Coarse Sheaf Cohomology for Foliations

James L. Heitsch ^{∗} Department of Mathematics (m/c 249) University of Illinois at Chicago 851 S. Morgan Street Chicago Illinois 60607-7045 USA e-mail: heitsch@math.uic.edu

(January 23, 2001)

Abstract

We show that coarse cohomology for foliations can be defined as a sheaf cohomology and use this fact to compute two examples.

1 Introduction

In [HH], we introduced a new invariant of a foliation of a compact manifold, its coarse cohomology. This cohomology combines the usual cohomology of the ambient manifold with the coarse cohomology (due to Roe [R]) of the holonomy covers of the leaves of the foliation. It is expected that this theory will play an important role in the theory of foliations. In particular, secondary classes, the coarse Chern character, higher torsion invariants, spectral invariants of leafwise operators, and the coarse index approach to the Miscenko-Kasparov theory and its applications to the Novikov Conjecture should all have natural expressions in terms of coarse cohomology. For a more complete discussion of this, see the introduction of [HH].

In this paper, we show that this cohomology is in fact a sheaf cohomology. The immediate advantage of this fact is a spectral sequence which converges to the coarse cohomology. We then use this spectral sequence to compute the coarse cohomology of two important foliations, the double Reeb foliation and the Sullivan foliation.

∗Partially supported by NSF Grant DMS 9400676

2 Coarse Sheaf Cohomology

In this section we define the differential sheaf over a compact foliated manifold whose associated cochain complex of continuous sections is the coarse de Rham complex of the foliation. We will freely use the notation and results of [HH].

Let F be a codimension q foliation of a compact n dimensional manifold M without boundary. Recall the holonomy groupoid G_{F} of F. A point y = [γ] ∈ G_{F} is the equivalence class of a path γ : [0, 1] ↦ M whose image is contained in a single leaf L. Two such leafwise paths γ_{1} and γ_{2} are equivalent provided γ_{1}(0) = γ_{2}(0), γ_{1}(1) = γ_{2}(1) and the holonomy along the two paths is the same on some transversal containing γ_{1}(0). There are natural maps s, r : G_{F} ↦ M defined by s(y) = γ(0), r(y) = γ(1). G_{F} is a (generally non-Hausdorff) 2n − q dimensional manifold with the local charts given as follows. Let U and V be foliation charts of s(y) and r(y) respectively and choose γ ∈ y. Then the local chart (U, γ, V) consists of all equivalence classes of leafwise paths which start in U, end in V and which are homotopic to γ through a homotopy of leafwise paths whose end points remain in U and V respectively. It is easy to see that if U, V ≃ R^{n−q} × R^{q}, then (U, γ, V) ≃ R^{n−q} × R^{n−q} × R^{q}. In order to apply the results of [HH], we assume that G_{F} is Hausdorff.

We now recall the coarse de Rham cohomology for F. For each x ∈ M, s^{−1}(x) ≃ L̃_{x}, the holonomy cover of the leaf L_{x} containing x. Denote by G_{ℓ} the submanifold of ×_{ℓ}G_{F} consisting of those points (y_{1}, …, y_{ℓ}) with s(y_{1}) = s(y_{j}) for j = 2, …, ℓ. We also denote by s the map s : G_{ℓ} ↦ M given by s(y_{1}, …, y_{ℓ}) = s(y_{1}). Note that s^{−1}(x) ≃ ×_{ℓ}L̃_{x}. Choose a metric on M. This induces a metric on each leaf L and so also on L̃ and ×_{ℓ}L̃ which makes them complete Riemannian manifolds. Their quasi isometry types are independent of the choice of metric on M since M is compact. Denote the metric on s^{−1}(x) by D_{x}. Given A ⊆ G_{ℓ} and r > 0, define

P en(A, r) = {(y′_{1}, …, y′_{ℓ}) ∈ G_{ℓ} | ∃ (y_{1}, …, y_{ℓ}) ∈ A with
s(y_{i}) = s(y′_{i}) and D_{s(y_{i})}(y_{i}, y′_{i}) < r for i = 1, …, ℓ }.

