On smooth foliations with Morse singularities
On smooth foliations with Morse singularities
Lilia Rosati
Università 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 let F be a codimension one, C∞ foliation on M , with isolated singularities
of Morse type. The study and classification of pairs (M,F) is a challenging (and difficult) problem. In this
setting, 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 if the foliation is given by a function. Along these lines
a result due to Eells and Kuiper [Ee-Kui] classify manifolds having a real-valued function admitting exactly
three non-degenerate singular points. In the present paper, we prove a generalization of the above mentioned
results. To do this, we first describe the possible arrangements of pairs of singularities and the corresponding
codimension one invariant sets, and then we give an elimination procedure for suitable center-saddle and some
saddle-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 singular
set, Sing(F) of the foliation F , we consider weakly stable components, that we define as those components
admitting a neighborhood where all leaves are compact. If Sing(F) admits only weakly stable components,
given by smoothly embedded curves diffeomorphic to S1, 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 Scárdua for proposing me such an interesting subject and for his valuable
advice. My hearthy good thanks to prof. Graziano Gentili for his suggestions on the writing of this article.
1 Foliations and Morse Foliations
Definition 1.1 A codimension k, foliated manifold (M,F) is a manifold M with a differentiable structure,
given by an atlas {(Ui, φi)}i∈I , satisfying the following properties:
(1) φi(Ui) = B
n−k × Bk;
(2) in Ui ∩ Uj 6= ∅, we have φj ◦ φ
i (x, y) = (fij(x, y), gij(y)),
where {fij} and {gij} are families of, respectively, submersions and diffeomorphisms, defined on natural
domains. Given a local chart (foliated chart) (U, φ), ∀x ∈ Bn−k and y ∈ Bk, the set φ−1(·, y) is a plaque and
the set φ−1(x, ·) is a transverse section.
The existence of a foliated manifold (M,F) determines a partition of M into subsets, the leaves, defined
by means of an equivalence relation, each endowed of an intrinsic manifold structure. Let x ∈ M ; we denote
by Fx or Lx the leaf of F through x. With the intrinsic manifold structure, Fx turns to be an immersed (but
not embedded, in general) submanifold of M .
In an equivalent way, a foliated manifold (M,F) is a manifold M with a collection of couples {(Ui, gi)}i∈I ,
http://arxiv.org/abs/0704.0164v1
where {Ui}i∈I is an open covering of M , gi : Ui → B
k is a submersion, ∀i ∈ I , and the gi’s satisfy the cocycle
relations, gi = gij ◦ gj , gii = id, for suitable diffeomorphisms gij : B
k → Bk, defined when Ui ∩ Uj 6= ∅.
Each Ui is said a foliation box, and gi a distinguished map. The functions γij = dgij are the transition maps
[Stee] of a bundle NF ⊂ TM , normal to the foliation. More completely, there exists a G-structure on M
[Law], which is a reduction of the structure group GL(n, R) of the tangent bundle to the subgroup of the
matrices
, where A ∈ GL(n− k, R) and C ∈ GL(k, R).
A codimension one, C∞ foliation of a smooth manifold M , with isolated singularities, is a pair F =
(F∗, Sing(F)), where Sing(F) ⊂ M is a discrete subset and F∗ is a codimension one, C∞ foliation (in the
ordinary sense) of M∗ = M \Sing(F). The leaves of F are the leaves of F∗ and Sing(F) is the singular set
of F . A point p is a Morse singularity if there is a C∞ function, fp : Up ⊂ M → R, defined in a neighborhood
Up of p, with a (single) non-degenerate critical point at p and such that fp is a local first integral of the foliation,
i.e. the leaves of the restriction F|Up are the connected components of the level hypersurfaces of fp in Up\{p}.
A Morse singularity p, of index l, is a saddle, if 0 < l < n (where n = dimM ), and a center, if l = 0, n. We
say that the foliation F has a saddle-connection when there exists a leaf accumulated by at least two distinct
saddle-points. A Morse foliation is a foliation with isolated singularities, whose singular set consists of Morse
singularities, 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 on M .
A less evident example is given by the foliation depicted in figure 2.
In the literature, the orientability of a codimension k (regular) foliation is determined by the orientability of
the (n− k)-plane field tangent to the foliation, x → TxFx. Similarly transverse orientability is determined by
the orientability of a complementary k-plane field. A singular, codimension one foliation, F , is transversely
orientable [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 on M , we may find
a global vector field transverse to the foliation, X = grad(ω), ωX ≥ 0, and ωxXx = 0 if and only if x is a
singularity for the foliation (ω(x) = 0). A transversely orientable, singular foliation F of M is a transversely
orientable (regular) foliation F∗ of M∗ in the sense of the classical definition. Viceversa, if F∗ is transversely
orientable, in general, F is not.
Thanks to the Morse Lemma [Mil 1], Morse foliations reduce to few representative cases. On the other
hand, Morse foliations describe a large class among transverseley orientable foliations. To see this, let F be a
foliation defined by an integrable one-form, ω ∈ Λ1(M), with isolated singularies. We proceed with a local
analysis; using a local chart around each singularity, we may suppose ω ∈ Λ1( Rn), ω(0) = 0, and 0 is the
only singularity of ω. We have ω(x) =
hi(x)dx
i and, in a neighborhood of 0 ∈ Rn, we may write ω(x) =
ω1(x) +O(|x|
2), where ω1 is the linear part of ω, defined by ω1(x) =
idxj , aij = ∂h
i(x)/∂xj . We
recall that the integrability of ω implies the integrability of ω1 and that the singularity 0 ∈ R
n is said to be non
degenerate if and only if (aij) ∈ R(n) is non degenerate; in this latter case (aij) is symmetric: it is the hessian
matrix of some real function f , 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 space σ of integrable one-forms in Rn, with a singularity at the origin, endowed with the
C1-Whitney topology, τ . If ω, ω′ ∈ σ, we say ω equivalent ω′ (ω ∼ ω′) if there exists a diffeomorphism
φ : Rn → Rn, φ(0) = 0, which sends leaves of ω into leaves of ω′. Moreover, we say ω is structurally stable,
if there exists a neighborhood V of ω in (σ, τ) such that ω′ ∼ ω, ∀ω′ ∈ V .
Theorem 1.2 (Wagneur)[Wag] The one-form ω ∈ σ is structurally stable, if and only if the index of 0 ∈
Sing(ω) is neither 2 nor n− 2.
Let us denote by S the space of foliations defined by non degenerate one-forms with singularities, whose
index is neither 2 nor n− 2. If S1 ⊂ S is the subset of foliations defined by linear one-forms, then we have:
Corollary 1.3 There exists a surjective map,
s : S1 → S/∼.
PSfrag replacementsF1
L0 L1
Figure 1: F1,F2 foliations on RP
Hol(L,F1) = {e}, Hol(L0,F1) = {e, g0},
g20 = e, Hol(L1,F2) = {e, g1}, g1
orientation reversing diffeomorphism,
Hol(L2,F2) = {e, g2}, g2 generator of
unilateral holonomy.
PSfrag replacements
Figure 2: A singular foliation of the sphere S2,
which does not admit a first integral. With the
same spirit, a singular foliation on S3 may be
given.
2 Holonomy and Reeb Stability Theorems
It is well known that a basic tool in the study of foliations is the holonomy of a leaf (in the sense of Ehresmann).
