A HOPF INDEX THEOREM FOR FOLIATIONS

VICTOR BELFI, EFTON PARK, AND KEN RICHARDSON

Abstract. We formulate and prove an analog of the Hopf Index Theorem for Riemannian foliations. We compute the basic Euler characteristic of a closed Riemannian manifold as a sum of indices of a non-degenerate basic vector field at critical leaf closures. The primary tool used to establish this result is an adaptation to foliations of the Witten deformation method.

1. Introduction

The Euler characteristic is one of the simplest homotopy invariants of a smooth, closed manifold. We begin by briefly reviewing some standard theorems of topology which establish the equivalence of three ways of computing it. Then we proceed to new analogs for Riemannian foliations, where more details will be supplied.

For a smooth, closed manifold M, we define the Euler characteristic as

χ(M) = ∑(−1)^{k} dim H^{k}(M),

where H^{k} is de Rham cohomology. According to classical results of de Rham the spaces H^{k}(M) are finite-dimensional and homotopy invariant; thus the definition of χ(M) makes sense and is homotopy invariant. An alternate route to the finite dimensionality of H^{k} and the topological invariance of χ goes by way of the Čech-de Rham double complex, which computes the de Rham cohomology and provides an isomorphism with the Čech cohomology of a finite good cover of M (here “good” means that non-empty finite intersections are diffeomorphic to ℝ^{n}). Using this isomorphism one can define the Euler class e(E) ∈ H^{n}(M) of an oriented (n−1)-sphere bundle E → M . The vanishing of the Euler class is a necessary, but not in general sufficient, condition for the bundle to have a section. In the case that e is the Euler class of the sphere bundle associated with the tangent bundle of M, ∫_{M} e = χ(M). A corollary of this is the Hopf Index Theorem: χ(M) is the sum of the indices of the singular points of any non-degenerate vector field on M. Details can be found in [2].

Suppose that a smooth, closed manifold M is endowed with a smooth foliation F. A form ω on M is basic if for every vector field X tangent to the leaves, i(X)ω = 0 and i(X)(dω) = 0, where i(X) denotes interior product with X. The exterior derivative of a basic form is again basic, so the basic forms are a subcomplex Ω_{B}^{∗} (M, F) (or Ω_{B}^{∗} (M)) of the de Rham complex Ω^{∗} (M). The cohomology of this subcomplex is the basic cohomology H_{B}^{∗} (M, F) .

Date: April, 2001.

1991 Mathematics Subject Classification. 53C12, 58G25.

Key words and phrases. foliation, basic, Euler characteristic, Hopf index, vector field.

Research at MSRI is supported in part by NSF grant DMS-9701755.

Suppose F has codimension q. The basic Euler characteristic is defined as

χ_{B} (M, F) = ∑_{k=0}^{q}(−1)^{k} dim H_{B}^{k} (M, F) ,

provided that all of the basic cohomology groups are finite-dimensional. Although H_{B}^{0} (M, F) and H_{B}^{1} (M, F) are always finite-dimensional, there are foliations for which higher basic cohomology groups can be infinite-dimensional. For example, in [6], the author gives an example of a flow on a 3-manifold for which H_{B}^{2} (M, F) is infinitedimensional. Therefore, we must restrict our investigation to a class of foliations for which the basic cohomology is finite-dimensional. A large class of foliations with this property are Riemannian foliations; a foliation is Riemannian if its normal bundle admits a holonomy-invariant Riemannian metric. There are various proofs that the basic cohomology of a Riemannian foliation on a closed manifold is finite-dimensional; see for example [4] for the original proof using spectral sequence techniques or [7] and [13] for proofs using a basic version of the Hodge theorem.

In Section 2 we develop a basic version of Čech-de Rham cohomology along the lines of [2]. The assumption that the foliation is Riemannian is needed to obtain basic partitions of unity and the basic Mayer-Vietoris sequence. For the basic Poincaré lemma we further assume that all of the leaves are closed. Examples are given to show that these conditions are necessary. The basic Čech-de Rham theorem establishes the equivalence of H_{B}^{∗} and Ȟ_{B}^{∗}, if all the leaves are closed. Examples show that the basic Čech cohomology and the basic de Rham cohomology are not necessarily isomorphic for Riemannian foliations in general.

A primary goal of this paper is to establish a foliation version of the Hopf Index Theorem. However, the standard proofs of this theorem do not carry over, even for Riemannian foliations. The problem is there are many Riemannian foliations that have a nonzero basic Euler characteristic, yet have trivial top-dimensional basic cohomology (for example, a non-taut Riemannian foliation ([17]) can possess this property). Thus it is impossible for these foliations to have any sort of basic Euler class that can be integrated to obtain the basic Euler characteristic (we do mention that for taut Riemannian foliations, one can define a nontrivial basic Euler class [19]).

To establish a Hopf Index Theorem for Riemannian foliations, another approach is required. The approach we use, which is in Section 3, involves a modification of the Witten deformation of the de Rham complex to the foliation case. Let V be a basic vector field on (M, F); i.e., a vector field on M whose flow maps leaves to leaves (see [12] or [17], for example). In analogy with what is needed in the classical Hopf Index Theorem, we require that V satisfy a transverse nondegeneracy condition, which we call F-nondegeneracy. We say that a leaf closure is critical for V if V is tangent to F at that leaf closure. When V is F-nondegenerate, the critical leaf closures are necessarily isolated. For each critical leaf closure L, we define the index ind_{L}(V) of V at L; just as in the classical case, this index is always ±1.

Next, let d_{B} be the restriction of the exterior derivative to basic forms, let δ_{B} denote the formal adjoint of d_{B}, and for each real number s, define the basic Witten differential

D_{B,s} = d_{B} + s i (V) + δ_{B} + sV^{♭}∧ : Ω_{B}^{∗} (M) → Ω_{B}^{∗} (M) .

We show that the index of D_{B,s} is independent of s, and examine the behavior of this operator as s goes to infinity. In the limit, the formula for the index of D_{B,s} concentrates at the critical leaf closures. We next establish the necessary analytic properties of the basic Witten deformation. This leads to Theorem 3.18: Let (M, F) be a Riemannian foliation, and let V be a basic vector field that is F-nondegenerate. Given a critical leaf closure L, let O_{L} = O_{L} (V) denote the orientation line bundle of V at L (Definition 3.1). Then

χ_{B} (M, F) = ∑_{L critical} ind_{L} (V) χ_{B} (L, F, O_{L}) .

where χ_{B} (L, F, O_{L}) is the alternating sum of the dimensions of the certain cohomology groups H_{B}^{∗} (L, F, O_{L}) associated to the foliation F restricted to L(Definition 3.17). We remark that in many simple cases, χ_{B} (L, F, O_{L}) = 1, whence our formula takes on the precise form of the classical Hopf Index Theorem.

One important implication (Corollary 3.20) of this result is that if (M, F) admits a basic vector field that is never tangent to the leaves, then χ_{B} (M, F) = 0. For example, this implies that if F has codimension 1 or 2 and has such a basic vector field, then M has infinite fundamental group (Corollary 3.21).

2. Basic Čech and de Rham Cohomologies

Let M be a compact manifold, and let F be a Riemannian foliation on M. Let J be a finite ordered set, and let U = {U_{α}}_{α∈J} be a finite basic open cover of M; i.e., a finite open cover in which each set U_{α} is a union of leaves. Such an open cover always exists in this case, because tubular neighborhoods of leaf closures are unions of leaves. For α_{0}, α_{1}, ⋯, α_{n} an increasing sequence of indices define U_{α_{0}α_{1}···α_{n}} = ⋂_{i=0}^{n} U_{α_{i}}, and let Ω_{B}^{p}(U_{α_{0}α_{1}···α_{n}}) be the collection of basic p-forms on U_{α_{0}α_{1}···α_{n}}. Define

δ : ∏ Ω_{B}^{∗}(U_{α_{0}α_{1}···α_{k}}) ⟶ ∏ Ω_{B}^{∗}(U_{α_{0}α_{1}···α_{k+1}})

by the formula

(δω)_{α_{0}α_{1}···α_{k+1}} = ∑_{i=0}^{k+1}(−1)^{i}ω_{α_{0}···α̂_{i}···α_{k+1}},

where each form is restricted to the appropriate subset. It is convenient to extend the index set of the product to include all sequences of the appropriate length from J, regardless of order or repetition, and to adopt the convention that interchanging indices of a component introduces a minus sign. So, for example, ω_{α_{2}α_{1}} = −ω_{α_{1}α_{2}} for all α_{1}and α_{2} in J. It is straightforward to check that the definition of δ respects the sign convention and δ^{2} = 0.

Definition 2.1. A basic partition of unity subordinate to a basic open cover {U_{α}} is a partition of unity {ρ_{α}} consisting of basic functions ρ_{α}; that is, functions that are constant on leaves.

Lemma 2.2. Every basic open cover of a Riemannian foliation admits a basic partition of unity.

Proof. Endow the manifold with a bundle-like metric, and choose any partition of unity subbordinate to the basic cover. Orthogonally project the functions in the partition of unity to the space of basic functions; the smoothness and other desired properties of the resulting functions are guaranteed by the results of [13].

Theorem 2.3. (Basic Mayer-Vietoris Sequence) The sequence

0 ⟶ Ω_{B}^{∗}(M) ⟶^{r} ∏ Ω_{B}^{∗}(U_{α_{0}}) ⟶^{δ} ∏ Ω_{B}^{∗}(U_{α_{0}α_{1}}) ⟶^{δ} ⋯

is exact, where r denotes the restriction map.

Proof. The proof of this theorem is the same as the proof of Theorem II.8.5 in [2], with forms replaced by basic forms and partitions of unity replaced by basic partitions of unity.

