\documentclass[JEP,XML,SOM,Unicode]{cedram}
\datereceived{2015-07-15}
\dateaccepted{2016-01-13}
\dateepreuves{2016-01-21}

\usepackage{mathrsfs}
\let\mathcal\mathscr

\newenvironment{enumeratei}
{\bgroup\def\theenumi{\roman{enumi}}\def\theenumii{\arabic{enumii}}\begin{enumerate}}
{\end{enumerate}\egroup}

\newcommand{\cont}{\mathrm{cont}}
\newcommand{\fol}{\mathrm{fol}}
\newcommand{\symp}{\mathrm{symp}}

\let\<\langle
\let\>\rangle

\DeclareMathOperator{\Id}{Id}
\DeclareMathOperator{\BGamma}{B\Gamma}
\DeclareMathOperator{\BSp}{BSp}
\DeclareMathOperator{\hol}{hol}
\DeclareMathOperator{\lk}{lk}
\DeclareMathOperator{\Star}{star}

\let\ra\rightarrow
\def\to{\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}}
\def\mto{\mathchoice{\longmapsto}{\mapsto}{\mapsto}{\mapsto}}
\def\hto{\mathrel{\lhook\joinrel\to}}
\def\To#1{\mathchoice{\xrightarrow{\textstyle\kern4pt#1\kern3pt}}{\stackrel{#1}{\longrightarrow}}{}{}}

\numberwithin{equation}{section}

\theoremstyle{plain}
\newtheorem{thm}[subsection]{Theorem}
\newtheorem{prop}[subsection]{Proposition}
\newtheorem{lemme}[subsection]{Lemma}
\newtheorem{cor}[subsection]{Corollary}
\newtheorem*{claim}{Claim}

\theoremstyle{definition}
\newtheorem{remarque}[subsection]{Remark}
\newtheorem{remarques}[subsection]{Remarks}

\newcommand{\be}{\begin{enumerate}}
\newcommand{\ee}{\end{enumerate}}
\newcommand{\bi}{\begin{itemize}}
\newcommand{\ei}{\end{itemize}}

\def\RR{\mathbb{R}}
\def\DD{\mathbb{D}}
\def\cF{\mathcal{F}}
\def\cG{\mathcal{G}}
\def\cX{\mathcal{X}}
\def\PP{\mathcal{P}}
\def\bbS{\mathbb{S}}

\let\ep\varepsilon

\begin{document}
\frontmatter
\title{Haefliger structures and symplectic/contact~structures}

\author[\initial{F.} \lastname{Laudenbach}]{\firstname{Fran{\c c}ois} \lastname{Laudenbach}}
\address{Universit{\'e} de Nantes, LMJL, UMR 6629 du CNRS\\
2, rue de la Houssini{\`e}re, 44322 Nantes Cedex 3, France}
\email{francois.laudenbach@univ-nantes.fr}
\urladdr{http://www.math.sciences.univ-nantes.fr/~laudenba/}
\thanks{FL is supported by ERC Geodycon}

\author[\initial{G.} \lastname{Meigniez}]{\firstname{Ga\"el} \lastname{Meigniez}}
\address{Universit{\'e} de Bretagne Sud, LMBA, UMR 6205 du CNRS\\
BP 573, F-56017 Vannes, France}
\email{Gael.Meigniez@univ-ubs.fr}
\urladdr{http://web.univ-ubs.fr/lmam/meigniez/}

\keywords{Foliations, Haefliger's $\Gamma$-structures, jiggling, inflation, symplectic structure, contact structure, submersion, immersion}

\subjclass{57R17, 57R30}

\begin{abstract}
For some geometries including symplectic and contact structures on an $n$\nobreakdash-dimensional manifold, we introduce a two-step approach to Gromov's $h$-principle. From formal geometric data, the first step builds a transversely geometric Haefliger structure of codimension~$n$. 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 parametric versions. The proofs borrow ideas from W.\,Thurston, like jiggling and inflation. Actually, we are using a more primitive jiggling due to R.\,Thom.
\end{abstract}

\alttitle{Structures de Haefliger et structures de contact/symplectiques}

\begin{altabstract}
Sur une vari{\'e}t{\'e} de dimension $n$, nous introduisons une approche en deux temps du $h$-principe de Gromov pour certaines structures g{\'e}om{\'e}triques incluant les structures symplectiques et les structures de contact. A partir de donn{\'e}es formelles d'une telle g{\'e}om{\'e}trie, la premi{\`e}re {\'e}tape construit une structure de Haefliger de codimension $n$ munie transversalement de cette g{\'e}om{\'e}trie. Cette construction vaut pour toutes les vari{\'e}t{\'e}s, m\^eme celles qui sont compactes {\`a} bord vide. La seconde {\'e}tape, qui ne vaut que pour les vari{\'e}t{\'e}s ouvertes mais pour n'importe quelle g{\'e}om{\'e}trie, consiste {\`a} r{\'e}gulariser une structure de Haefliger transversalement g{\'e}om{\'e}trique et ainsi produire une vraie structure g{\'e}om{\'e}trique sur la vari{\'e}t{\'e} consid{\'e}r{\'e}e. Les deux {\'e}tapes admettent des versions param{\'e}triques relatives. Les preuves empruntent des id{\'e}es de W.\,Thurston dans ses travaux sur les feuilletages. L'une d'elles, sous une forme {\'e}l{\'e}mentaire, remonte {\`a} R.\,Thom sous le nom de \og dents-de-scie\fg.
\end{altabstract}

\altkeywords{Feuilletages, $\Gamma$-structures de Haefliger, structure symplectique, structure de contact, submersion, immersion}

\maketitle
\tableofcontents
\mainmatter

\section{Introduction}\label{intro}

We consider geometric structures on manifolds, such as the following:
symplectic structure, contact structure, foliation of prescribed codimension,
immersion or submersion to another manifold.
We recall that, in order to provide
a given manifold $M$
with such a structure,
Gromov's $h$-principle
consists of starting from a \emph{formal}
version of the structure on $M$ (this means a {\it non-holonomic} --
that is, non-integrable -- section of some jet space) and deforming it until it becomes \emph{genuine}
(holonomic) \cite{gromov}. In the present paper, we 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 $M$, transversely geometric; this concept
will be explained below.
For each of the geometries above-mentioned,
the first step works for every manifold $M$, even closed.

The second step, which is 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 {\it jiggling.}
We recall that Thurston's work on foliations used his famous Jiggling Lemma \cite{thu74}.
As A.\,Haefliger told us
\cite{haefliger2},
Thurston himself was aware that this lemma applies for getting some $h$-principles
in the sense
of Gromov.
In a not very popular paper by R.\,Thom \cite{thom59}, we discovered
a more primitive jiggling lemma
that is
remarkably suitable for the needs of our approach.

\subsection{Groupoids and geometries}\label{list-geo}
According to
O.\,Veblen and J.H.C.\,Whitehead
\cite{veblen_31}, a geometry in dimension $n$ is defined by an
$n$-dimensional model manifold $X$ (often $\RR^n$) and by an open subgroupoid $\Gamma$
in the groupoid
$\Gamma(X)$ of the germs of local $C^\infty$-diffeomorphisms of $X$;
here the topology on $\Gamma(X)$ is meant to be the
{\it sheaf} topology.
In what follows, we use the classical notation $\Gamma_n:= \Gamma(\RR^n)$. Here are examples of such open
subgroupoids.

\begin{enumerate}
\item When $n$ is even, $\Gamma^\symp_n\subset \Gamma_n$ denotes the subgroupoid of germs
preserving the standard symplectic form of $\RR^n$.
\item When $n$ is odd, $\Gamma^\cont_n\subset \Gamma_n$ denotes the subgroupoid of germs
preserving the standard (positive) contact structure of $\RR^n$.
\item For $n=p+q$, one has the subgroupoid
$\Gamma^\fol_{n,q}\subset\Gamma_n$
preserving the standard foliation
of codimension $q$ (whose leaves are the $p$-planes parallel to $\RR^p$).
\item When $Y$ is any $q$-dimensional manifold and $X=\RR^p\times Y$,
one has the subgroupoid
$\Gamma_n^Y\subset\Gamma(X)$ of the germs of the form
$(x,y)\mto
(f(x,y),y)$.
\end{enumerate}

\subsection{$\Gamma$-foliations}
\label{gamma_fol}
The next concept goes back to A.\,Haefliger \cite{haefliger}. For an open subgroupoid $\Gamma$ of
$\Gamma(X)$, a \emph{$\Gamma$-foliation} on a manifold $E$ is meant to be a codimension-$n$ foliation
on $E$ equipped with a {\it transverse geometry} associated with $\Gamma$ and invariant by holonomy. More precisely,
this foliation is
defined by a maximal atlas of submersions $(f_i:U_i\to X)$ from open subsets
of $E$ into $X$ such that, for every $i, j$ and every $x\in U_i\cap U_j$, there is
a germ $\gamma_{ij}\in\Gamma$ at point $f_j(x)$ verifying
$$[f_i]_x=\gamma_{ij}\, [f_j]_x,
$$
where [-]$_x$ stands for the germ at $x$. When $n=\dim E$, one also speaks of
a \emph{$\Gamma$-geometry} on $E$.

Here are examples related to the previous list of groupoids.
The first two have been already considered by D.\,McDuff in \cite{mcduff}.
\begin{enumerate}
\item A $\Gamma^\symp_n$-foliation on $E$
amounts to a closed differential $2$-form $\Omega$ on $E$, whose kernel
is of codimension $n$ at every point. The closedness of $\Omega$ is equivalent to the conjunction of the next facts:
\begin{itemize}
\item the codimension-$n$ plane field $(x\in E\mto \ker \Omega_x)$ is integrable,
\item $\Omega$ is basic\footnote{We recall that a form $\alpha$ is said to be basic with respect to a foliation $ \mathcal F$ if the Lie derivative $L_X\alpha$ vanishes for every vector field $X$ tangent to $\mathcal F$; this is the infinitesimal version of the {\it invariance by holonomy}.} with respect to that foliation,
\item $\Omega$ is closed on a total transversal.
\end{itemize}

\item A $\Gamma_n^\cont$-foliation
on $E$ amounts to a codimension-one plane field $P$ on $E$ defined by
an equation $A=0$, where $A$ is a differential form of degree $1$, unique up to multiplying by
a positive\footnote{Here, we limit ourselves to co-orientable contact structures.} function, which satisfies
the next conditions:
\begin{itemize}
\item the $n$-form $A\wedge(dA)^{(n-1)/2}$ is closed and has a codimension-$n$ kernel $K_x$ at every point $x\in E$; in particular, the field $(x\mto K_x)$ is integrable, tangent to a codimension-$n$ foliation denoted by $\mathcal K$;
\item $K_x$ is a vector sub-space of $P_x$ for every $x$;
\item $P$ is invariant by the holonomy of $\mathcal K$.
\end{itemize}
\item A $\Gamma^\fol_{n,q}$-foliation on $E$
consists of a flag $\cF\subset\cG$ of
two nested foliations of respective codimensions $n$ and $q$ in $E$ with $n>q$.
\item A $\Gamma_n^Y$-foliation on $E$ consists of a codimension-$n$ foliation
and a submersion $w:E\to Y$ which is
constant on every leaf (\cite[p.\,145]{haefliger}).
\end{enumerate}

\subsection{Haefliger's $\Gamma$-structures} \label{gamma_struct}
A.\,Haefliger defined a \emph{$\Gamma$-structure} as some class of cocycles valued
in $\Gamma$ (see \cite[p.\,137]{haefliger}).
This definition, that
makes sense on every topological space and for every topological groupoid $\Gamma$,
allowed him
to build a classifying space $\BGamma$ for these structures (\cite[p.\,140]{haefliger}).
A second description (\cite[p.\,188]{haefliger70}) is
more suitable for our purpose
when the topological space is a manifold $M$ and when the groupoid is
an open subgroupoid $\Gamma$ in the
groupoid of germs $\Gamma(X)$ of a $n$-manifold~$X$. Here it is.

