Coarse Cohomology for Foliations: The General Case

James L. Heitsch

Department of Mathematics, Statistics, and Computer Science University of Illinois at Chicago 851 S. Morgan Street (MC 249) Chicago Illinois 60607-7045 USA e-mail: heitsch@math.uic.edu

March 14, 2003

Abstract

We show how to extend coarse cohomology for foliations to non-Hausdorff foliations and compute several examples of Reeb type.

2000 Mathematics Subject Classification. Primary 55N30; Secondary 57R30.

1 Introduction

Coarse cohomology for metric spaces was introduced by Roe in [R1, R2, R3]. It evolved from the study of the index theory of geometric (Dirac-type) operators on complete open manifolds, and it has yielded new invariants beyond the usual index. In [HH], we extended this theory to parametrized families of metric spaces, the most important examples being foliations of compact manifolds. Unfortunately, in order to apply our general theory to foliations, we had to assume that the graph of the foliation was Hausdorff, a very strong assumption. In this paper we show how to remove this assumption and so extend the notion of coarse cohomology to all foliated manifolds. We also compute several important examples. In a subsequent paper, we will show how this cohomology theory naturally pairs with leafwise-elliptic differential operators to yield new invariants for foliated manifolds. This gives a new and potentially important tool for understanding leafwise elliptic operators on foliated manifolds in general, not just those with Hausdorff graph.

Coarse cohomology for foliations combines the usual cohomology of the ambient manifold with the coarse cohomology of the holonomy covers of the leaves of the foliation. In particular, it is the cohomology of the differential forms on M with coefficients in the coarse de Rham cochains of the holonomy coverings of the leaves of F. When interpreted as a sheaf cohomology it yields a theory which is eminently computable. The results presented here essentially reduce the computation of this cohomology for any foliation to a good understanding of the local structure of the graph. Our main technical tool, the Controlled Poincaré Lemma should have numerous and wide applications in the computation of particular examples. As one such application, we compute the coarse cohomology for several foliations (all of Reeb type) which have non-Hausdorff graphs.

It is a pleasure to thank A. K. Bousfield for helpful conversations, and the referee for pointing out a hiatus in the construction of the examples.

2 Coarse de Rham Cohomology for Foliations

Let F be a codimension q foliation of a compact n dimensional manifold M without boundary. The holonomy groupoid G_{F} of F consists of equivalence classes y = [γ] of leafwise paths γ : [0, 1] → M. 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 germ along the two paths is the same at 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}.

When we defined coarse cohomology for metric families in [HH], we assumed that the family of metrics satisfied a semi-continuity property, which holds for a foliation only if its graph is Hausdorff. We now give an alternate definition of the coarse de Rham cohomology for F, which for a foliation with Hausdorff graph agrees with that given in [HH], and which is natural for all foliations whether or not their graphs are Hausdorff.

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 here 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}, and note that for foliations with non-Hausdorff graph, this metric is not continuous in x. Given A ⊆ G_{ℓ} and r > 0, define

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

For each x ∈ M let ∗x be the equivalence class of the constant path at x, and ∗M_{ℓ} the subset of G_{ℓ} given by

∗M_{ℓ} = ⋃_{x∈M}(∗x, …, ∗x).

Denote by A_{c}^{k,ℓ}(F) the space of smooth (that is, smooth in each coordinate chart) k forms ω on G_{ℓ+1} such that for all r > 0, there exists R > 0 so that

sup(ω) ∩ P en(Δ_{ℓ+1}, r) ⊆ P en(∗M_{ℓ+1}, R),

where Δ_{ℓ+1} ≃ G_{F} is the thin diagonal of G_{ℓ+1}.

When we defined the de Rham coarse cohomology of a foliation in [HH], we defined A_{c}^{k,ℓ}(F) by requiring that sup(ω) ∩ P en(Δ_{ℓ+1}, r) be relatively compact. Note that when G_{F} is Hausdorff, these two notions agree. For Hausdorff spaces, relative compactness is reasonably easy to check, but it is not so easy for non-Hausdorff spaces. For the application we have in mind (namely pairing coarse cohomology with leafwise elliptic operators), the crucial property is not the relative compactness of sup(ω) ∩ P en(Δ_{ℓ+1}, r), but rather the fact that it has finite volume so that integration over this intersection of a bounded differential form produces a finite quantity.

We define two differentials on A_{c}^{k,ℓ}(F) as follows. The usual exterior derivative d does not increase supports, so it is clear that it maps A_{c}^{k,ℓ}(F) to A_{c}^{k+1,ℓ}(F). Define δ : A_{c}^{k,ℓ}(F) → A_{c}^{k,ℓ+1}(F) by

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

where π_{j} : G_{ℓ+2} → G_{ℓ+1} deletes the j th entry. To see that δ actually does map A_{c}^{k,ℓ}(F) to A_{c}^{k,ℓ+1}(F) we have the following Proposition whose proof is immediate.

Proposition 2.1 If sup(ω) ∩ P en(Δ_{ℓ+1}, r) ⊆ P en(∗M_{ℓ+1}, R), then sup(π_{j}^{∗}ω) ∩ P en(Δ_{ℓ+2}, r) ⊆ P en(∗M_{ℓ+2}, 2R + 2r).

Definition 2.2 The coarse de Rham cohomology HX^{∗}(F) of F is the cohomology of the bicomplex {A_{c}^{∗,∗}(F), δ, d}.

As in [H], this cohomology theory is a sheaf cohomology theory given as follows. 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}).

This reduces the computation of the coarse cohomology of a foliation to understanding the local structure of the map s : G → M and the analysis of the spectral sequence given above.

3 Some Reeb Type Examples

In this section, we compute the coarse cohomology of several foliations of S^{3} of Reeb type.

1. The Reeb Foliation.

The Reeb foliation is the foliation F of S^{3} = 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 R^{2} 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.

Theorem 3.1 The coarse cohomology of the Reeb foliation is trivial, that is HX^{∗}(F) = 0.

