Riemannian manifolds not quasi-isometric to leaves in codimension one

Paul A. Schweitzer, S.J.^{∗}

February 2, 2008

Abstract

We show that no conditions on the local geometry of an open Riemannian manifold L suffice to make it quasi-isometric to a leaf of a codimension one foliation of a closed manifold. We introduce the ‘bounded homology property’, a semi-local property that is necessary for L to be a leaf in codimension one. An essential step involves a partial generalization of the Novikov closed leaf theorem to higher dimensions.

CONTENTS

1. Introduction

2. Definitions and statements

3. Leaves have the bounded homology property

4. Modifying the metric on a manifold by inserting balloons.

5. Proof of the Finiteness Lemma and Lemma 3.3

6. Novikov’s Theorem for embedded vanishing cycles References.

1 Introduction

In this paper we show that no set of local bounds on the geometry of an open Riemannian p-manifold L can be sufficient to guarantee that it be quasi-isometric to a leaf of a C^{2} codimension one foliation of a smooth closed (p + 1)-manifold. A bound on the geometry of L is local if it only depends on the Riemannian geometry of the ϵ-balls around all the points of L, for some constant ϵ > 0. For example, bounds on the various curvatures and on the injectivity radius of L are local geometric bounds.

∗Supported in early stages of this work by the CNPq, FAPERJ, PRONEX of the Ministry of Science and Technology of Brazil, and the Clay Mathematics Institute

We say that two Riemannian manifolds are quasi-isometric if there is a diffeomorphism f from one to the other such that both f and f^{−1} produce only bounded distortion of the metrics, as formulated precisely in Definition 2.1 below.

Our main result is the following.

Theorem 1.1 Every connected non-compact smooth p-manifold L with p ≥ 2 possesses a C^{∞} complete Riemannian metric g with bounded geometry that is not quasi-isometric to any leaf of a codimension one C^{1} foliation on any compact smooth (p + 1)-manifold.

Furthermore g can be chosen such that no end is quasi-isometric to an end of a leaf of such a foliation, and also to have any growth type compatible with bounded geometry. In fact, for each such L there are uncountably many quasi-isometry classes of such metrics g.

This theorem was proven in [18] for p = 2, so in this paper we shall assume that p ≥ 3. The proof for p ≥ 3 depends on Theorem 2.6, which states that the leaves of a codimension one foliation of a smooth compact manifold M of dimension greater than 4, endowed with the restriction of a Riemannian metric on M, have a certain bounded homology property, a property which we define for open Riemannian manifolds of dimension at least 3. In Theorem 2.8 we show how to modify a given complete Riemannian metric on a connected smooth open manifold, without changing its growth type, so that it does not have the bounded homology property. The construction involves introducing spherical ‘bubbles’ of arbitrarily large size, but with ‘necks’ of uniformly bounded size, as in [18].

The proof of Theorem 2.6 (in Section 3) involves a Finiteness Lemma (Section 5) and a partial extension to higher dimensions of Novikov’s theorem that vanishing cycles only occur on the boundary of Reeb components (Section 6). Definitions and statements of results are given in Section 2 and the construction of Riemannian manifolds that cannot be leaves is given in Section 4.

The three theorems of this paper constitute an extension to higher dimensions of three similar theorems for open surfaces in [18]. That paper used a different bounded homotopy property for surfaces, involving contractible loops rather than bounding submanifolds, to show that all open surfaces have complete Riemannian metrics with bounded geometry that cannot be leaves in C^{1} foliations of compact 3-manifolds.

2 Definitions and statements

In this section we give several definitions leading up to the definition of the bounded homology property, and then we state Theorem 2.6, which states that, under certain hypotheses, leave have this property.

Definition 2.1 Two Riemannian manifolds L and L′ are quasi-isometric if there exist a diffeomorphism f : L → L′ and a constant C > 0 such that the distance functions d and d′ on L and L′ satisfy

C^{−1}d′(f(x), f(y)) ≤ d(x, y) ≤ Cd′(f(x), f(y))

for all points x, y ∈ L.

For example, the identity map of any smooth compact manifold gives a quasiisometry of any two Riemannian metrics on the manifold.

Let L be a p-dimensional Riemannian manifold and let δ be a small positive number.

Definition 2.2 The δ-volume Vol_{δ}(S) ∈ ℕ ∪ {∞} of a set S ⊂ L is the smallest integer K such that S can be covered by K open balls of radius δ in L, or ∞ if no such finite number exists.

Definition 2.3 Let C be a compact Riemannian p-manifold with boundary B and let δ be a constant greater than 0. Then we define Morse δ-volume M(C, δ) to be the smallest positive constant M for which there is a Morse function f : C → [0, ∞) such that f(B) = 0 and each level set f^{−1}(t) for t ∈ [0, ∞) has Vol_{δ}(f^{−1}(t)) ≤ M.

This definition makes sense since it is evident that the δ-volumes of the level sets of a fixed Morse function f on a compact manifold are uniformly bounded, so that M(C, δ) is finite. For a set S contained in a Riemannian manifold L, the open δ-neighborhood of S, V_{δ}(S), is the set of all points in L whose distance from S, measured along geodesics in L, is less than δ. We now define the bounded homology property, the fundamental tool used in this paper.

Definition 2.4 A Riemannian p-manifold L (p ≥ 3) has the bounded homology property if for every pair of constants δ > 0 and k > 0 there is a constant K = K(δ, k) such that if

1. B ⊂ L is a 1-connected smooth closed oriented (p − 1)-submanifold embedded in L,

2. the δ-neighborhood V_{δ}(B) of B in L fibers over B as a tubular neighborhood,

3. B has δ′-volume Vol_{δ′}(B) < k on L, where δ′ = δ/4, and

4. B bounds a compact oriented 1-connected region C on L,

then the Morse δ-volume of C satisfies M(C, δ) < K.

Comments. Note that the diameter of B is the supremum of distances between two points of B (measured along geodesics in B that minimize the distance) and the δ-neighborhood V_{δ}(B) of B in L is the set of points of L with distance measured along L less than δ from B. When we affirm that V_{δ}(B) fibers over B as a tubular neighborhood, we mean that for each point in V_{δ}(B) there is a unique nearest point in B (with the distance measured along L), and the map which takes each point of V_{δ}(B) to the nearest point of B is a locally trivial fibration. It is well-known that for a smooth compact submanifold B of a Riemannian manifold this property holds for sufficiently small positive numbers δ. We require that B and C be 1-connected (i.e., connected and simply connected) so that they can be lifted along the leaves of a transverse 1-dimensional foliation T to leaves of F near to L. The following observation depends on the fact that we require a quasi-isometry to be a diffeomorphism.