If L is a leaf of a codimension k foliation (M,F), the holonomy Hol(L,F) = Φ(π1(L)), is the image of a
representation, Φ : π1(L) → Germ( R
k, 0), of the fundamental group of L into the germs of diffeomorphisms
of Rk, fixing the origin. Let x ∈ L and Σx be a section transverse to L at x; with abuse of notation, we will
write that a diffeomorphism g : 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}, with
g2 = e, g 6= e, (iii) Hol(L,F) = {e, g}, where gn 6= e, ∀n, and g is a (orientation preserving or reversing)
diffeomorphism. In particular, among orientation preserving diffeomorphisms, we might find a g : Σx → Σx,
such that g is the identity on one component of Σx \ {x} and it is not the identity on the other; in this case, we
say that L has unilateral 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) Let F be a C1, codimension k foliation of a manifold M andF a compact
leaf with finite holonomy group. There exists a neighborhood U of F , saturated in F (also called invariant), in
which all the leaves are compact with finite holonomy groups. Further, we can define a retraction π : U → F
such that, for every leaf F ′ ⊂ U , π|F ′ : F
′ → F is a covering with a finite number of sheets and, for each
y ∈ F , π−1(y) is homeomorphic to a disk of dimension k and is transverse to F . The neighborhood U can be
taken to be arbitrarily small.
The last statement means in particular that, in a neighborhood of the point corresponding to a compact leaf
with finite holonomy, the space of leaves is Hausdorff.
Under certain conditions the Reeb Local Stability Theorem may replace the Poincaré 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 in Sing(F).
Theorem 2.2 (Reeb Global Stability) Let F be a C1, codimension one foliation of a closed manifold, M . If
F contains a compact leaf F with finite fundamental group, then all the leaves of F are compact, with finite
fundamental group. If F is transversely orientable, then every leaf of F is diffeomorphic to F ; M is the total
space of a fibration f : M → S1 over S1, with fibre F , and F is the fibre foliation, {f−1(θ)|θ ∈ S1}.
This theorem holds true even when F is a foliation of a manifold with boundary, which is, a priori, tangent
on certain components of the boundary and transverse on other components [God]. In this setting, let H l =
{(x1, . . . , xl) ∈ Rl|xl ≥ 0}. Taking into account definition 1.1, we say that a foliation of a manifold with
boundary is tangent, respectively transverse to the boundary, if there exists a differentiable atlas {(Ui, φi)}i∈I ,
such that property (1) of the above mentioned definition holds for domains Ui such that Ui ∩ ∂M = ∅, while
φi(Ui) = B
n−k × H k, respectively, φi(Ui) = H
n−k × Bk for domains such that Ui ∩ ∂M 6= ∅. Moreover,
we ask that the change of coordinates has still the form described in property (2). Recall that F|∂M is a regular
codimension k − 1 (respectively, k) foliation of the (n− 1)-dimensional boundary. After this, it is immediate
to write the definition for foliations which are tangent on certain components of the boundary and transverse
PSfrag replacements b
Figure 3: n = 2: a singular foliation with center-
type singularities, having no first integral.
PSfrag replacements
Ws(q))
Wu(q)
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 replace S1 with [0, 1] in the second statement
of the Reeb Theorem 2.2 (see Lemma 5.6).
We say that a component of Sing(F) is weakly stable if it admits a neighborhood, U , such that F|U is a
foliation with all leaves compact. The problem of global stability for a foliation with weakly stable singular
components may be reduced to the case of foliations of manifolds with boundary, tangent to the boundary. It is
enough to cut off an invariant neighborhood of each singular component.
Holonomy is related to transverse orientability by the following:
Proposition 2.3 Let L be a leaf of a codimension one (Morse) foliation (M,F). If Hol(L,F) = {e, g}, where
g2 = e, g 6= e, then F is non-transversely orientable. Moreover, if π : M → M/F is the projection onto the
space of leaves, then ∂(M/F) 6= ∅ and π(L) ∈ ∂(M/F).
Proof. We choose x ∈ L and a segment Σx, transverse to the foliation at x. Then g : Σx → Σx turns out to
be g(y) = −y. Let y → Ny a 1-plane field complementary to the tangent plane field y → TyFy . Suppose
we may choose a vector field y → X(y) such that Ny = span{X(y)}. Then it shoud be X(x) = −X(x) =
(dg)x(X(x)), a contraddiction. Consider the space of leaves near L; this space is the quotient of Σx with
respect to the equivalence relation ∼ which identifies points on Σx of the same leaf. Then Σx/∼ is a segment
of type (z, x] or [x, z), where π−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 dimension n ≥ 3, having only centers as singularities, is homeomorphic to the n-sphere.
This result is proved by showing that the foliation considered must be given by a Morse function with only two
singular points, and therefore thesis follows by Morse theory. Notice that the theorem still holds true for n = 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 here
how 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 set C(F) ⊂ M as the union of center-type singularities
and leaves diffeomorphic to Sn−1 (with trivial holonomy if n = 2) and for a center singularity, p, we denote
by Cp(F) the connected component of C(F) that contains p.
Proposition 3.2 Let F be a Morse foliation on a manifold M . We have:
(1) C(F) and Cp(F) are open in M .
(2) Cp(F) ∩ Cq(F) 6= ∅ if and only if Cp(F) = Cq(F). Cp(F) = M if and only if ∂ Cp(F) = ∅. In this
case the singularities of F are centers and the leaves are all diffeomorphic to Sn−1.
(3) If q ∈ Sing(F) ∩ ∂ Cp(F), then q must be a saddle; in this case ∂ Cp(F) ∩ Sing(F) = {q}. Moreover,
for n ≥ 3 and F transversely orientable, ∂ Cp(F) 6= ∅ if and only if ∂ Cp(F) ∩ Sing(F) 6= ∅. In these
hypotheses, ∂ Cp(F) contains at least one separatrix of the saddle q.
(4) ∂ Cp(F) \ {q} is closed in M \ {q}.
PSfrag replacements
Figure 5: A saddle q of index 1 (n− 1), accumu-
lating one center p (Theorem 3.5, case (ii)).
PSfrag replacements
Figure 6: A saddle q of index 1 (n− 1), accumu-
lating one center p (Theorem 3.5, case (iii)).
Figure 7: Two saddles in trivial coupling for the
foliation defined by the function fǫ = −
ǫy + z
, (ǫ > 0).
PSfrag replacements
L2p q
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 saddle q, as there are no saddle connections. The second part follows by the Reeb Global
Stability Theorem for manifolds with boundary and the third by the Morse Lemma. (4) By the Transverse
Uniformity Theorem (see, for example, [Cam-LN]), it follows that the intrinsic topology of ∂ Cp(F) \ {q}
coincides with its natural topology, as induced by M \ {q}.
We recall the following (cfr., for example [Mor-Sc]):
Lemma 3.3 (Holonomy Lemma) Let F be a codimension one, transversely orientable foliation on M , let A
be a leaf of F and K be a compact and path-connected set. If g : K → A is a C1 map homotopic to a constant
in A, then g has a normal extension i.e. there exist ǫ > 0 and a C1 map G : K × [0, ǫ] → M such that
Gt(x) = G
x(t) = G(x, t) has the following properties: (i) G0(K) = g, (ii) Gt(K) ⊂ A(t) for some leaf A(t)
of F with A(0) = A, (iii) ∀x ∈ K the curve Gx([0, ǫ]) is normal to F .
