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Cohomology of horizontal forms. (English) Zbl 1271.58002

Given a connected \(C^\infty\) manifold \(M\) with a foliation \(F\subset T M\), an \(r\)-form \(\omega_r\in\bigwedge^rT^*M\) is said to be \(s\)-horizontal (for \(1\leq s \leq r\)) if the interior derivatives \(i_{X_0},\dots,i_{X_{r-s}}\omega_r\) vanish for all \(X_0,\dots,X_{r-s}\in F\). The \(C^\infty_M\)-module of \(s\)-horizontal \(r\)-forms is denoted by \(\Omega^{r,s}_F\).
In the paper under review, the authors prove a Poincaré Lemma for the complex of \(s\)-horizontal forms (cf. Theorem 2.1): The complex \[ \mathcal{C}^\bullet_s \;: \;0\rightarrow Z^s_F\rightarrow \Omega^{s,s}_F\rightarrow \Omega^{s+1,s}_F \rightarrow \Omega^{\mathrm{dim} M,s}_F\rightarrow 0 \] is exact for every \(s\), where \(Z^s_F\) denotes the sheaf of closed \(s\)-horizontal \(r\)-forms on \(M\). This is proved by adapting the usual proof of the classical Poincaré Lemma to show that the local primitive of a closed \(s\)-horizontal form can be taken to be \(s\)-horizontal. The authors also prove an explicit formula for the \(s\)-horizontal primitive of an \(s\)-horizontal closed form.
The authors then proceed to compare the cohomology groups \(H^{r,s}_F(M)\) of \(\mathcal{C}^\bullet_s\) to other cohomology groups which are associated to \(F\) in the literature. For example (cf. Theorem 3.1): if \(p:M\rightarrow N\) is a fibered manifold, and \(\Omega^\bullet_{M/N}\) denotes the sheaf of relative differentials, then the homology presheaves \(\mathcal{H}^i\) of \(p_*\Omega^\bullet_{M/N}\) are already sheaves, and moreover if \(p\) is a fiber bundle, there is a natural isomorphism of \(\mathbb{R}\)-vector spaces \(\mathcal{H}^i/\mathfrak{m}_x\mathcal{H}^i\simeq H^i(p^{-1}(x),\mathbb{R})\), where \(\mathfrak{m}_x\) denotes the ideal of \(C^\infty_N\) of functions vanishing at \(x\in N\).
The main goal of the paper is to give conditions under which a de Rham cohomology class may be represented by a horizontal closed form. The results in §5 address this problem by applying some technical results on \(H^{r,s}_F(M)\) proved in the previous sections. In §6, the cohomology groups \(H^{r,s}_F(M)\) are compared to the cohomology groups defined by means of an auxiliary Riemannian structure on \(M\). In §7, the authors give several applications and examples of computation.
The paper is well-written, with many examples to illustrate the theory and references to previous work.

MSC:

58A10 Differential forms in global analysis
18F20 Presheaves and sheaves, stacks, descent conditions (category-theoretic aspects)
53C12 Foliations (differential geometric aspects)
55N30 Sheaf cohomology in algebraic topology
58A30 Vector distributions (subbundles of the tangent bundles)
58J10 Differential complexes
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