Remark 2.5 It is easy to check that the bounded homology property is invariant under quasi-isometry.

Most of this paper is devoted to proving the following theorem.

Theorem 2.6 Let F be a codimension one foliation of dimension p ≥ 3 on a closed smooth manifold M. Suppose that the leaves of F have a differentiable structure that varies continuously in the transverse direction and have Riemannian metrics that also vary continuously in the transverse direction.

Then the leaves of F have the bounded homology property, and for every k > 0 and δ > 0, the same constant K = K(k, δ) > 0 can be chosen for all the leaves of F.

In particular, if F is a C^{1} foliation, then the smoothness hypothesis holds, since the differentiable structures can be smoothed to C^{∞} in the leaf direction. (This can be done locally and then globalized by using a partition of unity, although the smoothness class of the foliation itself cannot be increased, in general.) Hence we have the following corollary.

Corollary 2.7 If F is a C^{1} codimension one foliation of dimension p ≥ 3 of a closed smooth manifold M, then the leaves have the bounded homology property for any Riemannian metric induced on the leaves by a Riemannian metric on M.

The proof of Theorem 2.6 is given in Section 3 using results of Sections 5 and 6. In Section 4 we shall show that every smooth open manifold of dimension at least 3 admits Riemannian metrics such that the bounded homology property does not hold. These examples can be constructed to have bounded geometry, i.e., the sectional curvature is uniformly bounded and the injectivity radius is bounded below by a positive constant, giving the following result proven in Section 4.

Theorem 2.8 Let L be a complete open connected Riemannian p-manifold of dimension p ≥ 3 with bounded geometry. Then L has other complete Riemannian metrics g with bounded geometry and the same growth type that do not possess the bounded homology property. Furthermore g can be chosen such that no end of L has the bounded homology property, and there are uncountably many quasi-isometry classes of such metrics g.

Theorem 1.1 will follow immediately from Theorems 2.8 and 2.6.

Before closing this section, we shall state a weak generalization of Novikov’s theorem on the existence of Reeb components. We define a (generalized) Reeb component as follows, requiring that the boundary be connected.

Definition 2.9 A compact (p+1)-dimensional manifold with a codimension one foliation R is a (generalized) Reeb component if the boundary ∂R is a leaf and the interior Int(R) fibers over the circle with the leaves as fibers.

Let B be a compact connected (p − 1)-dimensional manifold and let H be the “horizontal foliation” of B × [0, 1] given by the leaves B × {t} for t ∈ [0, 1]. Consider a foliated map

h : (B × [0, 1], H) → (M, F)

that is a topological immersion (i.e., a local homeomorphism onto its image), and suppose that the map h_{t} : B → L_{t}, defined by h_{t}(b) = h(b, t), is an embedding for every t into a leaf L_{t}. Then we have the following weak generalization of the second part of Novikov’s theorem.

Theorem 2.10 An Extension of Novikov’s Theorem. If B_{t} = h_{t}(B) is the boundary of a compact 1-connected region C_{t} ⊂ L_{t} for every t ∈ (0, 1] but not for t = 0, then the leaf L_{0} is the boundary of a Reeb component R whose interior Int(R) is the union of the leaves L_{t} for t > 0.

The proof of this theorem is given in Section 6. We observe that this case, in which B is an embedded “vanishing cycle”, is much easier than the case when double points are allowed. (See Haefliger [10] for a good exposition of the proof of Novikov’s original theorem that treats the problem of double points very clearly.) We also give a construction that can be shown to produce all Reeb components, up to foliated homeomorphism, though since we do not use that fact, we do not prove it here.

3 Leaves have the bounded homology property

Here we prove Theorem 2.6 using the results of Sections 5 (the Finiteness Lemma 3.2) and 6 (the extension of Novikov’s Theorem, 2.10). The idea of the proof is to consider a (p − 1)-dimensional submanifold B that satisfies the hypotheses of Definition 2.4 and approximate it by a subcomplex X of a triangulation on the same leaf. Then we show that there is a finite set of “transverse families” of subcomplexes that suffice in this process (the Finiteness Lemma 3.2). Finally we show that there is a constant K that is an upper bound for all the resulting Morse δ-volumes. Theorem 2.6 will follow easily from the Finiteness Lemma 3.2 and Proposition 3.1.

Let F be a codimension one foliation of a compact manifold M of dimension at least 4, so that the leaf dimension p ≥ 3, and suppose that constants k > 0 and δ > 0 are given. Suppose that the leaves are smooth and their differentiable structure varies continuously in the transverse direction. As a consequence, it is possible to choose Riemannian metrics on the leaves of F which vary continuously in the transverse direction, and we fix such Riemannian metrics. Thus the hypotheses of Theorem 2.6, are satisfied. We also fix a transverse 1-dimensional foliation T (which always exists, see [12] or [20]). All these conditions remain fixed throughout this section.

By a region in a smooth manifold L we mean a compact connected submanifold with smooth boundary that is the closure of an open set in L. If L has a Riemannian metric, then the diameter of a smooth submanifold S in L (possibly with boundary and corners, such as a simplex), diam(S), is the supremum of distances between pairs of points of S, measured along geodesics in S. The mesh of a triangulation of a region is the maximum of the diameters of its simplexes. A triangulation is an ϵ-triangulation if its mesh is less than a positive number ϵ. All of the triangulations we shall consider will be δ′-triangulations with δ′ = δ/4.

The Simplicial Approximation Process (SAP). Let B be the set of all submanifolds B of leaves of F that satisfy the Conditions (1), (2), and (3) of Definition 2.4. Given B ∈ B on a leaf L of F, choose a smooth δ′-triangulation of a region in L that contains V_{δ}(B) and give B a smooth triangulation that is sufficiently fine so that the open star of each vertex v of B lies in the open star of some vertex g(v) of the triangulation of L. Then the function g taking the vertices of B to those of L extends to a simplicial map ḡ : B → L, which is a simplicial approximation to the inclusion B ↪ L. Note that the image X_{B} = ḡ(B) is a (p − 1)-dimensional complex contained in V_{δ′}(B). Say that X_{B} is obtained from B by the SAP, the simplicial approximation process.

Transverse families and the Finiteness Lemma. Recall that a product manifold X × Y has two product foliations H and V, the horizontal and vertical foliations, given respectively by the leaves X × {y} for y ∈ Y and {x} × Y for x ∈ X.