Denote by A_{c}^{k,ℓ}(F) the space of k forms ω on G_{ℓ+1} such that for all r > 0, sup(ω) ∩ P en(Δ_{ℓ+1}, r) is relatively compact, where Δ_{ℓ+1} is the diagonal of G_{ℓ+1}. We have two differentials defined on A_{c}^{k,ℓ}(F), the usual exterior derivative d : A_{c}^{k,ℓ}(F) ↦ A_{c}^{k+1,ℓ}(F), and δ : A_{c}^{k,ℓ}(F) ↦ A_{c}^{k,ℓ+1}(F) given by

δω = ∑_{j=1}^{ℓ+2}(−1)^{j+1}π_{j}^{∗}ω

where the π_{j} : G_{ℓ+2} ↦ G_{ℓ+1} deletes the j th entry. The cohomology of the bicomplex {A_{c}^{∗,∗}(F), δ, d} is the coarse de Rham cohomology HX^{∗}(F) of F.

Definition 2.1 The coarse presheaf L^{∗} of F is the differential presheaf which associates to each open set U ⊂ M and each non-negative integer q, the space

L^{q}(U) = ∑_{k+ℓ=q} A_{c}^{k,ℓ}(U) where A_{c}^{k,ℓ}(U) = {ω |_{s^{−1}(U)} | ω ∈ A_{c}^{k,ℓ}(F)}.

The differential D : L^{q}(U) ↦ L^{q+1}(U) is given by D | A_{c}^{k,ℓ}(U) = d + (−1)^{k}δ.

The coarse sheaf L^{∗} of F is the differential sheaf associated to the differential presheaf L^{∗}.

For each q, L^{q} is a fine sheaf, so the Čech bicomplex associated to L^{∗} computes the coarse de Rham cohomology of F. Theorem 2.1, page 132 of [B] gives a spectral sequence which converges to HX^{∗}(F). Its E_{2} term is

E_{2}^{p,q} = H^{p}(M; H^{q}(L^{∗}))

where H^{q}(L^{∗}) is the homology sheaf of the differential sheaf L^{∗}. In particular,

H^{q}(L^{∗}) = Ker(D : L^{q} ↦ L^{q+1})/Im(D : L^{q−1} ↦ L^{q}).

3 Two Examples

In this section, we compute the coarse cohomology of the double Reeb foliation and the Sullivan foliation using the above spectral sequence.

1. The Double Reeb Foliation.

The double Reeb foliation is the foliation F of M = S^{1} × S^{2} = S^{1} × D^{2} ∪ S^{1} × D^{2} where each copy of S^{1} × D^{2} is a Reeb component, [MS], p.41. This foliation has a single compact leaf which is diffeomorphic to T^{2} and all the other leaves are diffeomorphic to R^{2}. For each x ∈ T^{2}, the holonomy cover L̃_{x} is quasi isometric to S^{1} × R with the usual metric which makes it coarsely equivalent to R with the usual metric. For x ∈ M − T^{2}, L̃_{x} is isometric to R^{2} with a metric making it coarsely equivalent to [0, ∞) with the usual metric.

Proposition 3.1 Let R_{T^{2}} be the direct image under the inclusion T^{2} ↪ M of the constant sheaf T^{2} × R. The coarse cohomology of the double Reeb foliation F is

HX^{∗}(F) ≃ H^{∗−1}(M; R_{T^{2}}).

Proof: We need only show that H^{q}(L^{∗}) = 0 for q ≠ 1 and H^{1}(L^{∗}) = R_{T^{2}}.

Each x ∈ M − T^{2} has a neighborhood U so that the metric family {s^{−1}(U), d, s, U} is coarsely equivalent to the metric family {U × [0, ∞), d_{U}, π, U}. See [HH]. For each u ∈ U, d is the induced metric on L̃_{u} = s^{−1}(u), and d_{U} is the usual metric on {u} × [0, ∞). As they are coarsely equivalent, these two families have the same coarse cohomology. But the coarse cohomology of [0, ∞) is trivial, so the spectral sequence of [HH] gives that the coarse cohomology of {U × [0, ∞), d_{U}, π, U} is trivial. Thus the coarse cohomology of {s^{−1}(U), d, s, U} is also trivial, so for all x ∈ M − T^{2} and all q, H^{q}(L^{∗})_{x} = 0.

Each neighborhood of x ∈ T^{2} contains a neighborhood U ≃ D^{3} so that the metric family {s^{−1}(U), d, s, U} is coarsely equivalent to the metric family F_{x} = {U, d_{U}, π, U}, where U ⊂ D^{3} × R is the set

U = {(u_{1}, u_{2}, u_{3}, t) | t ≥ −u_{1}^{−2}},

π : U → U is the natural projection, and for each u ∈ U, the metric d_{U} on π^{−1}(u) = [−u_{1}^{−2}, ∞) is the usual metric. We use the convention here that if u_{1} = 0, then t > −∞.