{\it A $\Gamma$-structure
on $M$ consists of a pair $\xi= (\nu,\cF)$, where
\begin{itemize}
\item $\nu$ is a real vector bundle over $M$ of rank $n$, called the \emph{normal bundle};
its total space
is denoted by $E(\nu)$; and $Z:M\to E(\nu)$ denotes the zero section;
\item $\cF$ is a germ along $Z(M)$ of $\Gamma$-foliation
in $E(\nu)$ transverse to every fibre of $\nu$.
\end{itemize}
}

An important feature of $\Gamma$-structures is that pulling back by smooth maps
(in our restricted setting) is
allowed without assuming any transversality: if $f: N\to M$ is a smooth map and $\xi$
is a $\Gamma$-structure
on $M$,
one defines $f^*\xi$ as
the $\Gamma$-structure on~$N$ whose normal bundle is $f^*\nu$ equipped with the
$\Gamma$-foliation $F^{-1}(\cF)$, where $F$ is a
bundle morphism over $f$ which is a
fibre-to-fibre linear isomorphism.

Let $H^1(M;\Gamma)$ (\resp $H^1_\nu(M;\Gamma)$)
denote the space of the $\Gamma$-structures on $M$
(\resp those whose normal bundle is $\nu$).
It is a topological space since $\Gamma$ is a topological groupoid; their elements
are denoted by $\xi=(\nu,\cF)$.
In what follows, we are mainly
interested in the case where $\dim M=n$ and $\nu$ is isomorphic to the
tangent space $\tau M$; in that case, the elements are just denoted $\cF$.

In what follows, a $\Gamma$-structure on $M$ whose normal bundle
is the tangent bundle~$\tau M$
will be called a \emph{tangential} $\Gamma$-structure. For short, when it is not ambiguous,
we write~$\mathcal F$
for $(\tau M, \mathcal F)$. We introduced the general definition
of $\Gamma$-structure --
at least in the smooth case -- since we are going to refer to in several places.

\subsection{Underlying formal geometries}\label{formal}
In the cases of the geometries (1), (2) and~(4) given above, that is,
for $\Gamma=\Gamma_n^\symp$, $\Gamma_n^\cont$
or $\Gamma_n^Y$,
every $\Gamma$-structure has an
underlying
formal $\Gamma$-geometry in the sense of Gromov.
But, we do not intend to enter Gromov's generality. We just describe what they are.
In the case of the geometry (3), every $\Gamma^\fol_{n,q}$-structure has an
underlying object, somewhat formal, but more complicated than in the cases
(1), (2), (4).

\begin{enumerate}
\item Assume $n$ is even.
Given $\cF\in H^1_{\tau M}(M;\Gamma_n^\symp)$, one has
an
associated basic closed $2$-form $\Omega$ on a neighborhood of $Z(M)$
in the total space $TM$. Its kernel is everywhere transverse to the fibres. Therefore,
$\Omega$ defines
a non-singular $2$-form
$\omega$ on~$M$ by the formula $\omega_x:=\Omega_{Z(x)}\vert T_xM$ for every $x\in M$.
This is the underlying {\it formal symplectic structure}.
\item Assume $n$ is odd.
Given $\cF\in H^1_{\tau M}(M;\Gamma_n^\cont)$, one has a $(n-1)$-plane field $Q$
defined near $Z(M)$ with the following
properties:
\begin{itemize}
\item at each point $z$ near $Z(M)$, the plane $Q_z$ is vertical, meaning that
it is contained in the fibre of $TM$ passing through $z$;
\item $Q_z$ carries a symplectic bilinear form, well defined up to a positive factor;
\item as a (conformally) symplectic bundle over a neighborhood of $Z(M)$, the plane field $Q$
is invariant by the holonomy of $\cF$.
\end{itemize}
Then, there is a symplectic sub-bundle whose fibre at $x\in M$ is $S_x:= Q_{Z(x)}$ (the indeterminacy
by a positive factor is irrelevant here). This is the underlying {\it formal contact structure}.
Another way to say the same thing consists of the following: $S$ is the kernel of a $1$-form $\alpha$ and
there is a $2$-form $\beta$ on $M$ such that $\beta$ makes $S$ be a symplectic bundle; equivalently, it may be said that $\alpha\wedge\beta^{\sfrac{(n-1)}{2}}$ is a volume form.
\item Assume $n=p+q$.
Given $\cF\in H^1_{\tau M}(M;\Gamma_{n,q}^\fol)$, one has
an
associated
foliation~$\cG$ of codimension $q$ on a neighborhood of $Z(M)$
in the total space $TM$.
The foliation~$\cG$ induces on $M$ a $\Gamma_q$-structure $\gamma:=Z^*(\cG)$,
whose normal bundle is $\nu_\gamma:=Z^*(\nu_\cG)$;
and a monomorphism of vector bundles $\epsilon:\nu_\gamma\hto \tau M$.
Indeed, for every point $x\in M$,
the foliation $\cG$ being transverse to $T_xM$ at $Z(x)$,
the normal to $\cG$ at $Z(x)$ embeds into~$T_xM$. The pair $(\gamma,\epsilon)$,
an \emph{augmented $\Gamma_q$-structure} (according to the vocabulary from \cite{wrinkling3}), plays the role of a formal
geometry associated to $\cF$.
\item Assume $n\ge\dim Y$.
Given $\cF\in H^1_{\tau M}(M,\Gamma_{n}^{Y})$, one has
an
associated submersion $w$ from a neighborhood of $Z(M)$
in the total space $TM$ to $Y$. This submersion~$w$ induces a \emph{formal submersion} $(f,F)$ from $M$ to $Y$,
that is, a bundle epimorphism from~$TM$ to $TY$ whose value at every $x\in M$ is:
$f(x)=w(Z(x))$, $F_x:=Dw_{Z(x)}\vert T_xM$.
\end{enumerate}
Observe that all these spaces of formal geometries have natural topologies.
Our first theorem yields a converse: for the above geometries,
formal $\Gamma$-geometries lead
to $\Gamma$\nobreakdash-structures.

\begin{thm}\label{step_1_thm}
Let $M$ be an $n$-dimensional manifold, possibly closed.
Let $\Gamma$ be a groupoid in the set
of $n$-dimensional
geometries $\{\Gamma^\symp_n,\Gamma_n^\cont,\Gamma_{n,q}^\fol,
\Gamma_{n}^{Y} \}$. Then, the forgetful map from
$H^1_{\tau M}(M; \Gamma)$ to the corresponding space of formal $\Gamma$-geometries is a homotopy equivalence.
\end{thm}

\skpt
\begin{remarques}
\begin{enumerate}
\item
Actually, according to R.\,Palais \cite[Th.\,15]{palais}, the considered spaces have
the property that a weak homotopy equivalence is a genuine homotopy equivalence. Thus,
it is sufficient to prove that the mentioned forgetful map is a weak homotopy equivalence, meaning that
it induces an isomorphism of homotopy groups in each degree.

\item
In the case of symplectic/contact geometry, D.\,McDuff proved theorems of the same flavor
using the {\it convex integration} technique of Gromov (\cite{mcduff},
see also \cite[p.\,104, 138]{elias}).

\item
Let $\cF\in H^1_{\tau M}(M; \Gamma)$. By taking a section $s$ of $\tau M$ valued
in the domain
foliated by $\cF$ and generic with respect to $\cF$, there is an induced $\Gamma$-geometry
with singularities on $M\cong s(M)$.
This seems to be a very natural notion of singular symplectic/contact structure.
It follows from
Theorem \ref{step_2_thm} that the singular locus may be localized in a ball of $M$.
\end{enumerate}
\end{remarques}

\subsection{Homotopy and regularization}
Our second theorem will allow us to regularize every parametric family of $\Gamma$-structures
on every manifold $M$ which is \emph{open},
that is, which has no closed connected component; this terminology
will be permanently used in what follows.

A \emph{homotopy} (also called a {\it concordance}\footnote{This second word emphasizes the difference with a one-parameter family of $\Gamma$-structures.}) between two $\Gamma$-structures $(\nu_i,\cF_i)$ (\hbox{$i=0,1$})
on $M$ is a $\Gamma$-structure on $M\times[0,1]$
whose restriction to $M\times 0$ (\resp $M\times 1$) equals $(\nu_0,\cF_0)$
(\resp $(\nu_1,\cF_1)$). Of course, $\nu_0$ and $\nu_1$ must be isomorphic.

A $\Gamma$-structure $(\nu,\cF)$ is said to be
\emph{regular} if the foliation $\cF$ is transverse not only to the fibres of $\nu$ but also
to $Z(M)$ in $E(\nu)$. This bi-transversality of $\cF$
induces an isomorphism $\nu\cong\tau\bigl(Z(M)\bigr)$.
In that case, the pull-back $Z^*(\cF)$ is
a $\Gamma$-geometry on $M$, namely the foliation by points equipped with a
transverse $\Gamma$-geometry.

\subsection{The exponential $\Gamma_n$-structure}
Given a complete Riemannian metric on
the $n$-manifold $M$,
there is a well defined map
$$\exp: TM\to M.$$
When restricting $\exp$ to a small neighborhood $U$ of $Z(M)$ in $TM$, we get a submersion to $M$. The foliation defined by the level sets of $\exp\vert U$ represents a regular $\Gamma_n$-structure on $M$,
denoted by $\cF_{\exp}\in H^1_{\tau M}(M; \Gamma_n)$. Up to isomorphism (vertical isotopy in~$TM$),
$\cF_{\exp}$ does not depend on the Riemannian metric as it is shown by the next construction.

Consider the product $M\times M$ and its diagonal $\Delta\cong M$. We have two projections
$p_v, p_h: M\times M\to \Delta$, respectively the vertical and the horizontal projection. A small tube $U$
about $\Delta$ equipped with $p_v$ is isomorphic to $\tau M$ as micro-bundle. Then,
the same tube equipped with $p_h$ defines the $\Gamma_n$-structure $\cF_{\exp}$.