For the case of center-saddle pairings we prove the following descriptions of the separatrix:
Theorem 3.4 Let F be a C∞, codimension one, transversely orientable, Morse foliation of a compact n-
manifold, M , n ≥ 3. Let q be a saddle of index l /∈ {1, n− 1}, accumulating to one center p. Let L ⊂ Cp(F)
be a spherical leaf intersecting a neighborhood U of q, defined by the Morse Lemma. Then ∂ Cp(F) \ {q}
has a single connected component (see figure 13) and is homeomorphic to Sn−1/Sl−1. If F is a leaf such that
U \ Cp(F)
6= ∅, then F is homeomorphic to Bl×Sn−l−1∪φ B
l×Sn−l−1, where φ is a diffeomorphism
of the boundary (for example, we may have F ≃ Sl × Sn−l−1, but also F ≃ Sn−1, for l = n/2).
Proof. Let ω ∈ Λ1(M) be a one-form defining the transversely orientable foliation. We choose a riemannian
metric on M and we consider the transverse vector field Xx = grad(ω)x. We suppose ||X || = 1. In U , we
have X = h · grad(f) for some real function h > 0 defined on U . Further, we may suppose that ∂U follows
the orbits of X in a neighborhood of ∂ Cp(F).
The Morse Lemma gives a local description of the foliation near its singularities; in particular the local topology
of a leaf near a saddle of index l is given by the connected components of the level sets of the function f(x) =
−x21 − · · · − x
l + x
l+1 + · · · + x
n. If, for c ≥ 0, we write f
−1(c) = {(x1, . . . , xn) ∈ R
n|x21 + · · · +
x2l + c = x
l+1 + · · · + x
n}, it is easy to see that f
−1(0) is homeomorphic to a cone over Sl−1 × Sn−l−1
and f−1(c) ≃ Bl × Sn−l−1 (c > 0). Similarly, we obtain f−1(c) ≃ Bn−l × Sl−1 for c < 0. Therefore,
by our hypothesis on l, the level sets are connected; in particular the separatrix S ⊃ f−1(0) is unique and
∂ Cp(F) = S ∪ {q}; moreover U is splitted by f
−1(0) in two different components. A priori, a leaf may
intersect more than one component. As F is transversely orientable, the holonomy is an orientation preserving
diffeomorphism, and then a leaf may intersect only non adiacent components; then this is not the case, in our
hypotheses.
Let L be a spherical leaf ⊂ Cp(F) enough near q. Then L ∩ U 6= ∅ and it is not restrictive to suppose it
is given by f−1(c) for some c < 0. We define the compact set K = Sn−1 \ Bn−l × Sl−1 ≃ L \ U . As
n ≥ 3, the composition K
// L \ U
// L is homotopic to a constant in its leaf. By
the proof of the Holonomy Lemma 3.3, L \ U projects diffeomorphically onto A(ǫ) = ∂ Cp(F), by means
of 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) ≃ S
n−1/Sl−1, a set with the
homotopy type of Sn−1 ∨ Sl (where ∨ is the wedge sum), simply connected in our hypotheses. Consequently,
the map K×{ǫ} → ∂ Cp(F), obtained with the extension, admits, on turn, a normal extension. This completes
the piecewise description of F .
In case of presence of a saddle of index 1 or n− 1, we have:
Theorem 3.5 Let F be a C∞, codimension one, transversely orientable, Morse foliation of a compact n-
manifold, M , n ≥ 3. Let q be a saddle of index 1 or n − 1 accumulating to one center p. Let L ⊂ Cp(F) be
a spherical leaf intersecting a neighborhood U 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 to Sn−1; (ii) ∂ Cp(F) contains
both separatrices S1 and S2 of the saddle (see figure 5) and is homeomorphic to S
n−1/Sn−2 ≃ Sn−1 ∨Sn−1.
If this is the case, there exist two leaves Fi (i = 1, 2), such that Fi and L intersect different components of
U \ Si and we have that Fi is homeomorphic to S
n−1 (i = 1, 2); (iii) q is a self-connected saddle (see figure
6) and ∂ Cp(F) is homeomorphic to S
n−1/S0. In this case we will refer to the couple
Cp(F),F| Cp(F )
a singular Reeb component. Moreover, U \ ∂ Cp(F) has three connected components and L intersects two of
them. If F is a leaf intersecting the third component of U \ ∂ Cp(F), then F is homeomorphic to S
1 × 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 sketch
here. The three cases arise from the fact that q has two local separatrices, S1 and S2, but not necessarily
∂ Cp(F) contains both of them. When this is the case, we may have that S1 and S2 belong to distinct leaves,
or to the same leaf (in this case all spherical leaves contained in Cp(F) intersect two different components of
U \ (S1 ∪ S2) ). Using the Morse lemma, we construct the set K for the application of the Holonomy Lemma
3.3. We have, respectively: K = Bn−1, K = K1 ⊔K2 = S
0 ×Bn−1 (we apply twice the Holonomy Lemma),
K = B1 × Sn−2. In the first two cases, as K is simply connected, the map K → L, to be extended, is clearly
homotopic to a constant in its leaf. Then L \U projects onto ∂Cp(F) and on neighbour leaves. This completes
the 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} ≃ B
1 × Sn−2, simply connected for n 6= 3. With a second application of the
Holonomy Lemma (n 6= 3), K projects diffeomorphically onto any neighbour leaf, F . The same also happens
for n = 3, because a curve γ : S1 → ∂ Cp(F), as the one depicted in figure 6, is never a generator of the
holonomy, which is locally trivial (a consequence of the Morse lemma). Nevertheless, there are essentially two
ways to piece pieces together. We may have F ≃ S1 × Sn−2 or F ≃ R× Sn−2.
The last result gives the motivation for a new concept.
Definition 3.6 In a codimension one singular foliation F it may happen that, for some leaf L and q ∈ Sing(F),
the set L ∪ {q} is arcwise connected. Let C = {q ∈ Sing(F)|L ∪ {q} is arcwise connected}. If for some
leaf L the set C 6= ∅, we define the corresponding singular leaf [Wag] S(L) = L ∪ C. In particular, if F is a
transversely orientable Morse foliation, each singular leaf is given by S(L) = L∪{q}, for a single saddle-type
singularity q, either selfconnected or not.
In the case of a transversely orientable Morse foliation F on M (n = dimM ≥ 3), given a saddle q and
a separatrix L of q, we may define a sort of holonomy map of the singular leaf S(L). This is done in the
following way.
As the foliation is Morse, in a neighborhoodU ⊂ M of q there exists a (Morse) local first integral f : U → R,
with f(q) = 0. Keeping into account the structure of the level sets of the Morse function f (see Theorem 3.4
and Theorem 3.5) we observe that there are at most three connected components in U \ S(L) = U \ {f−1(0)}
(notice that the number of components depends on the Morse index of q).
