Codimension-one singular foliations and dynamics in dimension 3 (joint work with G. Meigniez)

The allowed singularities are those of functions. According to A. Haefliger (1958), such structures on manifolds, called $\Gamma_1$-structures, are objects of a cohomological theory with a classifying space $B\Gamma_1$. The problem of cancelling the singularities (or regularization problem) arose naturally. For a closed manifold, it was solved by W. Thurston in a famous paper (1976), with a proof relying on J. Mather's isomorphism (1971): $Diff^\infty(\mathbb R)$ as a discrete group has the same homology as the based loop space $\Omega B\Gamma_1^+$. For further extension to contact geometry, it is necessary to solve the regularization problem without referring to Mather's isomorphism. That is what we have done in dimension 3. Our result is the following.
If $\xi$ is a $\Gamma_1$-structure on a 3-manifold $M$ and if the normal bundle to $\xi$ embeds into the tangent bundle to $M$, then $\xi$ is homotopic to a regular foliation carried by a (possibily twisted) open book.
The proof is elementary and relies on the dynamics of a (twisted) pseudo-gradient of $\xi$.
All the objects will be defined in the talk, in particukar the notion of twisted open book which is a central object in the reported paper.