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Haefliger structures and symplectic/contact structures. (Structures de Haefliger et structures de contact/symplectiques.) (English. French summary) Zbl 1358.57028
A geometry in dimension $$n$$ is defined by an $$n$$-dimensional model manifold $$X$$ and by an open subgroupoid $$\Gamma$$ in the groupoid $$\Gamma(X)$$ of the germs of local $$C^\infty$$-diffeomorphisms of $$X$$ with the sheaf topology. If $$X=\mathbb R^n$$, some examples are the following subgroupoids of the groupoid $$\Gamma_n=\Gamma(\mathbb R^n)$$: (i) the subgroupoid $$\Gamma^{\mathrm{symp}}_n$$ of germs preserving the standard symplectic form of $$\mathbb R^n$$, (ii) the subgroupoid $$\Gamma^{\mathrm{cont}}_n$$ of germs preserving the standard contact structure of $$\mathbb R^n$$, (iii) the subgroupoid $$\Gamma^{\mathrm{fol}}_{n,q}$$ preserving the standard foliation of codimension $$q$$ whose leaves are the $$p$$-planes parallel to $$\mathbb R^n$$, and (iv) when $$Y$$ is any $$q$$-dimensional manifold and $$X=\mathbb R^p\times Y$$, the subgroupoid $$\Gamma^Y_n\subset\Gamma(X)$$ of the germs of the form $$(x,y)\mapsto(f(x,y),y)$$. In [Topology 9, 183–194 (1970; Zbl 0196.26901)], A. Haefliger defined the notion of a $$\Gamma$$-structure on a manifold $$M$$ (with $$\Gamma$$ an open subgroupoid in the groupoid of germs $$\Gamma(X)$$ of an $$n$$-manifold $$X$$), as a pair $$\xi=(\nu,{\mathcal F})$$, where the normal bundle $$\nu$$ is a real vector bundle over $$M$$ of rank $$n$$, its total space is denoted by $$E(\nu)$$, and $$Z:M\to E(\nu)$$ is the zero section, and $${\mathcal F}$$ is a germ along $$Z(M)$$ of a $$\Gamma$$-foliation on $$E(\nu)$$ transverse to every fibre of $$\nu$$. A $$\Gamma$$-structure leads to a $$\Gamma$$-geometry on $$E$$.
In this paper, the authors prove the converse result. To provide a given manifold $$M$$ with such a $$\Gamma$$-structure, Gromov’s $$h$$-principle is applied. It consists of starting from a formal version of the structure on $$M$$, which means a non-integrable section of some jet space, and deforming it until it becomes holonomic. The authors introduce a two-step approach to the $$h$$-principle for such structures.
From the formal data, the first step builds a Haefliger structure of codimension zero on a closed $$M$$, transversely geometric. The second step, which works for open manifolds only, regularizes the intermediate Haefliger structure, providing a genuine geometric structure. Both steps admit relative parametric versions. An essential tool in both steps consists of jiggling. First, the authors show that if $$M$$ is an $$n$$-dimensional manifold, possibly closed, and $$\Gamma$$ is a groupoid in the set of $$n$$-dimensional geometries $$\{\Gamma^{\mathrm{symp}}_n,\Gamma^{\mathrm{cont}}_n,\Gamma^{\mathrm{fol}}_{n,q},\Gamma^Y_n\}$$, then the forgetful map from $$H^1_\tau(M;\Gamma)$$ to the corresponding space of formal $$\Gamma$$-geometries is a homotopy equivalence. Finally, they show that if $$X$$ is an $$n$$-manifold, $$\Gamma\subset\Gamma(X)$$ is an open subgroupoid, $$M$$ is a connected open $$n$$-manifold, and $$s\mapsto\xi_s=(\tau M,{\mathcal F}_s):\mathbb D^k\to H^1_{\tau M}(M;\Gamma)$$ is a continuous family of tangential $$\Gamma$$-structures, parametrized by the compact $$k$$-disk, such that for every $$s\in\partial\mathbb D^k$$, the $$\Gamma$$-structure $$\xi_s$$ is regular and $${\mathcal F}_s$$ is tangent to $${\mathcal F}_{\mathrm{exp}}$$ along $$Z(M)$$, then there exists a continuous family of concordances $$s\mapsto \overline\xi_s=(\tau M\times[0,1],\overline{\mathcal F}_s):\mathbb D^k\to H^1_{\tau M}(M\times[0,1];\Gamma)$$ satisfying certain conditions.

##### MSC:
 57R17 Symplectic and contact topology in high or arbitrary dimension 57R30 Foliations in differential topology; geometric theory
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##### References:
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