We recall the fundamental property of the differential of $\exp$ (independent of any Riemannian metric):
$$d (\exp\vert T_xM)_{Z(x)}=\Id : T_xM\to T_xM.
$$
As a consequence, if $f: M\to Y$ is a smooth map and $v\in T_xM$, one has
\begin{equation}\label{diff}
f\circ \exp_x(v)-f(x)= df_x(v) +o(\Vert v\Vert).
\end{equation}

\begin{thm}\label{step_2_thm}
Let $X$ be an $n$-manifold, let $\Gamma\subset\Gamma(X)$ be an open subgroupoid and
let $M$ be a (connected) $n$-manifold.
Assume that $M$ is open (that is, no connected component is closed). Let
$$s\mto \xi_s=(\tau M, \cF_s):\DD^k\to H^1_{\tau M}(M;\Gamma)$$
be a continuous family of tangential $\Gamma$-structures, parametrized by the compact $k$-disk ($k\geq 0$),
such that for every $s\in\partial\DD^k$, the $\Gamma$-structure
$\xi_s$ is regular and $\cF_s$ is tangent to $\cF_{\exp}$ along $Z(M)$.

Then, there exists a continuous family of concordances
$$s\mto\bar\xi_s=(\tau M\times[0,1], \bar\cF_s):\DD^k\ra H^1_{\tau M}(M\times[0,1];\Gamma)$$
such that
\begin{itemize}
\item
$\bar\cF_s=pr_1^*(\cF_s)$ for every $s\in\partial\DD^k$, where $pr_1: M\times[0,1]\to M$ is the projection;
\item $\bar\cF_s\vert(M\times 0)=\cF_s$ for every $s\in\DD^k$;
\item for every $s\in\DD^k$, the $\Gamma$-structure $\bar\xi_s\vert(M\times 1)$
is regular and $\bar\cF_s$ is tangent to $\cF_{\exp}$ along $Z(M\times 1)$.
\end{itemize}
\end{thm}

\begin{remarque}
The Smale-Hirsch classification of immersions $S\to Y$ (see \cite{smale,hirsch}), where $S$ is a closed manifold
of dimension less than $\dim Y$, is covered by Theorem \ref{step_2_thm}; in particular, the famous {\it sphere
eversion} amounts to the case where $S$ is the $2$-sphere, $Y= \RR^3 $ and $k=1$. Let us show it.

Let $(f,F):TS\to TY$ be a formal immersion. Then thanks to $F$ we have a monomorphism
$ F_*: \tau S\to f^*\tau Y$ over $\Id_S$. Let $\nu$ be a
complementary sub-bundle to the image of $ F_*$; when $f$ is an immersion, $\nu$ is its normal bundle.
Let $\hat S$ be a disk bundle in $\nu$; it is a compact manifold with non-empty boundary and
$\dim \hat S=\dim Y$.
Thus, instead of immersing $S$ to $Y$ one tries to immerse $\hat S$ to $Y$; if it is done, the restriction to the
$0$-section yields an immersion of $S$ to $Y$ with normal bundle $\nu$. The
formal immersion $(f,F):TS\to TY$ easily
extends to a formal immersion $(\hat f, \hat F):\hat S\to Y$ in codimension $0$.
Since $\hat F: T_x\hat S\to T_{\hat f(x)} Y$ is a linear isomorphism for every $x\in \hat S$, the level sets of
$\exp_Y\circ F$ is a $\Gamma_n^Y$-foliation $\cF$ near $Z(\hat S)$, that is, a $\Gamma_n^Y$-structure on $\hat S$.
Moreover, thanks to Equation (\ref{diff}), if $\hat f$ is an immersion $\cF$ is tangent to $\cF_{\exp}$;
here $\exp$ stands for $\exp_{\hat S}$.
Then, Theorem \ref{step_2_thm} applies and yields the desired immersion (or family of immersions).
\end{remarque}

\begin{cor} Let $\Gamma$ be a groupoid as in Theorem \ref{step_2_thm} and $\xi=(\tau M,\mathcal F)$ be a
tangential $\Gamma$-structure on a closed manifold $M$. Then, after a suitable concordance, all singularities
(that is, the points where $\mathcal F$ is not transverse to $Z(M)\cong M$) are confined in a ball.
\end{cor}

\begin{proof}
Let $B\subset M$ be a closed $n$-ball. Apply Theorem \ref{step_2_thm} to
$\xi\vert (M\smallsetminus \mathrm{int}\,B)$. We are given a regularization concordance $C$ of this
restricted $\Gamma$-structure. Since this concordance is given on a manifold with boundary, it extends
to the whole manifold. Indeed, $B\times[0,1]$ collapses to
$\bigl(B\times\{0\}\bigr)\cup\bigl(\partial B\times [0,1]\bigr)$.
\end{proof}

\begin{remarque}
Y.\,Eliashberg \& E.\,Murphy (\cite[Cor.\,1.6]{murphy}) gave a similar result for symplectic structures on closed almost symplectic manifolds of dimension greater than~$4$. Moreover, in the confining ball $B$ their singular symplectic structure is the {\it negative} cone of an {\it overtwisted} contact structure on $\partial B$. Their proof is based on the new techniques in contact geometry initiated by E.\,Murphy \cite{murphy0} and developed in \cite{borman}.
\end{remarque}

\subsection{The classical $h$-principle for $\Gamma$-geometries}
For a groupoid $\Gamma$ as listed in
\ref{list-geo} the $h$-principle states the following:

\emph{If $M$ is an open $n$-manifold,
the space of $\Gamma$-geometries on $M$ has the
same (weak) homotopy type as the space of formal $\Gamma$-geometries on $M$.}\smallskip

This statement follows from Theorems \ref{step_1_thm} and \ref{step_2_thm}.

\begin{proof}
We start with a $k$-parameter family of formal $\Gamma$-geometries on
$M$, $k\geq 0$, which are genuine $\Gamma$-geometries when the parameter $s$ lies in $\partial \DD^k$.
Then, for every $s\in \DD^k$, the foliation $\cF_{\exp}$ is a $\Gamma$-foliation near $Z(M)$. Thus, Theorem
\ref{step_1_thm} applies and yields a $k$-parameter family of $\Gamma$-structures on $M$ which remains
unchanged when $s\in \partial\DD^k$. Now, since $M$ is open, Theorem \ref{step_2_thm} applies and all the relative homotopy groups of the pair $(\text{formal }\Gamma\text{-geometries}, \Gamma\text{-geometries})$ vanish.
\end{proof}

The article is organized as follows. In Section \ref{section_thom}, we detail the tool that goes back to
R.\,Thom \cite{thom59}
and we prove Theorem \ref{step_1_thm}
for submersion structures and for foliation structures.
The next sections are devoted to the proof of Theorem \ref{step_1_thm}
in the case of
transversely symplectic structures. The existence part is treated in Section \ref{section_symp}.
The family of such structures are considered in Section \ref{parametric_symp}; the proof of Theorem
\ref{step_1_thm} is completed there when the groupoid is $\Gamma_n^\symp$. In Section
\ref{section_contact},
we adapt the proof to the groupoid $\Gamma_n^\cont$.
Finally, in Section \ref{open}, we solve
the problem of regularizing the $\Gamma$-structures on every open manifold.

\subsubsection*{Acknowledgments}
We thank Howard Jacobowitz and Peter Landweber who have expressed
their interest towards our article and sent to us some valuable remarks.
We are deeply grateful to the anonymous referee for his exceptionally careful reading and for having suggested to us many improvements of writing.

\section{Thom's subdivision and jiggling}\label{section_thom}
Reference \cite{thom59} is the report of a lecture where R.\,Thom announced a sort of homological
$h$-principle (ten years before Gromov's thesis).
A statement and a sketch of proof are given there; the details never appeared. From this text, we extracted
an unusual subdivision process of the standard simplex and we derived two jiggling formulas.\footnote{Thom speaks of ``dents de scie'' (saw teeth); we keep the word {\it jiggling} that W.\,Thurston introduced in
\cite{thu74}.} Our jiggling will be vertical
while Thom's jiggling is transverse to the fibres in some jet bundle. Nevertheless, we shall speak of
Thom's jiggling for it mainly relies on Thom's subdivision. Actually, neither statement nor proof nor
formula were written in \cite{thom59}, only words
describing the object, a beautiful object indeed.

Here is a good occasion for mentioning that the famous {\it Holonomic Approximation Theorem}
by Y.\,Eliashberg and N.\,Mishachev (\cite[Chap.\,3]{elias}) is also based on a jiggling process, even
if that word is not used there. The difference between their jiggling and ours is that the first one takes place in
the manifold itself while the second one is somehow vertical in the total space of a fibre bundle.

\begin{prop} \label{thom_sub}
Let $\Delta^n$ denote the standard $n$-simplex. For every positive integer~$n$,
there exist a non-trivial subdivision
$K_n$ of $\Delta^n$ and a simplicial map \hbox{$\sigma_n: K_n\to \Delta^n$} such that:
\begin{enumerate}
\item
\emph{(non-degeneracy)} the restriction of $\sigma_n$ to any $n$-simplex of $K_n$ is surjective;

\item
\emph{(heredity)} for any $(n-1)$-face $F$ of $\Delta^n$,
the intersection $K_n\cap F$ is simplicially isomorphic to $K_{n-1}$
and $\sigma_n\vert F\cong \sigma_{n-1}$.
\end{enumerate}
\end{prop}

\begin{proof}
Condition (2)
implies $\sigma_n(v)=v$ for any
vertex of $\Delta_n$. For $K_1$, we may take $\Delta^1= [0,1]$ subdivided by two interior vertices:
$0<v_1<v_0<1$ and we define $\sigma_1$ by $\sigma_1(v_1)=1$ and $ \sigma_1(v_0)=0$.

For $n=2$, let $A,B,C$ denote the vertices of $\Delta^2$. The polyhedron $K_2$ will be built
in the following way: subdivide each edge of $\Delta_2$ as $K_1$ subdivides $\Delta_1$;
add an interior triangle with vertices $a,b,c$ so that the line supporting $[b,c]$ is parallel to $[B,C]$
and separates $A$ from $a$, etc.; join $a$ to
the four vertices of $[B,C]$, etc. The simplicial map~$\sigma_2$ is defined by
$a\mto A, b\mto B, c\mto C$ and by imposing to coincide with $\sigma_1$ on each
edge of $\Delta^2$. Condition (1) is easily checked.

This construction extends to any dimension. If $K_{n-1}$ and $\sigma_{n-1}$ are known, each facet
of $\Delta^n$ will be subdivided as $K_{n-1}$. Then, one puts a small $n$-simplex $\delta^n$
in the interior of $\Delta^n$ applying the same rules of parallelism and separation as for $n=2$.
Each vertex $v$ of $\delta^n$ will be joined to the vertices of the facet $F(v)$ of $\Delta^n$
in front of~$v$,
this facet being subdivided by $K_{n-1}$. The map $\sigma_n$
maps~$v$ to the vertex $V$ which is opposite to $F(v)$ in $\Delta^n$.
\end{proof}

\skpt
\begin{remarques}\label{iteration}
\begin{enumerate}
\item
This subdivision may be iterated $r$ times producing a subdivision $K_n^r$ which is arbitrarily fine and a simplicial map $\sigma_n^r: K_n^r\to \Delta^n$ fulfilling the two conditions
of Proposition \ref{thom_sub}. More precisely, thinking of $\sigma_n$ as a map from $\Delta^n$ to itself,
$\sigma_n^r$ will denote its $r$-th iterate and $K_n^r$ is defined by the next formula:
\begin{equation}\label{folding}
K_n^r=(\sigma_n^{r-1})^{-1}(K_n).
\end{equation}
We will call $\sigma_n^r$ an {\it $r$-folding map}.