Let γ : [0, 1] → S(L) be a C1 path through the singularity q. At first, we consider the case γ([0, 1]) ⊂ U ,
q = γ(t) for some 0 < t < 1. For a point x ∈ M \ Sing(F), let Σx be a transverse section at x. The
set Σx \ {x} is the union of two connected components, Σ
x and Σ
x that we will denote by semi-transverse
sections at x. For x = γ(0) ∈ S(L) we have f(x) = 0 and we can choose semi-transverse sections at x in a
way that f(Σ+x ) > 0 and f(Σ
x ) < 0. We repeat the construction for y = γ(1), obtaining four semi-transverse
sections, which are contained in (at most) three connected components of U \S(L). As a consequence, at least
two of them are in the same component. By our choices, this happens for Σ−x and Σ
y (but we cannot exclude it
happens also for Σ+x and Σ
y ). We define the semi-holonomy map h
− : Σ−
∪γ(0) → Σ−
∪γ(1) by setting
h−(γ(0)) = γ(1) and h−(z) = h(z) for z ∈ Σ−
, where h : Σ−
is a classic holonomy map (i.e.
such that for a leaf F , it is h(F ∩ Σ−
) = F ∩ Σ−
). In the same way, if it is the case, we define h+.
Consider now any curve γ : [0, 1] → S(L). As F is transversely orientable, the choice of a semi-transverse
section for the curve γ([0, 1]) ∩ U , may be extended continuously on the rest of the curve, γ([0, 1]) \ U ; with
this remark, we use classic holonomy outside U . To complete the definition, it is enough to say what a semi-
transverse section at the saddle q is. In this way we allow q ∈ γ(∂[0, 1]). To this aim, we use the orbits of
the transverse vector field, grad(f). By the property of gradient vector fields, there exist points t, v such that
α(t) = ω(v) = q. Let Σ+q (Σ
q ) be the negative (positive) semi-orbit through t (v). Each of Σ
q and Σ
transverse to the foliation and such that Σ+q ∩ Σ
q = {q}, is a semi-transverse section at the saddle q.
In this way, the semi-holonomy of a singular leaf Hol+(S(L),F) is a representation of the fundamental
group π1(S(L)) into the germs of diffeomorphisms of R≥0 fixing the origin, Germ( R≥0, 0).
Now we consider the (most interesting) case of a selfconnected separatrix S(L) = ∂ Cp(F), with ∂ Cp(F)
satisfying the description of Theorem 3.5, case (iii). The singular leaf ∂ Cp(F), homeomorphic to S
n−1/S0,
has the homotopy type of Sn−1 ∨ S1. We have Hol+(∂ Cp(F),F) = {e, h
γ }, where γ is the non trivial
generator of the homotopy, and h−γ is a map with domain contained in the complement ∁ Cp(F). The two
options h−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 delete
pairs of singularities in certain configurations, and, this way, to lead us back to simple situations as in the Reeb
Sphere Theorem (2.4). We need the following notion [Cam-Sc]:
Definition 4.1 Let F be a codimension one foliation with isolated singularities on a manifold Mn. By a
dead branch of F we mean a region R ⊂ 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 necessarily
distinct leaves in F ) and, respectively, of transverse sections, Σ ≈ {t} × B1, t ∈ Sn−2.
Let Σi, i = 1, 2 be two transverse sections. Observe that the holonomy from Σ1 → Σ2 is always trivial, in the
sense of the Transverse Uniformity Theorem [Cam-LN], even if Σi ∩ S(L) 6= ∅ for some singular leaf S(L).
In this case we refer to the holonomy of the singular leaf, in the sense above.
A first result includes known situations.
Proposition 4.2 Given a foliated manifold (Mn,F), with F Morse and transversely orientable, with Sing(F) ∋
p, q, where p is a center and q ∈ ∂ Cp(F) is a saddle of index 1 or n− 1, there exists a new foliated manifold
(M, F̃), such that: (i) F̃ and F agree outside a suitable region R of M , which contains the singularities p, q;
(ii) F̃ is nonsingular in a neighborhood of R.
Proof. We are in the situations described by Theorem 3.5. If we are in case (i), the couple (p, q) may be
eliminated with the technique of the dead branch, as illustrated in [Cam-Sc]. If we are in case (ii), we observe
that the two leaves Fi, i = 1, 2 bound a region, A, homeomorphic to an anulus, S
n−1 × [0, 1]. We may now
replace the singular foliation F|A with the trivial foliation F̃|A, given by S
n−1 × {t}, t ∈ [0, 1]. If we are in
case (iii), we may replace the singular Reeb component with a regular one, in the spirit of [Cam-Sc]. Even in
this case, we may think the replacing takes place with the aid of a new sort of dead branch, the dead branch
PSfrag replacements
Figure 9: On the left: the height function on the
plane V defines a foliation of the torus; on the
right: a possible description of the foliation.
PSfrag replacements
∂ Cp(F)
Figure 10: On the left, a dead branch for the self-
connected saddle q of figure 9; on the right, the
foliation obtained after the elimination of the two
couples of singularities.
of the selfconnected saddle, that we describe with the picture of figure 10, for the case of the foliation of the
torus of figure 9, defined by the height Morse function [Mil 1]. Observe that the couples (p, q) and (r, s) of this
foliation may be also seen as an example of the coupling described in Theorem 3.5, case (ii). In this case the
elimination 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 foliation F on an n-manifold Mn, there exists a new foliation F̃ on M , with
Sing(F̃) = Sing(F) ∪ {p, q}, where p and q are a couple of saddles of consecutive indices, connecting
transversely (i.e. such that the stable manifold of p, Ws(p), intersects transversely the unstable manifold of q,
Wu(q)).
Proof. We choose the domain of (any) foliated chart, (U, φ). Observe that R′ = U (≃ φ(U)) is a dead branch
for a foliation F ǫ′ , given (up to diffeomorphisms) by the submersion fǫ = −x
1/2− · · · − x
k−1/2 + (x
ǫxk) + xk+1/2+ · · ·+ x
n/2, for some ǫ = ǫ
′ < 0. We consider F ǫ′′ , given by taking ǫ = ǫ
′′ > 0 in fǫ, which
presents a couple of saddles of consecutive indices, and we choose a dead branch R′′ around them. We also
choose a homeomorphism between R′ and R′′ which sends invariant sets of F ǫ′ into invariant sets of F ǫ′′ in a
neighborhood of the boundary. With a surgery, we may replace F ǫ′ with F ǫ′′ .
The converse of the above poposition is preceded by the following
Remark 4.5 Given a foliation F on Mn with two complementary saddle singularities p, q ∈ Sing(F), having
a strong stable connection γ, there exist a neighborhood U of p, q and γ in Mn, a δ ∈ R+ and a coordinate
system φ : U → Rn taking p onto (0, . . . , φk = −δ, . . . , 0), q onto (0, . . . , φk = δ, . . . , 0), γ onto the
xk-axis, {xl = 0}l 6=k, and such that: (i) the stable manifold of p is tangent to φ
−1({xl = 0}l>k) at p, (ii)
the unstable manifold of q is tangent to φ−1({xl = 0}l<k) at q (we are led to the situation considered in
[Mil 2], A first cancelation theorem). So using the chart φ : U → Rn we may assume that we are on a
dead branch of Rn and the foliation F|U is defined by fǫ, for ǫ = δ
2. In this way the vector field grad(fǫ)
defines a transverse orientation in U . For a suitable µ > 0, the points r1 = (0, . . . , φ
k = −δ − µ, . . . , 0)
and r2 = (0, . . . , φ
k = δ + µ, . . . , 0) are such that the modification takes place in a region of U delimited by
Lri , i = 1, 2.
