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Take a smooth manifold $M^n$ with a smooth foliation $F$. Consider the sheaf $\cal F$ of $C^{\infty}$ functions on $M^n$, locally constant along the foliation $F$. What is known about Chech cohomology of such a sheaf?

I am pretty sure that such a question was studied (and maybe even has a complete answer), but I don't know a reference.

A more specific question is: what happen when $F$ is 1-dimensional, given by integral trajectories of a non-vanishing vector field? Or even more specifically, suppose $H^1(M^n)=0$ and we consider a Killing vector field $v$ on $M^n$ (i.e. $v$ is preserving a metric). Is it true the the sheaf of functions $\cal F$ locally constant along trajectories of $v$ is acyclic? (we need $H^1(M^n)=0$, otherwise $S^1$ will be an obvious counterexample).

An example of a foliation. Consider the unit sphere $S^3$ in $\mathbb C^2$ and conisder the action of $\mathbb R$ via diagonal matrixes : $(z,w)\to (e^{ita}z, e^{itb}w)$ with $\frac{a}{b}$ irrational.

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Does this have anything to do with the semicontinuity theorem for cohomology along fibers? Just curious. – Akhil Mathew Feb 27 '10 at 15:31
Is such a sheaf pulled back from the quotient space $M/F$? – Chris Schommer-Pries Feb 27 '10 at 15:36
Chris, notice, that the clouse of a leaf ot F can easily have dimension larger then the dimesnion of the leaf. So the quotient can be non Hasudorf.... Akhil, I don't know how to answer your question... – Dmitri Feb 27 '10 at 15:42
Right. After playing a bit more, I think that the following is plausible: If the leaves are simply connected (so that on each leaf locally constant sheaf = constant sheaf), then the sheaf is pulled back from some sheaf on the (bad) quotient space. This should be true in Dmitri's $S^3$ example above. I think that the Cech cohomlogy on M will then be the same as the sheaf cohomology on the quotient (Since the quotient can be bad, and this can be any sheaf, this could be really realy exotic). – Chris Schommer-Pries Feb 27 '10 at 16:35
I don't think it matters if the leaves are simply connected, as long as the sheaf is equivariant for the equivalence relation (ie is the constant sheaf along the fibers). In general (if F is just locally constant) I agree you'll get a sheaf (not a local system I don't think since the topology of leaves jumps) which is the (group algebra) of relative $\pi_1$ and you'll get a module over this, but at this point I'm not sure working downstairs gains you anything (ie you're basically saying, a sheaf on the fibers of a fibration - ie pushing forward the stack of sheaves rather than descending) – David Ben-Zvi Feb 27 '10 at 17:02
up vote 4 down vote accepted

Nikita Markarian just explained to me (if there is a mistake below, it is mine), that the last and more specific question about acyclicity has 100% negative answer. Namely, we can consider the case $M^3=S^3$ ($H^1(S^3)=0$) and the foliation is given by the fibers of the Hopf fibration $S^3\to S^2$. In this chase the sheaf of functions locally constant on the fibers has a two-term resolution (by soft sheaves). The first term is given by all functions on $S^3$ and the second by $1$-forms on $S^3$, restricted to fibers. The differential is just the differential along the fibers. In this case it is clear, that the first cohomology is huge, it is parameterised by all functions on the base $S^2$.

So this condition $H^1(M^n)=0$ does not help at all.

It is a good exercise to apply the same reasoning to the other foliation on $S^3$, described in the question.

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This may not be exactly what you are looking for, but your question rang a particular bell: namely the paper On the relative de Rham sequence by Buchdahl, which I read when I was a graduate student and I used in my own research. My motivation at the time was to understand so-called classical BRST cohomology, which is a homological approach to symplectic reduction. This procedure is a subquotient, whose last step is a quotient of the "constraint surface" by a foliation defined by the integral submanifolds of the hamiltonian vector fields corresponding to "first-class constraints". (The classical case is when the constraints are the components of an equivariant momentum mapping, but the general case of first-class constraints only yields a foliation which might not fiber.) One is interested therefore in functions which are locally constant on the leaves of the foliation. This can be identified with the zeroth Cech cohomology of the complex of "vertical forms" which is a special case of the relative de Rham complex of Buchdahl's.

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Jose, thanks a lot! I will have a look on this article once at work – Dmitri Feb 27 '10 at 20:18

I am joining this discussion a bit late, but let me add an example. If you consider a smooth minimal action of Z on the circle S^1 the suspension gives a flow on the torus. If the action is C^2 conjugate to an irrational rotation, then the transverse basic cohomology is finite dimensional. But if the action is only topologically conjugate to a rotation, then the basic cohomology may be infinite. The literature on this is quite a long time ago, in the 1970's perhaps. here is one reference

Haefliger, A.and Banghe, Li Currents on a circle invariant by a Fuchsian group. Geometric dynamics (Rio de Janeiro, 1981), 369–378, Lecture Notes in Math., 1007, Springer, Berlin, 1983.

Here is a more recent article

Avila, Artur and Kocsard, Alejandro Cohomological equations and invariant distributions for minimal circle diffeomorphisms. Duke Math. J. 158 (2011), no. 3, 501–536.

and there is one more artcile that is likely relevant to the question

Lott, John Invariant currents on limit sets. Comment. Math. Helv. 75 (2000), no. 2, 319–350.

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I am also very interested in "What is known about Chech cohomology of such a sheaf?", so if you got more information I would be interested.

A fine and torsionless resolution is given by foliated ("tangential" sometimes is used to refer to it) cohomology.

Transversal structures gives a great deal of information as well.

More related to what Chris said about $M/\mathcal{F}$ is Basic cohomology of foliated manifolds. Basic cohomology can be infinite dimensional, so it can or can not satisfy a Poincarè duality. It seems to be well cover in the literature the riemannian foliation case.

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It appears that the relation between basic cohomology (base-like cohomology as it was coined by Reinhart) and $M/\mathcal{F}$ is contained in this paper – R.S. Nov 29 '11 at 1:18

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