Definition 3.1 A triple (X, Ω, f) will be called a transverse family if Ω is a smooth compact triangulated p-manifold with boundary, X ⊂ Ω is a (p − 1)-dimensional subcomplex, and

f : (Ω × [0, 1], H, V) → (M, F, T )

is a smooth bifoliated embedding, such that if we set Ω_{t} = f(Ω × {t}) and X_{t} = f(X × {t}), then, for every t ∈ [0, 1], relative to the metric on the leaf L_{t} that contains Ω_{t} and X_{t},

1. Ω_{t} contains V_{2δ′}(X_{t}), the 2δ′-neighborhood of X_{t}, and

2. the triangulation induced on Ω_{t} by the triangulation on Ω is a smooth δ′-triangulation.

We say that a transverse family (X, Ω, f) is good if there is a submanifold B ⊂ L_{0} with B ∈ B and a smooth bifoliated embedding

h : (B × [0, 1], H, V) → (M, F, T )

such that for every t ∈ [0, 1] the submanifold B_{t} = h(B × {t}) is contained in L_{t}, B_{0} = B, and X_{t} can be obtained from B_{t} by the SAP described above.

The following Finiteness Lemma will be proven in Section 5.

Lemma 3.2 (Finiteness Lemma). There is a finite set of good transverse families (X_{i}, Ω_{i}, f_{i}), 1 ≤ i ≤ ℓ, such that, for each B ∈ B, the complex X_{B} obtained from B by the SAP can be chosen to be X_{i,t} = f(X_{i} × {t}) for some i ∈ {1, …, ℓ} and t ∈ [0, 1].

Let B_{i} be the set of B ∈ B for which the SAP can yield the complex X_{B} = X_{i,t} for some t ∈ [0, 1]. Then Theorem 2.6 is an obvious consequence of the Finiteness Lemma 3.2 and the following Proposition, since K = K(δ, k) = max_{1≤i≤ℓ}K_{i} will be a common upper bound for the Morse δ-volumes of the regions C corresponding to all the submanifolds B ∈ B.

Proposition 3.1 For each i, 1 ≤ i ≤ ℓ, there exists a constant K_{i} such that, for each B ∈ B_{i} that satisfies condition (4) of Definition 2.4, the region C with ∂C = B has Morse δ-volume M(C, δ) less than K_{i}.

To prepare for the proof of this Proposition we need some preliminary results. Fix an index i and consider the good transverse family (X_{i}, Ω_{i}, f_{i}), which for simplicity we denote by (X, Ω, f) (without indicating the index). Set

J = {t ∈ [0, 1]|∃ a 1 − connected region C_{t} ⊂ L_{t} such that ∂C_{t} = B_{t}}.

Thus the indices t ∈ J are those for which B_{t} = B_{i,t} satisfies condition (4) of Definition 2.4. It is clear that J is open in the interval [0, 1], since the region C_{t} is 1-connected by hypothesis, and therefore lifts along T to nearby leaves.

Note that in the Finiteness Lemma two distinct submanifolds B and B′ in B_{i} may both yield the same complex X_{t} under the SAP. Then there is an important relationship between them, according to the following lemma, whose proof is given in Section 5.

Lemma 3.3 If the same complex X_{t} can be obtained by the SAP from two submanifolds B, B′ ∈ B, and B satisfies condition (4), i.e., there is a compact 1-connected region C on the leaf L containing B such that ∂C = B, then B′ also satisfies condition (4), so there is a compact 1-connected region C′ ⊂ L whose boundary is B′, and

|M(C′, δ) − M(C, δ)| ≤ k.  (1)

Four types of subintervals. Next we show that the interval [0, 1] can be covered by intervals J_{0} of one of four types. For the sake of simplicity, we say that a submanifold B in a leaf L bounds if it is the boundary of a compact 1-connected region C ⊂ L.

Lemma 3.4 For every point t_{0} ∈ (0, 1), there is an open interval J_{0} ⊂ [0, 1] with compact closure J̄_{0} and t_{0} ∈ J_{0}, such that one of the following properties holds:

1. J̄_{0} ⊂ J, i.e. B_{t} bounds for every t ∈ J̄_{0};

2. J̄_{0} ∩ J = ∅, i.e. for every t ∈ J̄_{0}, B_{t} does not bound;

3. for some t_{1} ∈ J_{0}, J̄_{0} ∩J is J̄_{0} ∩(t_{1}, 1] or J̄_{0} ∩[0, t_{1}), i.e. for every t ≤ t_{1} in J̄_{0}, B_{t} does not bound, but B_{t} does bound for every t > t_{1} in J̄, or the analogous condition switching the left and right intervals around t_{1};

4. for some t_{1} ∈ J_{0}, J ∩ J̄_{0} = {t ∈ J̄_{0}|t ≠ t_{1}}, i.e. B_{t_{1}} does not bound, but B_{t} does bound for every t ≠ t_{1} in J̄.

Furthermore, if t_{0} = 0 or 1 is an endpoint of [0, 1] that has no neighborhood in J and no neighborhood disjoint from J, then there is a half-open interval J_{0} containing t_{0} such that J ∩ J̄_{0} = {t ∈ J̄_{0}|t ≠ t_{0}}, i.e. B_{t_{0}} does not bound, but B_{t} does bound for every t ≠ t_{0} in J̄_{0}.

Proof of the Lemma. First, consider an interior point t_{0} ∈ (0, 1). If a neighborhood of t_{0} is contained in J or is disjoint from J, then (1) or (2) holds. If not, there are both points in J and points not in J arbitrarily near to t_{0}. Since J is open, there must be an endpoint t_{1} of a component of J in (0, 1). Reversing the orientation of [0, 1] if necessary, so that t_{1} is a left endpoint, we get an open subinterval J_{0} ⊂ (0, 1) with t_{1} ∈ J_{0} such that t_{1} ∉ J but all t ∈ J_{0} with t > t_{1} are in J. This implies by the generalization of Novikov’s theorem, Theorem 2.10, that the leaf L_{t_{1}} is the boundary of a Reeb component whose interior is the union of the leaves L_{t} for all points t ∈ J_{0} with t > t_{1}. If there are no points t < t_{1} in a small left neighborhood of t_{1}, then (possibly diminishing J_{0}) we have conclusion (3). If, on the other hand, t_{1} is a limit of points t ∈ J on the left, then the whole interval {t ∈ J_{0}|t < t_{1}} must be in J, for otherwise there would be another left endpoint of a connected component of J, say t_{2} ∈ J, so that L_{t_{2}} would be the boundary of another Reeb component containing the compact leaf L_{t_{1}} in its interior, but this is impossible, since the interior leaves are non-compact; hence (4) holds in this case.

Now if t_{0} is an endpoint of [0, 1] and has a neighborhood in J or a neighborhood disjoint from J, then the result follows as in cases (1) and (2) above. Otherwise, t_{0} ∉ J, since J is open in [0, 1], and t_{0} must be the limit of points in J. As before, L_{t_{0}} is the boundary of a Reeb component and {t ∈ [0, 1] | t ≠ t_{0}} ⊂ J. □

The following Proposition, whose proof is given in Section 6, is an essential step in the Proof of Proposition 3.1.

