Coarse Sheaf Cohomology for Foliations: Corrigendum

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

(February 15, 2003)

Abstract

We point out an error in the computation of the coarse cohomology of the Sullivan foliation in our paper Coarse Sheaf Cohomology for Foliations.

In [H1], we claimed to have computed the coarse cohomology of the Sullivan foliation. There are two problems with this claim. First, the techniques of [H1] are for foliations with Hausdorff

graph, and it is a corollary of a result of Epstein, [E], see also [M], that the graph of the Sullivan foliation is not Hausdorff. This problem is addressed in the paper [H2]. The second and more serious problem is that Proposition 6 of [H1] contains an error, which makes the computation incomplete.

Recall that 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]. 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}. The computations H^{0}(L^{∗}) = R_{M−T} and H^{q}(L^{∗})_{x} = 0 for x ∈ M − T and q > 0 given in [H1] are correct. Proposition 6 of [H1] is now

Conjecture For all x ∈ T, and q > 0, H^{q}(L^{∗})_{x} = 0.

∗Partially supported by NSF Grant DMS 9400676

Given this, the isomorphism HX^{∗}(F) ≃ H^{∗}(M, T; R) follows immediately since T is closed in M.

In the proof of Proposition 6 of [H1], we claimed that we could assume that for x ∈ T, any element of H^{∗}(L^{∗})_{x} is represented by a cochain which is completely independent of u. This does not follow from our argument. What the construction in [H1] does is to eliminate any dependence on the T coordinates, but it does not eliminate dependence on those M coordinates transverse to T. This has the effect of reducing the computation of H^{∗}(L^{∗})_{x} to the computation of H^{∗}(L^{∗})_{(0,0)} for the metric family

F_{x} = {(U × R)/Z, d^{U}, π, U}.

Here U ≃ D^{2} with coordinates u_{1}, u_{2}, and if (u_{1}, u_{2}) ≠ (0, 0), then a generator of the Z action takes (u_{1}, u_{2}, r) to (u_{1}, u_{2}, r + (u_{1}^{2} + u_{2}^{2})^{−1}). There is no identification on the fiber (0, 0) × R. π : (U × R)/Z → U is the natural projection and for each u ∈ U, d_{u}^{U} is the metric on {u} × R/Z induced from the usual metric on R.

In support of the conjecture, we have the following result.

Proposition For all x ∈ T, H^{1}(L^{∗})_{x} = 0.

Proof: As stated above, this is equivalent to showing that H^{1}(L^{∗})_{(0,0)} = 0 for the metric family F_{x}. Let α = [ω_{1,0} + ω_{0,1}] ∈ HX^{1}(F_{x}). Then ω_{1,0} is a 1 form on G_{1} = (U × R)/Z which satisfies the support condition for coarse cochains, so it has compact support. By restricting to a subset of U if necessary, we may assume that its support is contained in the subset of G_{1} determined by the subset of U × R given by

{(u_{1}, u_{2}, t) | |t| ≤ 1/2(u_{1}^{2} + u_{2}^{2})}.

Apply the Controlled Poincaré Lemma of [H2] to ω_{1,0} to obtain a new representative of α which has ω_{1,0} completely independent of u. Then δω_{1,0} is also completely independent of u, and as δω_{1,0} = dω_{0,1} (since D(ω_{1,0} + ω_{0,1}) = 0), ω_{0,1} must also be independent of u. Thus ω_{0,1} is a function on G_{2} = (U × R × R)/(Z × Z), which is independent of u, and which satisfies the support condition for coarse cochains. Therefore we may view ω_{0,1} as a function on R^{2} which satisfies the support condition for coarse cochains (since it satisfies it on G_{2} at (0, 0)) and which is periodic in each variable of period (u_{1}^{2} + u_{2}^{2})^{−1} for all (u_{1}, u_{2}) ∈ U, with (u_{1}, u_{2}) ≠ (0, 0) (since it defines a function on G_{2}). The only such function is the zero function so we have ω_{0,1} = 0. Now write ω_{1,0} = ψ(t)dt. Then δω_{1,0}(t_{0}, t_{1}) = ψ(t_{1})dt_{1} − ψ(t_{0})dt_{0}. As δω_{1,0} = dω_{0,1}, we have δω_{1,0} = 0, which immediately gives ω_{1,0} = 0. □

References

[E] D. B. A. Epstein. Foliations with all leaves compact. Ann. Inst. Fourier (Grenoble), 26:265-282, 1976

[H1] James L. Heitsch. Coarse Sheaf Cohomology for Foliations. Illinois J. of Math., 44:860-867,

Coarse Sheaf Cohomology for Foliations: Corrigendum 3

[H2] James L. Heitsch. Coarse Cohomology for Foliations: The General Case. Illinois J. of Math., to appear

[M] Kenneth C. Millett. Compact Foliations. Lecture Notes in Math., 484:277-287, 1975.

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