Partial hyperbolicity and foliations in ${\mathbb{T}}^3$

We prove that dynamical coherence is an open and closed property in the space of partially hyperbolic diffeomorphisms of ${\mathbb{T}}^3$ isotopic to Anosov. Moreover, we prove that strong partially hyperbolic diffeomorphisms of ${\mathbb{T}}^3$ are either dynamically coherent or have an invariant two-dimensional torus which is either contracting or repelling. We develop for this end some general results on codimension one foliations which may be of independent interest.