Homotopy of codimension one foliations on 3-manifolds
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.