Proposition 3.2 Let R̄ be the Reeb component given by the conclusion of Theorem 2.10. Then there is a common upper bound for the Morse δ-volumes M(C_{t}, δ) for t ∈ (0, 1].

Proof of Proposition 3.1. First of all, if for some B ∈ B bounds compact 1-connected regions on both sides on its leaf L, then the leaf L must be simply connected, and by the Reeb Stability Theorem, all leaves of F are simply connected and form a fibration over the circle. Then the local product structure show that there is a local upper bound for the Morse δ-volumes of the regions. Since the circle is compact, this implies the existence of a global upper bound. Hence we may assume that no B ∈ B bounds compact 1connected regions on both sides.

Now consider the transverse family (X, Ω, f) = (X_{i}, Ω_{i}, f_{i}), given by the Finiteness Lemma 3.2 for one value of i, and consider the open set J ⊂ [0, 1] defined above. Since (X, Ω, f) is good, there is a 1-parameter family B_{t}, t ∈ [0, 1], such that X_{t} = X_{i,t} is obtained from B_{t} by the SAP. Then for any interval J′ in J, the regions C_{t} for t ∈ J′ form a continuous 1-parameter family, since the 1-connected regions C_{t} lift to nearby leaves, and no B_{t} bounds compact 1-connected regions on both sides in L_{t}.

In order to find a common upper bound for the Morse δ-volumes M(C_{t}, δ) for all the regions C_{t} whose respective boundaries are B_{t}, that is, for those B_{t} that satisfy condition (4), the four cases of the Lemma 3.4 must be considered. In case (1), for t ∈ J̄_{0} we get a continuous 1-parameter family of regions C_{t}. In view of the local product structure, it is obvious that the Morse δ-volumes M(C_{t}, δ) vary continuously, and so on the compact interval J̄_{0} there is a common upper bound. In case (2), there is nothing to prove. In case (3), if J̄_{0} ∩J = J̄_{0} ∩(t_{1}, 1], then by the generalized Novikov Theorem 2.10, L_{t_{1}} is the boundary of a Reeb component with the leaves for t ∈ J̄_{0} ∩ (t_{1}, 1] as interior leaves. We apply Proposition 3.2 to obtain a common upper bound for the Morse δ-volumes of the regions on interior leaves of the Reeb component. In case (4), there are Reeb components on both sides of the leaf L_{t_{1}}, and again Proposition 3.2 gives the desired upper bound. Finally, if t_{0} is 0 or 1, this case can be treated as one of the cases (1), (2), or (3). Thus in every case envisaged in Lemma 3.4, there is a common upper bound on the Morse δvolumes M(C_{t}, δ) for t ∈ J ∩ J̄_{0}. Next we use the compactness of [0, 1] to see that a finite number of such intervals J_{0} cover [0, 1], and so there is a common upper bound for the submanifolds in the transverse family (X_{i}, Ω_{i}, f_{i}) under consideration, i.e., for all B_{t} ∈ B_{i} that satisfy condition (4) of Definition 2.4.

Finally, if B′ ∈ B_{i}, by the Finiteness Lemma there is some B_{t} such that both B′ and B_{t} yield the same complex X_{t} by the SAP. Then by Lemma 3.3, B′ bounds a 1-connected region C′, and the Morse δ-volumes satisfy M(C′, δ) ≤ M(C_{t}, δ) + k, so by increasing the previous upper bound for all the regions C_{t} by k we get a common upper bound for all B ∈ B_{i}. This completes the proof of Proposition 3.1. □

4 Modifying the metric on a manifold by inserting balloons.

In this section we show how the Riemannian metric on any open Riemannian manifold L of dimension at least three can be modified without changing the growth type, so that L with the new metric does not have the bounded homology property, and hence cannot be a leaf in a codimension one foliation of a closed manifold. The construction is an obvious adaptation of the construction for surfaces in Section 2 of [18], which we follow closely. We insert p-dimensional “balloons” of unbounded size with “necks” of uniformly bounded size. The balloons are widely spaced so that the original growth type of L does not change.

Let L be a connected noncompact p-dimensional manifold and let g_{0} be a complete Riemannian metric on L with globally bounded sectional curvature, with injectivity radius bounded away from zero, and with a given growth type. Hence for some d > 0, the exponential map exp_{x} : B_{x}(0, d) → L is injective for every point x in L, where B_{x}(0, t) is the open ball of radius t centered at the origin in the tangent space at x. Consequently exp_{x} is a diffeomorphism from B_{x}(0, d) onto the ball B(x, d) in L.

We fix a basepoint x_{0} ∈ L and consider a sequence of positive numbers d_{n}, n = 1, …, with d_{n} + 2d < d_{n+1} for each n. Choose a sequence of points x_{1}, x_{2}, … so that dist(x_{0}, x_{n}) = d_{n}, so that in particular the balls B(x_{n}, d) are disjoint. Since the metric g_{0} has bounded geometry, there is no loss of generality in supposing, as we do, that the balls B(x_{n}, d) are isometric to the balls B(0, d) in Euclidian p-space ℝ^{p}. In fact, it suffices to modify g_{0} slightly (perhaps replacing d by a smaller radius) with care to preserve the growth type and bounded geometry globally.

Let S^{p}(n) be the the sphere of radius n in ℝ^{p+1} and S its basepoint at the south pole. Choose a diffeomorphism ϕ : B(x_{n}, d) → S^{p}(n) ∖ {S} by the formula

ϕ(exp_{x_{n}}(tv)) = exp_{S}((d − t)h(v))

for d/2 ≤ t < d and v a unit tangent vector to L at x_{n}, where h is a linear isometry from the tangent space to L at x_{n} to the tangent space to S^{p}(n) at S, and then extending ϕ as a diffeomorphism over the complementary disk B(x_{n}, d/2). Define a new metric g on B(x_{n}, d) by interpolating between the given metric g_{0} and the metric g_{1} obtained as the pullback under ϕ of the round metric on S^{p}(n), so that g coincides with g_{0} near the boundary of B(x_{n}, d) and with g_{1} on B(x_{n}, d/2). This defines the new “balloon” metric on the balls B(x_{n}, d), and outside these balls g is defined to coincide with the original metric g_{0}. The uniform manner of carrying out this construction as n varies ensures that (L, g) has bounded geometry. Note that the metric g on the closed ball C_{n} = \overline{B(x_{n}, d/2)} (as previously defined using the metric g_{0}) is now the round metric of S^{p}(n) from which a small neighborhood of S has been removed.