Here and in the succeeding examples, we will be computing the stalks of certain sheaves. In all the cases we consider the stalks will be either 0 or R, and it will be easy to see that there is no twisting in any of the sheaves. Thus all of the sheaves we consider will be trivial R sheaves over subsets of S^{3} extended by zero to all of S^{3}.

Recall, see [HH], that a metric family F = {G, d, π, U} consists of a paracompact Hausdorff space G, a metric space U, a continuous map π: G → U, and a family of fiberwise metrics {d_{x} | x ∈ U} satisfying certain continuity and properness conditions. In [HH], we also defined the notion of coarsely equivalent metric families. Roughly speaking two such families {G_{1}, d_{1}, π_{1}, U } and {G_{2}, d_{2}, π_{2}, U} with the same base space U are coarsely equivalent if there is a map ϕ : G_{1} → G_{2} covering the identity which induces a coarse equivalence on each of the fibers of the maps π_{j}: G_{j} → U. Coarse cohomology is defined for metric families and coarsely equivalent families have the same coarse cohomology. These notions and results extend (using our new definition of coarse cohomology) to the metric families defined taking U to be an open subset in any foliated manifold M (whether or not G_{F} is Hausdorff), G to be s^{−1}(U) ⊂ G_{F}, π = s and d_{x} the metric on L̃_{x} induced from a metric on M.

To prove Theorem 3.1 we need only show that for all x and q, H^{q}(L^{∗})_{x} = 0. There are two separate cases to consider, namely x ∈ S^{3} − T^{2}, and x ∈ T^{2}.

Proposition 3.2 For all x ∈ S^{3} − T^{2}, and for all q, H^{q}(L^{∗})_{x} = 0.

Proof: Each x ∈ S^{3} − T^{2} has a neighborhood U ⊂ S^{3} − T^{2}, U ≃ D^{3}, so that the metric family {s^{−1}(U), d, s, U} is coarsely equivalent to the product metric family {U × [0, ∞), d^{U}, π, U}, where for each u ∈ U, d_{u}^{U} is the usual metric on {u}×[0, ∞). 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 also trivial. Thus the coarse cohomology of {s^{−1}(U), d, s, U} is trivial, so for all x ∈ S^{3} − T^{2} and all q, H^{q}(L^{∗})_{x} = 0. □

We now compute H^{q}(L^{∗})_{x} for x ∈ T^{2}. 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} = {G, d^{U}, π, U}, where G = Ĝ/Z, and Ĝ ⊂ D^{3} × R^{2} is the set

Ĝ = {(u_{1}, u_{2}, u_{3}, t_{1}, t_{2}) | if u_{3} ≥ 0, t_{1} > −u_{3}^{−2}, and if u_{3} ≤ 0, t_{2} > −u_{3}^{−2}}.

We use the convention that if u_{3} = 0, then −u_{3}^{−2} = −∞. The action of Z is generated by

if u_{3} > 0, then (u_{1}, u_{2}, u_{3}, t_{1}, t_{2}) → (u_{1}, u_{2}, u_{3}, t_{1}, t_{2} + 1);
if u_{3} < 0, then (u_{1}, u_{2}, u_{3}, t_{1}, t_{2}) → (u_{1}, u_{2}, u_{3}, t_{1} + 1, t_{2}).

The map π : G → U is the natural projection and for each u ∈ U, the metric d_{u}^{U} on π^{−1}(u) is induced from the restriction of the natural metric on {u}×R^{2}. If u has u_{3} ≠ 0, then π^{−1}(u) ≃ S^{1}×(−u_{3}^{−2}, ∞) with the usual metric, while if u_{3} = 0, π^{−1}(u) ≃ R^{2} with the usual metric. Note that the set U ∩ T^{2} = {u | u_{3} = 0}.

Denote by Ĝ_{ℓ} ⊂ U × R^{2ℓ} the set

Ĝ_{ℓ} = {(u, t_{1}, …, t_{2ℓ}) | if u_{3} ≥ 0, t_{2j+1} > −u_{3}^{−2}; if u_{3} ≤ 0, t_{2j} > −u_{3}^{−2}},

and set

G_{ℓ} = Ĝ_{ℓ}/Z^{ℓ},

where the action of Z^{ℓ} is the natural extension of the action of Z above. As above, we have the natural projection π : G_{ℓ} → U. For each u ∈ U with u_{3} ≠ 0, π^{−1}(u) ≃ ×_{ℓ}(S^{1} × (u_{3}^{−2}, ∞)) with the usual metric, and for each u ∈ U with u_{3} = 0, π^{−1}(u) ≃ (R^{2})^{ℓ} with the usual metric.

Denote by Δ_{ℓ} ≃ G the thin diagonal of G_{ℓ}, and by ∗U_{ℓ} ⊂ G_{ℓ} the subset given by the image in G_{ℓ} of the set U × {(0…. , 0)} ⊂ U × R^{2ℓ}. 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 G_{ℓ} substituted for G_{ℓ}. In particular, we have the spaces A_{c}^{k,ℓ}(F_{x}) of k forms ω on the space G_{ℓ+1} such that for all r > 0, there exists R > 0 so that

sup(ω) ∩ P en(Δ_{ℓ+1}, r) ⊆ P en(∗U_{ℓ+1}, R).

The differentials d and δ are as above, and the coarse cohomology of F_{x} is the cohomology of the bicomplex {A_{c}^{∗,∗}(F_{x}), δ, d}.

Proposition 3.3 For all x ∈ T^{2}, and for all q, H^{q}(L^{∗})_{x} = 0.

Proof: HX^{0}(F_{x}) consists of constant functions on G_{1} with compact support. As G_{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 lemmas.

Lemma 3.4 For all ω ∈ A_{c}^{k,ℓ}(F_{x}), sup(ω) ∩ π^{−1}(U ∩ T^{2}) = ∅.

