Comment. (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.) By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy.

Consider sheaf cohomology defined by derived functor, which means use injective resolutions of sheaves to define cohomology group. Geometric dynamics (Rio de Janeiro, 1981), 369–378, Lecture Notes in Math., 1007, Springer, Berlin, 1983. Or even more specifically, suppose $H^1(M^n)=0$ and we consider a Killing vector field $v$ on $M^n$ (i.e. The first term is given by all functions on $S^3$ and the second by $1$-forms on $S^3$, restricted to fibers. 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.

If you consider a smooth minimal action of Z on the circle S^1 the suspension gives a flow on the torus. Thanks for contributing an answer to MathOverflow! A more specific question is: what happen when $F$ is 1-dimensional, given by integral trajectories of a non-vanishing vector field? Just curious. To subscribe to this RSS feed, copy and paste this URL into your RSS reader.

I am joining this discussion a bit late, but let me add an example. Math. If the action is C^2 conjugate to an irrational rotation, then the transverse basic cohomology is finite dimensional. 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. Take a smooth manifold $M^n$ with a smooth foliation $F$. Let A be an abelian category, that is, roughly, an additive category in which there exist well-behaved kernels and cokernels for each morphism, so that, for example, the notion of an exact sequence in A makes sense.

In mathematics, the constant sheaf on a topological space X associated to a set A is a sheaf of sets on X whose stalks are all equal to A. To learn more, see our tips on writing great answers.

Chris, notice, that the clouse of a leaf ot F can easily have dimension larger then the dimesnion of the leaf. It seems to be well cover in the literature the riemannian foliation case. $\begingroup$ 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). Now, if Z is the constant sheaf of … MathJax reference. Right. J.

I will have a look on this article once at work, 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 arxiv.org/PS_cache/arxiv/pdf/0903/0903.2871v1.pdf, Cohomology of a sheaf of functions locally constant along a foliation, Responding to the Lavender Letter and commitments moving forward, Cohomology of Structure Sheaves: Algebraic, Constructible and more, Equivalence of different cohomology groups. Basic cohomology can be infinite dimensional, so it can or can not satisfy a Poincarè duality.

Equivalently, S X is the sheaf whose sections are locally constant functions f: U!Sand also is equivalent to the shea cation of the constant presheaf which assigns Ato every open set. An example of a foliation. So this condition $H^1(M^n)=0$ does not help at all.

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Making statements based on opinion; back them up with references or personal experience. What is known about Chech cohomology of such a sheaf?

Avila, Artur and Kocsard, Alejandro

One answer is that f is injective if and only if the associated homomorphism on stalks Bx → Cx is injective for every point x in X.

This should be true in Dmitri's $S^3$ example above. Invariant currents on limit sets. Grothendieck cohomology. and there is one more artcile that is likely relevant to the question.

", so if you got more information I would be interested. In this case it is clear, that the first cohomology is huge, it is parameterised by all functions on the base $S^2$. Stack Exchange network consists of 176 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share … Is such a sheaf pulled back from the quotient space $M/F$? 158 (2011), no. 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. Does this have anything to do with the semicontinuity theorem for cohomology along fibers? But if the action is only topologically conjugate to a rotation, then the basic cohomology may be infinite. 75 (2000), no. MathOverflow is a question and answer site for professional mathematicians. The constant presheaf with value A is the presheaf that assigns to each non-empty open subset of X the value A, and all of whose restriction maps are the identity map A → A. Cohomological equations and invariant distributions for minimal circle diffeomorphisms. site design / logo © 2020 Stack Exchange Inc; user contributions licensed under cc by-sa. Is it true the the sheaf of functions $\cal F$ locally constant along trajectories of $v$ is acyclic?

The constant sheaf S X is de ned to be S X(U) = ff: U!Sjfis continuous and Shas the discrete topologyg Remark. It only takes a minute to sign up. sheaf cohomology using the derived functors of the global sections functor. In the case when $ {\mathcal F} $ is the constant sheaf corresponding to the Abelian group $ {\mathcal F} $, the groups $ \check{H} {} ^ {n} ( X , {\mathcal F} ) $ are the same as the Aleksandrov–Čech cohomology groups with coefficients in the group $ {\mathcal F} $.

Homology, Cohomology, and Sheaf Cohomology Jean Gallier and Jocelyn Quaintance Department of Computer and Information Science University of Pennsylvania Philadelphia, PA 19104, USA e-mail: jean@cis.upenn.edu c Jean Gallier Please, do not reproduce without permission of the authors September 24, 2020.

The differential is just the differential along the fibers. 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 literature on this is quite a long time ago, in the 1970's perhaps. The category of sheaves of abelian groups on a topological space X is an abelian category, and so it makes sense to ask when a morphism f: B → C of sheaves is injective or surjective.

My motivation at the time was to understand so-called classical BRST cohomology, which is a homological approach to symplectic reduction. 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 $S^3\to S^2$.

Duke Math. Transversal structures gives a great deal of information as well.

module, etc). rev 2020.10.9.37784, The best answers are voted up and rise to the top, MathOverflow works best with JavaScript enabled, Start here for a quick overview of the site, Detailed answers to any questions you might have, Discuss the workings and policies of this site, Learn more about Stack Overflow the company, Learn more about hiring developers or posting ads with us.

Currents on a circle invariant by a Fuchsian group. The constant sheaf associated to A is the sheafification of the constant presheaf associated to A. It is a good exercise to apply the same reasoning to the other foliation on $S^3$, described in the question. 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 fine and torsionless resolution is given by foliated ("tangential" sometimes is used to refer to it) cohomology.

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, 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). Lott, John 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. 2. One is interested therefore in functions which are locally constant on the leaves of the foliation. Helv. Sheaf-theoretically characterize a Riemannian structure? 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), Jose, thanks a lot! It is denoted by A or AX.

In this chase the sheaf of functions locally constant on the fibers has a two-term resolution (by soft sheaves).

Haefliger, A.and Banghe, Li Remark. 2, 319–350.

$v$ is preserving a metric). I am also very interested in "What is known about Chech cohomology of such a sheaf?

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