Homotopy of codimension one foliations on 3-manifolds
Abstract: We are interested in the topology of the space of smooth codimension one foliations on a given closed 3-manifold. In 1969, J. Wood proved that any smooth plane field on a closed 3-manifold can be deformed into a plane field tangent to a foliation. This fundamental result was then reproved and generalized by W. Thurston. It is natural then to wonder whether two foliations whose tangent plane fields are homotopic can be connected by a path of foliations, or, in other words, if there is actually a bijection between the (pathwise) connected components of the space of foliations on a given manifold and that of the space of plane fields. In this talk, we will show that the answer is essentially yes. To that end, we will first present Thurston's construction, along with later works by A. Larcanche, who answered the above question in the case of two sufficiently close taut foliations.