Denote by U_{ℓ+1} the set

U_{ℓ+1} = {(u_{1}, u_{2}, u_{3}, t_{0}, …, t_{ℓ}) ∈ U × R^{ℓ+1} | t_{i} ≥ −u_{1}^{−2}, i = 0, …, ℓ}.

Then the coarse cohomology of the metric family F_{x} may be computed just as the de Rham coarse cohomology for a foliation is computed, with U_{ℓ+1} substituted for G_{ℓ+1}. In particular, we have the spaces A_{c}^{k,ℓ}(F_{x}) of k forms ω on the space U_{ℓ+1} such that for all r > 0, sup(ω) ∩ P en(Δ_{ℓ+1}, r) is relatively compact. Also d : A_{c}^{k,ℓ}(F_{x}) ↦ A_{c}^{k+1,ℓ}(F_{x}) and δ : A_{c}^{k,ℓ}(F_{x}) ↦ A_{c}^{k,ℓ+1}(F_{x}) are as above, and the coarse cohomology of F_{x} is the cohomology of the bicomplex {A_{c}^{∗,∗}(F_{x}), δ, d}.

HX^{0}(F_{x}) consists of constant functions on U_{1} with compact support. As U_{1} is not compact, HX^{0}(F_{x}) = 0 which implies H^{0}(L^{∗})_{x} = 0.

To compute the higher coarse cohomology of the metric family F_{x}, we need the following lemma. Denote by d^{h} : A_{c}^{k,ℓ}(F_{x}) ↦ A_{c}^{k+1,ℓ}(F_{x}) the exterior derivative with respect to the u coordinates only.

Lemma 3.2 (Controlled Poincaré Lemma) Suppose k > 0 and that each term of ω_{k,ℓ} ∈ A_{c}^{k,ℓ}(F_{x}) contains at least one du_{i}. Further suppose that d^{h}(ω_{k,ℓ}) = 0. Then there is ω_{k−1,ℓ} ∈ A_{c}^{k−1,ℓ}(F_{x}) such that d^{h}(ω_{k−1,ℓ}) = ω_{k,ℓ}.

Proof: For a monomial ω_{k,ℓ} = f(u, t)du_{i_{1}} ∧ … ∧ du_{i_{r}} ∧ dt_{j_{1}} ∧ … ∧ dt_{j_{k−(r+1)}} ∈ A_{c}^{k−1,ℓ}(F_{x}), set

ϕ(ω)(u, t) = (∫_{0}^{1} s^{r−1}f(su, t)ds) du_{i_{1}} ∧ … ∧ du_{i_{r}} ∧ dt_{j_{1}} ∧ … ∧ dt_{j_{k−(r+1)}}.

Denote by i_{X} interior multiplication by the vector field X = u_{1}∂/∂u_{1} + u_{2}∂/∂u_{2} + u_{3}∂/∂u_{3} on U_{ℓ+1}. Then the proof of the usual Poincaré lemma, see [W] page 155, shows that the form ω_{k−1,ℓ} = ϕ(i_{X}ω_{k,ℓ}) satisfies the conclusion of the lemma. Extend to general ω_{k,ℓ} by linearity.