Proof: The Z^{ℓ+1} action on Ĝ_{ℓ+1} induces two different actions of Z^{ℓ+1} on π^{−1}(T^{2}) ≃ T^{2} × R^{2ℓ+2}, one being the natural action on the odd coordinates of R^{2ℓ+2}, called the odd action, and the other being the natural action on the even coordinates. Suppose there is z ∈ sup(ω) ∩ π^{−1}(T^{2}). Then there is a sequence, {z_{n}} in G_{ℓ+1}, which converges to z with ω(z_{n}) ≠ 0, and we may assume without loss of generality that u_{3}(z_{n}) < 0 for all n. Then the sequence {z_{n}} also converges to all the points in the orbit of z under the odd action of Z^{ℓ+1}. Let the coordinates of z be z = (y, (t_{1}, t_{2}), …, (t_{2ℓ+1}, t_{2ℓ+2})). Now z is in some bounded neighborhood of the diagonal in y × (R^{2})^{ℓ+1}, so there is r > 0 so that for all a, b ∈ {1, 2, …, ℓ + 1}, d_{y}^{U}((t_{2a−1}, t_{2a}), (t_{2b−1}, t_{2b})) < r. But then the sequence of points (y, (t_{1} + n, t_{2}), …, (t_{2ℓ+1} + n, t_{2ℓ+2})) also satisfies d_{y}^{U}((t_{2a−1}+n, t_{2a}), (t_{2b−1}+n, t_{2b})) < r, so it is also in the same bounded neighborhood of the diagonal. This sequence is in the orbit of z under the odd action, so it is also in sup(ω) ∩ π^{−1}(T^{2}). As this sequence increases without bound as n → ∞, the support condition is violated. □

Denote by d̂: A_{c}^{k,ℓ}(F_{x}) → A_{c}^{k+1,ℓ}(F_{x}) the exterior derivative with respect to the u coordinates only. Strictly speaking this does not make sense. However, we may pull back any form ω on G_{ℓ+1} to Ĝ_{ℓ+1} by the natural projection ρ : Ĝ_{ℓ+1} → G_{ℓ+1}, and d̂(ρ^{∗}(ω)) does make sense. As d̂(ρ^{∗}(ω)) is invariant under the action of Z^{ℓ+1} it induces a well defined form on G_{ℓ+1}. Define this to be d̂(ω).

Lemma 3.5 (Controlled Poincaré Lemma) Suppose x ∈ T^{2}, k > 0 and ω_{k,ℓ} ∈ A_{c}^{k,ℓ}(F_{x}). Assume that each term of ρ^{∗}(ω_{k,ℓ}) contains at least one du_{i} and that d̂(ω_{k,ℓ}) = 0. Then there is ω_{k−1,ℓ} ∈ A_{c}^{k−1,ℓ}(F_{x}) such that d̂(ω_{k−1,ℓ}) = ω_{k,ℓ}.

Proof: We will work on the space Ĝ_{ℓ+1} using objects which are invariant under the action of Z^{ℓ+1}. Let ω ∈ A_{c}^{k−1,ℓ}(F_{x}) and suppose that ρ^{∗}(ω) on Ĝ_{ℓ+1} is a monomial

ρ^{∗}(ω) = f(u, t)du_{i_{1}} ∧ … ∧ du_{i_{r}} ∧ dt_{j_{1}} ∧ … ∧ dt_{j_{k−(r+1)}}.

Set

ϕ(ρ^{∗}(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)}}.

Let i_{X} be interior multiplication by the vector field u_{1}∂/∂u_{1} + u_{2}∂/∂u_{2} + u_{3}∂/∂u_{3} on Ĝ_{ℓ+1}. Then the proof of the usual Poincaré lemma, see [W] page 155, shows that the form ω̂_{k−1,ℓ} = ϕ(i_{X}ρ^{∗}(ω_{k,ℓ})) satisfies d̂(ω̂_{k−1,ℓ}) = ρ^{∗}(ω_{k,ℓ}). In addition, ω̂_{k−1,ℓ} is invariant by the action of Z^{ℓ+1}, so it induces a well defined form ω_{k−1,ℓ} on G_{ℓ+1} and d̂(ω_{k−1,ℓ}) = ω_{k,ℓ}.

It remains to check that ω_{k−1,ℓ} satisfies the support condition required of an element of A_{c}^{k−1,ℓ}(F_{x}). First, note that for all 0 < s ≤ 1,

(u, t) ∈ ρ^{−1}(P en(Δ_{ℓ+1}, r)) ⇔ (su, t) ∈ ρ^{−1}(P en(Δ_{ℓ+1}, r))

and

(u, t) ∈ ρ^{−1}(P en(∗M_{ℓ+1}, R)) ⇔ (su, t) ∈ ρ^{−1}(P en(∗M_{ℓ+1}, R)).

Now suppose that ρ(u, t) ∈ sup(ω_{k−1,ℓ}) ∩ P en(Δ_{ℓ+1}, r). It follows immediately that (u, t) ∈ sup(ω̂_{k−1,ℓ}) ∩ ρ^{−1}(P en(Δ_{ℓ+1}, r)). This implies that there is 0 < s ≤ 1 so that (su, t) ∈ sup(ρ^{∗}ω_{k,ℓ}) ∩ ρ^{−1}(P en(Δ_{ℓ+1}, r)), for if not ω̂_{k−1,ℓ} would be zero on a neighborhood of (u, t). Because sup(ρ^{∗}ω_{k,ℓ})∩ρ^{−1}(P en(Δ_{ℓ+1}, r)) = ρ^{−1}(sup(ω_{k,ℓ}) ∩ P en(Δ_{ℓ+1}, r)) and ω_{k,ℓ} satisfies the support condition, we may choose R > 0, so that (su, t) ∈ ρ^{−1}(P en(∗M_{ℓ+1}, R)). Thus we have that (u, t) ∈ ρ^{−1}(P en(∗M_{ℓ+1}, R)), so ρ(u, t) ∈ P en(∗M_{ℓ+1}, R) and the support condition is satisfied.