Remark 2.4. If the foliation is not Riemannian, it may not admit basic partitions of unity, and in such cases, the Mayer-Vietoris sequence may fail to be exact. To describe such an example, we begin with some notation. For each real number t and open interval (a, b), let L_{(a,b)}^{t} be the curve \{(x, \frac{1}{x−a} + \frac{1}{b−x} + t) | a < x < b\} in ℝ^{2}. Foliate [0, 1) × ℝ by the lines x = \frac{1}{3} and x = \frac{2}{3}, along with Reeb components L_{(0, \frac{1}{3})}^{t}, L_{(\frac{1}{3} , \frac{2}{3})}^{t}, and L_{(\frac{2}{3} ,1)}^{t}, where t ranges over all real numbers. Let 𝕋^{2} ⊂ ℂ^{2} be the 2-

torus, and consider the open map p : [0, 1)×ℝ ⟶ 𝕋^{2} given by p(x, y) = (e^{2πix}, e^{2πiy}). Then the image under p of our foliation on [0, 1)×ℝ determines a foliation on 𝕋^{2}. Let U_{0} = (0, \frac{2}{3})×ℝ, U_{1} = (\frac{1}{3}, 1)×ℝ ,and U_{2} = ([0, \frac{1}{3}) ∪ (\frac{2}{3}, 1))×ℝ. There does not exist a basic partition of unity subordinate to the basic open cover {p (U_{0}) , p (U_{1}) , p (U_{2})} of 𝕋^{2}, because the only basic functions are the constant ones. To see that the Mayer-Vietoris sequence is not exact in this example, first note that the intersection of all three sets in our basic open cover is empty. Therefore, for basic functions, we have the following piece of the Mayer-Vietoris sequence:

C_{B}^{∞} (p (U_{0})) ⊕ C_{B}^{∞} (p (U_{1})) ⊕ C_{B}^{∞} (p (U_{2})) ⟶^{δ}

C_{B}^{∞} (p (U_{0} ∩ U_{1})) ⊕ C_{B}^{∞} (p (U_{1} ∩ U_{2})) ⊕ C_{B}^{∞} (p (U_{0} ∩ U_{2})) ⟶ 0.

The map δ here is given by the formula δ(c_{0}, c_{1}, c_{2}) = (c_{1} −c_{0}, c_{2} −c_{1}, c_{2} −c_{0}), which is not surjective since there is a nontrivial relation among the three components of the image of δ.

Remark 2.5. Even if a basic open cover of a (necessarily) non-Riemannian foliation does not admit basic partitions of unity, it is possible for the Mayer-Vietoris sequence to be exact. One such example is as follows; we maintain the notation of the previous remark. Foliate [0, 1) × ℝ by leaves of two sorts: straight lines L_{a} = {(a, y) | y ∈ ℝ} for 0 ≤ a ≤ \frac{1}{2}, and a Reeb component L_{(\frac{1}{2} ,1)}^{t}, t ∈ ℝ. As in the previous remark,

the image under p of this foliation of [0, 1) × ℝ determines a foliation on 𝕋^{2}. Let U_{0} = (0, 1) × ℝ and U_{1} = ([0, \frac{1}{4}) ∪ (\frac{1}{2}, 1)) × ℝ. Then there does not exist a basic partition of unity subordinate to the basic open cover {p (U_{0}) , p (U_{1})} of 𝕋^{2}, because every basic function on the Reeb component must be constant. In spite of this, it is straightforward to check that the Mayer-Vietoris sequence associated to this basic open cover is exact.

There are various cohomology theories that one can associate to the foliation F. Note that the exterior derivative dω of a basic form ω is basic, and therefore the collection of basic forms of F is a subcomplex of the de Rham complex. The cohomology of this subcomplex is denoted H_{B}^{∗}(M, F), and is called the basic de Rham cohomology of (M, F).

Another cohomology theory can be defined for any basic open cover U of (M, F). For each p ≥ 0, define

C_{B}^{p}(U, ℝ) = ker (d : ∏ Ω_{B}^{0}(U_{α_{0}···α_{p}}) ⟶ ∏ Ω_{B}^{1}(U_{α_{0}···α_{p}})) .

It is easy to check that (C_{B}^{∗}(U, ℝ), δ) is a cochain complex; we declare its cohomology Ȟ_{B}^{p}(U, ℝ) to be the basic Čech cohomology of the basic open cover U.

Our immediate goal is to show that for “nice” covers U, basic de Rham cohomology and basic Čech cohomology are isomorphic. To do this, we need a foliation version of the Poincaré lemma.

Proposition 2.6. (Basic Poincaré Lemma) Let F be a Riemannian foliation of M , suppose that all of the leaves of F are closed, and equip M with a bundle-like metric. Let L be a leaf; for ε > 0, let U be the tubular neighborhood of L consisting of points that are a distance less than ε from L. Then, for ε sufficiently small and for k > 0, every closed basic k-form on U is exact.

Proof. First observe that since we have given M a bundle-like metric, the tubular neighborhood U is a union of leaves, and so the restriction F |_{U} of F to U makes sense. Choose ε small enough so that U misses the cut locus of L; since M is compact and L is closed, this can always be done. Fix x ∈ L, and let D be the exponential image of the ball of radius ε in the normal space N_{x}L. Then

Ω_{B}^{k}(U) ≅ {η ∈ Ω^{k}(D) | η is holonomy invariant}.

The holonomy of F acts by a finite subgroup Γ of the orthogonal group ([12]), so we have

Ω_{B}^{k}(U) ≅ {η ∈ Ω^{k}(D) |g^{∗}η = η for all g in Γ},

and this isomorphism commutes with the exterior derivative. Suppose ω ∈ Ω_{B}^{k}(U) is closed, and let η ∈ Ω^{k}(D) be the closed form associated to ω via the isomorphism above. Since D is diffeomorphic to Euclidean space, there exists by the ordinary Poincaré lemma a form μ ∈ Ω^{k−1}(D) such that dμ = η. Now, μ may not be Γinvariant, but the averaged form ζ = \frac{1}{|Γ|} ∑_{g∈Γ} g^{∗}μ is, and dζ also equals η. Therefore η, and hence ω, is exact.

Definition 2.7. A basic good cover of (M, F) is a basic open cover U of (M, F) with the feature that the basic cohomology of each intersection is trivial.

When all of the leaves of F are closed, the basic Poincaré lemma implies that we can obtain a basic good cover by covering each leaf by a sufficiently small tubular neighborhood. In fact, since M is compact, we can choose a finite basic good cover of M.

Theorem 2.8. (Basic Čech-de Rham Theorem) Suppose F is Riemannian and consists entirely of closed leaves, and suppose that U is a basic good cover of M. Then

H_{B}^{∗} (M, F) ≅Ȟ_{B}^{∗} (U, ℝ) .

Proof. For each p, q ≥ 0, let C^{p} (U, Ω_{B}^{q}) be the collection of basic q-forms restricted to (p + 1)-fold intersections, and define a double complex

C^{∗} (U, Ω_{B}^{∗}) = ⨁_{p,q≥0} C^{p} (U, Ω_{B}^{q}) .

The horizontal differential is the map δ defined above, and the vertical differential is (−1)^{p}d, where d is the exterior derivative. We compute the cohomology H^{∗} (U, Ω_{B}^{∗}) of this double complex in two ways.

First, augment C^{∗} (U, Ω_{B}^{∗}) by the column ⨁_{q≥0} Ω_{B}^{q}(M), and then map into ⨁_{q≥0} C^{1} (U, Ω_{B}^{q}) by the restriction map r. By Theorem 2.3, the rows of the augmented double complex are exact. By [2, p. 97], the cohomology H^{∗} (U, Ω_{B}^{∗}) of the original double complex is isomorphic to the cohomology of the initial column, which is precisely the basic de Rham cohomology H_{B}^{∗}(M, F) of (M, F).

Second, augment the original double complex with the row ⨁_{p≥0} C_{B}^{p} (U, ℝ). Then the columns of the new double complex are exact, and thus Ȟ_{B}^{∗} (U, ℝ) is also isomorphic to H^{∗} (U, Ω_{B}^{∗}).

Corollary 2.9. The group Ȟ_{B}^{p} (U, ℝ) is independent of the choice of basic good cover U.

Corollary 2.10. If F is a Riemannian foliation and consists entirely of closed leaves, then H_{B}^{∗}(M, F) is finite-dimensional.

In fact, the group H_{B}^{∗}(M, F) is finite-dimensional for any Riemannian foliation F, as mentioned in the introduction.

Remark 2.11. The basic Čech-de Rham theorem is not necessarily true if the Riemannian foliation F has some non-closed leaves. Consider the foliation of the 2-torus 𝕋^{2} foliated by translates of a line of irrational slope. In this case, the only nonempty basic open set is 𝕋^{2} itself, so there is only one basic open cover U to choose. Clearly Ȟ_{B}^{1} (U, ℝ) ≅ 0, whereas it is straightforward to check that H_{B}^{1}(M, F) ≅ ℝ. Note also that since H_{B}^{1}(M, F) is nontrivial, the basic Poincaré lemma fails in this example.

3. Witten deformation of the basic de Rham complex

For the remainder of this paper, F is a Riemannian foliation on a smooth compact manifold M, and M is equipped with a bundle-like metric; ω_{vol} will denote the volume form associated to the metric.

Let V be a smooth vector field on (M, F). We say that V is a basic vector field if for every vector field X in T F, [V, X] is in T F. If in addition V (x) is in N_{x}F (the normal space to the leaf at x) for every x ∈ M, we will say that V is a basic normal vector field. Basic normal vector fields always exist; the projection of any basic vector field onto NF is such a vector field. Associated to a basic vector field V is a one-parameter family of diffeomorphisms of M that preserves the foliation F.

Let V be a basic vector field. Let L be a leaf closure with the property that V is tangent to every leaf in L; such a leaf closure will be called a critical leaf closure for V. We note in passing that if a basic vector field is tangent to the foliation at any point, it is in fact tangent to every leaf in the leaf closure. Define the linear part of V at x ∈ L to be the linear map V_{L} : N_{x}L → N_{x}L defined by X ↦ π [V, \overline{X}]_{x}, where \overline{X} is any vector field that restricts to X at x, and π : T_{x}M → N_{x}L is the projection map. The basic vector field V will be called F-nondegenerate if the linear part of V is an isomorphism at each point of every critical leaf closure. Every Riemannian foliation F admits F-nondegenerate basic vector fields; simple examples are gradients of basic Morse functions (see [1]). A critical leaf closure for a F-nondegenerate basic vector field V is necessarily isolated. We say that a nondegenerate vector field V has index 1 (respectively, index −1) at a critical leaf closure L if the determinant of the linear transformation V_{L} is everywhere positive (respectively, negative) on L. Clearly, if V is F-nondegenerate, then at each critical leaf closure, V must have either index 1 or −1.

Given any point x_{0} of a critical leaf closure L, choose orthonormal coordinates \overline{y} = (y_{1}, ..., yq̄) for the normal space N_{x}\overline{F} = N_{x}L, and extend these coordinates to orthonormal coordinates y = (y_{1}, ..., yq̄, y_{q̄+1}, ..., y_{q}) for N_{x}F. Let x = (x_{1}, ..., x_{p}) be geodesic normal coordinates for the leaf near x_{0}. The coordinates (x, y) parametrize a tubular neighborhood of L near x_{0} via the normal exponential map. It is elementary to check that we may write V near x_{0} as an orthogonal sum V = V_{1}+V_{2}+V_{3}, where V_{1} is tangent to F, V_{2} (y) = ∑q̄ ; i=1 α^{i} (y) \frac{∂}{∂y_{i}}, and V_{3} (y) = ∑_{j=_{q̄+1}}^{q} β^{j} (y) \frac{∂}{∂y_{j}}. The linear

transformation V_{L} is given by multiplication by the q̄× q̄ matrix (\frac{∂α^{i}}{∂y_{m}} (y))_{1≤i,m≤q̄} on

N_{x}L, so that the index of V is simply the sign of the determinant of that matrix. We remark that if V is the gradient of a basic function f, then V_{L} is the Hessian of f restricted to the normal ball, and V_{1} = V_{3} = 0.