Proposition 4.6 Given a foliation F on Mn with a couple of saddles p, q of complementary indices, having
a strong stable connection, there exists a dead branch of the couple of saddles, R ⊂ M and we can obtain
a foliation F̃ on M such that: (i) F̃ and F agree on M \ R; (ii) F̃ is nonsingular in a neighborhood of R;
indeed F̃ |R is conjugated to a trivial fibration; (iii) the holonomy of F̃ is conjugate to the holonomy of F in
the following sense: given any leaf L of F such that L ∩ (M \R) 6= ∅, then the corresponding leaf L̃ of F̃ is
such that Hol(L̃, F̃) is conjugate to Hol(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 of M = Sn, invariant for the splitting Sn = Bn−l × Sl ∪φ S
n−l−1 × Bl+1, where φ is a
diffeomorphism of the boundary. In fact, by theorem 3.4 or 3.5, case (iii), Bn−l × Sl admits a foliation with
one center and one saddle of index l. Similarly, Sn−l−1 × Bl+1 admits a foliation with a saddle of index
n− l− 1, actually a saddle of index l+ 1, after the attachment. We may eliminate the trivial couple of saddles
and we are led to the well-known foliation of Sn, with a couple of centers and spherical leaves.
Remark 4.8 The elimination of saddles of consecutive indices is actually a generalization of the elimination of
couples center-saddle, (p, q) with q ∈ ∂ Cp(F). Indeed, we may eliminate (p, q) only when the saddle q has
index 1 or n−1. This means the singularities of the couple must have consecutive indices and, as q ∈ ∂ Cp(F),
there exists an orbit of the transverse vector field having p as α-limit (backward) and q as ω-limit (forward), or
viceversa. 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 a C∞, codimension one foliation F on M is a stable
singularity, if there exists a neighborhood U of p in M and a C∞ function, f : U → R, defining the foliation
in U , such that f(p) = 0 and f−1(a) is compact, for |a| small. The following characterization of stable
singularities can be found in [Cam-Sc].
Lemma 5.2 An isolated singularity p of a function f : U ⊂ Rn → R defines a stable singularity for df ,
if and only if there exists a neighborhood V ⊂ U of p, such that, ∀x ∈ V , we have either ω(x) = {p} or
α(x) = {p}, where ω(x) (respectively α(x)) is the ω-limit (respectively α-limit) of the orbit of the vector field
grad(f) through the point x.
In particular it follows the well-known:
Lemma 5.3 If a function f : U ⊂ Rn → R has an isolated local maximum or minimum at p ∈ U then p is a
stable singularity for df .
The converse is also true:
Lemma 5.4 If p is a stable singularity, defined by the function f , then p is a point of local maximum or minimum
for f .
Proof. It follows immediately by Lemma 5.2 and by the fact that f is monotonous, strictly increasing, along
the orbits of grad(f).
With this notion, we obtain
Lemma 5.5 Let F be a codimension one, singular foliation on a manifold Mn. In a neighborhood of a stable
singularity, the leaves of F are diffeomorphic to spheres.
Proof. Let p ∈ Sing(F) be a stable singularity. By Lemma 5.4, we may suppose p is a minimum (otherwise
we use −f ). Using a local chart around p, we may suppose we are on Rn and we may write the Taylor-
Lagrange expansion around p for an approximation of the function f : U → R at the second order. We
have f(p + h) = f(p) + 1/2〈h,H(p + θh)h〉, where H is the Hessian of f and 0 < θ < 1. It follows
〈h,H(p+ θh)h〉 ≥ 0 in U . Then f is convex and hence the sublevels, f−1(c), are also convex.
We consider the flow φ : D(φ) ⊂ R × U → U of the vector field grad(f). By the properties of gradient
vector fields, in our hypothesis, D(φ) ⊃ (−∞, 0] × U and ∀x ∈ U there exists the α-limit, α(x) = p. For
any x ∈ f−1(c), the tangent space, Txf
−1(c), to the sublevels of f does not contain the radial direction, −→px.
This is obvious otherwise, for the convexity of f−1(c), the singularity p should lie on the sublevel f−1(c), a
contraddiction because, in this case, p should be a saddle. Equivalently, the orbits of the vector field grad(f)
are transverse to spheres centered at p. An application of the implicit function theorem shows the existence
of a smooth function x → tx, that assigns to each point x ∈ f
−1(c) the (negative) time at which φ(t, x)
intersects Sn−1(p, ǫ), where ǫ is small enough to have Bn(p, ǫ) ( R(f−1(c)), the compact region bounded by
f−1(c) . The diffeomorphism between the leaf f−1(c) and the sphere Sn−1(p, ǫ) is given by the composition
x → φ(tx, x). The lemma is proved.
Lemma 5.6 Let F be a codimension one, transversely orientable foliation of M , with all leaves closed,
π : M → M/F the projection onto the space of leaves. Then we may choose a foliated atlas on M and
a differentiable structure on M/F , such that M/F is a codimension one compact manifold, locally diffeomor-
phic to the space of plaques, and π is a C∞ map.
Proof. At first we notice that the space of leaves M/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 and
with no holonomy, we may choose a foliated atlas {(Ui, φi)} such that, for each leaf L ∈ F , L∩Ui consists, at
most, of a single plaque. Let π : M → M/F be the projection onto the space of leaves and πi : Ui → R the
projection onto the space of plaques. With abuse of notation, we may write πi = p2 ◦ φi, where p2 is the pro-
jection on the second component. As there is a 1-1 correspondence between the quotient spaces π|Ui(Ui) and
πi(Ui), then, are homeomorphic. Let V ⊂ M/F be open. The set π
−1(V ) is an invariant open set. We may
find a local chart (Ui, φi) such that π(Ui) = V . We say that (V, πi ◦ (π|Ui )
−1) is a chart for the differentiable
atlas with the required property. To see this, it is enough to prove that, if (V, πj ◦(π|Uj )
−1) is another chart with
the same domain, V , there exists a diffeomorphism between the two images of V , i.e. between πi◦(π|Ui)
−1(V )
and πj ◦ (π|Uj )
−1(V ). This is not obvious when Ui ∩ Uj = ∅. Indeed, the searched diffeomorphism exists,
and it is given by the Transverse Uniformity Theorem [Cam-LN]. Observe that, in coordinates, π coincides
with the projection on the second factor.
Lemma 5.7 Let n ≥ 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 of p with all leaves compact. We need a
local first integral near p. As a consequence of the Reeb Local Stability Theorem 2.1, we can find an (invari-
ant) open neighborhood V ⊂ U of p, whose leaves have all trivial holonomy. The set V \ {p} is open in
M∗ = M \ Sing(F). Let F∗ = F \ Sing(F); the projection π∗ : M∗ → M∗/F∗ is an open map (see, for
example [Cam-LN]). As a consequence of Lemma 5.6, the connected (as n ≥ 2) and open set π∗(V \ {p})
is a 1-dimensional manifold with boundary, i.e. it turns out to be an interval, for example (0, 1). Now, we
extend smoothly π∗ to a map π on U . In particular, let W ( V be a neighborhood of p. If (for example)
π∗(W \ {p}) = (0, b) for some b < 1, we set π(p) = 0. Thesis follows by lemma 5.3.
Theorem 5.8 Let Mn be a closed n-dimensional manifold, n ≥ 3. Suppose that M supports a C∞, codimen-
sion one, transversely orientable foliation, F , with non-empty singular set, whose elements are, all, weakly
stable singularities. Then M 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 lemma
5.5, in an invariant neighborhood Up of p, the leaves are diffeomorphic to spheres. Now we can proceed as in
the proof of the Reeb Sphere Theorem 2.4.