\item
Thanks to heredity (condition that the barycentric subdivision does not fulfill), this subdivision of the standard simplex
and the $r$-folding map apply to any polyhedron.

\item
It is worth noticing that Thom's subdivision is not {\it crystalline} in the sense of H.\,Whitney \cite[App.\,II]{whitney}. Thus, it does not fit Thurston's techniques of {\it jiggling} (compare \cite{thu74}).
\end{enumerate}
\end{remarques}

Actually, the above construction has an unfolding property which is stated in the next proposition.

\begin{prop} \label{unfolding}
With the above notations, for every $n$-simplex $\tau$ of $K_n^{r-1}$,
the restriction $\sigma_n^r\vert\tau$ is homotopic to $\sigma_n^{r-1}\vert\tau$ among piecewise linear
maps $\tau\to \Delta^n$ which are compatible with the face operators.
\end{prop}

\begin{proof}
According to
formula (\ref{folding})
it is sufficient to prove the proposition
for $\sigma_n\equiv\nobreak\sigma_n^1$. In that case, $\sigma_n^{r-1}= \Id$. The homotopy is obvious for $n=1$; it consists of
shrinking the middle interval $\delta^1$ to the barycenter of $\Delta^1$ and shrinking its image
at the same time. Recursively, the homotopy of $\sigma_n$ is known on the faces of~$\Delta^n$. Then, it is sufficient
to define the homotopy on the interior small $n$-simplex $\delta^n$. As when $n=1$, the homotopy consists
of shrinking $\delta^n$ and its image simultaneously to the barycenter of $\Delta^n$.
\end{proof}

\subsection{First jiggling formula}
Let $M$ be an $n$-manifold and $\tau M=(TM
\To{p} M)$ be its tangent bundle. Choose an auxiliary Riemannian metric
on $\tau M$ and an arbitrarily small open disk sub-bundle $U$ so that, for every $x\in M$,
the exponential map \hbox{$\exp_x: U_x\to M$} is an embedding. Take a combinatorial triangulation
$T$ of $M$ so fine that every $n$-simplex $\tau$ of $T$ is covered by $\exp_x(U_x)$ for every $x\in \tau$.
Let $T^r$ be the $r$-th Thom subdivision of $T$ and $\sigma^r: T^r\to T$ be the
corresponding simplicial folding map. The $r$-th {\it jiggling map} $j^r:M\to TM$
is defined in the following way. For each $x\in M$, the point $j^r(x)$ is the unique point in $U_x$
such that
\begin{equation}\label{1-formula} \exp_x(j^r(x))=\sigma^r(x).
\end{equation}
This formula defines $j^r$ as a piecewise smooth section $M\to TM$.
We have the following properties.

\skpt
\begin{prop}\label{jig}
\begin{enumerate}
\item
Let $\tau$ be an $n$-simplex of $T$, let $x$ be a point in $\tau$ and let $\delta$ be an $n$-simplex of~$T^r$ passing through $x$.
Then, $j^r(\delta)$ goes to $\exp_x^{-1}(\tau)$ as $r\to\infty$. The convergence is uniform for $x\in \tau$.

\item
The map $j^r$ is homotopic to the $0$-section
$Z$ among $PL$ maps which are transverse to the exponential
foliation $\cF_{\exp}$ on each $n$-simplex of their domain.
\end{enumerate}
\end{prop}

\skpt
\begin{proof}
\begin{enumerate}
\item
The diameter of the simplices of $T^r$ goes to $0$ as $r$ goes to $\infty$. Then, for $y\in \delta$, the point
$j^r(y):= \exp_y^{-1}(\sigma^r(y))$ is close to $\exp_x^{-1}(\sigma^r(y))$.
Since $\sigma^r$ is a surjective simplicial map onto $\tau$,
we have the $C^0$ closeness of $j^r(\delta)$ and $\exp_x^{-1}(\tau)$. A similar argument holds for the derivatives.

\item
On the one hand, the leaves of $\cF_{\exp}$ in $U$ are $n$-disks. We define $\exp^u: U\to U$,
$u\in [0,1]$, to be
the map which is the homothety by $u$ in each fibre of $\exp$. It is a homotopy from $\Id_U$
to $\exp\vert U$ which restricts to a homotopy from $j^r$ to $\sigma^r$. On the other hand,
according to Proposition \ref{unfolding},
$\sigma^r$ is homotopic to $\Id_M$ through $PL$ maps which are non-degenerate on each $n$-simplex
of their domain,
hence transverse to $\cF_{\exp}$.\qedhere
\end{enumerate}
\end{proof}

\skpt
\begin{remarques}\label{smoothness}
\begin{enumerate}
\item
Any piecewise smooth map defined on an $n$-manifold
$M$ and smooth on each $n$-simplex of a triangulation $T$ may be approximated
by a smooth map with the same polyhedral
image. It is sufficient to precompose with a smooth homeomorphism such that, for every simplex
$\tau$ in the $(n-1)$-skeleton of $T$ and every $x\in \tau$,
all partial derivatives in directions transverse $\tau$ vanish at $x$.
Then, even if the concept of $\Gamma$-structure is restricted to the smooth category,
there is no trouble to pull-back a $\Gamma$-structure by $j^r$;
it will be well defined up to homotopy.

\item
In general a jiggling, for instance based on the iterated barycentric triangulation,
does not share the properties stated in Proposition \ref{jig} (non-degeneracy
and $PL$-homotopy).
\end{enumerate}
\end{remarques}

\subsection{Second jiggling formula}
Here, we consider a trivial bundle $\ep^n$ of rank $n$
whose base is an $n$-manifold $M$ equipped
with a {\it colored} triangulation.\footnote{A triangulation $T$ of dimension $n$ is colored when each vertex has a color in $\{0,1,\ldots,n\}$ such that two vertices of the same simplex have different colors. The first barycentric subdivision of any triangulation is colored.}
Let $\Delta^n\subset \RR^n$ be a non-degenerate $n$-simplex whose vertices are colored. The coloring
defines a simplicial map $c:T\to \Delta^n$.
We have a first jiggling $j^1: M\to M\times\RR^n$, $x\mto \bigl(x,c(x)\bigr)$. Then, the Thom process defines a $r$-th jiggling
\begin{equation}\label{2-formula}
j^r(x)= \bigl(x, c\circ \sigma^r(x)\bigl).
\end{equation}
The first item of Proposition \ref{jig} holds true for this formula: $j^r(\delta)$ tends to $\{x\}\times\Delta^n$ when
$n$ goes to $\infty$.

\subsection{Proof of Theorem \ref{step_1_thm} in the easy cases}
For two of the four geometries considered, namely the submersion geometry
$\Gamma^Y_n$ and the foliation geometry $\Gamma^\fol_{n,q}$, the jiggling
method yields directly a simple proof of Theorem \ref{step_1_thm}.

For the {\it submersion geometry}, we begin by proving that
the forgetful map is $\pi_0$\nobreakdash-surjective.
One is given an $n$-manifold $M$, a $q$-manifold $Y$
and a {\it formal submersion} (in the sense of Subsection \ref{formal}(4)),
that is, a pair $(f,F)$, where $f:M\to Y$ is a smooth map and
where $F:TM\to TY$ is a bundle epimorphism above $f$.
One seeks for a one-parameter family $(f_u,F_u)$ of formal
submersions, $u\in[0,1]$, such that $(f_0,F_0)=(f,F)$, and such that $(f_1,F_1)$ underlies
some $\Gamma_n^Y$-structure $\xi=(\tau M,\cF)$. According the definitions given
in Subsections \ref{gamma_fol} and \ref{formal},
we have to find a pair $(w,\cF)$ formed with a submersion
valued in $Y$ and a codimension-$n$ foliation, both defined near $Z(M)$ in $TM$, such that:
\begin{itemize}
\item $w$ is constant on each leaf of $\cF$;
\item $f_1(x)= w(Z(x))$ and $F_1= Dw_{Z(x)}\vert T_xM$.
\end{itemize}
This will work by taking $w= \exp\circ F$ which is clearly a submersion on some neighborhood
$U$ of $Z(M)$ in $TM$; here,
$Y$ is endowed with some auxiliary Riemannian metric and $\exp:TY\to Y$ is the associated exponential map. The only somehow delicate point is to find $\cF$ as
a subfoliation of the foliation $\mathcal W$ whose leaves are $w^{-1}(y)$, $ y\in Y$.
Let $P$ be an $n$-dimensional plane field on $U$ transverse to every fibre $T_xM$ and contained in
the kernel of the differential of $w$.

Let $T$ be a triangulation of $M$. Consider the iterated Thom subdivisions $T^r$.
By Proposition
\ref{jig} (1), for $r$ large enough, the $r$-th
Thom jiggling $j^r$ maps every $n$-simplex of $T^r$ into $U$
and transversely to $P$. Fix such an $r$.
Then, on some small open neighborhood
$V$ of $j^r(T^r)$ in $TM$, there is a $C^0$-small perturbation
of $P$, among the $n$-plane fields tangent to $\mathcal W$,
yielding an integrable plane field on $V$. In the present situation where
the dimension of the simplicial complex $j^r(T^r)$ is not larger
than the codimension of $P$, the wanted integrating perturbation can be easily constructed
by induction on the dimension of the simplices (see the very beginning of Section 6 in \cite{thu74}).

Let $S:M\to TM$
be a smooth section so close to $j^r$, that $S(M)\subset V$. For every $u\in[0,1]$,
one has the section $S_u:=uS$ valued in $U$.
Set $f_u:=w\circ S_u$,
and $F_u(v_x):=Dw_{uS(x)}v_x$. The structure $S^*(w, \cF)$ is really a $\Gamma_{n}^{Y}$-structure
whose underlying formal structure is $(f_1,F_1)$.
The $\pi_0$-surjectivity is proved.

More generally, one is given a parametric family of formal submersions
$(f_s,F_s)$, $s\in\DD^k$, which are underlying some
$\Gamma_n^Y$-structures $\cF_s\in H^1_{\tau M}(M;\Gamma_n^Y)$ for every
$s\in\partial\DD^k$. First, one
constructs,
on some open neighborhood $U$
of the zero section in $TM$ and for every $s\in\DD^k$, an $n$-plane field
$P_s$ as above:
$P_s$ is contained in $\ker D(\exp\circ F_s)$, and transverse to every fibre $T_xM$.
One arranges that $P_s$ depends smoothly on $s$, and coincides with the tangent space
to $\cF_s$ for every $s\in\partial
\DD^k$. The construction of such a family is easy, by convexity\footnote{Compare Footnote \ref{convex-note}.} of
the space of the $n$-plane fields on $U$ contained in $\ker(D(\exp\circ F_s))$
and transverse to the fibres.

Let $T$ be a triangulation of $M$, and let $T^r$ be a Thom
subdivision whose order $r$ is large enough so that the same jiggling $j^r(T^r)$
is transverse to $P_s$ for every $s\in \DD^k$; since the considered family is compact,
such an $r$ certainly exists.
Then, the integrating
perturbation can be chosen smoothly with respect to $s$
and coinciding with the identity for every $s\in\partial\DD^k$. A single neighborhood $V$
and a single section $S$ fit all parameters $s\in\DD^k$.
We get on $V$ a parametric family $(\PP_s)$ of $\Gamma_n^Y$-foliations transverse to every fibre $T_xM$; and $\PP_s=\cF_s\vert V$
for every $s\in\partial\DD^k$.

