transverse to the leaves{f−1(m1)}, {f

−1(m2)}.With this result, we may rephrase the previous theorem.

Corollary 7.3 LetF be a codimension one,C∞ foliation on a3-manifoldM , such thatSing(F) is regularwith stable components. Then(i) there are no closed transversals, or equivalently,F is a foliation by compactleaves,(ii) there exists a closed transversal, or equivalently,F has a Novikov component.Remark 7.4In the situation we are considering, we cannot state a singular version of Auxiliary Theorem I (see,for example [Mor-Sc]). In fact, even though a singular version of Haefliger Theorem is given, the existence of

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a closed curve transverse the foliation, homotopic to a constant, does not lead, in general, to the existence of avanishing cycle, as it is shown by the following counterexample.Example 7.5We consider the foliation ofS3 given by a Reeb component,ST1, glued (through a diffeomor-phism of the boundary which interchanges meridians with parallels) to a solid torusST2 = S1 × D2 =T 2 × (0, 1) ∪ S1. The torusST2 is endowed with the singular trivial foliationF|ST2

= T 2 × {t}, fort ∈ (0, 1), whereSing(F|ST2

) = S1 = Sing(F). As a closed transversal to the foliation, we considerthe curveγ : S1 → ST1 ⊂ S3, drawed in figure 14. Letf : D2 → S3 be an extension ofγ; the extensionf is assumed to be in general position with respect toF , as a consequence of proposition 6.5. Asγ(S1) islinked to the singular componentS1 ⊂ ST2, thenf(D2) ∩ Sing(F) 6= ∅. As a consequence, we find adecreasing sequence of cycles,{βn}, (the closed curves of the picture) which does not admit a cycle,β∞, suchthatβn > β∞, for all n. In fact the “limit” of the sequence is not a cycle, but the point f(D2) ∩ Sing(F).Example 7.6The different situations of Theorem 7.1 or Corollary 7.3 maybe exemplified as follows. It iseasy to see thatS3 admits a singular foliation with all leaves compact (diffeomorphic toT 2) and two singular(stable) components linked together, diffeomorphic toS1. In fact one can verify thatS3 is the union of twosolid tori,ST1 andST2, glued together along the boundary, both endowed with a singular trivial foliation.We construct another foliation onS3, modifying the previous one. We setST1 = S1 × {0} ∪ T 2 × (0, 1/2].In this way,ST1 = ST1 ∪ T 2 × (1/2, 1]. We now modify the foliation inST1 \ ST1, by replacing the trivialfoliation with a foliation with cylindric leaves accumulating to the two components of the boundary.

References

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[Mor-Sc] C.A. Morales, B. Scardua: Geometry and Topology of foliated manifolds.

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[Pal-deM] J. Palis, jr., W. de Melo: Geometric theory of dinamical systems: an introduction, New-York,Springer,1982.

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[Ros-Rou] H. Rosemberg, R. Roussarie: Some remarks on stability of foliations, J. Diff. Geom. 10, 1975,207-219.

[Stee] N. Steenrod: The topology of fiber bundles, Princeton, NJ, Princeton University Press, 1951

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