Theorem 5.9 (Classification of codimension one foliations with all leaves compact) Let F be a (possibly
singular, with isolated singularities) codimension one foliation of M , with all leaves compact. Then all pos-
sible singularities are stable. If F 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) = ∅. In
all the other cases, denoting by π : M → [0, 1] the projection onto the space of leaves, it is Hol(π−1(x),F) =
{e}, ∀x ∈ (0, 1). Moreover, if x = 0, 1, we may have: (i) π−1(x) ⊂ ∂M 6= ∅ and Hol(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 leaf π−1(y) is a two-sheeted covering of π−1(x).
Proof. If F is transversely orientable, by the Reeb Global Stability Theorem 2.2 and Lemma 5.6, the space of
leaves is either diffeomorphic to S1 or to [0, 1]. In particular, M/F ≈ S1 if and only if M is closed and F non
singular. When this is not the case, M/F ≈ [0, 1], and there are exactly two points (∂[0, 1]) which come from
a 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 in M/F to (by Proposition 2.3). By the proof of Lemma 5.6, the projection is not
differentiable and the space of leaves M/F , a Hausdorff topological 1-dimensional space, turns out to be an
orbifold (see [Thu]). We pass to the transversely orientable double covering, p : (M̃, F̃) → (M,F). The fo-
liation F̃ , pull-back of F , has all leaves compact, and singular set empty or with stable components; therefore
we apply the first part of the classification to M̃/F̃ . Both if M̃/F̃ is diffeomorphic to S1 or to [0, 1], M/F is
homeomorphic 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 similar
result of Camacho and Scárdua [Cam-Sc] concerning the case n = 3, we need to recall another result, that we
are going to generalize.
As we know, the Reeb Sphere Theorem, in its original statement, consideres the effects (on the topology of a
manifold M ) determined by the existence, on M , of a real valued function with exactly two non-degenerate
singular points. A very similar problem was studied by Eells and Kuiper [Ee-Kui]. They considered manifolds
admitting 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 of M , which must
be even and assume one of the values n = 2m = 2, 4, 8, 16. Moreover, the homotopy type of the manifold
turns out to vary among a finite number of cases, including (or reducing to, if n = 2, 4) the homotopy tupe of
the projective plane over the real, complex, quaternion or Cayley numbers.
Definition 5.10 In view of the results of Eells and Kuiper [Ee-Kui], if a manifold M admits a real-valued func-
Figure 11: Elimination technique applied in case
(ii) (Theorem 3.5) for the foliation of figure 9.
PSfrag replacements q
Figure 12: A foliation of RP2 with three singular
points.
tion with exactly three non-degenerate singular points, we will say that M is an Eells-Kuiper manifold.
We have (see [Cam-Sc] for the case n = 3):
Theorem 5.11 (Center-Saddle Theorem) Let Mn be an n-dimensional manifold, with n ≥ 2 such that
(M,F) is a foliated manifold, by means of a transversely orientable, codimension-one, Morse, C∞ folia-
tion F . Moreover F is assumed to be without holonomy if n = 2. Let Sing(F) be the singular set of F , with
#Sing(F) = k + l, where k, l are the numbers of, respectively, centers and saddles. If we have k ≥ l + 1,
then there are two possibilities:
(1) k = l + 2 and M is homeomorphic to an n-dimensional sphere;
(2) k = l + 1 and M is an Eells-Kuiper manifold.
Proof. If l = 0, assertion is proved by the Reeb Sphere Theorem 2.4. Let l ≥ 1; we prove our thesis by
induction on the number l of saddles. We set F l = F .
So let l = 1 and F1 = F . By hypothesis, in the set Sing(F) there exist at least two centers, p1, p2, with
p1 6= p2, and one saddle q. We have necessarily q ∈ ∂ Cp1(F) ∩ ∂ Cp2(F). In fact, if this is not the case and,
for example q /∈ ∂ Cp1(F), then (keeping into account that for n = 2, the foliation F is assumed to be without
holonomy) ∂ Cp1 = ∅ and M = Cp1(F). A contraddiction. Let i(q) the Morse index of the saddle q.
For n ≥ 3 we apply the results of Theorems 3.4 and 3.5 to the couples (p1, q) and (p2, q). In particular, by
Theorem 3.5, (iii), it follows that the saddle q 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.
For n = 2, we have necessarily i(q) = 1 and, in our hypotheses, q is always selfconnected. With few changes,
we adapt Theorem 3.5, to this case, obtaining ∂ Cp(F) ≃ S
1 or ∂ Cp(F) ≃ S
1 ∨ S1; in this latter case we will
say that the saddle q is selfconnected with respect to p. We obtain:
(a’) (p1, q) or (and) (p2, q) is a trivial couple;
(b’) q is selfconnected both with respect to p1 and to p2.
In cases (a) and (a’) we proceed with the elimination of a trivial couple, as stated in Proposition 4.2, and then
we obtain the foliated manifold (M,F0), with no saddle-type and some center-type singularities. We apply the
Reeb Sphere Theorem 2.4 and obtain #Sing(F) = 2 and M ≃ Sn.
In case (b) (n ≥ 3), as a consequence of Theorem 3.4, we necessarily have i(q) = n/2 (and therefore n must
be even!). Moreover Cp1(F) ≈ Cp2(F) and M = 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 diffeomorphism φ.
In case (b’) (n = 2), we obtain the same result as above, i.e. Cp1(F) ≈ Cp2(F) and M = Cp1(F)∪φ Cp2(F).
We notice that case (b’) occurs when the set Cpi(F) ≃ B
2/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 by
the projection of M onto the space of (possibly singular) leaves. In fact, by Lemma 5.6, the space of leaves is
diffeomorphic to a closed interval of R. In this way M turns out to be an Eells-Kuiper manifold. This ends the
case l = 1.
Let l > 1 (and #Sing(F) > 3). As above, in Sing(F) there exist at least one saddle q and two (distinct) cen-
ters, p1, p2 such that q ∈ ∂ 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, otherwise M = Cp1(F) ∪φ Cp2(F) and #Sing(F) = 3,
a contraddiction. Then we may proceed with the elimination of a trivial couple. In this way we obtain the
foliated manifold (M,F l−1), which we apply the inductive hypothesis to. The theorem is proved, observing
that, a posteriori, case (1) holds if k = l + 2 and case (2) if k = l + 1.
6 Haefliger-type theorems
In this paragraph, we investigate the existence of leaves of singular foliations with unilateral holonomy. Keep-
ing into account the results of the previous paragraph, for Morse foliations, we may state or exclude such an
occurrence, according to the following theorem:
Theorem 6.1 Let F be a C∞, codimension one, Morse foliation on a compact manifold Mn, n ≥ 3, assumed
to be transversely orientable, but not necessarily closed. Let k be the number of centers and l the number of
saddles. We have the following possibilities: (i) if k ≥ l + 1, then all leaves are closed in M \ Sing(F); in
particular, if ∂M 6= ∅ or k ≥ l + 2 each regular (singular) leaf of F , is diffeomorphic (homeomorphic) to a
sphere (in the second option, it is diffeomorphic to a sphere with a pinch at one point); (ii) if k = l there are
two possibilities: all leaves are closed in M \Sing(F), or there exists some compact (regular or singular) leaf
with unilateral holonomy.