Define $S$ and $S_u$ as above.
Set $\cF_s:=S^*(\PP_s)\in H^1_{\tau M}(M,\Gamma_{n}^{Y})$.
Set $f_{s,u}:=\exp\circ F_s\circ S_u$
and $F_{s,u}(v_x):=D(\exp\circ F_s)_{uS(x)}v_x$. This is a one-parameter family
of $\DD^k$-parametrized families of formal submersions,
between $(f_{s,0},F_{s,0})=(f_s,F_s)$ and $(f_{s,1},F_{s,1})$, which is
the formal submersion underlying $\cF_s$.

Finally, the families $(\cF_s)$ and $(\cF_s)\vert\partial\DD^k$
are homotopic as mappings
$\partial\DD^k\to H^1_{\tau M}(M,\Gamma_{n}^{Y})$. The homotopy
consists of pulling $\cF_s$ back through $S_u$.
The proof of Theorem \ref{step_1_thm} is complete for the groupoid $\Gamma_n^{Y}$.\qed

\medskip
In the case of the {\it foliation geometry} on a manifold $M$ of dimension $n=p+q$,
we are given
a parametric family of augmented $\Gamma_q$-structures
$(\xi_s,\epsilon_s)$, $s\in\DD^k$. Moreover, for every $s\in\partial\DD^k$, the augmented
$\Gamma_q$-structure $(\xi_s,\epsilon_s)$ is underlying some
$\Gamma^\fol_{n,q}$-structure $\cF_s\in H^1_{\tau M}(M;\Gamma^\fol_{n,q})$.

Denote $\xi_s:=(\nu,
\cX_s)$ this family of $\Gamma_q$-structures. Of course, the normal vector bundle $\nu$ over $M$ does not depend on
$s\in\DD^k$. Recall that $\epsilon_s:\nu\hto TM$ is a monomorphism
of vector bundles.

For every $s\in\partial\DD^k$, denote by
$\cG_s$ the foliation
of codimension $q$ tangent to $\cF_s$ on a neighborhood $U$ of $Z(M)$ in $TM$;
that is, if $\cF_s$ is viewed as a codimension-$n$ foliation, we have $\cF_s\subset \cG_s$.
For every $x\in M$, define
$\tau_s(x):=(T\cG_s)_{Z(x)}\cap T_xM$ to be the $p$-plane tangent
to the foliation $\cG_s\cap T_xM$ at $Z(x)$.
Thus, $\tau M=\tau_s\oplus\epsilon_s(\nu)$.
The family~$(\tau_s)$ extends to a $\DD^k$-parametrized
family $(\tau_s)$ of $p$-plane fields
on $M$ complementary to~$\epsilon_s(\nu)$.

For every $s\in\partial\DD^k$,
after pushing $\cG_s$ by a vertical isotopy in $TM$, whose $1$-jet at every point of
the zero section is the identity, and after restricting to some smaller neighborhood,
one can moreover assume that
the trace $\cG_s\cap (T_xM\cap U)$ is the restriction to $T_xM\cap U$ of
the linear $p$-dimensional foliation parallel to $\tau_s(x)$. Then, the family
$(\cG_s)$ extends to a $\DD^k$-parametrized family $(\cG_s)$ of foliations of codimension $q$,
transverse to the fibres of $TM\to M$ and
defined on some neighborhood $U$ of $Z(M)$
independent of $s$.
Indeed, for every $s\in\DD^k
\setminus\partial\DD^k$,
we define $\cG_s$ as the pullback of $\cX_s$ through the linear projection of
$TM=\tau_s\oplus\epsilon_s(\nu)$ onto $\epsilon_s(\nu)$ parallel to $\tau_s$.

Just as in the case of the submersion geometry, one constructs a
$\DD^k$-parametrized smooth family $(P_s)$
of $n$-plane fields on $U$, transverse to the fibres $T_xM$,
and contained in $T\cG_s$. For $s\in\partial\DD^k$, one has $P_s=T\cF_s$.
For a large enough integer $r$, a $C^0$-small perturbation
of $P_s$ yields a foliation $\PP_s$ of codimension $n$,
contained in $\cG_s$, on some open
neighborhood $V$ of the jiggled zero section $j^r(T^r)$.
For $s\in\partial\DD^k$, one has $\PP_s=\nobreak\cF_s$.

Define sections $S$, $S_u$ as above.
For every $s\in\DD^k$, set
\[
\cF'_s:=S^*(\PP_s)\in H^1_{\tau M}(M,\Gamma_{n,q}^\fol).
\]
The underlying augmented $\Gamma_q$-structure is homotopic to the given one
$(\xi_s,\epsilon_s)$. The homotopy consists of the
$1$-parameter family of $\DD^k$-parametrized families of augmented $\Gamma_q$-structures
$(\nu,S_u^*(\cG_s),
\epsilon_s)$ (note that $\cG_s$ is defined on the whole of the open set $U$).
Finally, the $\partial\DD^k$-parametrized families $\cF_s$ and $\cF'_s$
of $\Gamma_{n,q}^\fol$-structures are homotopic:
the homotopy consists of $\cF_{s,u}:=S_u^*(\cF_s)$.
The proof of Theorem \ref{step_1_thm} is complete for the groupoid $\Gamma_{n,q}^\fol$.\qed

\section{Existence of transversely symplectic \texorpdfstring{$\Gamma_n$}{Gn}-structures}\label{section_symp}

In this section we prove a slightly more general statement than the existence part of Theorem \ref{step_1_thm} for
the groupoid $\Gamma_n^\symp$: we consider any symplectic bundle of rank~$n$.
We are going to use a more
informative notation: a $\Gamma_n^\symp$-structure on $M$ will be denoted by $\xi=(\nu,\cF,\Omega)$, where $\Omega$ is a closed $2$-form whose kernel is $\cF$.

\begin{thm}\label{existence}
Let $\nu=(E\to M)$ be a symplectic bundle of even rank $n$ over a manifold $M$ of dimension
$\leq n+1$.
Then there exists a $\Gamma_n^\symp$-structure $\xi$
on $M$ whose normal
bundle $\nu(\xi)$ is isomorphic to $\nu$ as a symplectic bundle.

Moreover, if a real cohomology class $\bar a\in H^2(M,\RR)$ is given, $\xi$ can be chosen
so that
the cohomology class $[Z^*\Omega]$ equals $\bar a$, where $\Omega$ is the closed $2$-form underlying $\xi$.
\end{thm}

We think of this problem as a lifting problem that we attack by obstruction theory. Let us
explain how it works. As for any groupoid of germs,
there are a classifying space\footnote{The contravariant homotopy
functor $\Gamma_n^\symp(-)$ satisfies the axiom of gluing (Mayer-Vietoris) and wedge sum; then, the classifying space exists according to E.\,Brown's Theorem \cite{brown}.} $\BGamma_n^\symp$ and
a canonical isomorphism $$\Gamma_n^\symp(M)\cong[M,\BGamma_n^\symp], $$
where $[-,-]$ stands for the set of homotopy classes of maps.

This classifying space is
the source of two maps. The first one is $\beta:\BGamma_n^\symp\to \BSp(n;\RR) $:
if $f:M\to \BGamma_n^\symp$ classifies a $\Gamma_n^\symp$-structure $\xi=(\nu,\cF,\Omega)$
up to {concordance},
$\beta\circ f$ classifies its normal bundle $\nu$. The second one is
$\kappa: \BGamma_n^\symp\to K(\RR,2) $, where the target is the Eilenberg-MacLane space
classifying the functor $H^2(-,\RR)$: the composed map $\kappa\circ f$ classifies
the cohomology class of the closed
2-form $Z^*\Omega$.
Finally, the {pair}
$(\beta,\kappa)$ defines
a map
$$\pi^\symp: \BGamma_n^\symp\to \BSp(n;\RR)\times K(\RR,2),
$$
that we see as a homotopy fibration. For Theorem \ref{existence}, we are given a map $M\to \BSp(n;\RR)\times K(\RR,2)$
and we have to lift this map to $\BGamma_n^\symp$.
Since $M$ is $(n+1)$-dimensional, Theorem \ref{existence} is a direct corollary
of the next statement (the $(n-1)$-connectedness would be sufficient for Theorem \ref{step_1_thm}).
Indeed, thanks to the long exact sequence associated with $\pi^\symp$, this map induces
a monomorphism up to the $n$-th homotopy group (see, for instance Hatcher's book \cite[\S4.3]{hatcher}).

\begin{thm}[Haefliger, McDuff]\label{trans_symp}
The homotopy fibre of $\pi^\symp$, denoted by $F\pi^\symp$, is $n$-connected.
\end{thm}

A.\,Haefliger \cite[\S6]{haefliger} showed that the $(n-1)$-connectedness of this homotopy fibre is a consequence of the $h$-principle.
D.\,McDuff \cite[Th.\,6.1]{mcduff} proved the $n$-connectedness thanks to the convex integration
technique.

\subsection{What do we have to prove for Theorem \ref{trans_symp}?}
We have to prove that the $k$-th homotopy group $\pi_k(F\pi^\symp)$ vanishes
when $k\leq n$. An element of this group is represented by a $\Gamma_n^\symp$-structure
$\xi =(\ep^n,\cF,\Omega)$
on the $k$-sphere with the following properties:
\begin{itemize}
\item The normal bundle is trivial as a symplectic bundle; this means that its underlying
symplectic bilinear form is the standard form $\omega_0$ of $\RR^n$ on each fibre.
\item The closed $2$-form $\Omega$, which is defined in a neighborhood of
the $0$-section $Z(S^k)$ in $\bbS^k\times\RR^n$,
is assumed to be exact.
\end{itemize}

Let $(p_1,p_2)$ denote the two projections of $\bbS^k\times \RR^n$ onto its factors.
Recall that the kernel of $\Omega$ is the tangent space to the codimension-$n$
foliation $\cF$ and that $\cF$ is transverse to the fibres of $p_1$.

We have to
extend this structure $\xi$ over the $(k+1)$-ball $\DD^{k+1}$
or, equivalently, to show that it is homotopic to the
trivial structure $\xi_0:= (\ep^n,\cF_0, \Omega_0)$, where $\Omega_0=p_2^*\,\omega_0$.

According to Moser's Lemma with $k$
parameters \cite{moser},
there exists a vertical isotopy of $\bbS^k\times\RR^n$, keeping $Z$ fixed,
which reduces us to the case
where the germ at $Z(x)$ of the form induced by $\Omega$ on $p_1^{-1}(x)$ equals
$\omega_0$ for every $x\in\bbS^k$. After this vertical Moser isotopy, take a trivial tube $U=\bbS^k\times B^n$
in the domain of $\Omega$, where $B^n$ is an $n$-ball of small radius. Now, Theorem \ref{trans_symp} directly follows from the next lemma,
as we will see just after its statement.\enlargethispage{-\baselineskip}%

\begin{lemme}\label{vertical} Given the above-mentioned data,
there exist a section $s:\bbS^k\to U$, a neighborhood $W$ of $s(\bbS^k)$ in $U$
and an ambient diffeomorphism $\psi$ such that:
\begin{enumerate}
\item $\psi$ is the time-$1$ map of a \emph{vertical} isotopy $(\psi_t)$; set $W_0:= \psi(W)$;
\item $\psi$ sends the pair $(W, \Omega)$ to $(W_0,\Omega_0)$;
\item the isotopy $\psi_t$ is Hamiltonian with respect to $\omega_0$ in each fibre.
\end{enumerate}
\end{lemme}
Here, ``vertical'' means that the isotopy preserves each fibre of $p_1$.

