Haefliger structures and symplectic/contact structures.
(Structures de Haefliger et structures de contact/symplectiques.)

*(English. French summary)*Zbl 1358.57028A 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.

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.

Reviewer: Andrew Bucki (Edmond)

##### MSC:

57R17 | Symplectic and contact topology in high or arbitrary dimension |

57R30 | Foliations in differential topology; geometric theory |

##### Keywords:

foliations; Haefliger’s \(\Gamma\)-structures; jiggling; inflation; symplectic structure; contact structure; submersion; immersion
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\textit{F. Laudenbach} and \textit{G. Meigniez}, J. Éc. Polytech., Math. 3, 1--29 (2016; Zbl 1358.57028)

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