Example 6.2 The foliation of example 4.7 is an occurrence of theorem 6.1, case (ii) with all leaves closed. The
Reeb foliation of S3 and each foliation we may obtain from it, with the introduction of l = k trivial couples
center-saddle, are examples of theorem 6.1, case (ii), with a leaf with unilateral holonomy.
Now we consider other possibilities for Sing(F).
Definition 6.3 Let F be a C∞, codimension one foliation on a compact manifold Mn, n ≥ 3, with singular
set Sing(F) 6= ∅. We say that Sing(F) is regular if its connected components are either isolated points or
smoothly embedded curves, diffeomorphic to S1. We extend the definition of stability to regular components,
by saying that a connected component Γ ⊂ Sing(F) is (weakly) stable, if there exists a neighborhood of Γ,
where the foliation has all leaves compact (notice that we can repeat the proof of Lemma 5.7 and obtain that a
weakly stable component is a stable component).
In the case Sing(F) is regular, with stable isolated singularities, when n ≥ 3 we may exclude a Haefliger-
type result, as a consequence of Lemma 5.5 and the Reeb Global Stability Theorem for manifolds with bound-
ary. Then we study the case Sing(F) regular, with stable components, all diffeomorphic to S1. Let J be a set
such that for all j ∈ J , the curve γj : S
1 → M , is a smooth embedding and Γj := γj(S
1) ⊂ Sing(F) is
stable. Then J is a finite set. This is obvious, otherwise ∀j ∈ J , we may select a point xj ∈ Γj and obtain
that the set {xj}j∈J has an accumulation point. But this is not possible because the singular components are
separated. We may regard a singular component Γj , as a degenerate leaf, in the sense that we may associate to
it, a single point of the space of leaves.
We need the following definition
Definition 6.4 Let F be a C∞, codimension one foliation on a compact manifold M . Let D2 be the closed
2-disc and g : D2 → M be a C∞ map. We say that p ∈ D2 is a tangency point of g with F if (dg)p( R
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 a C∞ map, such that the restriction A|∂D2 is transverse to F , i.e.
∀x ∈ ∂D2, (dA)x(Tx(∂D
2)) + TA(x)FA(x) = TA(x)M . Then, for every ǫ > 0 and every integer r ≥ 2,
there exists a C∞ map, g : D2 → M , ǫ-near A in the Cr-topology, satisfying the following properties: (i)
g|∂D2 is transverse to F . (ii) For every point p ∈ D
2 of tangency of g with F , there exists a foliation box
U of F with g(p) ∈ U and a distinguished map π : U → R such that p is a non-degenerate singularity of
π ◦ g : g−1(U) → R. In particular there are only a finite number of tangency points of g with F , since they
are isolated, and they are contained in the open disc D2 = {z ∈ R2 : ||z|| < 1}. (iii) If T = {p1, . . . , pt}
is the set of tangency points of g with F , then g(pi) and g(pj) are contained in distinct leaves of F , for every
i 6= j. In particular, the singular foliation g∗(F) has no saddle connections.
We are now able to prove a similar result, in the case of existence of singular components.
Proposition 6.6 Let F be a codimension one, C∞ foliation on a compact manifold Mn, n ≥ 3, with regular
singular set, Sing(F) = ∪j∈JΓj 6= ∅, where Γj are all stable components diffeomorphic to S
1 and J is finite.
Let A : D2 → M be a C∞ map, such that the restriction A|∂D2 is transverse to F . Then, for every ǫ > 0 and
every integer r ≥ 2, there exists a C∞ map, g : D2 → M , ǫ-near A in the Cr-topology, satisfying properties
(i) and (iii) of proposition 6.5, while (ii) is changed in: (ii’) for every point p ∈ D2 of tangency of g with F , we
have two cases: (1) if Lg(p) is a regular leaf of F , there exists a foliation box, U of F , with g(p) ∈ U , and a
distinguished map, π : U → R, satisfying properties as in (ii) of Proposition 6.5; (2) if Lg(p) is a degenerate
leaf of F , there exists a neighborhood, U of p, and a singular submersion, π : U → R, satisfying properties
as in (ii) Proposition 6.5.
Proof. We start by recalling the idea of the classical proof.
We choose a finite covering of A(D2) by foliation boxes {Qi}
i=1. In each Qi the foliation is defined by
a distinguished map, the submersion πi : Qi → R. We choose an atlas, {(Qi, φi)}
i=1, such that the last
component of φi : Qi → R
n is πi, i.e. φi = (φ
i , φ
i , . . . , φ
i , πi). We construct the finite cover of D
{Wi = A
−1(Qi)}
i=1; the expression of A in coordinates is A|Wi = (A
i , . . . , A
i , πi ◦A). We may choose
covers of D2, {Ui}
i=1, {Vi}
i=1, such that Ui ⊂ Vi ⊂ Vi ⊂ Wi, i = 1, . . . , r; then we proceed by induction
on the number i. Starting with i = 1 and setting g0 = A, we apply a result ([Cam-LN], Cap. VI, §2, Lemma
1, pag. 120) and we modify gi−1 in a new function gi, in a way that gi(Wi) ⊂ Qi and πi ◦ gi : Wi → R is
Morse on the subset Ui ⊂ Wi. At last we set g = gr.
In the present case, essentially, it is enough to choose a set of couples, {(Uk, πk)}k∈K , where {Uk}k∈K is
an open covering of M , πk : Uk → R, for k ∈ K , is a (possibly singular) submersion and, if Uk ∩ Ul 6= ∅
for a couple of indices k, l ∈ K , there exists a diffeomorphism plk : πk(Uk ∩ Ul) → πl(Uk ∩ Ul), such
that πl = plk ◦ πk. By hypothesis, there exists the set of couples {(Ui, πi)}i∈I , where {Ui}i∈I , is an open
covering of M \ Sing(F), and, for i ∈ I , the map πi : Ui → R, is a distinguished map, defining the foliated
manifold (M \ Sing(F),F∗). Let y ∈ Sing(F), then y ∈ Γj , for some j ∈ J . As y ∈ M , there exists
a neighborhood C ∋ y, homeomorphic to an n-ball. Let h : C → Bn be such a homeomorphism. As the
map γj : S
1 → Γj is a smooth embedding, we may suppose that, locally, Γj is sent in a diameter of the
ball Bn, i.e. h(C ∩ Γj) = {x2 = · · · = xn = 0}. For each singular point z = h
−1(b, 0, . . . , 0), the set
D = h−1(b, x2, . . . , xn), homeomorphic to a small (n−1)-ball, is transverse to the foliation at z. Moreover, if
z1 6= z2, then D1 ∩D2 = ∅. The restriction F|D is a singular foliation with an isolated stable singularity at z.
By lemma 5.5, the leaves of F|D are diffeomorphic to (n− 2)-spheres. It turns out that y has a neighborhood
homeomorphic 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}. Let πy : Uy → [0, 1)
be the projection. If, for a couple of singular points y, w ∈ Sing(F), we have Uy ∩Uw 6= ∅, we may suppose
they belong to the same connected component, Γj . We have πw ◦ π
y (0) = 0 and, as a consequence of lemma
5.6, there exists a diffeomorphism between πy(Uy ∩ Uw \ Γj) and πw(Uy ∩ Uw \ Γj). The same happens if
Uy ∩Ui 6= ∅ for some Ui ⊂ M \Sing(F). It comes out that πy is singular on Uy ∩Sing(F) and non-singular
on Uy \ Sing(F), i.e. (dπy)z = 0 ⇔ z ∈ Uy ∩ Sing(F). At the end, we set K = I ∪ Sing(F).
