For some geometries including symplectic and contact structures on an n-dimensional manifold, we introduce a two-step approach to Gromovs h-principle. From formal geometric data, the first step builds a transversely geometric Haefliger structure of c
odimension n. This step works on all manifolds, even closed. The second step, which works only on open manifolds and for all geometries, regularizes the intermediate Haefliger structure and produces a genuine geometric structure. Both steps admit relative parametri
Many compactly generated pseudo-groups of local transformations on 1-manifolds are realizable as the transverse dynamic of a foliation of codimension 1 on a compact manifold of dimension 3 or 4.
We consider singular foliations of codimension one on 3-manifolds, in the sense defined by A. Haefliger as being Gamma_1-structures. We prove that under the obvious linear embedding condition, they are Gamma_1-homotopic to a regular foliation carried
by an open book or a twisted open book. The latter concept is introduced for this aim. Our result holds true in every regularity C^r, r at least 1. In particular, in dimension 3, this gives a very simple proof of Thurstons 1976 regularization theorem without using Mathers homology equivalence.
We exhibit a pseudogroup of smooth local transformations of the real line which is compactly generated, but not realizable as the holonomy pseudogroup of a foliation of codimension 1 on a compact manifold. The proof relies on a description of all fol
iations with the same dynamic as the Reeb component.
For $Gamma_1$-structures on 3-manifolds, we give a very simple proof of Thurstons regularization theorem, first proved in cite{thurston}, without using Mathers homology equivalence. Moreover, in the co-orientable case, the resulting foliation can be
chosen of a precise kind, namely an open book foliation modified by suspension. There is also a model in the non co-orientable case.
We prove the existence of a minimal (all leaves dense) foliation of codimension one, on every closed manifold of dimension at least 4 whose Euler characteristic is null, in every homotopy class of hyperplanes distributions, in every homotopy class of
Haefliger structures, in every differentiability class, under the obvious embedding assumption. The proof uses only elementary means, and reproves Thurstons existence theorem in all dimensions. A parametric version is also established.