Finally, extend this construction 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}). We first show that we may assume the ω_{i,j} are independent of u. Since D(∑_{i+j=k} ω_{i,j}) = 0, dω_{k,0} = 0. Write ω_{k,0} = ω_{u} + ω_{t} where ω_{u} is the part of ω_{k,0} of highest u form degree. (Strictly speaking, we should say this about their pullbacks by ρ. We will no longer make this distinction.) The fact that d(ω_{k,0}) = 0 implies that d̂(ω_{u}) = 0. The Controlled Poincaré Lemma gives a k−1 form ω_{k−1,0} ∈ A_{c}^{k−1,0}(F_{x}) so that d̂(ω_{k−1,0}) = ω_{u}. As α = [∑_{i+j=k} ω_{i,j} − D(ω_{k−1,0})], we may replace ∑_{i+j=k} ω_{i,j} by ∑_{i+j=k} ω_{i,j} − D(ω_{k−1,0}). This has the effect of reducing the highest u form degree of ω_{k,0} by 1. By induction, we may assume that the u form degree of ω_{k,0} is 0. The fact that dω_{k,0} = 0 combined with the fact that ω_{k,0} contains no du_{i} s, means that ω_{k,0} is completely independent of u.

By induction, we have a representative ∑_{i+j=k} ω_{i,j} of α which has ω_{i,j} completely independent of u for i > r. The fact that D(∑_{i+j=k} ω_{i,j}) = 0, implies that dω_{r,s} = ±δω_{r+1,s−1}, which is entirely independent of u. This implies that d̂ω_{r,s} = 0. Proceed exactly as above to replace ∑_{i+j=k} ω_{i,j} by a new representative of α with ω_{i,j} completely independent of u for i ≥ r. Thus we may assume that ∑_{i+j=k} ω_{i,j} is completely independent of u. But on π^{−1}(U ∩ T^{2}), ∑_{i+j=k} ω_{i,j} = 0 so ∑_{i+j=k} ω_{i,j} = 0, and α = 0. □

This completes the proof of Theorem 3.1.

We now compute the coarse cohomology for three other foliations derived from the Reeb foliation. We do this by replacing the compact leaf T^{2} by T^{2} × I foliated in three different ways: first by a foliation with all leaves diffeomorphic to T^{2}; second by foliations with all leaves diffeomorphic to S^{1} ×R; and third by foliations with all leaves diffeomorphic to R^{2}. These will be called the T^{2} Reeb foliation, the S^{1} × R Reeb foliations, and the R^{2} Reeb foliations respectively.

2. The T^{2} Reeb Foliation.

To obtain this foliation, replace the T^{2} leaf of the Reeb foliation by a family of such leaves, i.e. by the manifold T^{2}×[0, 1] foliated by the fibers of the fibration T^{2} × [0, 1] → [0, 1]. Any point in the interior of T^{2} × [0, 1] has a neighborhood U ≃ D^{3} so that the metric family {s^{−1}(U), d, s, U} is coarsely equivalent to the product metric family {U × T^{2}, d^{U}, π, U}, where the metric on each {u} × T^{2} is the usual metric. As the fibers of the projection are compact, this family is coarsely equivalent to the trivial metric family {U, d, π, U}, with points as fibers. The coarse cohomology of a point is just its usual cohomology, so the spectral sequence of [HH] gives

Proposition 3.6 For all x ∈ T^{2} × (0, 1), H^{0}(L^{∗})_{x} = R, otherwise H^{q}(L^{∗})_{x} = The computation of H^{q}(L^{∗})_{x} for x ∈ S^{3} − (T^{2} × [0, 1]) is the same as in the Reeb case so we have

Proposition 3.7 For all x ∈ S^{3} − (T^{2} × [0, 1]), H^{q}(L^{∗})_{x} = 0 for all q.

For x in the boundary of T^{2} × [0, 1], we have

Proposition 3.8 For all x ∈ ∂(T^{2} × [0, 1]), and for all q, H^{q}(L^{∗})_{x} = 0.

Proof: Any such x has a neighborhood U ≃ D^{3} with the metric family {s^{−1}(U), d, s, U} coarsely equivalent to the metric family F_{x} = {G, d^{U}, π, U}, where G = Ĝ/Z^{2}, and Ĝ ⊂ D^{3} × R^{2} is the set

Ĝ = {(u_{1}, u_{2}, u_{3}, t_{1}, t_{2}) | if u_{3} ≥ 0, t_{1} > −u_{3}^{−2}}.

We continue to use the convention that if u_{3} = 0, then −u_{3}^{−2} = −∞. The action of Z^{2} is given by

if u_{3} ≥ 0, then (a, b)((u_{1}, u_{2}, u_{3}, t_{1}, t_{2})) = (u_{1}, u_{2}, u_{3}, t_{1}, t_{2} + b);
if u_{3} < 0, then (a, b)((u_{1}, u_{2}, u_{3}, t_{1}, t_{2})) = (u_{1}, u_{2}, u_{3}, t_{1} + a, t_{2} + b).

The map π : G → U is the natural projection. For each u ∈ U with u_{3} ≥ 0, π^{−1}(u) ≃ S^{1}×(−u_{3}^{−2}, ∞) with the usual metric, and for each u ∈ U with u_{3} < 0, π^{−1}(u) ≃ T^{2} with the usual metric. Finally note that U ∩ ∂(T^{2} × [0, 1]) = {u | u_{3} = 0}.

Since G_{1} is not compact, we have as above that H^{0}(L^{∗})_{x} = 0.

Lemma 3.9 Any ω ∈ A_{c}^{k,ℓ}(F_{x}) has C^{∞} contact with 0 on the set π^{−1}(U ∩ ∂(T^{2} × [0, 1])).

Proof: If not, there is a sequence {z_{n}} with u_{3}(z_{n}) < 0 and ω(z_{n}) ≠ 0. Proceeding as in the proof of Lemma 3.4, we have that the support condition is violated. □

Thus any coarse class α may be written as α = α_{1} + α_{2} , where α_{1} has a representative which is zero for u_{3} ≤ 0 and α_{2} has a representative which is zero for u_{3} ≥ 0.