\pagebreak[2]
\skpt
\begin{remarques}
\begin{enumerate}
\item
The statement holds true for every symplectic vector bundle of rank $n$,
equipped (near the $0$-section)
with two forms exact forms $\Omega$ and $\Omega_0$ which define $\Gamma_n^\symp$-foliations and induce
the same symplectic form on each fibre.

\item
Moreover,
the two first items are valid for a pair of $\Gamma_n$-foliations without any transverse geometry.
\end{enumerate}
\end{remarques}

\begin{proof}[Proof of Theorem \ref{trans_symp}]
Since $\psi$ is vertical, $s_0:= \psi\circ s$ is a section
of the trivial bundle~$\ep^n$ over $\bbS^n$. Also, recall the $0$-section $Z$.
Then, we have a sequence of homotopies of $\Gamma_n^\symp$-structures on $\bbS^k$:
\begin{itemize}
\item a first homotopy from $Z^*\Omega$ to $s^*\Omega$;
\item then, a homotopy from $s^*\Omega$ to $s_0^*\Omega_0$ defined by the isotopy $(\psi_t)$;
\item a last homotopy from $s_0^*\Omega_0$ to $Z^*\Omega_0$.
\end{itemize}
The last structure obviously extends to the $(k+1)$-ball.
\end{proof}

Shortly said, the proof of Lemma \ref{vertical} will consist of taking a jiggled section in the sense of
formula (\ref{2-formula})
whose simplices are very vertical, then
covering it by boxes which {\it trivialize} the kernel of $\Omega$ and pushing
these boxes by some vertical Hamiltonian isotopy
until $\ker\Omega $ becomes horizontal. This isotopy is done
recursively on the boxes. There are two main problems:
\begin{enumerate}
\item
rectifying the $(j+1)$-th box should not destroy what was gained for the $j$-th box;
\item
manage the vertical isotopies to be Hamiltonian and not just symplectic; if not, they could not extend.
\end{enumerate}

\begin{proof}[Proof of Lemma \ref{vertical}]
We limit ourselves to $k= n$; for $k<n$, it is the same argument
by replacing the base $\bbS^k$ with an $n$-dimensional base $\bbS^k\times \DD^{n-k}$.
Let $B^n:= p_2(U)$ and let $\Delta^n$ be a non-degenerate and
colored $n$-simplex in the interior of $B^n$.

Take a decreasing sequence
$$\ep_0>\cdots> \ep_j>\cdots >\ep_{n-1}.
$$
When $\alpha$ is a strict closed $j$-face of $\Delta^n$, let $N(\alpha)$ denote
the closed $\ep_j$-neighborhood of $\alpha$ in $\RR^n$. Set
$$
N(\Delta^n):= \Delta^n\mathop{\cup}\limits_\alpha N(\alpha),
$$
where the union is taken over all faces of $\Delta^n$. For a suitable choice of the sequence~$(\ep_j)$
we may arrange that :
\begin{enumerate}
\item $N(\alpha)\cap N(\beta)=\emptyset$ if $\alpha$ and $\beta$ are two disjoint faces;
\item if $\alpha\cap\beta\neq\emptyset$ and if $\alpha$ and $\beta$
are not nested, then $N(\alpha)\cap N(\beta)$ is interior
to $N(\alpha\cap\beta)$;
\item $N(\Delta^n)\subset B^n$.
\end{enumerate}
Now, take a colored triangulation $T$ of the base $\bbS^n$, its Thom subdivision~$T^r$ and the associated
jiggling $j^r$ given by formula (\ref{2-formula}).
We are going to construct bi-foliated {\it boxes} associated with each simplex of
$T^r$ whose plaques are respectively contained in the leaves of $\cF$ and in the fibres of $p_1$;
the boundary of a box has a part tangent to $\cF$ and another part tangent to the fibres.
Let $\tau$ be a $k$-simplex of~$T^r$; with~$\tau$, the coloring of $T$
associates some face $\tau^\perp$ of $\Delta^n\subset B^n$.
The box $B(\tau)$ is defined in the following way. Its base $p_1(B(\tau))$
equals $\Star(\tau)$,
the star of $\tau$ in $T^r$. In the fibre over the barycenter $b(\tau)$, we take the domain $N(\tau^\perp)$. Finally, $B(\tau)$
is the union of all plaques of $\cF$ passing through $N(\tau^\perp)$ and contained in $p_1^{-1}(\Star(\tau))$.
If the diameter of the base
is small enough, that is, if the order $r$ of the subdivision is large enough, the holonomy of $\cF$
over the base is $C^0$ close to {\it Identity}.
Therefore,
each plaque in~$B(\tau)$ cannot get out of $U$;
thus, it covers $\Star(\tau)$.

Look at two faces $\tau$ and $\tau'$ of the same simplex $\sigma$ of $T^r$. Assume first that $\tau$ and $\tau'$
are disjoint. Apply the above condition (1) to $\tau^\perp$ and $\tau'^\perp$;
by the holonomy argument, if $r$ is large enough, the boxes $B(\tau)$ and $B(\tau')$ are disjoint.
Assume now that $\tau$ and~$\tau'$ are not disjoint but not nested. Then, by (2), we have
$$B(\tau)\cap B(\tau')\subset B(\tau\cap \tau').
$$
Nevertheless, if $\tau$ and $\tau'$ do not belong to the same $n$-simplex and if
\[
\partial \Star(\tau)\cap \partial \Star(\tau')\neq\emptyset,
\]
then
$B(\tau)$ and $B(\tau')$ could intersect badly. This is corrected in the following way.

Again, for $r$ large enough, the leaves of $\cF$ meeting $j^r(\tau)$ intersect the fibre over
$b(\tau)$ in $N(\tau^\perp)$. This guarantees that $j^r(T^r) $ is covered by the interior of the boxes.
From now on, $r$ is fixed. For $1>\eta>0$, the {\it $\eta$-reduced box}
associated with $\tau$ is defined~by
$$B_\eta(\tau):= B(\tau)\cap p_1^{-1}\bigl((1-\eta)\Star(\tau)\bigr),$$
where the homothety is applied from the barycenter $b(\tau)$. Fix $\eta>0$ small enough
so that the $\eta$-reduced open
boxes
still cover the jiggling. Now, we are sure that $B_\eta(\tau)$ and $B_\eta(\tau')$
are disjoint once $\tau$ and $\tau'$ are disjoint.

The desired open set $W$ is the union $V_0\cup \cdots\cup V_k\cup \cdots\cup V_{n-1}$,
where $V_k$ denotes the interior of the $\eta$-reduced
boxes associated with each $k$-simplex; the section $s$ is any smooth
approximation of $j^r$ valued in $W$. We are ready to perform the isotopy. It is done step by step,
in the boxes associated with the vertices of $T^r$ first, then with the edges etc. For $x\in \Star(\tau)$,
lifting the segment $[x,b(\tau)]$ to $\cF$ yields
a holonomy diffeomorphism between fibres of box
\begin{equation}\label{hol}
(\hol\cF)_x^{b(\tau)}: B(\tau)_x\to B(\tau)_{b(\tau)},
\end{equation}
which is an $\omega_0$-symplectomorphism since $\Omega$ is closed. Similarly, we have the
holonomy of $\cF_0$ which also give an $\omega_0$-symplectomorphism. The steps are numbered from $0$ to~$n$.

If $v$ is a vertex in $T^r$, we define $\psi^0$ in $B_\eta(v)$ by the next formula. For $z\in B_\eta(v)$
and $x=p_1(z)$,
\begin{equation}\label{step1}
\psi^0(z)= (\hol \cF_0)_{v}^x\circ (\hol \cF)_x^{v}(z).
\end{equation}
Since the reduced boxes are disjoint, this formula simultaneously applies to the reduced boxes associated
with all vertices. By shrinking the segment $[x, v]$ to \hbox{$[x, x+t(v-x)]$} and by replacing $v$ with $x+t(v-x)$ in formula (\ref{step1}), we define an interpolation between $\psi^0(z)$ and $z$.
As a consequence $\psi^0$ is the time-1 map of a vertical isotopy
of embeddings $(\psi^0_t)$
which is easily checked to be symplectic. Since the components of the domain of $(\psi^0_t) $ are contractible,
this is actually a Hamiltonian isotopy\footnote{The infinitesimal generator $X_t$
of an $\omega_0$-symplectic isotopy satisfies that $\iota(X_t)\omega_0$ is a closed $1$-form; it is said to be Hamiltonian if this form is the differential of a function.}
which therefore extends to a global Hamiltonian isotopy
supported in $U$, still denoted by $(\psi^0_{t})$. Let $\cF_1$ (\resp $\Omega_1$) be the
direct image of $\cF$ (\resp $\Omega$) by $\psi^0_{1}$; all reduced
boxes are transported in this way, becoming $B^1_\eta(\tau)$ for each $\tau\in T^r$.
Observe that $\cF_1$ is horizontal in the reduced new boxes associated with vertices.

The next step (numbered 1) deals with the edges. Let $e$ be an edge in $T^r$ with end points $v_0,v_1$. For
$z\in B^1_\eta(e)$ and $x=p_1(z)$, define $\psi^1(z)$ by:
\begin{equation}\label{step2}
\psi^1(z)= (\hol \cF_0)_{b(e)}^x\circ (\hol \cF_1)_x^{b(e)}(z).
\end{equation}
Observe that $\psi^1(z)=z$ when $z\in B^1_\eta(v_i),\ i= 0,1${; indeed, this box covers the barycenter $b(e)$ and $\cF_1$ is horizontal there}. Moreover, $\psi^1$ is the time-1 map of
a symplectic isotopy $(\psi^1_{t})$
relative to the reduced boxes of the vertices; this isotopy, called
the {\it step-1 isotopy},
follows from an interpolation formula
analogous to the one defining $(\psi^0_{t})$.

If $e$ and $e'$ are two edges, after Condition (2), the domain where their $\eta$-reduced boxes
could intersect
is contained in a domain where $ \cF_1$ is horizontal and, hence, $\psi^1_{t}=\nobreak\Id$ on this domain. Therefore,
$\psi^1_{t}$
is well defined on the union $V_1$ of closed $\eta$-reduced boxes associated
with the vertices and edges. Unfortunately, it is not a Hamiltonian isotopy of embeddings;
some vertical loops in $V_1$ may sweep out some non-zero $\omega_0$-area.
Thus, it could not extend to an ambient vertical symplectic isotopy. The needed correction is offered by the next claim, following well-known ideas (compare V.\,Colin \cite[Lem.\,4.4]{colin}).

