Helene Eynard-Bontemps

We are interested in the connected components of the space of smooth codimension one foliations on a closed 3-manifold. In 1969, J. Wood showed that any smooth plane field on a closed 3-manifold can be deformed into the tangent plane field to a foliation. This fundamental result was then reproved and generalized by W. Thurston. It is quite natural then to wonder whether two foliations whose tangent plane fields are homotopic can be connected by a continuous path of foliations, or in other words if there is actually a bijection between the (pathwise) connected components of the space of foliations of a given manifold and those of the space of plane fields. In this talk, we will show that the answer is "yes", provided one is not too picky about the regularity of the intermediate foliations. To that purpose, we will first present Thurston's construction, along with later works by A. Larcanche, who answered the above question in the particular case of sufficiently close taut foliations.