□

Let α = [∑_{i+j=k} ω_{i,j}] be a k > 0 dimensional coarse class for F_{x}, where ω_{i,j} ∈ A_{c}^{i,j}(F_{x}). Then ω_{k,0} is a closed k form on U_{1}. We may write ω_{k,0} = ω_{k,0}^{h} + ω_{k,0}^{f} where ω_{k,0}^{h} is a k form involving only the du_{i} and ω_{k,0}^{f} is a k form with each term having a dt_{0}. The fact that d(ω_{k,0}) = 0 implies that d^{h}(ω_{k,0}^{h}) = 0. The Controlled Poincaré Lemma gives a k − 1 form ω_{k−1,0} ∈ A_{c}^{k−1,0}(F_{x}) such that d^{h}(ω_{k−1,0}) = ω_{k,0}^{h}. Recall that the differential of the bicomplex {A_{c}^{∗,∗}(F_{x}), δ, d} is D = d + (−1)^{k}δ. As α = [∑_{i+j=k} ω_{i,j} − D(ω_{k−1,0})], we may assume that ω_{k,0}^{h} = 0. Now ω_{k,0}^{f} = ω_{k−1,0}^{f} ∧ dt_{0} for some element ω_{k−1,0}^{f} ∈ A_{c}^{k−1,0}(F_{x}), and ω_{k−1,0}^{f} has no terms involving a dt_{0}. In addition, as d(ω_{k,0}^{f}) = d^{h}(ω_{k,0}^{f}) = 0, we have d^{h}(ω_{k−1,0}^{f}) = 0 also. If k > 1, the Controlled Poincaré Lemma again implies that there is an element ω_{k−2,0} ∈ A_{c}^{k−2,0}(F_{x}) with d^{h}(ω_{k−2,0}) = ω_{k−1,0}^{f}. Thus d(ω_{k−2,0} ∧ dt_{0}) = ω_{k−1,0}^{f} ∧ dt_{0} = ω_{k,0}^{f}. As α = [∑_{i+j=k} ω_{i,j} − D(ω_{k−2,0} ∧ dt_{0})], we may assume that ω_{k,0}^{f} and so ω_{k,0} is 0. If k = 1 we have that ω_{k,0} is a form of pure fiber type, i.e. it contains no du_{i}. As d(ω_{k,0}) = 0, it must in fact be entirely independent of u. Thus, in both the k > 1 and k = 1 cases, we may assume that ω_{k,0} is completely independent of u, and we may regard it as an element of A_{c}^{k,0}(R), i.e. as a coarse cochain on R. Here we use the extended coarse cohomology of [HH] to compute the coarse cohomology of R.

Now consider ω_{k−1,1}, a k − 1 form on U_{2}. As d(ω_{k−1,1}) = δ(ω_{k,0}), which is independent of u, we have that d^{h}(ω_{k−1,1}) = 0 and proceeding as above, we may assume ω_{k−1,1} contains no forms which contain no dt_{j}. As d^{h}(ω_{k−1,1}) is still zero, proceeding as above, we may assume that it contains no forms which contain one dt_{j}, and then if k > 2 that it contains no forms which contain dt_{0} ∧ dt_{1}. If k = 2, the fact that ω_{k−1,1} = ϕ(u, t)dt_{0} ∧ dt_{1} and d^{h}(ω_{k−1,1}) = 0 means that we may assume that ω_{k−1,1} is entirely independent of u, and so may be regarded as an element of A_{c}^{k−1,1}(R).

Continuing in this manner, we may assume that ∑_{i+j=k} ω_{i,j} is entirely independent of u and so α may be viewed as a k dimensional coarse class on R. Now the coarse cohomology of R is zero in all dimensions except in dimension one where it is R. In fact, one can show that α ↦ ∫_{−∞}^{∞} ω_{1,0}^{f} gives an isomorphism between H^{1}(L^{∗})_{x} and R.

Thus we have for q ≠ 1, H^{q}(L^{∗}) = 0. For q = 1 and x ∈ T^{2}, H^{1}(L^{∗})_{x} = R, and for x ∈ M − T^{2}, H^{1}(L^{∗})_{x} = 0. It is not difficult to see that there is no twisting in the sheaf H^{1}(L^{∗}) so it is in fact R_{T^{2}}. □

The point here is that the metric family F_{x} is sufficiently similar to the trivial metric family (U ×R, d, π, U) that they have the same coarse cohomology. Thus for the double Reeb foliation we have that H^{q}(L^{∗}) = HX^{q}(L̃) where HX^{q}(L̃) is the sheaf which assigns HX^{q}(L̃_{x}) to each x. Below, we will see that the Sullivan foliation does not satisfy this property.

To finish the computation of the coarse cohomology of the double Reeb foliation, we compute H^{∗}(M; R_{T^{2}}). To do so, we tensor R_{T^{2}} with the fine, torsionless resolution of the constant R sheaf over M given by R ↦ C^{0} ↦ C^{1} ↦ C^{2} ↦ ⋯ where C^{k} is the sheaf of germs of smooth k forms on M. This gives us the differential sheaf R_{T^{2}} ⊗ C^{∗} on M, whose stalk at x ∈ M − T^{2} is 0, and at x ∈ T^{2} is C_{x}^{∗}. The set of continuous sections Γ(R_{T^{2}} ⊗ C^{k}) of R_{T^{2}} ⊗ C^{k} is thus the germs on T^{2} of smooth k forms on M, and the cohomology of the cochain complex Γ(R_{T^{2}} ⊗C^{∗}) is H^{∗}(M; R_{T^{2}}). Now the normal bundle of T^{2} in M is trivial, so the cohomology of this cochain complex is the same as the de Rham cohomology of T^{2} × D^{1}, which is the de Rham cohomology of T^{2}. Thus we have the following.