Observe that a basic normal vector field V (hence V_{1} = 0) is determined on a tubular neighborhood U of the leaf closure L by its values on a transverse q̄-dimensional ball. Let B (δ) be the image of a ball of radius δ under the normal exponential map exp_{x}^{⊥} : N_{x}L → U at x ∈ L, so that y = (y_{1}, ..., yq̄) are exponential coordinates for this ball. The holonomy group at x is represented on B (δ) by a group Γ of orthogonal transformations (see [12]), so that we may identify basic normal vector fields on U with normal vector fields restricted to B (δ) that are equivariant with respect to the action of Γ. That is, for any g ∈ Γ, every basic normal vector field restricted to B (δ) has the form

V (y) = ∑_{i=1}^{q} v^{i} (\overline{y}) \frac{∂}{∂y_{i}} = ( v^{1} (\overline{y}) ; ⋮ ; v^{q} (\overline{y}) )

and satisfies

g ( v^{1} (\overline{y}) ; ⋮ ; v^{q} (\overline{y}) ) = ( v^{1} (g\overline{y}) ; ⋮ ; v^{q} (g\overline{y}) ) .

Note that the notation g\overline{y} denotes the restriction of the action of g on N_{x}F to the subspace N_{x}L. This makes sense, because elements of the holonomy group map vector fields orthogonal to L to other vector fields orthogonal to L. As a consequence, the action of Γ commutes with the action of V_{L} on N_{x}L.

Definition 3.1. Let V be a basic vector field that has a nondegenerate zero on a critical leaf closure L. At each point x ∈ L, let V_{L,x} : N_{x}L → N_{x}L be the linear part of V at x. Since V_{L,x} is invertible, it can be written as the (uniquely determined) polar decomposition V_{L,x} = P_{x}Θ_{x}, where P_{x} = \sqrt{V_{L,x}^{∗}V_{L,x}} is positive and symmetric, and Θ_{x} = P_{x}^{−1}V_{L,x} is an isometry. The orientation line bundle O_{L} = O_{L} (V) of V at L is the orientation bundle of the subbundle of NL spanned by the eigenvectors of Θ_{x} corresponding the eigenvalue −1.

Remark 3.2. It can be shown that O_{L} is a smooth line bundle. Observe that in the context of this paper, the singularity of the subbundle is not an issue, since the eigenvalues of Θ_{x} are constant as x moves along the leaves. We also observe that if V = ∇f for a Bott-Morse function f, then this orientation line bundle corresponds to the orientation line bundle of the subbundle of negative directions of f.

We now prove a result that puts a basic vector field into standard form.

Lemma 3.3. Let V be a basic vector field that is F -nondegenerate, and let L be a critical leaf closure. Let T_{δ} (L) denote the tubular neighborhood of radius δ around L, and assume that δ is chosen so that T_{δ} (L) does not contain any other critical leaf closures and misses the cut locus of L. Then there is a basic normal vector field Ṽ and a δ̃ with 0 < δ̃ < δ such that:

1. Ṽ = V outside T_{δ} (L);

2. L is the only critical leaf closure of Ṽ in T_{δ} (L);

3. The index of Ṽ at L is the same as the index of V at L;

4. Ṽ = ∇f on Tδ̃ (L) , where f is a basic function whose Morse index (restricted to a normal ball at a point of L) is even if the index of V is 1 at L and is odd if the index of V is −1 at L;

5. The orientation line bundle of Ṽ at L is identical to the orientation line bundle of V at L.

Proof. Choose coordinates (x, y) = (x_{1}, ..., x_{p}, y_{1}, ..., yq̄, y_{q̄+1}, ..., y_{q}) near a point x ∈ L as described in the paragraphs above. As before, we write V in the form V = V_{1}+V_{2}+

V_{3}, where V_{1} is tangent to F, V_{2} (y) = ∑q̄ ; i=1 α^{i} (y) \frac{∂}{∂y_{i}}, and V_{3} (y) = ∑_{j=q̄+1}^{q} β^{j} (y) \frac{∂}{∂y_{j}}.

Given any δ′ such that 0 < δ′ < δ, we may multiply the tangent component V_{1} by a radial basic function that is zero in T_{δ′} (L) and is 1 outside T_{δ} (L). In doing so, we preserve the index and orientation line bundle and yet restrict to the case where V is a basic normal vector field. We now assume that V has already been modified in this way. Next, the basic normal vector field V is determined on T_{δ′} (L) by its restriction to a normal ball B (δ′) with the \overline{y} = (y_{1}, ..., yq̄) coordinates, and we may write V (\overline{y}) = ∑q̄ ; i=1 v^{i} (\overline{y}) \frac{∂}{∂y_{i}} + ∑_{j=q̄+1}^{q} v^{j} (\overline{y}) \frac{∂}{∂y_{j}} on B (δ′). Again, we may multiply the component ∑_{j=q̄+1}^{q} v^{j} (\overline{y}) \frac{∂}{∂y_{j}} by a similar radial basic function if necessary so that this component vanishes on a given B (δ′′) such that 0 < δ′′ < δ′, without changing the relevant properties of V. For the remainder of this proof, assume that we have already modified V so that it is in the form V = ∑q̄ ; i=1 v^{i} (\overline{y}) \frac{∂}{∂y_{i}} restricted to B (δ), choosing a slightly smaller δ if necessary. It follows that V (\overline{y}) = V_{L} (\overline{y})+O (∥\overline{y}∥^{2}) on B (δ). Given x ∈ L, we write V_{L} : N_{x}L → N_{x}L in terms of its polar decomposition V_{L} = P Θ. Let Γ be the closed subgroup of isometries on N_{x}L induced from the representation of the holonomy group at x on N_{x}L. For every g ∈ Γ, g V_{L} = V_{L} g, whence V_{L}^{∗} g^{−1} = g^{−1}V_{L}^{∗}. This in turn implies that every g ∈ Γ commutes with P and Θ. Let P_{t} be defined by P_{t}v_{i} = λ_{i} (t) v_{i} on the λ_{i} -eigenspace of P, where each λ_{i} (t) is any smooth positive function such that λ_{i} (0) = λ_{i} and λ_{i} (1) = 1. Since every g ∈ Γ and P have simultaneous eigenspaces, every g ∈ Γ also commutes with each P_{t}. Thus, the smooth, one-parameter family of transformations {T_{t} = P_{t}Θ} is a deformation of V_{L} (t = 0) to an orthogonal transformation Θ (t = 1) that has constant index (that is, the sign of the determinant of the linear transformation does not change). Next, since Θ is orthogonal, there is a complex orthogonal basis {w_{k}} of N_{x}L ⊗ ℂ consisting of eigenvectors such that Θw_{k} = e^{iθ_{k}}w_{k}, where 0 ≤ θ_{k} < 2π. If θ_{k} is 0, then Θ acts by the identity on span {Rew_{k}, Imw_{k}} . If θ_{k} is π, then Θ multiplies each vector in span {Rew_{k}, Imw_{k}} by −1. If θ_{k} ≠ 0 and θ_{k} ≠ π, then Θ acts by a rotation of θ_{k} on span {Rew_{k}, Imw_{k}}, which in this case is necessarily 2-dimensional. We let Θ_{0} = Θ and define the transformation Θ_{t} for 0 < t ≤ 1 by

Θ_{t} (w_{k}) = \{ \begin{matrix}w_{k} & if θ_{k} = 0 \\ −w_{k} & if θ_{k} = π \\ e^{i(1−t)θ_{k}}w_{k} & otherwise. \end{matrix}

The smooth, one-parameter family of transformations {Θ_{t}} is a deformation of the orthogonal transformation Θ (t = 0) to a transformation Θ_{1} (t = 1) that has constant index. Observe that since each g ∈ Γ commutes with Θ_{0}, the obvious action of g on N_{x}L ⊗ ℂ satisfies g (w_{k}) = e^{iα_{k}}w_{k} for some α_{k}; it follows that g commutes with each Θ_{t}. The final transformation Θ_{1} may be described in a real orthogonal basis as a diagonal matrix whose diagonal consists of 1’s and (−1)’s; this transformation is the linear part of a vector field of the form ∑q̄ ; i=1 ±y^{i} \frac{∂}{∂y_{i}}, where the y_{i} are geodesic normal coordinates on exp^{⊥} (N_{x}L) corresponding to that particular basis. We also observe that f (\overline{y}) := \frac{1}{2} ∑q̄ ; i=1 ± (y^{i})^{2} is then a basic function that is well-defined on a small neighborhood of L, and the linear part of ∇f (\overline{y}) at x ∈ L is Θ_{1} (\overline{y}) + O (∥\overline{y}∥^{2}). Note that we may extend f to be a basic function on all of (M, F) by multiplying by a radial cutoff function and extending by zero. Combining the two deformations described above, we see that V_{L} may be smoothly deformed to a Γ-equivariant transformation of the form Θ_{1} in such a way that the index is unchanged throughout the deformation.

The argument that follows is somewhat similar to that found in [5]. Let {Y_{t} : N_{x}L → N_{x}L} be a smooth, one-parameter family of Γ-equivariant transformations constructed above such that Y_{0} = V_{L} and Y_{1} is the linear part of ∑q̄ ; i=1 ±y^{i} \frac{∂}{∂y_{i}}. Thus each Y_{t} (\overline{y}) is a vector field that is well-defined in a sufficiently small tubular neighborhood of the leaf closure L. Next, let Z_{t} (\overline{y}) = h (t) (∇f (\overline{y}) − Y_{1} (\overline{y})),with Y_{1} and ∇f as above and h : ℝ →ℝ a smooth positive function such that h (t) = 0 for t ≤ 0 and h (t) = 1 for t ≥ 1. Because of the remarks above, Z_{t} (\overline{y}) = O (∥\overline{y}∥^{2}) for all t. Let η^{1} be a radial (and therefore basic) cutoff function that is 1 in an r_{1}-neighborhood of L and 0 outside an 2r_{1} -neighborhood of L (r_{1} will be chosen shortly). Then for r_{1} sufficiently small, η^{1} (\overline{y}) (Y_{t} (\overline{y}) + Z_{t} (\overline{y})) is a well-defined basic vector field on (M, F). Let the basic vector field X_{t}^{1} be defined by

X_{t}^{1} = η^{1} (Y_{t} + Z_{t}) + (1 − η^{1}) V,

so that

X_{t}^{1} (\overline{y}) = η^{1} (\overline{y}) (Y_{t} − Y_{0}) (\overline{y}) + Y_{0} (y) + O (∥\overline{y}∥^{2}) .