Choose a small δ > 0 less than the new injectivity radius and also less than d/4. Let B_{n} = ∂C_{n} be the boundary of C_{n}. Then for each n the δneighborhood V_{δ}(B_{n}) fibers over (B_{n}) as a tubular neighborhood. Letting δ′ = δ/4 as before, the spheres B_{n} have uniformly bounded δ′-volume, say less than some k > 0. Hence the hypotheses (2) and (3) of Definition 2.4 are satisfied, and the other two hypotheses are evident.

We claim that the new metric g does not have the bounded homology property. In fact, let f : C_{n} → [0, ∞) be a Morse function with f(B_{n}) = 0. As t increases from 0, there is a continuous 1-parameter family of complementary regions A_{n}(t) of f^{−1}(t), beginning with the interior of A_{n}(0) = C_{n} and varying continuously. By continuity, we see that f(A_{n}(t)) ⊂ [t, ∞). Consequently if t_{n} is the maximum of f on C_{n}, A_{n}(t_{n}) has empty interior and thus volume zero. For some value of t, say t′_{n}, A_{n}(t′_{n}) will have p-volume a_{p}n^{p}/2, where a_{p} is the p-volume of the unit p-sphere, i.e. A_{n}(t′_{n}) will have half p-volume of the round sphere of radius n, which is a_{p}n^{p}. Then, by the isoperimetric inequality on the sphere, the boundary ∂A_{n}(t′_{n}) must have (p − 1)-volume at least equal to a_{p−1}n^{p−1}, the (p − 1)-volume of the equator. As n increases these (p−1)-volumes tend to infinity, so the δ-volumes V_{δ}(C_{n}) also diverge to infinity. It follows that the Morse δ-volumes of the regions C_{n} are unbounded, and (L, g) does not have the bounded homology property.

The points d_{n} can be chosen so that no end of L has the bounded homology property. In fact, if the points x_{r} have been chosen for r ≤ n, let k be the number of noncompact connected components of the complement of B(x_{0}, d_{n}), and choose x_{n+1}, …, x_{n+k} so that one is in each of these components. Continue inductively, repeating this procedure with n + k in place of n. This guarantees that each end will contain an infinite number of the points x_{n} and therefore does not have the bounded homology property.

Furthermore, if the sequence {d_{n}} is chosen to grow sufficiently fast so that ∑_{r=1}^{n} a_{p}r^{p} is less than the the p-volume Vol(B(x_{0}, d_{n}), then it is easy to check that the growth functions f_{0}(r) = Vol(B(x_{0}, r)) in the metric g_{0} and f(r) = Vol(B(x_{0}, r) in the metric g satisfy f_{0}(r) ≤ f(r) ≤ 2f_{0}(r), so that they have the same growth type.

Finally, suppose that each d_{n} in the strictly increasing sequence {d_{n}} is an integral power of 2. There are uncountably many such sequences satisfying the above inequality. If the metrics obtained from two such sequences {d_{n}} and {d′_{n}} are quasi-isometric, then d_{n} = d′_{n} for all sufficiently large n. Hence each quasi-isometry class of such metrics is countable, and so there must be uncountably many quasi-isometry equivalence classes. This completes the proof of Theorem 2.8. □

Remark 4.1 It seems likely that the example of Tsuboi in Section 5 of [18] can be adapted to give an example of a codimension two smooth foliation of the compact (p + 2)-manifold S^{1} × S^{1} × S^{p−1} × S^{1} with p-dimensional leaves that do not possess the bounded homology property, for every p > 2. To adapt the construction, just replace the third factor of (S^{1})^{4} in the construction of [18] by S^{p−1}.

5 Proof of the Finiteness Lemma and Lemma 3.3

In this Section we prove the Finiteness Lemma 3.2 and Lemma 3.3. We assume all the conditions indicated in the first two paragraphs of Section 3.

Proof of the Finiteness Lemma 3.2. Given a point x ∈ M in a leaf L_{x}, let D_{x} = \overline{V_{ϵ}(x)} be the closed p-dimensional ball centered on x of radius ϵ for some positive ϵ less than the injectivity radius at x, so that D_{x} is a closed ball. We may lift D_{x} along T to disks D_{y} on nearby leaves L_{y} for y ∈ J, where J is an open interval embedded in a leaf of T and containing x, thus obtaining a bifoliated map

h : (D_{x} × J, H, V) → (M, F, T )

such that h(x, y) = y. Fix a larger smooth compact region E ⊂ L_{x} that contains V_{d}(D_{x}) where d = (2k + 1)δ and give E a δ′-triangulation.

If B ∈ B meets D_{x}, then V_{δ}(B) ⊂ E since B is connected and has δ-volume less than k and therefore diameter less than 2kδ. Then the SAP applied to B will yield a (p−1)-dimensional subcomplex X_{B} ⊂ E. Let Ω_{B} be a smooth compact region containing \overline{V_{2δ′}(B)} in its interior and contained in V_{3δ′}(B), and give Ω_{B} a smooth δ′-triangulation that agrees with the original triangulation on E for all the simplices that meet V_{2δ′}(B). Since V_{3δ′}(B) has a compact 1-connected closure that contains Ω_{B}, Ω_{B} can be lifted along T to Ω_{B,y} on every leaf L_{y} sufficiently close to L_{x}. Hence we get a bifoliated embedding

h_{B} : (Ω_{B} × J′, H, V) → (M, F, T )

such that f_{B}(x′, y) = y ∈ J′ for a sufficiently small subinterval J′ ⊂ J containing x and for some x′ ∈ B ∩ D_{x}.

Now there are only finitely many (p − 1)-dimensional subcomplexes of E; let X_{1}, … X_{m} be those that are obtained as X_{B} for some B ∈ B that meets D_{x}. Choose a submanifold B_{i} for each X_{i}, so that X_{i} = X_{B_{i}}, ie., X_{i} is obtained from B_{i} by the SAP. We may choose J′ small enough so that for every B ∈ {B_{,} …, B_{m}}, the lifted triangulation on Ω_{B,y} = h_{B}(Ω_{B} × {y}) has mesh less than δ′ and the lifted complex X_{B,y} = h_{B}(X_{B} × {y}) is contained in the 2δ′-neighborhood V_{2δ′}(B_{y}) of the lift B_{y} = h_{B}(B × {y}) of B. Now if we let g : [0, 1] → J′ be an embedding such that x = g(t) for some t ∈ (0, 1), then the triple (X_{B}, Ω_{B}, f_{B}), where f_{B}(t, y) = h_{B}(g(t), y), will be a good transverse family. Thus we get m good transverse families corresponding to D_{x}. Every B ∈ B that meets D_{y} for y ∈ g([0, 1]) will yield an X_{B,y} under the SAP.

