Regularization and minimization of codimension-one Haefliger structures

Abstract

On compact manifolds of dimensions $4$ and more, we give a proof of Thurston’s existence theorem for foliations of codimension one; that is, they satisfy some $h$-principle in the sense of Gromov. Our proof is an explicit construction not using the Mather homology equivalence. Moreover, the produced foliations are minimal, that is, all leaves are dense. In particular, there exist minimal, $C^{\infty}$, codimension-one foliations on every closed connected manifold of dimension at least $4$ whose Euler characteristic is zero.