Observe that X_{t}^{1} is a smooth, one-parameter family of basic vector fields that agree with V outside a 2r_{1}-neighborhood of L, and X_{1}^{1} = ∇f inside a r_{1}-neighborhood of L. Let m_{1} = inf_{∥\overline{y}∥=1,0≤t≤1} ∥Y_{t} (\overline{y})∥ , and let m_{2} = sup_{∥\overline{y}∥=1,0≤t≤1} ∥\frac{d}{dt}Y_{t} (\overline{y})∥ . Observe that 0 < m_{1} < ∞ and 0 ≤ m_{2} < ∞. Therefore, if t_{1} < \frac{m_{1}}{m_{2}} ≤ ∞, then

∥η^{1} (\overline{y}) (Y_{t} − Y_{0}) (\overline{y}) + Y_{0} (\overline{y})∥ ≥ ∥Y_{0} (\overline{y})∥ − ∥(Y_{t} − Y_{0}) (\overline{y})∥
≥ ∥Y_{0} (\overline{y})∥ − t m_{2}
> 0

for t ≤ t_{1}, and hence η^{1} (\overline{y}) (Y_{t} − Y_{0}) (\overline{y}) + Y_{0} (\overline{y}) is invertible. Choose r_{1} > 0 so small that O (∥\overline{y}∥^{2}) < η^{1} (\overline{y}) (Y_{t} − Y_{0}) (\overline{y}) + Y_{0} (\overline{y}) for ∥\overline{y}∥ < 2r_{1}; then the leaf closure L is the only critical leaf closure of the basic vector field X_{t}^{1} (\overline{y}) in the 2r_{1}-neighborhood of L. We continue by defining for t_{k} ≤ t ≤ t_{k+1} := t_{k} + t_{1}

X_{t}^{k} = η^{k} (Y_{t} + Z_{t}) + (1 − η^{k}) X_{t}^{k−1}
= η^{k} (\overline{y}) (Y_{t} − Y_{t_{k}}) (\overline{y}) + Y_{t_{k}} (\overline{y}) + O (∥\overline{y}∥^{2}) ,

where η^{k} is 1 in an r_{k}-neighborhood of L and 0 outside an 2r_{k}-neighborhood of L (where 0 < r_{k} ≤ r_{k−1} is chosen as above). Eventually, we will find a k such that X_{t}^{k} = V outside a 2r_{1} -neighborhood of L and is X_{t}^{k} = Y_{t} + Z_{t} inside a r_{k} -neighborhood of L. Furthermore, L is the only critical leaf closure of X_{t}^{k} in a 2r_{1} -neighborhood of L, and the index of X_{t}^{k} at L is the same as the index of V at L for 0 ≤ t ≤ 1. Finally, X_{1}^{k} = ∇f inside a r_{k}-neighborhood of L.

Observe that if we let 2r_{1} ≤ δ, δ̃ = r_{k}, and Ṽ = X_{1}^{k}, then the first four properties are satisfied. Moreover, since the orientation line bundles of Y_{t} and X_{t}^{k} at L are constant in t, the last property is satisfied as well.

We now proceed with a modified version of Witten’s deformation of the de Rham complex (see [18] and [15]). Let V be a basic normal vector field, and let i (V) : Ω_{B}^{∗} (M) → Ω_{B}^{∗} (M) denote interior multiplication with V. For a given s > 0, let d_{B,s} = d_{B} + s i (V) : Ω_{B}^{∗} (M) → Ω_{B}^{∗} (M). Note that d_{B,s} is the restriction of the differential operator d_{s} = d + s i (V) : Ω^{∗} (M) → Ω^{∗} (M). The formal adjoint of d_{B,s} is δ_{B,s} = δ_{B} + sV^{♭}∧ : Ω_{B}^{∗} (M) → Ω_{B}^{∗} (M), where δ_{B} is the basic adjoint of d_{B} and V^{♭} is the basic one-form ⟨V, ·⟩. From [13], we know that δ_{B} is the restriction of the differential operator δ + ε : Ω^{∗} (M) → Ω^{∗} (M) to basic forms, where ε : Ω^{i} (M) → Ω^{i−1} (M) is a zeroth-order operator that involves mean curvature and Rummler’s formula. The operator ε has the additional property that P εP = 0, where P : L^{2} (Ω^{∗} (M)) → L^{2} (Ω_{B}^{∗} (M)) is the orthogonal projection. We define

D_{B,s} = d_{B,s} + δ_{B,s} : Ω_{B}^{∗} (M) → Ω_{B}^{∗} (M) .

Letting D_{B} = d_{B} + δ_{B} and H = i (V) + V^{♭}∧ , observe that

D_{B,s}^{2} = (D_{B} + sH)^{2}
= D_{B}^{2} + s (HD_{B} + D_{B}H) + s^{2}H^{2}.

The operator H^{2} = (i (V) + V^{♭}∧)^{2} acts by multiplication by the basic function ∥V ∥^{2}. A simple calculation shows Z′ := H(d + δ + ε) + (d + δ + ε)H satisfies

Z′ = L_{V} + (L_{V})^{∗,B} + (d (V^{♭})) ∧ + ((d (V^{♭})) ∧)^{∗,B}
(3.1)  = L_{V} + (L_{V})^{∗} + Z + (d (V^{♭})) ∧ + (((d (V^{♭})) ∧)^{∗}

where L_{V} = i (V) d + d i (V) denotes the Lie derivative in the V direction, the superscript ∗, B denotes the adjoint restricted to basic forms, and Z := ε ∘ V^{♭} ∧ +V^{♭} ∧ ∘ε is a zeroth order operator. One can show using the Leibniz rule that L_{V} + (L_{V})^{∗} commutes with multiplication by a function, so this operator is also an operator of order zero. Thus Z′ is a differential operator of order zero, and it agrees with HD_{B} + D_{B}H on basic forms. Also note that Z′ maps odd forms to odd forms and even forms to even forms.

Observe that D_{B}^{2} = Δ_{B}^{j} = d_{B}δ_{B} + δ_{B}d_{B} on basic j-forms, the basic Laplacian. By the results of [13], this operator is essentially self-adjoint and has eigenvalues 0 ≤ λ_{1}^{B,j} ≤ λ_{2}^{B,j} ≤ λ_{3}^{B,j} ≤ ... with the property that λ_{k}^{B,j} ≥ Ck^{2/n} for some positive constant C and sufficiently large k (see [14] for more precise asymptotics). Furthermore, the basic Hodge decomposition theorem (see [13], [7] ) states that ker Δ_{B} is finite-dimensional and that the space of basic j-forms decomposes orthogonally as im d_{B} ⊕ im δ_{B} ⊕ ker Δ_{B}^{j}. Letting Δ_{B}^{j} denote the Laplacian on j-forms, we have ker Δ_{B}^{j} ≅ H_{B}^{j} (M, F). This implies

χ_{B} (M, F) = ∑_{j=0}^{q} (−1)^{j} dim H_{B}^{j} (M, F)

= ∑_{j=0}^{q} (−1)^{j} dim ker Δ_{B}^{j}
= index (D_{B} : Ω_{B}^{even} (M) → Ω_{B}^{odd} (M)) .

Next, let K_{B}^{j} (t, x, y) denote the basic heat kernel on j -forms (see [7],[13],[14]), which is the fundamental solution of the basic heat equation. More specifically, consider the bundle E → [0, ∞)×M×M where E_{(t,x,y)} = Hom (⋀^{j} (N_{y}F)^{∗} , ⋀^{j} (N_{x}F)^{∗}). The basic heat kernel is a section of E that is basic in x and y and satisfies

(\frac{∂}{∂t} + Δ_{B,x}^{j}) K_{B}^{j} (t, x, y) = 0 for t > 0;
lim_{t→0^{+}} ∫_{M_{y}} K_{B}^{j} (t, x, y) β (y) ω_{vol} (y) = β (x) for all basic j-forms β.

In [13, Theorem 3.5], the authors show that K_{B}^{j} (t, x, y) exists, is smooth in x, y, and t, is unique, and satisfies

K_{B}^{j} (t, x, y) = P_{x}P_{y}K̃^{j} (t, x, y) = ∑_{k=1}^{∞} e^{−λ_{k}^{B,j}t}α_{k} (x) ⊗ α_{k}^{∗} (y) ,

where the set of j-forms {α_{1}, α_{2}, ...} is an orthonormal basis of basic eigenforms corresponding to the eigenvalues {λ_{1}^{B,j}, λ_{2}^{B,j}, ...}, and K̃^{j} (t, x, y) is the heat kernel corresponding to the strongly elliptic operator Δ^{j} + δε^{∗} + ε^{∗}δ. The operator Δ^{j} is the ordinary Laplacian on j-forms. Note that this sum is finite if the leaves are dense. The map P_{x} is the orthogonal projection from L^{2} (Ω^{∗} (M)) to L^{2} (Ω_{B}^{∗} (M)) in the x -variable. The map P_{y} is the induced basic projection on duals of forms in the y-variable. In [13], the authors show that this basic projection P (or P_{x} or P_{y}) maps smooth forms to smooth basic forms, and the results in [13] also imply that the map β ⟼ P β is smooth. The main other fact used in the proofs of the results concerning the basic Laplacian and the basic heat kernel is that Δ_{B}^{j} is the restriction of Δ^{j} + εd + dε , which is a strongly elliptic operator defined on all forms.

The standard heat kernel approach to index calculations carries over to the basic case. Since D_{B} maps an eigenspace of Δ_{B} in Ω_{B}^{even} (M) isomorphically onto an eigenspace in Ω_{B}^{odd} (M), we have

χ_{B} (M, F) = index (D_{B} : Ω_{B}^{even} (M) → Ω_{B}^{odd} (M))
= ∑_{j=0}^{q} (−1)^{j} tr (e^{−tΔ_{B}^{j}})

= ∑_{j=0}^{q} (−1)^{j} ∫_{M} trK_{B}^{j} (t, x, x) ω_{vol} (x)

= ∫_{M} trK_{B}^{even} (t, x, x) ω_{vol} (x) − ∫_{M} trK_{B}^{odd} (t, x, x) ω_{vol} (x) .