Suppose ∑_{i+j=k} ω_{i,j} is a representative of α_{1} which is zero for u_{3} ≤ 0. The argument for the Reeb foliation that ∑_{i+j=k} ω_{i,j} may be assumed to be entirely independent of u works equally well here, and so ∑_{i+j=k} ω_{i,j} = 0 and α_{1} = 0.

Now suppose ∑_{i+j=k} ω_{i,j} represents α_{2} and is zero for u_{3} ≥ 0. If the support of ∑_{i+j=k} ω_{i,j} intersects π^{−1}(U ∩ ∂(T^{2} × [0, 1])) non-trivially, then the support condition will be violated by the same argument as in the Reeb case. If the support of ∑_{i+j=k} ω_{i,j} does not intersect π^{−1}(U ∩ ∂(T^{2} × [0, 1])) non-trivially, then because of the compactness of T^{2}, there is ϵ < 0 so that for all z with u_{3} > ϵ, ∑_{i+j=k} ω_{i,j}(z) = 0. As we are computing the sheaf cohomology at x, the coarse class determined by ∑_{i+j=k} ω_{i,j} is the same as that determined by its restriction to u_{3} > ϵ. Thus we may suppose that ∑_{i+j=k} ω_{i,j} = 0 so α_{2} = 0 also and α = 0. □

Given any subset A ⊂ S^{3} denote by R_{A} the trivial R sheaf over A extended by 0 to all of S^{3}. Combining Propositions 3.6, 3.7, and 3.8 we have

Proposition 3.10 For the T^{2} Reeb foliation, H^{0}(L) = R_{T^{2}×(0,1)}, otherwise H^{q}(L) = 0.

Now the spectral sequence of [B], Theorem 2.1 p. 132, applied to this foliation has E_{2} term

E_{2}^{p,q} = H^{p}(S^{3}; H^{q}(L^{∗})) = 0 for all q > 0;
E_{2}^{p,0} = H^{p}(S^{3}; H^{0}(L^{∗})) = H^{p}(S^{3}; R_{T^{2}×(0,1)}).

Thus the coarse cohomology of the T^{2} Reeb Foliation is

HX^{∗}(F) = H^{∗}(S^{3}; R_{T^{2}×(0,1)}).

Now consider the short exact sequence R_{T^{2}×(0,1)} → R_{S^{3}} → R_{B}, where B is the complement of the open subset T^{2} × (0, 1). Since B is closed, the map H^{∗}(S^{3}; R_{S^{3}}) → H^{∗}(S^{3}; R_{B}) in the associated long exact sequence is the usual restriction homomorphism H^{∗}(S^{3}; R) → H^{∗}(B; R). Thus H^{∗}(S^{3}; R_{T^{2}×(0,1)}) is the same as the third term of the long exact sequence for the inclusion B → S^{3}, namely the relative cohomology H^{∗}(S^{3}, B; R) ≃ H^{∗−1}(T^{2}; R) and we have

Theorem 3.11 For the T^{2} Reeb foliation of S^{3}, HX^{∗+1}(F) ≃ H^{∗}(T^{2}; R).

An isomorphism is given as follows. Choose a smooth non-negative function ϕ(x) on (0, 1) which is positive on some subset of the interior and zero on a neighborhood of the boundary. On T^{2} × (0, 1) we have coordinates (θ_{1}, θ_{2}, x). Over T^{2}×(0, 1), G_{1} ≃ T^{2}×(0, 1)×T^{2}, with coordinates (θ_{1}, θ_{2}, x, t_{1}, t_{2}). Let ξ be a differential form on T^{2} representing a class in H^{k}(T^{2}; R) and consider the form s^{∗}(ϕ ξ dx) ∈ A^{k+1,0}(F), where we extend it to all of G_{1} by defining it to be zero off s^{−1}(T^{2} × (0, 1)). Then d(s^{∗}(ϕ ξ dx)) = 0, and since s^{∗}(ϕ ξ dx) is independent of (t_{1}, t_{2}), δ(s^{∗}(ϕ ξ dx)) = 0 also, so D(s^{∗}(ϕ ξ dx)) = 0. The isomorphism we seek is A([ξ]) = [s^{∗}(ϕ ξ dx)] where [ ] indicates taking cohomology classes.

Note that if [ξ] ∈ H^{k}(T^{2}; R), where k = 0, 1, there is a k-form β on S^{3} so that d(β) = ϕ ξ dx. Consider the form s^{∗}(β) on G_{1}. It satisfies d(s^{∗}(β)) = s^{∗}(ϕ ξ dx). In addition, as s^{∗}(β) is independent of (t_{1}, t_{2}), it follows that δ(s^{∗}(β)) = 0. Thus D(s^{∗}(β)) = s^{∗}(ϕ ξ dx). This does not contradict the fact that A is an isomorphism, since the form s^{∗}(β) in not in A_{c}^{k,0}(F) as it will not satisfy the support condition off T^{2} × (0, 1) where the leaves of F are not compact.

3. The S^{1} × R Reeb Foliations.

To obtain these foliations, replace the T^{2} leaf of the Reeb foliation by T^{2} × [0, 1] with a foliation which has the two boundary components as leaves and the interior foliated by leaves diffeomorphic to S^{1} × R. There are two distinct ways to foliate S^{1} × R in this way . The first is obtained by foliating the strip {(x, y) ∈ R^{2} | 0 ≤ y ≤ 1} by horizontal lines, taking the product of this with S^{1}, and then identifying (x, y, θ) with (x + 1, φ(y), θ). The map φ : [0, 1] → [0, 1] is a diffeomorphism so that on (0, 1), φ(y) > y. The second way is to foliate {(x, y) ∈ R^{2} | 0 ≤ y ≤ 1} by copies of R so that the boundary components are leaves, and the leaves in the interior have both their ends at x = +∞. See the picture at the top of p. 41 of [MS]. We assume that the foliation is invariant under translations in the x variable. Product this foliation with S^{1} and identify (x, y, θ) with (x + 1, y, θ) to obtain a foliation of T^{2} × [0, 1]. The coarse cohomology of the foliations we construct below does not depend on which of these two foliations we use.