Let g : D2 → M be a map. Then g defines the foliation g∗(F), pull-back of F , on D2. Observe that if
Sing(F) = ∅, then Sing(g∗(F)) = {tangency points of g with F}, but in the present case, as Sing(F) 6= ∅,
we have Sing(g∗(F)) = {tangency points of g with F} ∪ g∗(Sing(F)). Either if p is a point of tangency of
g with F or if p ∈ g∗(Sing(F)), we have d(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) Let F be a codimension one, C2, possibly singular
foliation of an n-manifold M , with Sing(F), (empty or) regular and with stable components diffeomorphic to
S1. Suppose there exists a closed curve transverse to F , homotopic to a point. Then there exists a leaf with
unilateral 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 of
stable singular set. To this aim, we premise the following remark. Novikov’s statement establishes the existence
of a compact leaf for foliations on 3-manifolds with finite fundamental group. This result actually proves the
existence of an invariant submanifold, say N ⊂ M , with boundary, such that F|N contains open leaves whose
universal covering is the plane. Moreover these leaves accumulate to the compact leaf of the boundary. In what
follows, a submanifold with the above properties will be called a Novikov component. In particular a Novikov
component may be a Reeb component, i.e. a solid torus endowed with its Reeb foliation. We recall that two
PSfrag replacements
∂ Cp(F)
Figure 13: p − q is not a trivial coupling when
1 < l < n− 1, where l is the index of the saddle
PSfrag replacements ST1
Figure 14: A singular foliation of S3, with no van-
ishing cycles.
Reeb components, glued together along the boundary by means of a diffeomorphism which sends meridians in
parallels and viceversa, give the classical example of the Reeb foliation of S3.
If F is a Morse foliation of a 3-manifold, as all saddles have index 1 or 2, we are always in conditions of
proposition 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 all
leaves are compact; in case (ii), all leaves may be open and dense, as it is shown by an example of a foliation
of S3 with Morse singularities and no compact leaves [Ros-Rou].
As in the previous paragraph, we study the case in which Sing(F) is regular with stable components,Γj , j ∈ J ,
where J is a finite set. We have:
Theorem 7.1 Let F be a C∞, codimension one foliation on a closed 3-manifold M3. Suppose Sing(F) is
(empty or) regular, with stable components. Then we have two possibilities: (i) all leaves of F are compact;
(ii) F has a Novikov component.
Proof. If Sing(F) = ∅, thesis (case (ii)) follows by Novikov theorem. Let Sing(F) 6= ∅. We may suppose
that F is transversely orientable (otherwise we pass to the transversely orientable double covering). If Sing(F)
contains an isolated singularity, as we know, we are in case (i). Then we suppose Sing(F) contains no isolated
singularity, i.e. Sing(F) =
Γj . Set D(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 in
case (i), or ∂D(F) 6= ∅, and in this case it contains a leaf with unilateral holonomy, F . It is clear that F
bounds a Novikov component, and then we are in case (ii); in fact, from one side, F is accumulated by open
leaves. If F ′ is one accumulating leaf, then its universal covering is p : R2 → F ′. Suppose, by contraddiction,
that the universal covering of F ′ is p : S2 → F ′. By the Reeb Global Stability Theorem for manifolds with
boundary, all leaves are compact, diffeomorphic to p(S2). This concludes the proof since F must have infinite
fundamental group.
The last result may be reread in terms of the existence of closed curves, transverse to the foliation. We have:
Lemma 7.2 LetF be a codimension one, C∞ foliation on a closed 3-manifoldM , with singular set, Sing(F) 6=
∅, regular, with stable components. Then F is a foliation with all leaves compact if and only if there exist no
closed transversals.
Proof. (Sufficiency) If the foliation admits an open (in M \ Sing(F)) leaf, L, it is well known that we may
find a closed curve, intersecting L, transverse to the foliation. Viceversa (necessity), let F be a foliation with
all leaves compact. If necessary, we pass to the transversely orientable double covering p : (M̃, F̃) → (M,F).
In this way, we apply Lemma 5.6 and obtain, as Sing(F̃) 6= ∅, that the projection onto the space of leaves is
a (global) C∞ first integral of F̃ , f : M̃ → [0, 1] ⊂ R. Suppose, by contraddiction, that there exists a C1
closed transversal to the foliation F , the curve γ : S1 → M . The lifting of γ2 is a closed curve, Γ : S1 → M̃ ,
transverse to F̃ . The set f(Γ(S1)) is compact and then has maximum and minimum, m1,m2 ∈ R. A contrad-
diction, because Γ cannot be transverse to the leaves {f−1(m1)}, {f
−1(m2)}.
With this result, we may rephrase the previous theorem.
Corollary 7.3 Let F be a codimension one, C∞ foliation on a 3-manifold M , such that Sing(F) is regular
with stable components. Then (i) there are no closed transversals, or equivalently, F is a foliation by compact
leaves, (ii) there exists a closed transversal, or equivalently, F has a Novikov component.
Remark 7.4 In 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
a closed curve transverse the foliation, homotopic to a constant, does not lead, in general, to the existence of a
vanishing cycle, as it is shown by the following counterexample.
Example 7.5 We consider the foliation of S3 given by a Reeb component, ST1, glued (through a diffeomor-
phism of the boundary which interchanges meridians with parallels) to a solid torus ST2 = S
1 × D2 =
T 2 × (0, 1) ∪ S1. The torus ST2 is endowed with the singular trivial foliation F|ST2 = T
2 × {t}, for
t ∈ (0, 1), where Sing(F|ST2) = S
1 = Sing(F). As a closed transversal to the foliation, we consider
the curve γ : S1 → ST1 ⊂ S
3, drawed in figure 14. Let f : D2 → S3 be an extension of γ; the extension
f is assumed to be in general position with respect to F , as a consequence of proposition 6.5. As γ(S1) is
linked to the singular component S1 ⊂ ST2, then f(D2) ∩ Sing(F) 6= ∅. As a consequence, we find a
decreasing sequence of cycles, {βn}, (the closed curves of the picture) which does not admit a cycle, β∞, such
that βn > β∞, for all n. In fact the “limit” of the sequence is not a cycle, but the point f(D2) ∩ Sing(F).
Example 7.6 The different situations of Theorem 7.1 or Corollary 7.3 may be exemplified as follows. It is
easy to see that S3 admits a singular foliation with all leaves compact (diffeomorphic to T 2) and two singular
(stable) components linked together, diffeomorphic to S1. In fact one can verify that S3 is the union of two
solid tori, ST1 and ST2, glued together along the boundary, both endowed with a singular trivial foliation.
We construct another foliation on S3, modifying the previous one. We set S̃T1 = S
1 × {0} ∪ T 2 × (0, 1/2].
In this way, ST1 = S̃T1 ∪ T
2 × (1/2, 1]. We now modify the foliation in ST1 \ S̃T1, by replacing the trivial
foliation with a foliation with cylindric leaves accumulating to the two components of the boundary.
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	Foliations and Morse Foliations
	Holonomy and Reeb Stability Theorems
	Arrangements of singularities
	Realization and elimination of pairings of singularities
	Reeb-type theorems
	Haefliger-type theorems
	Novikov-type theorems
