An \(h\)-principle for symplectic foliations (Q2882368)

In this paper, the authors prove an \(h\)-principle for symplectic foliations, namely, a \(q\)-codimensional almost symplectic foliation on an open manifold is regularly homotopic to a symplectic foliation. Concerning the problem of deforming a \(q\)-codimensional almost symplectic distribution to an integrable one they show that an almost symplectic distribution on an open manifold with a nondegenerate \(2\)-form is homotopic to a symplectic foliation if and only if the distribution is homotopic to a foliation.NEWLINENEWLINEThis result can be formulated in terms of Poisson geometry saying that a regular bivector \(\pi_0\) on an open manifold is regularly homotopic to a Poisson bivector if and only if the distribution \(Im \pi_0^{\sharp}\) is homotopic to an integrable one.NEWLINENEWLINEIn the end, starting with an example of Bott and modifying it they construct a regular bivector on an open manifold which is not regularly homotopic to a Poisson structure.