For x ∉ ∂(T^{2} × [0, 1]), we have

Proposition 3.12 If x ∈ S^{3} − (T^{2} × [0, 1]), then for all q, H^{q}(L)_{x} = 0. If x ∈ T^{2} × (0, 1), then H^{1}(L)_{x} = R, otherwise H^{q}(L)_{x} = 0.

Proof: The first statement follows from the proof in the Reeb case. For the second, note that any point x ∈ T^{2} × (0, 1) has a neighborhood U ≃ D^{3} so that the metric family {s^{−1}(U), d, s, U} is coarsely equivalent to the product metric family F_{x} = {U × R, d^{U}, π, U}, where the metric on each {u} × R is the usual metric. This is because s^{−1}(U) ≃ U × (R × S^{1}) and each {u} × R × S^{1} has a metric coarsely equivalent to the usual metric which makes it coarsely equivalent to R with the usual metric. HX^{1}(R) = R and is 0 otherwise, so the spectral sequence of [HH] gives that HX^{1}(F_{x}) = R and is 0 otherwise. Thus for x ∈ T^{2} × (0, 1), H^{1}(L)_{x} = R and is 0 otherwise. □

We now compute H^{q}(L)_{x} for x ∈ ∂(T^{2} × [0, 1]). There are infinitely many ways we can glue the T^{2}×[0, 1] to the two copies of S^{1}×D^{2} to obtain S^{3}. In fact, by using all possible gluings, we obtain foliations of all three manifolds which can be obtained from S^{3} by Dehn surgery on the unknot. The computations here can be adapted to give the coarse cohomologies for all these foliations. We leave the details to the reader.

Let ϕ be the generator of π_{1}(T^{2}) which has trivial holonomy (for the S^{1} × R foliation of T^{2}×[0, 1]). Let α and β be the canonical generators of π_{1}(T^{2}) so that β has trivial holonomy (for the Reeb foliation of S^{1} × D^{2}). Suppose the gluing map takes ϕ to aα + bβ. If a = 0, there is no one-sided holonomy so the graph of the foliation is Hausdorff at all points in this component of ∂(T^{2} × [0, 1]). On the other hand, if a ≠ 0, the graph of the foliation is non-Hausdorff at all points in this component of ∂(T^{2} × [0, 1]).

The Hausdorff case: a = 0.

Any such x has a neighborhood U ≃ D^{3}with the associated metric family {s^{−1}(U), d, s, U} coarsely equivalent to the metric family {G, d^{U}, π, U}, G = Ĝ/Z, and Ĝ ⊂ D^{3} × R^{2} is the set

Ĝ = {(u_{1}, u_{2}, u_{3}, t_{1}, t_{2}) | if u_{3} > 0, t_{1} > −u_{3}^{−2}}.

The action of Z is generated by

(u_{1}, u_{2}, u_{3}, t_{1}, t_{2}) → (u_{1}, u_{2}, u_{3}, t_{1}, t_{2} + 1).

The map π : G → U is the natural projection. For each u ∈ U with u_{3} > 0, π^{−1}(u) ≃ S^{1}×(−u_{3}^{−2}, ∞) with the usual metric, and for each u ∈ U with u_{3} ≤ 0, π^{−1}(u) ≃ S^{1} × R with the usual metric. This in turn is coarsely equivalent to the metric family {G_{1}, d^{U}, π, U}, where G_{1} ⊂ U × R and

G_{1} = {(u_{1}, u_{2}, u_{3}, t) | if u_{3} > 0, t > −u_{3}^{−2}},

the projection is the natural one, and the fibers have the metric induced from U ×R. In [H], we computed the coarse cohomology of the Double Reeb foliation of S^{1}×S^{2} and in so doing, showed that for all x in the single compact leaf (= T^{2}), that H^{1}(L)_{x} = R and for all other q, H^{q}(L)_{x} = 0. The proof there works here also, so we have

Proposition 3.13 If x is in a component of ∂(T^{2} × [0, 1] where the graph is Hausdorff, H^{1}(L)_{x} = R, otherwise H^{q}(L)_{x} = 0.

Note that if one of the gluing maps satisfies a = 0, the other cannot satisfy this condition and still have the resulting manifold be S^{3}. This is because the composition of the two gluing maps must send β in the first ∂(S^{1} × D^{2}) to ±α in the second ∂(S^{1} × D^{2}) if the resulting manifold is S^{3}. Thus the second map must sent ϕ to ±α. So for S^{1} × R foliations of S^{3}, it is only possible to have one component of ∂(T^{2} × [0, 1]) over which the graph is Hausdorff. We call such foliations half-Hausdorff S^{1} × R foliations of S^{3}. Foliations where the graph is non-Hausdorff on both components of ∂(T^{2} ×[0, 1]) will be called non-Hausdorff S^{1} × R foliations of S^{3}.

The non-Hausdorff case: a ≠ 0.

Here x has 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} = {G, d^{U}, π, U}, where G = Ĝ/Z, and Ĝ ⊂ D^{3} × R^{2} is the set

Ĝ = {(u_{1}, u_{2}, u_{3}, t_{1}, t_{2}) | if u_{3} > 0, t_{1} > −u_{3}^{−2}}.

The action of Z is generated by

if u_{3} > 0, then (u_{1}, u_{2}, u_{3}, t_{1}, t_{2}) → (u_{1}, u_{2}, u_{3}, t_{1}, t_{2} + 1);
if u_{3} < 0, then (u_{1}, u_{2}, u_{3}, t_{1}, t_{2}) → (u_{1}, u_{2}, u_{3}, t_{1} + a, t_{2} + b).