A similar analysis may be applied to D_{B,s} = D_{B} + sH. The operator D_{B,s}^{2} = D_{B}^{2} + s (HD_{B} + D_{B}H) + s^{2} ∥V ∥^{2} is the restriction of the strongly elliptic operator

Δ′_{s} = Δ + εd + dε + sZ′ + s^{2} ∥V ∥^{2} (3.2)

to basic forms, and the terms involving s are zeroth order. By a proof similar to that in [13], this operator is essentially self-adjoint and has eigenvalues that grow at a rate similar to those of Δ_{B}. The basic heat kernel K_{B,s}^{even} (t, x, y) corresponding to D_{B,s}^{2} exists, is smooth in x, y, and t, is unique, and satisfies

K_{B}^{even} (t, x, y) = P_{x}P_{y}K̃_{s}^{even} (t, x, y) = ∑_{k=1}^{∞} e^{−t λ_{k}^{B,s,even}}α_{k}^{s} (x) ⊗ (α_{k}^{s} (y))^{∗} ,

where the set {α_{1}^{s}, α_{2}^{s}, ...} is an orthonormal basis of even basic eigenforms corresponding to the eigenvalues {λ_{1}^{B,s,even}, λ_{2}^{B,s,even}, ...} of D_{B,s}^{2}, and K̃_{s}^{even} (t, x, y) is the heat kernel corresponding to the strongly elliptic operator Δ^{even} + δε^{∗} + ε^{∗}δ + s (Z′)^{∗} +s^{2} ∥V ∥^{2}. Similar results are true for D_{B,s}^{2} on odd forms. Again, we have that D_{B,s} maps an eigenspace of D_{B,s}^{2} in Ω_{B}^{even} (M) isomorphically onto an eigenspace in Ω_{B}^{odd} (M), so that

index (D_{B,s} : Ω_{B}^{even} (M) → Ω_{B}^{odd} (M))
(3.3)  = ∫_{M} trK_{B,s}^{even} (t, x, x) ω_{vol} (x) − ∫_{M} trK_{B,s}^{odd} (t, x, x) ω_{vol} (x) .

The above results about D_{B,s}^{2} imply results about the first order operator D_{B,s}. The spectrum of D_{B,s}^{2} is nonnegative; let E_{λ^{2}} be an eigenspace of D_{B,s}^{2} with eigenvalue λ^{2}. Then (D_{B,s} + λ) E_{λ^{2}} and (D_{B,s} − λ) E_{λ^{2}} are finite-dimensional orthogonal eigenspaces of D_{B,s} with eigenvalues λ and −λ, respectively. Since D_{B,s} is the restriction of the first order elliptic operator

D_{s} = d + δ + ε + s (i (V) + V^{♭}∧) ,

the subspaces (D_{B,s} ± λ) E_{λ^{2}} are spanned by smooth eigenforms. Moreover, there is an orthonormal basis of L^{2} (Ω_{B}^{∗} (M)) consisting of such smooth eigenforms.

The operators \frac{∂}{∂t} − iD_{s}^{∗} and \frac{∂}{∂t} + iD_{s}^{∗} are strongly hyperbolic, so our initial value problem has a unique solution (see [3, sections 69-74], [8, chapters IV-V], [11, chapter 6], [16, section 6.5]). Thus, we define the function e^{itD_{s}^{∗}}u to be the unique solution to the initial value problem (the generalized traveling wave equation)

(\frac{∂}{∂t} − iD_{s}^{∗}) f (t, x) = 0 for all t > 0, x ∈ M
f (0, x) = u (x) for all x ∈ M.

Since D_{s} maps basic forms to basic forms, P D_{s}P = D_{s}P, and P D_{s}^{∗}P = P D_{s}^{∗}; the form e^{itD_{s}}u is basic for a given basic form u. Moreover,

\frac{∂}{∂t} (e^{itD_{s}}u) = iD_{s} (e^{itD_{s}}u) = iD_{B,s} (e^{itD_{s}}u) ,

so that e^{itD_{B,s}}u := e^{itD_{s}}u is a solution to the basic traveling wave equation with the appropriate initial condition:

(\frac{∂}{∂t} − iD_{B,s}) β (t, x) = 0 for all t > 0, x ∈ M
(3.4)  β (0, x) = u (x) ∈ Ω_{B}^{∗} (M) .

Then e^{itD_{B,s}}u and e^{−itD_{B,s}}u are both solutions of the basic wave equation

(\frac{∂^{2}}{∂t^{2}} + D_{B,s}^{2}) β (t, x) = 0 . (3.5)

Because of the results concerning \frac{∂}{∂t} −iD_{s}, the solutions to these wave equations exist and are unique with respect to the appropriate initial conditions (both β (0, x) and \frac{∂}{∂t}β (0, x) must be specified for the basic wave equation).

We now prove some analytic results about the operators D_{B,s} and similar operators; we include proofs only where the standard proofs do not translate directly to the basic case. Let L^{p} = L^{p} (Ω_{B}^{∗} (M)) denote the L^{p}−norm closure of the space of smooth basic forms, and let W^{k} = W^{k} (Ω_{B}^{∗} (M)) denote the closure of the space of these basic forms under the Sobolev (k, 2)-norm. Let ∥·∥_{p} and ∥·∥_{k,2} denote the norms on these spaces.

Lemma 3.4. Let D be a strongly elliptic, first order operator on Ω^{∗} (M) that restricts to an operator on Ω_{B}^{∗} (M). Suppose that the restriction of D is formally self-adjoint on Ω_{B}^{∗} (M). Then, for some c > 0, ∥(D^{2} + 1) α∥_{k,2} ≥ c ∥α∥_{k+1,2} for every basic form α.

Proof. We use induction on k. For k = 0, we clearly have that

∥(D^{2} + 1) α∥_{0,2} = (⟨D^{2}α, D^{2}α⟩ + 2 ⟨Dα, Dα⟩ + ⟨α, α⟩)^{1/2}
≥ ∥α∥_{0,2} .

Similarly, ∥(D^{2} + 1) α∥_{0,2} ≥ \sqrt{2} ∥Dα∥_{0,2}. By ellipticity, there exist c_{1} > 0 and c_{2} ≥ 0 (independent of α) such that \sqrt{2} ∥Dα∥_{0,2} ≥ c_{1} ∥α∥_{1,2} − c_{2} ∥α∥_{0,2}. Thus,

∥(D^{2} + 1) α∥_{0,2} ≥ (\frac{c_{1}}{1 + c_{2}} ) ∥α∥_{1,2} .

Assume that the conclusion is true for 0 ≤ k ≤ m. Then

∥(D^{2} + 1) α∥_{m+1,2} ≥ c_{1} ∥(D^{2} + 1)^{2} α∥_{m−1,2} by definition of Sobolev norm
≥ c_{2} ∥(D^{2} + 1) α∥_{m,2} by the induction hypothesis
≥ c_{3} ∥α∥_{m+2,2} − c_{4} ∥α∥_{m+1,2} by ellipticity,

for positive constants c_{1}, c_{2}, and c_{3} and for c_{4} ≥ 0, all independent of α. Also, it follows from our induction hypothesis that ∥(D^{2} + 1) α∥_{m,2} ≥ c_{5} ∥α∥_{m+1,2} for some c_{5} > 0. Finally, by the definition of the Sobolev norm, there exists a positive constant c_{6} such that

∥(D^{2} + 1) α∥_{m+1,2} ≥ c_{6} ∥(D^{2} + 1) α∥_{m+1,2} + \frac{c_{6}c_{4}}{c_{5}} ∥(D^{2} + 1) α∥_{m,2}
≥ c_{6}c_{3} ∥α∥_{m+2,2} ,

whence the result follows.

Lemma 3.5. Let D be as in Lemma 3.4. Then there exists a constant c > 0 such that ∥(D^{2} + 1) α∥_{k,2} ≥ c ∥α∥_{k+2,2} for every basic form α.

Proof. By ellipticity, there exist c_{1} > 0 and c_{2} ≥ 0 such that ∥(D^{2} + 1) α∥_{k,2} ≥ c_{1} ∥α∥_{k+2,2}−c_{2} ∥α∥_{k+1,2}. By Lemma 3.4, there exists c_{3} > 0 such that ∥(D^{2} + 1) α∥_{k,2} ≥ c_{3} ∥α∥_{k+1,2}. Thus

∥(D^{2} + 1) α∥_{k,2} ≥ (\frac{c_{1}}{1 + \frac{c_{2}}{c_{3}}} ) ∥α∥_{k+2,2} .

Remark 3.6. Observe that if the coefficients of the operator D depend polynomially on a parameter s, then the constants in the inequalities of the lemmas above can be chosen to depend polynomially in s . Also, the results of Lemma 3.4 and Lemma 3.5 extend to much more general situations; the operator D may be any strongly elliptic, first-order operator acting on smooth sections of a vector bundle over a compact manifold such that the restriction of D to a subspace consisting of smooth sections is formally self-adjoint.

Lemma 3.7. (Basic Sobolev Embedding Theorem) If a > \frac{n}{2}, then W^{a} ⊂ L^{∞}. Also, W^{a+k} ⊂ C^{k} for k > 0.

Proof. The standard proof (see [15]) works for the basic case.

Lemma 3.8. Define L_{β} (α) = ∫_{M} (α, β) ω_{vol}. Then the map β ↦ L_{β} defines a normpreserving injection of L^{1} (Ω_{B}^{∗} (M)) into (L^{∞} (Ω_{B}^{∗} (M)))^{∗}.

Proof. The standard proof (see [15]) works for the basic case.

Recall that a subset of a foliation is called saturated if it is a union of leaves.

Lemma 3.9. Let U be a saturated open set in a Riemannian foliation (M, F). Let P : L^{2} (Ω^{∗} (M)) → L^{2} (Ω_{B}^{∗} (M)) be the orthogonal projection. Then for every f ∈ C^{∞} (M) ,

∫_{U} f ω_{vol} = ∫_{U} P f ω_{vol}.

Proof. Let {η_{ε}} be a family of smooth basic functions approximating (in L^{2}) the characteristic function of U as ε → 0 (see the results of [13] and [9]). Then ∫_{M} fη_{ε} ω_{vol} = ∫_{M} P (fη_{ε}) ω_{vol} = ∫_{M} (P f) η_{ε} ω_{vol} since η_{ε} is basic ([13]). We then apply the dominated convergence theorem.

Let (·, ·) denote the pointwise inner product of forms. The following two results apply to D_{B,s}, since D_{B,s} is the restriction of D_{s} = d + δ + ε + s (i (V) + V^{♭}∧) to basic forms and since P εP = 0 (see [13]).