Theorem 3.3 Let F be the double Reeb foliation of S^{1} × S^{2}. Then

HX^{∗}(F) ≃ H^{∗−1}(T^{2}).

The isomorphism is effected as follows. Let (θ, x_{1}, x_{2}, x_{3}) be coordinates on M = S^{1} × S^{2}. Then T^{2} is given by T^{2} = {(θ, 0, x_{2}, x_{3})}. Note that G_{F} is diffeomorphic to (M ×R^{2})−(T^{2} × {(0, 0)}). Define ϕ_{1} : G_{F} ↦ R by

ϕ_{1}(θ, x_{1}, x_{2}, x_{3}, t_{1}, t_{2}) = ln(t_{1}^{2} + t_{2}^{2} + e^{−x_{1}^{−2}})

and ϕ : G_{F} ↦ M × R by ϕ(y) = (s(y), ϕ_{1}(y)). The image of ϕ is {(x, t) | t ≥ −x_{1}^{−2}} where we have the convention that if x_{1} = 0, then t > −∞. The map Φ = (ϕ, Id) is a leafwise map, [HH], between the metric families {G_{F} , d, s, M} and M = {M × R, d^{r}, π, M} where d^{r} is the usual metric on each {x} × R, and π : M × R ↦ M is the projection. The induced map Φ^{∗} on coarse cochains is the map of interest to us. Let ψ : R ↦ R be a smooth, non negative function with non empty support contained in [−2, −4]. The closed coarse one cochain ω = ω_{1,0} + ω_{0,1}, where ω_{1,0} = ψ(t)dt and ω_{0,1}(t_{0}, t_{1}) = ∫_{t_{0}}^{∞} ψ(t)dt − ∫_{t_{1}}^{∞} ψ(t)dt, defines a non zero class in HX^{1}(M). Set M_{2} = {(θ, x_{1}, x_{2}, x_{3}) | x_{1}^{2} > 1/2}. Note that on s^{−1}(M_{2}) ⊂ G_{1}, Φ^{∗}(ω_{1,0}) = 0, and on s^{−1}(M_{2}) ⊂ G_{2}, Φ^{∗}(ω_{0,1}) = 0. Consider the forms on M, α_{1} = dθ, and α_{2} = ρ^{∗}(dθ), where ρ : M ↦ S^{1} is the map (θ, x_{1}, x_{2}, x_{3}) ↦ (x_{2}^{2} + x_{3}^{2})^{−1/2}(x_{2}, x_{3}). Now the map ρ is not defined on S^{1} × {(±1, 0, 0)} ⊂ M_{2}, so α_{2} is not defined on all of M. However, s^{∗}(α_{2}) ∧ Φ^{∗}(ω_{0,1}) is a globally well defined one form on G_{2} and s^{∗}(α_{2}) ∧ Φ^{∗}(ω_{1,0}) is also a globally well defined two form on G_{1}. Thus s^{∗}(α_{2}) ∧ Φ^{∗}(ω) is a well defined coarse one cochain for F. We leave it to the reader to check that D(s^{∗}(α_{2}) ∧ Φ^{∗}(ω)) = 0, and that D(s^{∗}(α_{1}) ∧ Φ^{∗}(ω)) = 0. The cohomology of T^{2} is generated by the classes of the forms 1, dθ_{1}, dθ_{2} and dθ_{1} ∧ dθ_{2}. Under the isomorphism H^{∗−1}(T^{2}) ≃ HX^{∗}(F), these classes map to the classes of Φ^{∗}(ω), s^{∗}(α_{1}) ∧ Φ^{∗}(ω), s^{∗}(α_{2}) ∧ Φ^{∗}(ω), and s^{∗}(α_{1} ∧ α_{2}) ∧ Φ^{∗}(ω).