Now the original point x is in the open set h(Int(D_{x}) × g(0, 1)), and such open sets cover M, so a finite set of them suffice to cover M, say for x_{1}, …, x_{n}. The union of the sets of triples (X_{B}, Ω_{B}, f_{B}) where B varies over all the B_{i} for all the points x_{1}, …, x_{n}, will be a finite set of good transverse families. It satisfies the conclusion of the Finiteness Lemma since every B ∈ B will meet one of the sets h(Int(D_{x}) × g(0, 1)), for some x ∈ {x_{1}, …, x_{n}}, and so will yield one of the complexes X_{B_{i}} for that x. □

Proof of Lemma 3.3. Observe that the SAP moves points of each of B and B′ a distance less than δ′, since the triangulation on the leaf has mesh less than δ′. Hence B ⊂ V_{δ′}(X_{t}) and X_{t} ⊂ V_{δ′}(B), and similarly for B′. Thus B′ ⊂ V_{2δ′}(B). Now each connected submanifold B and B′ separates the leaf L into two connected components. For B they are the interior of C and L ∖ C. Since B′ ⊂ V_{2δ′}(B), one of the two connected components of L ∖ B′ must be contained in C ∪ V_{2δ′}(B); call its closure C′. Clearly ∂C′ = B′ and C′ is connected, so it only remains to show that its fundamental group is trivial. Now we have C′ ⊂ V_{2δ′}(C) ⊂ V_{δ}(C′), since δ = 4δ′. By hypothesis, the δ-neighborhood of B fibers over B as a tubular neighborhood, as does V_{2δ′}(B) over B, since 2δ′ < δ, and the same holds for B′. Hence π_{1}(V_{2δ′}(B)) ≈ π_{1}(B) = 1. For the same reason the inclusion C′ ⊂ V_{δ}(C′) induces an isomorphism on fundamental groups, but this inclusion factors through V_{2δ′}(B), which as we have just seen has trivial fundamental group. Thus C′ is 1-connected.

We want to show that M(C′, δ) ≤ M(C, δ) + k, which by symmetry will establish the second conclusion. Let f : C → [0, ∞) be a Morse function on C such that f(B) = 0 and every level set f^{−1}(t) has δ-volume Vol_{δ}(f^{−1}(t)) ≤ M = M(C, δ).

By an isotopy with compact support in V_{δ}(B), which is diffeomorphic to B × (−1, 1), we can take C to the diffeomorphic region C_{1} = C ∖ V_{2δ′}(B) whose boundary B_{1} is one of the two components of the boundary of the closure of V_{2δ′}(B). The isotopy that takes C to C_{1} will also modify f + 1 to a Morse function f_{1} : C_{1} → [1, ∞) with f(B_{1}) = 1. Since B′ ⊂ V_{2δ′}(B), it follows that B′ and B_{1} are disjoint, and so C_{1} is contained in the interior of C′. Hence we may extend the Morse function f_{1} to a Morse function f′ : C′ → [0, ∞) with f′(B′) = 0 by sending C′ ∖ C_{1} to [0, 1) appropriately. Now a level set f^{′−1}(t) of f′ that is entirely outside V_{δ}(B) (i.e, t is sufficiently large) coincides with the level set f_{1}^{−1}(t) = f^{−1}(t + 1) of f, so its δ-volume will be unchanged. Every other level set f^{′−1}(t) (for smaller t) will consist of a part of f^{−1}(t + 1) outside V_{δ}(B) and a part inside that set. The outside part can be covered by at most M balls of radius δ, by hypothesis. The inside part can be covered by at most k δ-balls, since by condition (3) of Definition 2.4, the δ′-volume of B on L satisfies Vol_{δ′}(B) < k, i.e., B can be covered by at most k balls of radius δ′, and so all of V_{δ}(B) (which contains the inside part) can be covered by at most k δ-balls. Hence the total number of δ-balls covering f^{′−1}(t) is at most M + k, showing that M(C′, δ) ≤ M(C, δ) + k. □

6 Novikov’s Theorem for embedded vanishing cycles

In this Section we prove the extension of Novikov’s Theorem (Theorem 2.10) on the existence of (generalized) Reeb components and Lemma 3.2 which asserts that for every 1-parameter family of connected (p − 1)-submanifolds embedded in the leaves in the interior of a Reeb component and for every δ > 0, there is a common upper bound for the Morse δ-volumes of the regions that they bound on the leaves. Recall (Definition 2.9) that we define a Reeb component to be a compact (p+1-dimensional manifold with connected nonempty boundary with a codimension one foliation such that the interior fibers over the circle with the leaves as fibers. (A more general definition would allow the boundary to consist of more than one leaf, but we do not need that situation, which is more complicated.) This section also contains the proof of Lemma 3.2, at the end.

First, we give a generalization of the second half of Novikov’s Theorem on the existence of Reeb components, Theorem 2.10, which is the main result of this Section. We observe that our case, with an embedded vanishing cycle, is much easier than the case when double points are allowed. (See Haefliger [10] for a good exposition of the proof of Novikov’s original theorem that treats the problem of double points very clearly.)

Proof of the generalization of Novikov’s Theorem, Theorem 2.10. We have a compact (p + 1)-dimensional manifold with a codimension one foliation F. We do not need to suppose any smoothness, but we fix a transverse 1-dimensional foliation T , which always exists. We re given a connected (p−1)-manifold B with empty boundary and a topological foliated immersion h : B × I → M taking leaves of H into leaves of F. By a small isotopy (possibly reducing the interval I = [0, 1] to a smaller interval [0, ϵ], if necessary), we can easily modify h to make it a bifoliated immersion

h : (B × I, H, V) → (M, F, T )

that embeds B × {t} into a leaf L_{t} for every t ∈ I = [0, 1]. We define h_{t} : B → L_{t} by setting h_{t}(x) = h(x, t) for x ∈ B and t ∈ I, and let B_{t} = h_{t}(B) ⊂ L_{t}. Then the essential hypothesis is that for t > 0, B_{t} is the boundary of a 1-connected region C_{t} ⊂ L_{t}, but for t = 0, B_{0} does not bound such a region on L_{0}. We must show that L_{0} is the boundary of a Reeb component R contained in M, and that the interior of R consists of the leaves L_{t} for t > 0.

Remark. If B is orientable with fundamental class [B] ∈ H_{(p−1)}(B), then its image h_{0∗}[B] ∈ H_{(p−1)}(L_{0}) is a homological vanishing cycle. Such homological vanishing cycles were introduced by Sullivan [21], in order to prove a generalization of Novikov’s Compact Leaf Theorem, but he did not suppose that h_{0} is the embedding of a submanifold, as we do. The assumption that h_{0} is an embedding simplifies the proof enormously. A paper showing that under convenient hypotheses every homological vanishing cycle for a codimension one foliation of a compact manifold lies on the boundary of a Reeb component is in preparation ([1]).