Lemma 3.10. Let D be a first order operator on Ω^{∗} (M) of the form D = d + δ + Z_{1} + Z_{2}, where

1. D restricts to an operator on Ω_{B}^{∗} (M);

2. The operator Z_{1} is zeroth order and satisfies P Z_{1}P = 0, where P is the orthogonal projection from L^{2} (Ω^{∗} (M)) to L^{2} (Ω_{B}^{∗} (M));

3. The operator Z_{2} is zeroth order and is formally self-adjoint with respect to the pointwise inner product of forms.

Let B (L, R) denote the set of points of distance less than R from a fixed leaf closure L; assume that R is sufficiently small to avoid the cut locus of L. Let β be a basic form, and let β_{t} = e^{itD}β be the corresponding solution to the basic traveling wave equation as in (3.4). Then, for 0 ≤ t < R, the function

f (t) = ∫_{B(L,R−t)} (β_{t}, β_{t}) ω_{vol}

is a decreasing function of t.

Proof. We have

\frac{df}{dt} = ∫_{B(L,R−t)} ((iDβ_{t}, β_{t}) + (β_{t}, iDβ_{t})) − ∫_{S(L,R−t)} (β_{t}, β_{t}) ,

where S (L, r) is the set of points of distance R from the leaf closure L. Observe that

(iDβ_{t}, β_{t}) + (β_{t}, iDβ_{t}) = ((i (d + δ + Z_{1} + Z_{2}) β_{t}, β_{t}) + (β_{t}, i (d + δ + Z_{1} + Z_{2}) β_{t}))
= i [((d + δ) β_{t}, β_{t}) − (β_{t}, (d + δ) β_{t})]
+i [(Z_{1}β_{t}, β_{t}) − (β_{t}, Z_{1}β_{t})] ,

since Z_{2} is formally self-adjoint with respect to (·, ·). By [15, proof of Proposition 2.9],

i [((d + δ) β_{t}, β_{t}) − (β_{t}, (d + δ) β_{t})] = iδω_{t},

where ω is the one-form defined by ω_{t} (X) = − (X.β_{t}, β_{t}) := − ((X^{♭} ∧ −i (X)) β_{t}, β_{t}). Then

∫_{B(L,R−t)} ((iDβ_{t}, β_{t}) + (β_{t}, iDβ_{t})) = ∫_{B(L,R−t)} iδω_{t} + i [(Z_{1}β_{t}, β_{t}) − (β_{t}, Z_{1}β_{t})]
= ∫_{B(L,R−t)} iδω_{t} + iP [(Z_{1}P β_{t}, β_{t}) − (β_{t}, Z_{1}P β_{t})] ,

which follows from Lemma 3.9 and the fact that β_{t} is basic. Furthermore, by the results of [13], P [(Z_{1}P β_{t}, β_{t}) − (β_{t}, Z_{1}P β_{t})] = (P Z_{1}P β_{t}, β_{t}) − (β_{t}, P Z_{1}P β_{t}). Since P Z_{1}P = 0 by hypothesis, the divergence theorem yields

\frac{df}{dt} = i ∫_{B(L,R−t)} δω_{t} − ∫_{S(L,R−t)} (β_{t}, β_{t})
= i ∫_{S(L,R−t)} ω_{t} (N) − ∫_{S(L,R−t)} (β_{t}, β_{t}) ,

where N is the unit vector field normal to S (L, R − t) with orientation chosen compatibly with the choice of orientation of S (L, R − t). Locally ∥ω_{t} (N)∥^{2} = ∥(N.β_{t}, β_{t})∥^{2} ≤ (β_{t}, β_{t}) (N.β_{t}, N.β_{t}) = (β_{t}, β_{t})^{2}. Since \frac{df}{dt} is real, we conclude that

\frac{df}{dt} ≤ 0.

Proposition 3.11. (Unit Propagation Speed) Let D be as in Lemma 3.10.Then for any β ∈ Ω_{B}^{∗} (M), the support of e^{itD}β lies within a distance |t| of the support of β.

Proof. One easily checks that such an operator D is formally self-adjoint on the space of basic forms, and e^{itD}e^{±isD}β = e^{i(t±s)D}β; thus, it is sufficient to prove the result for small positive t . Since β is basic, the support of β and its complement are saturated. Since (M, F) is a Riemannian foliation and since M is compact, for every leaf closure L, there exists a tubular neighborhood and R > 0 such that for every leaf closure L_{x} in that tubular neighborhood, the set B (L_{x}, R) of points of distance less than R from L_{x} misses the focal locus and cut locus of L_{x}. Choose any x ∈ M that is at a distance R or more from the support of β; let L_{x} denote the leaf closure containing x. Since (M, F) is Riemannian, the set B (L_{x}, R) is also disjoint from the support of β. Then

∫_{B(L_{x},R)} (β, β) = 0 ≥ ∫_{B(L_{x},R−t)} (e^{itD}β, e^{itD}β) for 0 < t < R,

by Lemma 3.10. Hence e^{itD}β = 0 at x for 0 < t < R.

Lemma 3.12. Let D be as in Lemma 3.10, and let ϕ : ℝ → ℝ be a rapidly decreasing even function. Then ϕ (D) has a smooth kernel k (x, y) that is basic in each factor and satisfies k (x, y) = \overline{k (y, x)}.

Proof. The operator ϕ (D) is a bounded, self-adjoint operator on L^{2} (Ω_{B}^{∗}). By Lemma 3.4 and Lemma 3.5, ϕ (D) maps L^{2} (Ω_{B}^{∗}) into W^{k} for every k > 0. By Lemma 3.7, this implies ϕ (D) maps L^{2} (Ω_{B}^{∗}) to smooth basic forms. Using the fact that D is formally self-adjoint on Ω_{B}^{∗} (M), the standard proofs are easily adapted to show the existence and properties of k (x, y) (see [15, Lemma 5.6]).

The index of a Fredholm operator is a homotopy invariant; we prove a similar result for the operators D_{B,s} on basic forms. The operators (Δ′_{s})^{∗} = Δ^{j} + δε^{∗} + ε^{∗}δ + s (Z′)^{∗} + s^{2} ∥V ∥^{2} depend smoothly on the parameter s, the vector field V, and the metric. Therefore the kernels K̃_{s}^{even} (t, x, y) and K̃_{s}^{odd} (t, x, y) are smooth in these parameters as well. Since the map β ⟼ P β is continuous, K_{B,s}^{even} (t, x, y) and K_{B,s}^{odd} (t, x, y) are continuous in s, V, and the metric. By (3.3) and the fact that the index is an integer, we have shown the following:

Proposition 3.13. χ_{B} (M, F) = index (D_{B,s} : Ω_{B}^{even} (M) → Ω_{B}^{odd} (M)) for all s ∈ ℝ.

Let ϕ be a smooth, rapidly decreasing function on [0, ∞) with ϕ (0) = 1. Then ϕ (D_{B}^{2}) is a trace class operator. Let

μ_{j} = tr (ϕ (D_{B}^{2})|_{L^{2}(Ω_{B}^{j}(M))}) .

By adapting the argument of [15, Chapter 12] for the standard case, we obtain

Theorem 3.14. (Basic Morse Inequalities) The numbers μ_{j} and β_{j} = dim H_{B}^{j} (M, F) satisfy the following system of inequalities:

β_{0} ≤ μ_{0},
β_{1} − β_{0} ≤ μ_{1} − μ_{0},
β_{2} − β_{1} + β_{0} ≤ μ_{2} − μ_{1} + μ_{0,}

etc., and the equality

χ_{B} (M, F) = index (D_{B} : Ω_{B}^{even} (M) → Ω_{B}^{odd} (M))
= ∑_{j=0}^{q} (−1)^{j} β_{j} = ∑_{j=0}^{q} (−1)^{j} μ_{j}.

Proof. The proof is identical to the proof of the standard case in [15, Chapter 12], replacing d with d_{B} and the standard Hodge theorem with the basic Hodge theorem.

Let

μ_{s}^{even} = tr (ϕ (D_{B,s}^{2})|_{L^{2}(Ω_{B}^{even}(M))})
μ_{s}^{odd} = tr (ϕ (D_{B,s}^{2})|_{L^{2}(Ω_{B}^{odd}(M))}) .

Combining the proof of the standard Morse inequalities with Proposition 3.13, we have the following result.

Proposition 3.15. χ_{B} (M, F) = μ_{s}^{even} − μ_{s}^{odd} for all s ∈ ℝ.

Fix a number ρ > 0, and choose a positive, even Schwarz function ϕ with ϕ (0) = 1 and such that the Fourier transform ϕ̂ (ξ) = ∫_{ℝ} e^{−ixξ}ϕ (x) dx is supported in the interval [−ρ, ρ]. Since ϕ is even, ϕ (D_{B,s}) makes sense as a smoothing operator, and the basic Euler characteristic satisfies Proposition 3.15 with

μ_{s}^{even} = tr (ϕ (D_{B,s})|_{L^{2}(Ω_{B}^{even}(M))}) and
μ_{s}^{odd} = tr (ϕ (D_{B,s})|_{L^{2}(Ω_{B}^{odd}(M))}) .

Let Crit (V) be the (finite) union of critical leaf closures of V in M.

Lemma 3.16. On the complement of a 2ρ-neighborhood of Crit (V), the basic kernel of ϕ (D_{B,s}) satisfies k_{B,s} (x, y) → 0 uniformly as s → ∞.

Proof. This proof is the same as [15, Lemma 12.10], but we include some details here because of subtleties. Choose a constant C so that ∥V (x)∥ ≥ C > 0 for all x in the complement of a ρ-neighborhood of Crit (V). By formula (3.2) and the remarks following that equation, ⟨D_{B,s}^{2}β, β⟩ ≥ \frac{C^{2}s^{2}}{2} ⟨β, β⟩ for every β ∈ Ω_{B}^{∗} (M) that is supported on the complement of such a neighborhood and for sufficiently large s. Let H denote the Hilbert space of L^{2} basic forms that vanish on a ρ-neighborhood of Crit (V). Then D_{B,s}^{2} is a positive symmetric operator on a dense subset of H, so it extends to a self-adjoint operator A on H satisfying the same inequality above.

Let ω be a basic form supported on the complement of a 2ρ -neighborhood of Crit (V), and let

ω_{t} = cos (tD_{B,s}) ω = \frac{1}{2} (e^{itD_{B,s}} + e^{−itD_{B,s}}) ω,

which is a solution to the generalized wave equation (3.5) corresponding to the operator Δ′_{s} with initial conditions ω_{0} = ω, \frac{∂}{∂t}ω_{0} = 0 . The family of forms ω_{t} is the unique solution to this generalized wave equation as well, by the statements before and after (3.5). Note that the formula above implies that ω_{t} is basic.

By the unit propagation speed property of the basic wave equation (Proposition 3.11), ω_{t} is identically zero on the ρ-neighborhood of Crit (V) if |t| < ρ. This implies that D_{B,s}^{2}ω_{t} = Aω_{t} for |t| < ρ, so that ω_{t} is the unique solution to the system

\frac{∂^{2}}{∂t^{2}} ω_{t} + Aω_{t} = 0; ω_{0} = ω, \frac{∂}{∂t}ω_{0} = 0.