\skpt
\begin{claim}
\begin{enumerate}
\item
There is a real combinatorial cocycle $\mu=\mu_{\Omega_1}$ of the triangulation $T^r$
such that, for each
triangle $\tau$, the real number $\<\mu,\tau\>$ measures the $\omega_0$-area swept out by the loop
$\{x\}\times (\partial\tau^\perp)$ through the isotopy $(\psi^1_{t})$
for every $x\in (1-\eta)\Star(\tau)$;
in particular, this area does not depend on $x$.

\item
When $\Omega_1$ is exact, $\mu$ is a coboundary.

\item
There is an ambient vertical $\omega_0$-symplectic isotopy $(g_t)_{t\in[0,1]}$,
supported in $U$,
which is stationary on $V_0$
and such that
$\mu_{g_1^*\Omega_1}=0$.
\end{enumerate}
\end{claim}

The third item, together with the first item, means that
the step-1 isotopy $(\psi^1_{t})$
becomes Hamiltonian when $\cF_1$ stands for the foliation tangent to
$\ker g_1^*\Omega_1$ instead of $\ker\Omega_1$.

The proof of the claim is postponed to the end of the section.
We first finish the proof of Lemma \ref{vertical}
by applying the claim in the next way.

After the step-0 isotopy, the cocycle $\mu_{\Omega_1}$ is calculated and the Hamiltonian
isotopy $(g_t)$ is derived.
Let $\tilde\cF_1$ denote the foliation tangent to $\ker g_1^*\Omega_1$;
let \hbox{$\tilde B^1_\eta(\tau):=(g_t)^{-1}\bigl(B^1_\eta(\tau)\bigr)$}.
Now, the straightening formula \ref{step2} of the box $\tilde B^1_\eta(e)$ is applied
$\tilde\cF_1$ instead of $\cF_1$. The associated isotopy $(\psi^1_t)$
becomes $\omega_0$-Hamiltonian. Hence, it extends
to a vertical isotopy supported in $U$, denoted likewise,
which is $\omega_0$-Hamiltonian on each fibre $U_x$. This finishes
step 1 of the isotopy.

The next steps of the induction
are similar, except
that the question of being a Hamiltonian isotopy is not raised again since, up to homotopy, every loop
in $W$ is already contained in $V_0\cup V_1$. In the end of this induction, we have a proof of Lemma
\ref{vertical} by taking $\psi=\psi^n_1$.
\end{proof}

\skpt
\begin{proof}[Proof of the claim]
\begin{enumerate}
\item
Let $e$ be an edge of $T^r$; its end points are denoted $v_0$ and $v_1$.
Let $x$ be a point in the base of $B_\eta(e)$. Set $\gamma:=\{x\}\times e^\perp$.
We first compute the $\omega_0$-area swept out by the vertical arc $(\psi^1_{1})^{-1}(\gamma)$
through the step-1 isotopy. Denote this area by $\mathcal A(x,e)$;
any other arc with the same end points would give the same area.

There are two natural ``squares'', $C$ and $C_0$, appearing for this computation. The square $C$
(\resp $C_0$)
is generated by the holonomy of $\cF_1$ (\resp $\cF_0$) over $[b(e),x]$ with initial vertical arc $e^\perp$
in the fibre $U_{b(e)}$. They have common horizontal edges: $\beta_i:=e\times v_i^\perp$ for $i=0,1$.
Orient $e^\perp$ from $v_0^\perp$ to $v_1^\perp$; thus, $\gamma$ and $(\psi^1_{1})^{-1}(\gamma)$
are oriented
by carrying the orientation of $e^\perp$ by the respective holonomies; and also
$C_0$ and $C$ are oriented by requiring $\{b(e)\}\times e^\perp$ to define the boundary orientation.
Then, we have
\begin{equation}
\mathcal A(x,e)= \int_C\Omega_0-\int_{C_0}\Omega_0.
\end{equation}
The second summand is $0$ by construction. Similarly, we have $\int_C\Omega_1=0$.
Then, if $\Lambda$ is any primitive of
$\Omega_1-\Omega_0$, we derive
\begin{equation}\mathcal A(x,e)= -\int_C d\Lambda\, .
\end{equation}
We now use a specific choice of primitive. Recall the zero-section $Z: \bbS^n\to U$.
For $t\in [0,1]$, let $c_t$
denote the contraction $(x,v)\mto (x, tv)$ and let $c: U\times [0,1]\to U$
be the corresponding homotopy
from $Z\circ p_1$ to $\Id_U$. This yields
the next formula:
\begin{equation}
\Omega_1-\Omega_0= d\biggl[ p_1^*\theta+ \int_0^1\! \iota\!({\partial_t})c^*(\Omega_1-\Omega_0)\biggr],
\end{equation}
where $\theta$ is a primitive of the exact form $Z^*\Omega_1$ (observe that $Z^* \Omega_0=0$);
the integral is just the mean value of a one-parameter family of $1$-forms. This primitive
of $\Omega_1-\Omega_0$ also reads
\begin{equation}
\Lambda_0:=p_1^* \theta +\int_0^1c_t^*\iota(v\partial_v)(\Omega_1-\Omega_0),
\end{equation}
which vanishes on every vertical vector since $\Omega_1$ and $\Omega_0$ coincide on the fibres.
\hbox{Orient}~$\beta_0$
as the horizontal lift of $[b(e),x]$ and $\beta_1$ as the opposite of the oriented horizontal lift.
We have
\begin{equation}\int_C d\Lambda_0= \int_{\beta_1}\Lambda_0+\int_{\beta_0}\Lambda_0.
\end{equation}

Now, we consider a triangle $\tau$ in $T^r$ and we look at the $\omega_0$-area $\mathcal A(x,\partial\tau)$
swept out by
$\{x\}\times(\partial\tau)^\perp$ when $x$ belongs to $ (1-\delta)\Star(\tau)$.
The vertices of $\tau$ are denoted by $v_i,\ i= 0, 1,2,$ cyclically ordered;
the oriented edges are $e_j:=[v_{j-1},v_j ]$, where $j-1$ is taken modulo $3$.
There are two particular horizontal lifts of $[b(e_{j}), x]$, denoted by~$\beta_{j,k}$ with $k=j$ or $j-1$
depending on whether its origin is $(b(e_{j}), v_j^\perp)$ or $(b(e_{j}), v_{j-1}^\perp)$. If $k=j-1$,
it is oriented as $[b(e_{j}), x]$; if $k=j$, it has the opposite orientation.
By summing up the area swept out by each edge of $\{x\}\times(\partial\tau)^\perp$, we have
\begin{equation}
\mathcal A(x,\partial\tau)=
\<\Lambda_0, \beta_{1,1}+\beta_{2,1}+\beta_{2,2}+\beta_{0,2}+\beta_{0,0}+\beta_{1,0}\>,
\end{equation}
where the bracketing stands for the integration over chain.

Since $\Omega_1-\Omega_0$
vanishes on $(1-\eta)\Star(\tau)\times\{v_i^\perp\}$, we have
$$
\<\Lambda_0, \beta_{i,i}+\beta_{i+1,i}\>=\<\Lambda_0, [b(e_i),b(e_{i+1})]\times v_i^\perp\>.
$$
By summation, we have
\begin{equation}
\mathcal A(x,\partial\tau)=\sum_i \<\Lambda_0, [b(e_i),b(e_{i+1})]\times v_i^\perp\>,
\end{equation}
which implies that $\mathcal A(x,\partial\tau)$ does not depend on $x$.
The combinatorial cochain $\mu$ is now defined by the next formula:
\begin{equation}\label{cochain}
\<\mu,\tau\>=\sum_i \<\Lambda_0, [b(e_i),b(e_{i+1})]\times v_i^\perp\>.
\end{equation}
If an arbitrary primitive $\Lambda$ of $\Omega-\Omega_0$ is used, the above formula
becomes
\begin{equation}\label{cochain-2}
\<\mu,\tau\>=\sum_i \<\Lambda, \{b(e_i)\}\times [v_{i-1}^\perp,v_i^\perp]\> +\<\Lambda, [b(e_i),b(e_{i+1})]\times v_i^\perp\>.
\end{equation}
Indeed, a change of primitive consists of adding a closed $1$-form; and the integral~of this on
the polygon $P$ considered in formula (\ref{cochain-2}) is zero since $P$ bounds a $2$-cell.\footnote{The cochain $\mu$ is a cocycle. Regarding the second item, this fact is not important and left to the reader. Note that the previous calculation uses a local primitive of $\Omega$ only.}

\item
Since $T^r$ is a finite simplicial set, we only have to prove that $\<\mu,\Sigma\>= 0$ for every $2$-cycle
$\Sigma$
of $T^r$. Here, the exactness of $\Omega_1$ is used.
Summing formula (\ref{cochain-2}) over all triangles of $\Sigma$ yields a sum of integrals
of $\Lambda$ over horizontal polygons in regions where $d\Lambda=0$ (one polygon for each vertex of $\Sigma$).
Then, these integrals are null.
Therefore, there exists a combinatorial $1$-cochain $\alpha$ of $T^r$ such that $\mu=\partial^*\alpha$, where~$\partial^*$ stands for the combinatorial co-differential.

\item
We are going to use this $1$-cochain $\alpha$ in order to correct $\Omega_1$ by a certain vertical isotopy.
Let $e$ be an oriented edge in $T^r$ with origin $v_-$ and extremity $v_+$. The value~$\alpha(e)$ is used
in the following way. In the fibre over $b(e)$, we find an $\omega_0$\nobreakdash-Hamiltonian isotopy
$(g_t^e)_{t\in [0,1]}$, compactly supported in $U_{b(e)}$ and
fixing $(V_0)_{b(e)}$,
such that the area swept out by the arc $\gamma_e:= b(e)\times [v_-^\perp,v_+^\perp]$
is $-\alpha(e)$.\footnote{In dimension $n=2$, this is possible only if $\vert \lambda(e)\vert$ is less than
the $\omega_0$-area of $U_{b(e)}$. This last condition is satisfied when $r$ is large enough.}
Observe that the Hamiltonian function is not required to vanish in the fixed domain, but only to
be constant on each connected component of the fiber $(V_0)_{b(e)}$ over $b(e)$.

Then, the infinitesimal generator $X_t$ of the desired isotopy $(g_t)$ is chosen in finitely many fibres. By
a suitable partition of unity there is an extension which is Hamiltonian in each fibre, compactly
supported and vanishing in $V_0$. Note that the Hamiltonian has to be constant in each connected
component of the fibre $(V_0)_{x}$, but these constants may vary with $x$.

The $2$-form $g_1^*\Omega_1-\Omega_1$ has a primitive associated with the isotopy, named
the {\it Poincar{\'e}} primitive,
\begin{equation}
A= \int_0^1 g_t^*\left(\iota(X_t) \Omega_1\right)dt.
\end{equation}
Since $X_t$ vanishes on $V_0$,
the $1$-form $A$ vanishes over here and we have:
\begin{equation}
\<A, \gamma_e\>= - \alpha(e).
\end{equation}
Now, $\Lambda+A$ is a primitive of $g_1^*\Omega_1-\Omega_0$. According to formula (\ref{cochain-2}),
the combinatorial cochain $\mu_{g_1^*\Omega_1}$ associated with the $2$-form $g_1^*\Omega_1$ vanishes
and the claim is proved.\qedhere
\end{enumerate}
\end{proof}

\section{Parametric family of transversely symplectic \texorpdfstring{$\Gamma_n$}{Gn}-structures}\label{parametric_symp}

In this section, we prove the parametric version of Theorem \ref{step_1_thm}
for the groupoid $\Gamma=\Gamma_n^\symp$. We emphasize
that the required $k$-connectedness of the homotopy fibre $F\pi^\symp$ depends only on the
dimension of $M$ and not on the number of parameters in the family. Indeed, there is no integrability
condition with respect to the parameter.
Moreover, we insist that a common jiggling will be used in the proof; its order is bounded by compactness of the parameter space.