2. The Sullivan Foliation.

The Sullivan foliation F is a one dimensional foliation of the compact manifold M = T^{1}S^{2} × S^{2} with all leaves compact, but with the volumes of the leaves unbounded. See [S]. Here T^{1}S^{2} is the unit tangent bundle of S^{2}. For −1 ≤ t ≤ 1 set T_{t} = T^{1}S^{2}×{(x, y, z) | z = t}, where (x, y, z) are global coordinates on S^{2} ⊂ R^{3}. Then the foliation on T_{±1} is given by the fibers of π : T^{1}S^{2} ↦ S^{2}. For −1 < t < 1, T_{t} is saturated by the leaves of F and the length of each leaf in T_{t} is f(t) where f : (−1, 1) ↦ R is a smooth positive function with lim_{t→±1} f(t) = ∞. Set T = T_{1} ∪ T_{−1}. Then H^{∗}(M, T; R) is the cohomology of Λ^{∗}(M, T), the forms on M which are zero on some neighborhood of T, and s : G_{F} ↦ M induces a well defined map s^{∗} : Λ^{∗}(M, T) ↦ A_{c}^{∗,0}(F). Recall the maps π_{1}, π_{2} : G_{2} → G_{1} and note that since π_{1} ∘ s = π_{2} ∘ s, δ ∘ s^{∗} = 0. In addition, d ∘ s^{∗} = s^{∗} ∘ d, so we have a well defined natural map s^{∗} : H^{∗}(M, T; R) ↦ HX^{∗}(F).

Theorem 3.4 For the Sullivan foliation F, the natural map

s^{∗} : H^{∗}(M, T; R) ↦ HX^{∗}(F)

is an isomorphism.

Proof: Note that on M − T the length of the leaves changes smoothly, which implies that there is no leafwise holonomy there and so the graph of F over M − T, G_{M−T} is locally a product bundle with fiber S^{1}. In particular, if −1 < r_{1} < r_{2} < 1 then s^{−1}(∪_{r_{1}≤t≤r_{2}}T_{t}) ⊂ G_{F} = G_{1} is compact. Using this fact, the argument after the proof of Proposition 4.2 of [HH] shows that for x ∈ M − T and q > 0, H^{q}(L^{∗})_{x} = 0. We will show below that for x ∈ T, and all q, H^{q}(L^{∗})_{x} = 0. Thus for q > 0, H^{q}(L^{∗}) = 0.

The argument of [HH] also shows that the sheaf H^{0}(L^{∗}) restricted to M − T consists of germs of constant functions on G_{M−T} , so it is the trivial R sheaf. Proposition 3.5 below gives H^{0}(L^{∗})_{x} = 0 for x ∈ T. As the codimension of T in M is two, any twisting in the global sheaf H^{0}(L^{∗}) must take place in M − T, so there is no twisting and we have that H^{0}(L^{∗}) is the direct image of the trivial R sheaf over M − T.

Let M^{∗} be the differential sheaf over M of germs of differential forms on M which are zero on some neighborhood of T. The usual Poincaré lemma gives that for x ∈ M − T, H^{0}(M^{∗})_{x} = R and for q > 0, H^{q}(M^{∗})_{x} = 0. Note that H^{0}(M^{∗})_{x} is just germs of constant functions on M − T. For x ∈ T, M_{x}^{∗} = 0, so for x ∈ T and all q, H^{q}(M^{∗})_{x} = 0. As above, H^{0}(M^{∗}) is the direct image of the trivial R sheaf over M − T.

It is now obvious that the map of differential sheaves s^{∗} : M^{∗} ↦ L^{∗} obtained from s^{∗} : Λ^{∗}(M, T) ↦ A_{c}^{∗,0}(F) induces an isomorphism s^{∗} : H^{∗}(M^{∗}) ↦ H^{∗}(L^{∗}). As both L^{∗} and M^{∗} are fine sheaves, Theorem 2.2 of [B], page 132, then gives the Theorem. □

Note that H^{p}(M, T; R) ≃ H_{c}^{p}(M − T; R) ≃ H^{p−1}(T^{1}S^{2} × S^{1}; R). As T^{1}S^{2} ≃ SO_{3}, we have that HX^{p}(F) ≃ R for p = 1, 2, 4 and 5 and is zero otherwise.