We may therefore write ω_{t} = cos (t\sqrt{A}) ω.

Let ϕ be a real-valued function with the following properties:

1. ϕ is a positive even Schwarz function;

2. ϕ (0) = 1; 3. the Fourier transform ϕ̂ (t) is supported in the interval [−ρ, ρ].

For each nonnegative integer m, define ϕ_{m} by the formula

ϕ_{m}(λ) = (1 + λ^{2})^{2m}ϕ(λ);

note that each ϕ_{m} satisfies (1) - (3).

For a basic form ω that is supported on the complement of the 2ρ -neighborhood of Crit(V),

ϕ_{m} (D_{B,s}) ω = \frac{1}{2π} ∫_{−ρ}^{ρ} ϕ̂_{m} (t) (e^{itD_{B,s}}ω) dt
= \frac{1}{2π} ∫_{−ρ}^{ρ} ϕ̂_{m} (t) (cos (tD_{B,s}) ω) dt since ϕ̂_{m} is even
= \frac{1}{2π} ∫_{−ρ}^{ρ} ϕ̂_{m} (t) (cos (t\sqrt{A}) ω) dt
(3.6)  = ϕ_{m} (\sqrt{A}) ω.

The operator \sqrt{A} is positive and has operator norm is bounded below by \frac{Cs}{\sqrt{2}} for s

sufficiently large. Thus, the operator norm of ϕ_{m} (\sqrt{A}) (as an operator from H to itself) is bounded above by

c_{m} (s) = sup \{|ϕ_{m} (λ)| : λ ≥ \frac{Cs}{\sqrt{2}}\} .

It is clear that c_{m} (s) is rapidly decreasing as s → ∞. By (3.6),

∥ϕ_{m} (D_{B,s}) ω∥_{2} ≤ c_{m} (s) ∥ω∥_{2}

for every basic form ω supported on the complement of a 2ρ -neighborhood of Crit (V).

Next, let L^{p} = L^{p} (Ω_{B}^{∗} (M)) denote the L^{p}−norm closure of the space of smooth basic forms, and let W^{k} = W^{k} (Ω_{B}^{∗} (M)) denote the closure of the space of such basic forms under the Sobolev (k, 2)-norm. By the basic elliptic estimates (Lemma 3.5), the operator norm of (1 + D_{B,s}^{2})^{−1} : W^{k} ⟶ W^{k+2} is bounded by a polynomial in s. The basic version of the Sobolev imbedding theorem (Lemma 3.7) implies that (1 + D_{B,s}^{2})^{−k} : L^{2} ⟶ L^{∞} is a bounded map whose operator norm is bounded by a polynomial in s if k > \frac{n}{4} . Using basic duality (Lemma 3.8) and essential selfadjointness of D_{B,s}, we see that (1 + D_{B,s}^{2})^{−k} : L^{1} ⟶ L^{2} is also a bounded map whose operator norm is bounded by a polynomial in s whenever k > \frac{n}{4}. Note that all of the statements above hold for the operator A as well as for D_{B,s}^{2}. Now, given a basic form ω supported on the complement of a 2ρ -neighborhood of Crit (V) and k > \frac{n}{4},

∥ϕ (D_{B,s}) ω∥_{∞} = ∥ϕ (\sqrt{A}) ω∥_{∞}

∥(1 + A)^{−k} (1 + A)^{k} ϕ (\sqrt{A}) (1 + A)^{k} (1 + A)^{−k} ω∥_{∞}
≤ ∥(1 + A)^{−k}∥_{L^{2}→L^{∞}} c_{k} (s) ∥(1 + A)^{−k} ω∥_{2}
≤ ∥(1 + A)^{−k}∥_{L^{2}→L^{∞}} c_{k} (s) ∥(1 + A)^{−k}∥_{L^{2}→L^{1}} ∥ω∥_{1}
≤ p (s) c_{k} (s) ∥ω∥_{1}

where p (s) is a polynomial in s. Next, since ϕ is rapidly decreasing, ϕ (D_{B,s}) has a continuous basic kernel k_{B,s} (x, y) (Lemma 3.12), and we have the inequality

∥k_{B,s} (x, ·)∥_{∞} ≤ sup ; ∫ ∥ω∥=1,x∈M ∥∫_{M} k_{B,s} (x, y) ω (y) ω_{vol} (y)∥ ≤ p (s) c_{k} (s)

from the above. Thus, k_{B,s} (x, y) → 0 uniformly as s → ∞.

Let (M, F) be a Riemannian foliation, let V be a F-nondegenerate basic vector field, let L be a leaf closure of F, and let O_{L} = O_{L} (V) denote the orientation line bundle of V at L. We denote by Ω^{∗} (L, F, O_{L}) the space of differential forms on L with values in O_{L}; there is a well-defined differential d on this space that is simply the exterior derivative when O_{L} is trivial (see [2, Section I.7]). Locally an element of Ω^{∗} (L, F, O_{L}) can be written as a sum of tensors ω ⊗ s, where ω is an ordinary differential form on M and s is a smooth section of O_{L}. Let X be a vector field on L. Then we can locally define interior multiplication i(X) on Ω^{∗} (L, F, O_{L}) by decreeing that i(X)(ω ⊗ s) = (i(X)ω) ⊗ s and extending linearly. It is straightforward to check that this definition of interior product is independent of the trivialization of O_{L} (V), and so we may therefore make the following definitions:

Definition 3.17. Let M, F, V, and L have the properties listed in the previous paragraph. The space of basic differential forms with values in O_{L} is denoted Ω_{B}^{∗} (L, F, O_{L}), and is defined to be the subcomplex of forms α in Ω^{∗} (L, F, O_{L}) with the property that i(X)α = 0 and i(X)dα = 0 for every vector field X on L tangent to the leaves of F restricted to L. The cohomology of this subcomplex is denoted H_{B}^{∗} (L, F, O_{L}) and called the basic deRham cohomology of L with values in O_{L}. Finally, the basic Euler characteristic of L with values in O_{L} is defined by the formula χ_{B} (L, F, O_{L}) = ∑_{k}(−1)^{k}dim H_{B}^{k} (L, F, O_{L}) .

Theorem 3.18. (Basic Hopf Index Theorem) Let M, F, and V be as in Definition 3.17. For each critical leaf closure L, let ind_{L} (V) be the index of V at L. Then

χ_{B} (M, F) = ∑_{L critical} ind_{L} (V) χ_{B} (L, F, O_{L}) .

Proof. Without loss of generality, we may assume that V is a basic normal vector field (otherwise, project to NF). By Proposition 3.15, we have that χ_{B} (M, F) = lim_{s→∞} (μ_{s}^{even} − μ_{s}^{odd}), which may be obtained by integrating the traces of the corresponding kernels of ϕ (D_{B,s}^{2})|_{L^{2}(Ω_{B}^{even}(M))} and ϕ (D_{B,s}^{2})|_{L^{2}(Ω_{B}^{odd}(M))}. By Proposition 3.16, the kernels of these operators go to zero uniformly on the complement of a fixed but arbitrarily small neighborhood of the critical leaf closures of V. For each critical leaf closure L, let ψ_{L} be a smooth, radial, basic function that is identically 1 in a tubular neighborhood of radius 2ρ around L and supported within a tubular neighborhood of radius 3ρ (assume that we have chosen ρ small enough so that this is possible for each L). Then we have that

χ_{B} (M, F) = ∑_{L critical} lim_{s→∞} tr (ψ_{L}ϕ (D_{B,s}^{2})|_{L^{2}(Ω_{B}^{even}(M))}) − tr (ψ_{L}ϕ (D_{B,s}^{2})|_{L^{2}(Ω_{B}^{odd}(M))}) .

We now use Lemma 3.3 to observe that V may be deformed without changing ind_{L} (V) or O_{L} so that within a tubular neighborhood of radius 4ρ around L, V = ∇f for a basic function f, such that if ind_{L} (V) = +1 then f has even Morse index and if ind_{L} (V) = −1 then f has odd Morse index; again we possibly decrease ρ so that the conclusion of this proposition holds. A unit propagation speed argument shows that the traces are independent of the choice of V with those properties, and thus we may calculate the contributions from each tubular neighborhood as if V = ∇f. By the results of [1], if D_{L}^{2} is the basic Laplacian on L with coefficients in O_{L} , then

lim_{s→∞} tr (ψ_{L}ϕ (D_{B,s}^{2})|_{L^{2}(Ω_{B}^{even}(M))}) = tr (ϕ (D_{L}^{2})|_{L^{2}(Ω_{B}^{even+ \frac{ind_{L}(V )−1}{2}} (L,F,O_{L}))}) ,

with the analogous result for the odd case. Thus,

χ_{B} (M, F) = ∑_{L critical} tr (ϕ (D_{L}^{2})|_{L^{2}(Ω_{B}^{even+ \frac{ind_{L}(V )−1}{2}} (L,F,O_{L}))}) −

tr (ϕ (D_{L}^{2})|_{L^{2}(Ω_{B}^{odd+ \frac{ind_{L}(V )−1}{2}} (L,F,O_{L}))})

= ∑_{L critical} dim H_{B}^{even+ \frac{ind_{L}(V )−1}{2}} (L, F, O_{L}) − dim H_{B}^{odd+ \frac{ind_{L}(V )−1}{2}} (L, F, O_{L})
= ∑_{L critical ind_{L}(V)=+1} χ_{B} (L, F, O_{L}) − ∑_{L critical ind_{L}(V)=−1} χ_{B} (L, F, O_{L}) .

Remark 3.19. If all of the bundles O_{L} are trivial and the critical leaf closures are leaves, then the formula simplifies to

χ_{B} (M, F) = ∑_{L critical} ind_{L} (V) ,

a formula which has the precise form of the ordinary Hopf Index Theorem.

Corollary 3.20. Suppose that there exists a basic vector field on a Riemannian foliation (M, F) that is nowhere tangent to F. Then χ_{B} (M, F) = 0.

Corollary 3.21. Suppose that there exists a basic vector field on a Riemannian foliation (M, F) that is nowhere tangent to F and that the codimension of F is less than 3. Then M has infinite fundamental group.

Proof. The hypotheses imply

χ_{B} (M, F) = dim H_{B}^{0} (M, F) − dim H_{B}^{1} (M, F) + dim H_{B}^{2} (M, F)
= 0.

Since dim H_{B}^{0} (M, F) ≥ 1, dim H_{B}^{1} (M, F) ≥ 1. Since H_{B}^{1} (M, F) injects into H^{1}(M) [17, Proposition 4.1], Poincaré duality and the Universal Coefficient Theorem imply that rank H_{1}(M) ≥ 1 . The result follows.

