The Poincaré lemma for $d\omega =F(x,\omega)$

Higher Poincaré lemma and integrability

“…We speculate about this in appendix A, where we show how differential ideals are related to certain L ∞ -structures on multivector fields, but the picture remains incomplete. Fortunately, the reformulation of the Frobenius theorem as an equation in differential forms has a generalization due to Jacobowitz [8]. This generalization is sufficient to establish a first proof of the higher Poincaré lemma for principal 2-and 3bundles.…”

“…The usual Poincaré lemma states that the equation dα = β involving some p-and p + 1forms α and β can be solved in an open, contractible region if and only if dβ = 0. In [8], Jacobowitz presented a generalization of this statement which we briefly review below. The precise definition of having local solutions is as follows.…”