We establish a form of the h–principle for the existence of foliations of codimension at least which are quasicomplementary to a given one. Roughly, “quasicomplementary” means that they are complementary except on the boundaries of some kind of Reeb components. The construction involves an adaptation of W Thurston’s “inflation” process. The same methods also provide a proof of the classical Mather–Thurston theorem.
"Quasicomplementary foliations and the Mather–Thurston theorem." Geom. Topol. 25 (2) 643 - 710, 2021. https://doi.org/10.2140/gt.2021.25.643