The map π : G → U is the natural projection. For each u ∈ U with u_{3} > 0, π^{−1}(u) ≃ S^{1} × (−u_{3}^{−2}, ∞) with the usual metric; for each u with u_{3} < 0, π^{−1}(u) ≃ S^{1} × R with the usual metric; and for each u ∈ U with u_{3} = 0, π^{−1}(u) ≃ R^{2} with the usual metric. This is essentially the same situation we encountered in computing H^{q}(L)_{x} for x ∈ T^{2} for the Reeb foliation, and the result is the same.

Proposition 3.14 If x is in a component of ∂(T^{2} × [0, 1]) where the graph is non-Hausdorff, H^{q}(L)_{x} = 0 for all q.

Combining Propositions 3.12, 3.13, and 3.14 we have

Proposition 3.15 For any half-Hausdorff S^{1}×R Reeb foliation of S^{3}, H^{1}(L) = R_{T^{2}×[0,1)}, otherwise H^{q}(L) = 0. For any non-Hausdorff S^{1} × R Reeb foliation of S^{3}, H^{1}(L) = R_{T^{2}×(0,1)}, otherwise H^{q}(L) = 0.

Applying the spectral sequence as above, we get that the coarse cohomology of any half-Hausdorff S^{1} × R Reeb foliation of S^{3} is

HX^{∗}(F) = H^{∗−1}(S^{3}; R_{T^{2}×[0,1)}),

and for any non-Hausdorff S^{1} × R Reeb foliation of S^{3} it is

HX^{∗}(F) = H^{∗−1}(S^{3}; R_{T^{2}×(0,1)}).

To compute H^{∗}(S^{3}; R_{T^{2}×[0,1)}), write S^{3} as the disjoint union S^{3} = B_{0} ∪ T^{2} ×

[0, 1) ∪ B_{1} where B_{1} ≃ D^{2} × S^{1}, B_{0} ≃ D_{0}^{2} × S^{1}, and D_{0}^{2} is the interior of the closed two disc D^{2}. Denote the complement of B_{0} by CB_{0} and consider the short exact sequence R_{T^{2}×[0,1)} → R_{CB_{0}} → R_{B_{1}}. As both B_{1} and CB_{0} are closed, H^{∗}(S^{3}; R_{CB_{0}}) → H^{∗}(S^{3}; R_{B_{1}}) in the associated long exact sequence is just the restriction homomorphism H^{∗}(CB_{0}; R) → H^{∗}(B_{1}; R), which is an isomorphism. Thus, H^{∗}(S^{3}; R_{T^{2}×[0,1)}) = 0. Combining this with the computation for H^{∗}(S^{3}; R_{T^{2}×(0,1)}) above we have

Theorem 3.16 If F is a half-Hausdorff S^{1} × R Reeb foliation of S^{3}, then HX^{∗}(F) = 0. If F is a non-Hausdorff S^{1} × R Reeb foliation of S^{3}, then HX^{∗+2}(F) ≃ H^{∗}(T^{2}; R).

Representatives of the classes determined by the elements of H^{∗}(T^{2}; R) are given as follows. The coarse cohomology of R is non-zero only in dimension 1 where it is isomorphic to R. A generator (in the extended coarse cohomology of R) is given by [ω_{1,0} + ω_{0,1}] where ω_{i,j} ∈ A_{c}^{i,j}(R), and

ω_{1,0}(t) = ψ(t)dt, ω_{0,1}(t_{1}, t_{2}) = ∫_{t_{1}}^{t_{2}} ψ(z)dz,

where ψ(t) is any non-negative smooth function on R with compact support and positive total integral. Let ξ represent a class in H^{k}(T^{2}; R), and ϕ and dx be as in the T^{2} Reeb foliation. Then s^{∗}(ϕξdx)ω_{1,0} defines an element of A_{c}^{k+2,0} over T^{2} × (0, 1) which is zero in a neighborhood of the boundary, so extending it by zero defines a class in A_{c}^{k+2,0} over all of S^{3}. Similarly, s^{∗}(ϕξdx)ω_{0,1} defines an element of A_{c}^{k+1,1} over T^{2} × (0, 1) which may be extended by zero to all of S^{3}. The coarse class corresponding to the class of ξ is then the class of s^{∗}(ϕξdx)ω_{1,0} + s^{∗}(ϕξdx)ω_{0,1}.

4. The R^{2} Reeb Foliations.

To obtain these foliations, replace the T^{2} leaf of the Reeb foliation by T^{2} × [0, 1] with a foliation which has the two boundary components as leaves and the interior foliated by leaves all diffeomorphic to R^{2}. There are essentially two distinct ways to do this also. The first is obtained by foliating the set {(x, y, z) ∈ R^{3} | 0 ≤ y ≤ 1} by planes parallel to the y = 0 plane and then identifying (x, y, z) with (x + 1, φ(y), z) and with (x, ψ(y), z + 1). The maps φ, ψ : [0, 1] → [0, 1] are commuting diffeomorphisms so that on (0, 1), φ(y) > y, ψ(y) > y, and if 0 < y < 1, φ^{k}(y) = ψ^{ℓ}(y) ⟺ k = ℓ = 0. The second way is to foliate {(x, y) ∈ R^{2} | 0 ≤ y ≤ 1} as above by copies of R so that the boundary components are leaves, the leaves in the interior have both their ends at x = +∞, and the foliation is invariant by translation in the x coordinate. Product this foliation with R to obtain a foliation of {(x, y, z) ∈ R^{3} | 0 ≤ y ≤ 1}. Now identify (x, y, z) with (x+1, y, z) and with (x+α, y, z +1), where α ∈ (0, 1) and is irrational. As above, both types of foliations of S^{3} have the same coarse cohomology.

For x ∉ ∂(T^{2} × [0, 1]), we have

Proposition 3.17 If x ∈ S^{3} − (T^{2} × [0, 1]), then for all q, H^{q}(L)_{x} = 0. If x ∈ T^{2} × (0, 1), then H^{2}(L)_{x} = R, otherwise H^{q}(L)_{x} = 0.