Since the regions C_{t} for t > 0 are compact and 1-connected, they can be lifted transversely to give local 1-parameter families. Now it is impossible for one of the submanifolds B_{t} to bound compact 1-connected regions on both sides, for then the leaf L_{t} would be simply connected, and by the Reeb Stability Theorem, all the leaves would be 1-connected and M would fiber over the circle with the leaves as fibers, but then all the leaves would lift transversely, and B_{0} would also bound a compact 1-connected region on L_{0}, contrary to hypothesis. Consequently each B_{t} bounds C_{t} only on one side, and these regions fit together to provide a 1-parameter family, i.e., a bifoliated immersion

H : (C × (0, 1], H, V) → (M, F, T )

which does not extend over C ×[0, 1] (for if it did, then B_{0} would bound such a region on L_{0}, contrary to hypothesis). In fact, since h_{0} is an embedding, by passing to a smaller interval [0, ϵ] we may assume that h itself is an embedding on B × [0, 1], after reparametrizing the interval.

Let x_{0} ∈ C ∖ B be a point such that H cannot be extended to the point (x_{0}, 0). The image of the curve t ↦ h(x_{0}, t) lies on a leaf of T , and as t ↘ 0, the curve must have a limit point y_{0} ∈ M. Hence we may find a strictly decreasing sequence of times t_{n} such that h(x_{0}, t_{n}) → y_{0} as n → ∞. Let L be the leaf containing y_{0} and V be a small neighborhood of y_{0} on L. By slightly adjusting the values of the times t_{n} (and possibly passing to a subsequence), we may suppose that all the points h(x_{0}, t_{n}) lie on V, and also that y_{0} ∉ B_{t} for any t > 0 (possibly passing to a smaller interval [0, ϵ] again, if necessary).

Now setting y_{n} = h(x_{0}, t_{n}) and C_{n} = H(C × {t_{n}}), we have y_{n} ∈ C_{n}. For each n, C_{n} is compact and the submanifolds B_{m} = h(B × {t_{n}}) are disjoint, so only finitely many can be contained in C_{n}. Consequently there exists n_{m} > n such that B_{n_{m}} ∩ C_{n} = ∅, which implies that C_{n} ⊂ C_{n_{m}}. Now the set C_{n_{m}} ∪C_{n}∪h(B×[t_{n_{m}}, t_{n}]) is an embedded closed connected submanifold with corners in M, so it separates M into two complementary regions. Clearly, for every n′ such that n < n′ < n_{m}, C_{n′}, which contains B_{n′} ⊂ h(B ×[t_{n_{m}}, t_{n}]), is contained in the complementary region whose closure is H(C × [t_{n_{m}}, t_{n}]). In fact, for the same reason, every C_{n′} ⊂ L for which B_{n′} meets h(B × [t_{n_{m}}, t_{n}]) is contained in H(C × [t_{n_{m}}, t_{n}]), and then C_{n} ⊂ C_{n′} ⊂ C_{n_{m}}.

Next, we show that the leaf L_{0} is compact. If not, then there will be a simple closed curve γ transverse to F in the positive direction that intersects L_{0} in one point γ(0) = z; we can choose z to be near to a point h(b, 0) ∈ L_{0} (for some b ∈ B) but outside the set h(B × {0}). Then we may isotope γ slightly so that near z, say for s ∈ [−ϵ, ϵ], the segment γ([−ϵ, ϵ]) lies on a leaf of T and is disjoint from h(B, [0, ϵ′]) for some ϵ′ > 0. Now for some sufficiently large n, γ will enter into the region H(C × [t_{n_{m}}, t_{n}]) at a point γ(s) for some s ∈ (−ϵ, ϵ), and then γ can never exit from that region, for it cannot exit along C_{m_{n}} ∖ C_{n} (where the transverse orientation enters) and we have chosen it to be disjoint from h(B, [0, ϵ′]). Hence L_{0} must be compact.

Now let {r_{i}}, i = 1, 2, …, be a strictly decreasing sequence in (0, 1] which takes on exactly those values t ∈ (0, 1] for which H(C × {t}) ∩ L ≠ ∅. Let C(i) = H(C × {r_{i}}). As before, we must have an embedding C(i) ⊂ C(i + 1). Then the only identifications made by the quotient map H on C × [r_{i+1}, r_{i}] correspond to that embedding; H identifies C × {r_{i}} with a subregion of Int(C) × {r_{i+1}} by setting (x, r_{i}) ∼ (ϕ(x), r_{i+1}), x ∈ C, for a certain embedding ϕ : C → Int(C). It is easy to check that the embedding ϕ is independent of i, since it just pushes the image of C along leaves of T until it meets L again. We also see that H(C × [r_{i+1}, r_{i}]) ⊂ H(C × [r_{i+2}, r_{i+1}]) for each i. Furthermore ∪_{i=1}^{∞}H(C ×[r_{i+1}, r_{i}]) is a complete 1-sided neighborhood of L_{0}, so every leaf L_{t} must meet H(C × [r_{i+1}, r_{i}]). On the other hand, beginning with L′ = L_{t} instead of L and following the same argument, we see that there is only one component of L_{t}∩H(C ×[r_{i+1}, r_{i}]), for otherwise no part of the leaf L would be between the two or more components, contrary to what we have just seen.

Thus what we have shown is that each leaf L_{t} contributes exactly one connected component to H(C × [r_{i+1}, r_{i}]); the leaf L contributes both C(i) and C(i + 1), but the identification makes them into one component. This means that H(C × [r_{i+1}, r_{i}]) fibers over S^{1} = [r_{i+1}, r_{i}]/{r_{i+1} ∼ r_{i}} by taking the leaf H(C × {t}) to t ∈ [r_{i+1}, r_{i}]/{r_{i+1} ∼ r_{i}}. As i increases, we see that these fibrations fit together to give a fibration R = ∪_{i=1}∞H(C ×[r_{i+1}, r_{i}]) → S^{1}.

It follows that the closure R̄ is a Reeb component, for the boundary leaf L_{0} is compact and connected, so that we can choose a transverse orientation pointing inwards. Furthermore if we take a small compact one-sided neighborhood N of L_{0} on the side where the leaves L_{t} with t > 0 are located, then for a sufficiently large i we get that R̄ = N ∪ H(C × [r_{i+1}, r_{i}]), showing that R̄ itself is compact. □

In the preceding proof, note that if we take C(r_{i+1}) ∖ Int(C(r_{i})) and isotope it along T into L_{0}, then it maps onto L_{0}, and the two boundary components of C(r_{i+1}) ∖ Int(C(r_{i})) are identified, essentially by the map ϕ : ∂C → ϕ(∂C). Thus the boundary leaf is diffeomorphic to D/{x ∼ ϕ(x)} where D = C ∖ ϕ(Int(C)).