We now illustrate Theorem 3.18 in the following examples. In each of these examples, there does not exist a basic vector field that is nowhere tangent to the foliation.

Example 3.22. Consider the sphere S^{2} in spherical coordinates (θ, φ) ∈ [0, 2π] × [0, π]. Let α be a fixed irrational multiple of π, and let M = ℝ × 𝕊^{2}/ ∼, where (t, θ, φ) ∼ (t + 1, θ + α, φ). The t-parameter curves make M into a codimension-2 foliation F, which is Riemannian with respect to the standard metric. Observe that the leaf closures are level sets where φ is constant. Consider the vector field

V = (cos φ) ∂_{θ} + (sin φ cos φ) ∂_{φ}.

This vector field is invariant under rotations in θ and is smooth on S^{2}, since it is the restriction of

(xz^{2} − yz) ∂_{x} + (yz^{2} + xz) ∂_{y} + (−x^{2}z − y^{2}z) ∂_{z}

on ℝ^{3}; therefore, it is a smooth basic vector field on (M, F). The critical leaf closures for this vector field correspond to the poles (z = ±1) and the equator (φ = \frac{π}{2}). At the

north pole leaf L_{1} (z = 1), the matrix for V_{L_{1}} is ( \begin{matrix}1 & −1 \\ 1 & 1 \end{matrix} ) in ( \begin{matrix}x \\ y \end{matrix} ) coordinates,

so that ind_{L_{1}} (V) = 1. At the south pole L_{−1} (z = −1), the matrix for V_{L_{−1}} is

( \begin{matrix}−1 & 1 \\ −1 & −1 \end{matrix} ) in ( \begin{matrix}x \\ y \end{matrix} ) coordinates, so that ind_{L_{−1}} (V) = 1. For these leaf closures,

we have χ_{B} (L_{±1}, F, O_{L_{±1}}) = 1 since O_{L_{±1}} and F are trivial. At the equator L_{0}, V_{L_{0}} is multiplication by −1 on each 1-dimensional normal space to L_{0}, so that ind_{L_{0}} (V) = −1. The orientation bundle is the trivial co-normal bundle. Next, observe that this leaf closure is a flat torus, on which the foliation restricts to be the irrational flow. Since the vector field ∂_{θ} is basic, nonsingular, and orthogonal to the foliation on this torus,

χ_{B} (L_{0}, F, O_{L_{0}}) = χ_{B} (L_{0}, F) = 0

by Corollary 3.20. By Theorem 3.18, we conclude that

χ_{B} (M, F) = ∑_{L critical ind_{L}(V)=+1} χ_{B} (L, F, O_{L}) − ∑_{L critical ind_{L}(V)=−1} χ_{B} (L, F, O_{L})

= (1 + 1) − (0) = 2.

In this example, one may independently verify that dim H_{B}^{k} (M, F) = 1 for k = 0 or k = 2, since M is a transversally oriented, taut, codimension-2 Riemannian foliation (see [17]). Also, every closed basic one-form can be written as g (φ) dφ for a smooth function g such that \frac{∂^{m}g}{∂φ^{m}} = 0 for even m at φ = 0 or π. Every exact basic oneform is the differential of a smooth function h such that \frac{∂^{m}h}{∂φ^{m}} = 0 for odd m at φ = 0 or π; thus, dim H_{B}^{1} (M, F) = 0. We have therefore directly computed that χ_{B} (M, F) = 1 − 0 + 1 = 2. Observe that Corollary 3.20 implies that there does not exist a basic vector field on (M, F) that is nowhere tangent to the leaves.

Example 3.23. Let X be any smooth, closed manifold whose fundamental group is the free product ℤ ∗ ℤ (such as the connected sum of two copies of S^{2} × S^{1}); choose a metric so that the volume of X is one. Let X̃ denote the universal cover of X. Suspend an action of the fundamental group on the torus to get the manifold Y = X̃ × S^{1} × S^{1}/ ∼, where the equivalence relation is defined as follows. Let a and b be the generators of π_{1} (X) , which act on the right by isometries on X̃. Let α be an irrational multiple of 2π, and let

(x, θ_{1}, θ_{2}) ∼ (xa, θ_{1} + α, θ_{2}) , and
(x, θ_{1}, θ_{2}) ∼ (xb, 2π − θ_{1}, 2π − θ_{2}) .

These actions on the torus (with the flat metric) are orientation-preserving isometries, so that the x-parameter (immersed) submanifolds form a transversally oriented, Riemannian foliation (Y, F). The leaf closures are parametrized by θ_{2} ∈ [0, π] ; observe that the leaf closures corresponding to θ_{2} = 0 and θ_{2} = π are not transversally oriented, even though F is.

We first calculate χ_{B} (Y, F) directly. As in the previous example, we observe that dim H_{B}^{k} (Y, F) = 1 for k = 0 or k = 2, since M is a transversally oriented, taut, codimension-2 Riemannian foliation. Each closed, basic one-form can be written as g (θ_{2}) dθ_{2}, where g is a smooth function on S^{1}such that g (2π − θ_{2}) = −g (θ_{2}). Since this is the differential of the well-defined basic function f (θ_{2}) = ∫_{0}^{θ_{2}} g (t) dt, we see that dim H_{B}^{1} (Y, F) = 0. Therefore, χ_{B} (Y, F) = 1 − 0 + 1 = 2.

Next, consider the basic vector field W = sin (θ_{2}) ∂_{θ_{2}}. This vector field is singular at the leaf closures θ_{2} = 0 and θ_{2} = π (both of codimension 1), and it is orthogonal to the leaves everywhere. The index of W at θ_{2} = 0 is 1, and its index at θ_{2} = π is −1. The polar decomposition of the linearization of W at θ_{2} = 0 is simply 1 ∗ 1, so there are no negative eigenvectors of the orthogonal part. Thus, the orientation line bundle is trivial at θ_{2} = 0. Therefore,

χ_{B} ({θ_{2} = 0} , F, O_{L}) = χ_{B} ({θ_{2} = 0} , F) = 1 − 0 = 1;

note that there are no closed one-forms on this leaf closure, because it is not transversally oriented. The polar decomposition of the linearization of W at θ_{2} = π is 1∗(−1), and the orientation line bundle O_{π} is simply the normal bundle to the leaf closure. This bundle is nontrivial and has no basic sections, so dim H_{B}^{0} ({θ_{2} = 0} , F, O_{L}) = 0. On the other hand, let s denote a basic section of the pullback of O_{π} via the lift of the leaf closure θ_{2} = π to X̃ × S^{1} × S^{1}; such a section exists because the pullback of O_{π} is trivial. The basic one-form dθ_{1} ⊗ s is closed and descends to a basic one-form on Y with values in O_{π}, because the orientation-reversing action of b changes the sign of both dθ_{1} and s. It follows that dim H_{B}^{1} ({θ_{2} = 0} , F, O_{L}) = 1. Thus,

χ_{B} ({θ_{2} = 0} , F, O_{L}) = 0 − 1 = −1.

By Theorem 3.18,

χ_{B} (Y, F) = ∑_{L critical ind_{L}(V)=+1} χ_{B} (L, F, O_{L}) − ∑_{L critical ind_{L}(V)=−1} χ_{B} (L, F, O_{L})

= (1) − (−1) = 2.

These calculations show that there does not exist a basic vector field on (Y, F) that is nowhere tangent to the foliation; in fact, any F-nondegenerate basic vector field on (Y, F) must be tangent to the foliation on at least two distinct leaf closures.

Example 3.24. Let N be a smooth, four-dimensional closed manifold with finite fundamental group. Let (M, F) be any foliation that is obtained by suspending a discrete subgroup Γ of a compact Lie group of diffeomorphisms of N. That is, choose a manifold X along with a surjective homomorphism ϕ : π_{1} (X) → Γ, and let M = X̃ × N/π_{1} (X), where π_{1} (X) acts on the universal cover X̃ by deck transformations and on N via ϕ. The foliation F is locally given by the X̃-parameter submanifolds. Choose metrics for X and N; by averaging the metric on N over the Lie group of diffeomorphisms, we may and do assume that π_{1} (X) acts on N by isometries. The metric on M defined locally as the product of these metrics is bundle-like for the foliation F. Furthermore, this foliation is taut, so the standard form of Poincaré duality holds for basic cohomology (see [17]). The basic forms of (M, F) are given by forms on N that are invariant under the discrete group of isometries, and the basic cohomology is isomorphic to the cohomology of invariant forms on N. Since π_{1} (N) is finite, H_{B}^{1} (M, F) is trivial. We conclude that

χ_{B} (M, F) = ∑_{j=0}^{4} (−1)^{j} dim H_{B}^{j} (M, F)
= dim H_{B}^{0} (M, F) + dim H_{B}^{2} (M, F) + dim H_{B}^{4} (M, F)
= 2 + dim H_{B}^{2} (M, F) ≥ 2.

Theorem 3.18 implies that every F-nondegenerate basic vector field on (M, F) must have at least two distinct critical leaf closures.

References

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[9] J. M. Lee, K. Richardson, Riemannian foliations and eigenvalue comparison, Ann. Glob. Anal. Geom. 16(1998), 497–525.

[10] X. M. Masa, Duality and minimality in Riemannian foliations, Comment. Math. Helv. 67(1992), 17–27.

[11] S. Mizohata, The theory of partial differential equations, London: Cambridge University Press, 1973.

[12] P. Molino, Riemannian foliations, Boston: Birkhä user, 1988.

[13] E. Park, K. Richardson, The basic Laplacian of a Riemannian foliation, Amer. J. Math. 118(1996), 1249–1275.

[14] K. Richardson, Traces of heat operators on Riemannian foliations, preprint.

[15] J. Roe, Elliptic operators, topology, and asymptotic methods, Pitman Research Notes in Math. 179, Longman Scientific and Technical: Harlow, 1988.

[16] M. E. Taylor, Partial Differential Equations I, New York: Springer-Verlag, 1996.

[17] Ph. Tondeur, Geometry of Foliations, Basel: Birkhä user, 1997.

[18] E. Witten, Supersymmetry and Morse theory, J. Diff. Geom. 17(1982), 661-692.

[19] A. Zeggar, Nombre de Lefschetz basique pour un feulletage riemannien, Ann. Fac. Sci. Toulouse Math., 1(1992), 105–131.

Victor Belfi, Department of Mathematics, Texas Christian University, Box 298900, Fort Worth, Texas 76129

  E-mail address: v.belfi@tcu.edu

Efton Park, Department of Mathematics, Texas Christian University, Box 298900, Fort Worth, Texas 76129

  E-mail address: e.park@tcu.edu

Ken Richardson, Department of Mathematics, Texas Christian University, Box 298900, Fort Worth, Texas 76129

  E-mail address: k.richardson@tcu.edu, krichar@msri.org