Proof: As above, the first statement follows from the proof in the Reeb case. For the second, note that any point x ∈ T^{2} × (0, 1) has a neighborhood U ≃ D^{3} so that the metric family {s^{−1}(U), d, s, U} is coarsely equivalent to the product metric family F_{x} = {U ×R^{2}, d^{U}, π, U}, where the metric on each {u}×R^{2} is the usual metric. The spectral sequence of [HH] gives that the coarse cohomology of the metric family F_{x} is the same as the coarse cohomology of R^{2}, namely R in dimension two and 0 otherwise, so for x ∈ T^{2} × (0, 1), H^{2}(L)_{x} = R and is 0 otherwise. □

We now compute H^{q}(L)_{x} for x ∈ T^{2} × {0, 1}. The result here does not depend on which gluing maps we use. Each such x has 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} = {G, d^{U}, π, U}, G = Ĝ/Z, and Ĝ ⊂ D^{3} × R^{2} is the set

Ĝ = {(u_{1}, u_{2}, u_{3}, t_{1}, t_{2}) | if u_{3} > 0, t_{1} > −u_{3}^{−2}}.

The action of Z is generated by

if u_{3} > 0, then (u_{1}, u_{2}, u_{3}, t_{1}, t_{2}) → (u_{1}, u_{2}, u_{3}, t_{1}, t_{2} + 1).

The map π : G → U is the natural projection. For each u ∈ U with u_{3} > 0, π^{−1}(u) ≃ S^{1} × (−u_{3}^{−2}, ∞) with the usual metric, and for each u ∈ U with u_{3} ≤ 0, π^{−1}(u) ≃ R^{2} with the usual metric.

As G_{1} is not compact, HX^{0}(F_{x}) = 0 as usual, which implies H^{0}(L^{∗})_{x} = 0.

To compute the higher coarse cohomology of the metric family F_{x}, first note that G_{ℓ+1} ⊂ U × (R^{2})^{ℓ+1} is the set

G_{ℓ+1} = {(u_{1}, u_{2}, u_{3}, t_{1}, …, t_{2ℓ+2}) | if u_{3} > 0, t_{2m+1} > −u_{3}^{−2}}/Z^{ℓ+1},

where the Z^{ℓ+1} action is given by: if u_{3} > 0, then

(a_{1}, …, a_{ℓ+1})((u_{1}, u_{2}, u_{3}, t_{1}, t_{2}, t_{3}, …, t_{2ℓ}, t_{2ℓ+1}, t_{2ℓ+2})) =
(u_{1}, u_{2}, u_{3}, t_{1}, t_{2} + a_{1}, t_{3}, …, t_{2ℓ} + a_{ℓ}, t_{2ℓ+1}, t_{2ℓ+2} + a_{ℓ+1}).

We will view an element ω_{i,j} ∈ A_{c}^{i,j}(F_{x}) as a differential i form on

Ĝ_{ℓ+1} = {(u_{1}, u_{2}, u_{3}, t_{1}, …, t_{2ℓ+2}) ∈ U × (R^{2})^{ℓ+1} | if u_{3} > 0, t_{2m+1} > −u_{3}^{−2}},

which is invariant under the action of Z^{ℓ+1}, that is which is periodic in the even t variables of period 1 for all u with u_{3} > 0. Note that this condition is preserved by the construction given in the Controlled Poincaré Lemma.

Now 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}). Proceeding just as we did for the Reeb foliation, we may assume that ∑_{i+j=k} ω_{i,j} is completely independent of u. Thus we may view ω_{i,j} as being an element of A_{c}^{i,j}(R^{2}), (i. e. an extended coarse cochain for R^{2}, see [HH]) because this is what it is for u_{3} = 0. It must also be invariant under the action of Z^{j+1}, because it is invariant under this action when u_{3} > 0. An argument similar to that given in the proof of Lemma 3.4 shows that the only form satisfying these conditions is the zero form. Thus α = 0 and we have

Proposition 3.18 For x ∈ T^{2} × {0, 1}, H^{q}(L)_{x} = 0 for all q.

And so also

Proposition 3.19 For any R^{2} Reeb foliation, H^{2}(L) = R_{T^{2}×(0,1)}, otherwise H^{q}(L) = 0.

The spectral sequence applied to this example has E_{2}^{p,2} = H^{p}(S^{3}; R_{T^{2}×(0,1)}) ≃ H^{p−1}(T^{2}; R), and all other E_{2} terms zero. Thus

Theorem 3.20 For any R^{2} Reeb foliation of S^{3}, HX^{∗+3}(F) ≃ H^{∗}(T^{2}; R).

16 J. L. Heitsch

Representatives of the coarse classes are given as follows. The coarse cohomology of R^{2} is non-zero only in dimension 2 where it is isomorphic to R. A generator (in the extended coarse cohomology of R^{2}) is given by [ω_{2,0} + ω_{1,1}] where ω_{i,j} ∈ A_{c}^{i,j}(R^{2}), and

ω_{2,0}(t_{1}, t_{2}) = ψ(t_{1}, t_{2})dt_{1} ∧ dt_{2},
ω_{1,1}(t_{1}, t_{2}, t_{3}, t_{4}) = [∫_{t_{1}}^{∞} ψ(z, t_{2})dz]dt_{2} − [∫_{t_{3}}^{∞} ψ(z, t_{4})dz]dt_{4},

where ψ(t_{1}, t_{2}) is any non-negative smooth function on R^{2} with compact support and positive total integral. Let ξ represent a class in H^{k}(T^{2}; R), and ϕ and dx be as in the T^{2} Reeb foliation. Then, just as in the non-Hausdorff S^{1} × R case, s^{∗}(ϕξdx)ω_{2,0} defines an element of A_{c}^{k+3,0} and s^{∗}(ϕξdx)ω_{1,1} defines an element of A_{c}^{k+2,1}. The coarse class corresponding to the class of ξ is then the class of s^{∗}(ϕξdx)ω_{2,0} + s^{∗}(ϕξdx)ω_{1,1}.

References

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