The Reeb component R̄ we have just shown to exist can be obtained, up to foliated homeomorphism, by the construction which we shall now describe. Furthermore this construction gives all Reeb components, up to foliated homeomorphism (See [1] for the proof, which is not difficult), though we shall not use that fact here.

Construction of Reeb components. Let C be a connected compact pmanifold C with connected boundary ∂C = B and let ϕ : C → Int C be an embedding of C into its own interior. Consider the product C × [0, 1] with the product foliation whose leaves are C × {t}. On the submanifold

C′ = C × [0, 1] ∖ ϕ(Int C) × {0}

let ∼ be the equivalence relation generated by setting (x, s) ∼ (ϕ(x), f(s)) for every (x, s) ∈ C′, where f : [0, 1] → [0, 1] is an embedding such that f(0) = 0 and f(s) < s for every s > 0. Then the quotient R̄ = C′/ ∼ is a compact (p + 1)-manifold with boundary endowed with a codimension one foliation R induced by the product foliation. (Note that the foliated homeomorphism type is independent of the choice of the embedding f.)

Proof of Lemma 3.2. Consider the Reeb component we have just constructed above in the proof of Theorem 2.10, which is obtained essentially by the construction using C and ϕ. For convenience, we first consider Morse functions with values in S^{1} rather than in ℝ. The boundary of R̄ is diffeomorphic to D/{x ∼ ϕ(x)}, where ∂D has two components, corresponding to ∂C = B and ∂ϕ(C) = ϕ(B), both of which are connected. Hence we may find a smooth map f : R̄ → S^{1} that is non-singular in a neighborhood of f^{−1}(1) and such that f^{−1}(1) = ∂R̄ . By a small perturbation we may make f a Morse function. Since L_{0} = ∂R̄ is compact, there is an upper bound on the δ-volumes V_{δ}(f^{−1}(z)) for z ∈ S^{1}. Extend f to F : N → S^{1} where N is a small tubular neighborhood of L_{0} in R̄, so that F is constant along intervals of leaves of T in N. Thus the Morse singularities on the leaves in N′ = N ∖ L_{0} correspond to Morse singularities on L_{0}.

Now lift F |_{N′} to F̃ : N′ → ℝ under the covering map ℝ → S^{1}, s ↦ exp(2πis). Let F̃_{t} be the restriction of F̃ to L_{t} ∩ N′. Then it is clear that there is a common upper bound K_{1} for the δ-volumes V (F̃_{t}^{−1}(s)) for all s and t. On the other hand, the interior R fibers over S^{1}, the closure T of R̄ ∖ N is compact, and the intersection of each leaf L_{t} with R̄ ∖ N is also compact, so there is a common upper bound K_{2} for the Morse δ-volumes of L_{t} ∩ T, for all t. Putting these two parts of L_{t} together we find that K_{1} + K_{2} is a common upper bound for the Morse δ-volumes of all the sets C_{t} for all t ∈ (0, 1]. □

References

[1] F. Alcalde, G. Hector and P.A. Schweitzer, The structure of generalized Reeb components, preprint, 2008.

[2] O. Attie and S. Hurder, Manifolds which cannot be leaves of foliations, Topology 35 (1996), 335-353.

[3] C. Camacho and A. Lins Neto, Geometric Theory of Foliations Birkhäuser Verlag (1984), translation of Teoria Geométrica das Folheações Projeto Euclides, IMPA (Rio de Janeiro, 1981).

[4] J. Cantwell and L. Conlon, Every surface is a leaf, Topology, 26 (1987), 265-285.

[5] A. Candel and L. Conlon, Foliations I, Graduate Studies in Mathematics vol. 23, Amer. Math. Soc. (Providence, 2000).

[6] A. Candel and L. Conlon, Foliations II, Graduate Studies in Mathematics vol. 60, Amer. Math. Soc. (Providence, 2003).

[7] E. Ghys, Une variété qui n’est pas une feuille, Topology 24 (1985), 67-73.

[8] E. Ghys, Topologie des feuilles gnriques, Annals of Math. (2) 141 (1995), 387-422.

[9] A. Haefliger, Variétés feuilletées. Ann. Scuola Norm. Sup. Pisa, 16 (1962), 367-397.

[10] A. Haefliger, Travaux de Novikov sur les feuilletages. Séminaire Bourbaki, 339 (1968).

[11] G. Hector and U. Hirsch, Introduction to the geometry of foliations, A. Vieweg, Braunschweig, 1981.

[12] G. Hector and U. Hirsch, Introduction to the geometry of foliations, B. Vieweg, Braunschweig, 1983.

[13] T. Inaba, T. Nishimori, M. Takamura and N. Tsuchiya, Open manifolds which are non-realizable as leaves. Kodai Math. J., 8 (1985), 112-119.

[14] S.P. Novikov, Topology of foliations. Trans. Moscow Math. Soc., 14 (1965), 268-304.

[15] A. Phillips and D. Sullivan, Geometry of leaves, Topology 20 (1981), 209-218.

[16] G. Reeb , Sur certaines propriétés topologiques des variétés feuilletées. Actual. Sci. Ind. 1183, Hermann, Paris, 1952.

[17] P.A. Schweitzer, Codimension one foliations without compact leaves. Comment. Math. Helvetici, 70 (1995), 171-209.

[18] P.A. Schweitzer, Surfaces not quasi-isometric to leaves of foliations of compact 3-manifolds, in Analysis and geometry in foliated manifolds, Proceedings of the VII International Colloquium on Differential Geometry, Santiago de Compostela, 1994. World Scientific, Singapore, (1995), 223-238.

[19] J. Sondow, When is a manifold a leaf of some foliation? Bull. Amer. Math. Soc., 81 (1975), 622-624.

[20] V.V. Solodov, Components of topological foliations (Russian). Mat. Sb. (N.S.), 119 (1982), 340-354.

[21] D. Sullivan, Cycles for the dynamical study of foliated manifolds and complex manifolds. Inventiones Math., 36 (1976), 225-255.

[22] W.P. Thurston, Existence of codimension-one foliations. Ann. of Math., 104 (1976), 249-268.

[23] P. Walczak, A virtual leaf, Int. J., Bifur. and Chaos, 9 (1996), 1845-1852.

[24] A. Zeghib, An example of a 2-dimensional no leaf, Geometric Theory of Foliations, Proceedings of the 1993 Tokyo Foliations Symposium, World Scientific (Singapore, 1994), 475-477.