To finish proof of the Theorem, we have

Proposition 3.5 For all x ∈ T, and all q, H^{q}(L^{∗})_{x} = 0

Proof: We may assume that x ∈ T^{1}S^{2} × {(0, 0, 1)}. It has a neighborhood U ≃ D^{5} with coordinates u_{1}, …, u_{5} so that T ∩ U = {u | u_{4} = u_{5} = 0} and if T_{t} ∩ U ≠ ∅,then T_{t} ∩ U = {u | u_{4}^{2} + u_{5}^{2} = 1 − t^{2}}. Sullivan’s example is actually given by the flow Φ_{r} of a unit vector field on M, so s^{−1}(U), the graph of F over U, admits a surjection U × R ↦ s^{−1}(U). This surjection is given by mapping the point (u, r_{0}) to the class of the curve r ↦ Φ_{r}(u) where the domain of the curve is [0, r_{0}]. Consider the metric family

F_{x} = {U × R, d_{U}, π, U}

where π : U ×R → U is the natural projection and for each u ∈ U, d_{U} is the usual metric on {u} × R. The metric family F_{x} surjects onto the metric family {s^{−1}(U), d, s, U} where the map is given by identifying each (u, r) ∈ U × R with (u, r + f(1 − u_{4}^{2} − u_{5}^{2})). If u_{4}^{2} + u_{5}^{2} = 0 we make no identifications.

The zero dimensional coarse cohomology of {s^{−1}(U), d, s, U} consists of constant functions on s^{−1}(U) which have compact support. As s^{−1}(U) is non-compact, this cohomology group is trivial so for all x ∈ T, H^{0}(L^{∗})_{x} = 0.

Now suppose α = [∑_{i+j=k} ω_{i,j}] is a k > 0 dimensional coarse class for {s^{−1}(U), d, s, U}. Using the above surjection, we may regard ω_{i,j} as a form on U × R^{j+1} whose restriction to {u | u_{4}^{2} + u_{5}^{2} = 1 − t^{2}} × R^{j+1} is periodic in each R variable with period f(t). Note that ω_{i,j} is not necessarily a coarse cochain for F_{x}, since in general it will not satisfy the coarse support condition. We now proceed just as we did in the case of the double Reeb foliation to show that we may assume ∑_{i+j=q} ω_{i,j} is completely independent of u. However, we must take care that the construction we use in the Controlled Poincaré Lemma respects the identification.The flow of the vector field X = u_{1}∂/∂u_{1} +u_{2}∂/∂u_{2} +u_{3}∂/∂u_{3} on U ×R^{j+1} preserves the set {u | u_{4}^{2} +u_{5}^{2} = 1−t^{2}}×R^{j+1}, so if we integrate along the flow lines of X, the form ω_{i−1,j} we construct with d^{h}(ω_{i−1,j}) = ω_{i,j} will be a form on U × R^{j+1} whose restriction to {u | u_{4}^{2} + u_{5}^{2} = 1 − t^{2}} × R^{j+1} is also periodic in each R variable with period f(t), i.e. ω_{i−1,j} will define a coarse cochain for {s^{−1}(U), d, s, U}. More specifically, in the definition of ϕ(ω)(u, t) in the Controlled Poincaré Lemma, replace the integral (∫_{0}^{1} s^{r−1}f(su, t)ds) by the

integral (∫_{0}^{1} s^{r−1}f(su_{1}, su_{2}, su_{3}, u_{4}, u_{5}, t)ds). Then we may assume that α is represented by a cochain ∑_{i+j=k} ω_{i,j} which is completely independent of u. In particular each ω_{i,j} is a form on U × R^{j+1} which does satisfies the coarse support condition since it is independent of u and it must do so at u_{4}^{2} + u_{5}^{2} = 0. At the same time it must be periodic in each R variable of period f(t) for t ∈ (s, 1) for some s < 1. It is clear that the zero form is the only such form, so α = 0. □

References

[B] Glen E. Bredon. Sheaf Theory. McGraw-Hill, New York, 1967.

[HH] James L. Heitsch and Steven Hurder. Coarse cohomology for families. To appear, Illinois J. of Math.

[MS] Calvin C. Moore and Claude Schochet. Global Analysis on Foliated Spaces. MSRI Pub. Vol. 9, Springer-Verlag, New York, 1988.

[R] John Roe. Coarse cohomology and index theory on complete Riemannian manifolds. Memoirs of AMS, 104(497), 1993.

[S] Dennis Sullivan. A counterexample to the periodic orbit conjecture. Publ. Math. Inst. Hautes Etudes Sci., 46:5–14, 1976.

[W] Frank W. Warner. Foundations of Differentiable Manifolds and Lie Groups. Scott, Foresman and Company, Glenview, Illinois, 1971.