We consider the same setting as in Theorem \ref{existence}: $\nu=(E\to M)$ is a bundle of
even rank $n$ over a manifold $M$ of dimension $\leq n+1$ equipped with a
$k$-parameter family $(\omega_u)_{u\in \DD^k}$
of symplectic bilinear forms $\omega_u$ on $E$. It is understood that $k$ is positive.

\begin{thm}
Assume there is a family $(\xi_u)_{u\in \partial\DD^k}$
of $\Gamma^\symp$-structures, namely a~family $(\Omega_u)_{u\in \partial\DD^k}$
of closed $2$-forms defined near the zero section $Z$ of $E$,
such that~$\Omega_u$ induces $\omega_u$ on the fibres of $\nu$ for every $u\in \partial\DD^k$.\footnote{In other words, the symplectic normal bundles equal $(\nu,\omega_u)_{u\in \partial\DD^k}$.}

Then,
this family extends over the whole $\DD^k$
such that $\Omega_u$ induces $\omega_u$ on the fibres of $\nu$ for every $u\in\DD^k$.
Moreover, the family of cohomology classes $[Z^*\Omega_u]_{u\in \DD^k}$ may be arbitrarily chosen among
those which extend the boundary data.
\end{thm}

\begin{proof}
We start with a cell decomposition $\mathcal C$ of $M$ fine enough so that,
for every $u\in \partial\DD^k$
and every cell $C \in \mathcal C$, there is a fibered isotopy of $E_{\vert C}$
(depending smoothly on~$u$) whose time-1 map $\psi_u$ satisfies:
$(\psi_u)_*\Omega_u=\theta_u^*\Omega_0$, where $\theta_u$ is a linear symplectic trivialization
of $(\nu_{\vert C},\omega_u)$, depending smoothly on $u\in \DD^k$, and where $\Omega_0$ stands
for the pull-back of $\omega_0$ by the projection $C\times\RR^n\to\RR^n$.

The theorem will be proved by induction on an order of the simplices of $\mathcal C$ for which their
dimension is
a non-decreasing function. Skipping the intermediate dimensions we jump to the $(n+1)$-cells.
Thus, we are reduced to consider the $n$-trivial bundle over~$\bbS^n$ and a family
$(\Omega_u)$ of exact $2$-forms on a small disk bundle $U$ about the zero section~$Z$,
which induce the standard form $\omega_0$ on each fibre $U_x$, $x\in\bbS^n$, (here a parametric version
of Moser's lemma is applied again). This family fulfills the condition that $\Omega_u=\Omega_0$ for every
$u\in \partial \DD^k$. Let $T$ be a triangulation of $\bbS^n$ and let $T^r$ be a Thom
subdivision whose order $r$ is large enough so that the same jiggling $j^r(T^r)$
fits the proof of Lemma \ref{vertical} for every $u\in \DD^k$; since the considered family is compact,
such an~$r$ certainly exists.

Each step of that proof may be performed with parameters using this fixed jiggling. Here it is worth
noticing that the vertical isotopy given by Lemma \ref{vertical} is stationary when $\Omega_u=\Omega_0$,
in particular when $u\in \partial \DD^k$.

The proof of Theorem \ref{step_1_thm} is now completed for the groupoid $\Gamma_n^\symp$.
\end{proof}

\section{Transversely contact \texorpdfstring{$\Gamma_n$}{Gn}-structures}\label{section_contact}
Here, we prove a theorem which implies Theorem \ref{step_1_thm} for $\Gamma_n^\cont$-structures.
Our setting is not the one of tangential $\Gamma$-structures. It is the following.
Given an odd natural integer $n$, a manifold $M$ and a vector bundle $\nu= (E\to M)$ of rank $n$, we
recall that a $\Gamma_n^\cont$-structure on $M$ with normal bundle $\nu= (E\to M)$ is given by
$\xi=(A,\mathcal K)$, where $A$ is a $1$-form and $\mathcal K$ is a codimension-$n$ foliation,
both defined near the $0$-section $Z$ in $E$, such that:
\begin{itemize}
\item $A\wedge dA^{\sfrac{(n-1)}{2}}$ induces a germ of volume form on $E_x$ for every $x\in M$;
\item $\ker(A\wedge dA^{\sfrac{(n-1)}{2}})= T\mathcal K$;
\item $\ker \!A$ contains $T\mathcal K$ and is invariant by the holonomy of $\mathcal K$.
\end{itemize}

As in the symplectic case,
the next Theorem was known to A.\,Haefliger \cite{haefliger} when $\dim M<n+1$ and to
D.\,McDuff \cite{mcduff} when $\dim M=n+1$.

\begin{thm} \label{cont-structures}
Assume $M$ is a manifold of dimension not greater than the rank of the vector bundle $\nu$.
Let $(\alpha,\beta)$ be \emph{formal contact data}, that is, a section $\alpha $ of $\nu^*$ and a section $\beta$
of $\wedge^2\nu^*$ such that
$\alpha\wedge \beta^{\sfrac{(n-1)}{2}}$ is a non-vanishing section of $\wedge ^n \nu^*$.
Then, there exists a
$\Gamma_n^\cont$-structure $\xi= (A, \mathcal K)$ on $M$ with normal bundle $\nu$ such that,
for all $x\in M$, the next two conditions are fulfilled:
\begin{equation}\label{cont_equation}
\begin{cases}
\ker \! A_{Z(x)}\cap \nu_x= \ker \alpha(x)\\
(dA)_{Z(x)}= \beta(x).
\end{cases}
\end{equation}

Moreover, this statement holds true in a relative parametric version.
\end{thm}

\begin{proof}
For simplicity, we do not formulate any homotopy statement at the level of classifying spaces. Nevertheless, the strategy of proof is similar to the one we used for $\Gamma_n^\symp$-structures.
It is even simpler since every contact isotopy is Hamiltonian.

Let us first consider the non-parametric version. The construction of $\xi$ is performed step by
step over each cell of a cell
decomposition of $M$. We are looking on the last $n$-cell $e^n$ only.
Let $C:= \partial e^n\times[0,1]\cong \bbS^{n-1}\times [0,1]$ be a collar neighborhood
of the boundary, on which we are given a $\Gamma_n^\cont$-structure $\xi= (A,\mathcal K)$ which fulfills
(\ref{cont_equation}).

Since the formal data $(\alpha,\beta) $ extends over $e^n$, there is a trivialization of $\nu$ over $e^n$,
$(p_1,p_2): E\vert e^n\to e^n\times \RR^n$,
in which $(\alpha(x),\beta(x))$ is independent of $x\in e^n$. On $\RR^n$ equipped with $(\alpha,\beta)$,
we may think of $\beta $ as a closed differential form with constant coefficients; by taking a primitive,
we have a unique contact form $\alpha_0$ such that :
\begin{equation}
\begin{cases}
\alpha_0(0)=\alpha\\
d\alpha_0=\beta.
\end{cases}
\end{equation}

We derive a {\it trivial} $\Gamma_n^\cont$-structure $\xi_0=(A_0,\mathcal K_0)$ on $e_n$
such that
\begin{equation}
\begin{cases}
A_0=p_2^*(\alpha_0)\\
dA_0=p_2^*(d\alpha_0).
\end{cases}
\end{equation}
Hence, $\mathcal K_0$ is the horizontal foliation.
Now, there is a Moser type lemma\footnote{The statement comes from Eliashberg-Mishachev's book
\cite{elias} where the proof is left to the reader. We only add the relative
and parametric version.}
which we are going to present below.
This allows us to perform some vertical isotopy which reduces to the case where,
in a small tube $U$ about the zero section and for every point $x\in C$,
we have
\begin{equation}\label{moser_cont-eq}
A_{\vert U_x}= {A_0}_{\vert U_x}.
\end{equation}

\begin{lemme}\label{moser_cont}
Let $(\alpha_t)_{t\in [0,1]}$ be a path of contact forms in a manifold $V^{n}$.
Let $L$ be a hypersurface in $V$. It is assumed that the Reeb vector field $R_t$
of $\alpha_t$ is never tangent to $L$. Then, we have the following:

\begin{enumerate}
\item
The next equation whose unknown is $X_t$ can be solved near $L$:
\begin{equation}\label{5.1}
L_{X_t}\alpha_t +\dot\alpha_t=0.
\end{equation}

\item
Let $(\alpha_t)_{t\in [0,1]}$ be a path of germs in $(\RR^{2p+1}, 0)$ of contact forms. If
$\ker\alpha_t(0)$ is independent of $t$, then these germs are isotopic.
\item
The previous statements hold true with parameters and in a relative version.
\end{enumerate}
\end{lemme}

\skpt
\begin{proof}
\begin{enumerate}
\item
The vector $X_t$ decomposes as $X_t=Y_t+Z_t$ with $Y_t\in \ker\alpha_t$
and
$Z_t= \alpha_t(X_t)\, R_t$. Let us recall that $R_t$ generates in each point the
kernel of $d\alpha_t$. Then, Equation (\ref{5.1}) becomes the following system:
\begin{equation}\label{xx}
\begin{cases}
R_t\cdot\left(\alpha_t(X_t)\right)+\dot \alpha_t(R_t)=0\\
\iota(Y_t)\left({d\alpha_t}_{\vert\ker\alpha_t}\right)+ d \left(\alpha_t(X_t)\right)_{\vert \ker\alpha_t}+\dot{\alpha_t}_{\vert\ker\alpha_t}=0.
\end{cases}
\end{equation}
Fix $t\in [0,1]$. The first equation of this system is a differential equation along the orbits of $R_t$
whose unknown function is $\alpha_t(X_t)$. It has a unique solution if $\alpha_t(X_t)$ is required to equal $0$
along $L$ (here the transversality assumption is used). Then, the component $Z_t$ of $X_t$ is determined.
Once $\alpha_t(X_t)$ is known, the second equation of~(\ref{xx}) has a unique solution since the form induced
by $d\alpha_t$ on $\ker\alpha_t$
is symplectic.

\item
Here $L$ is the
hyperplane which is the common kernel of the contact forms in the considered path. Replace the germs
with genuine representatives. Following the solution of (1), we have
$X_t(0)=0$ for every
$t\in [0,1]$. Therefore, the flow $\varphi_t$ of $X_t$ keeps the origin fixed. It is well defined on some
neighborhood of the origin
up to $t=1$ and it is the identity on $L$. We deduce from
Equation (\ref{5.1}) that the following is satisfied near the origin for every $t\in[0,1]$:
\begin{equation}\label{5.2}
\varphi_t^*(L_{X_t}\alpha_t +\dot\alpha_t)=0,
\end{equation}
and the latter is obtained by taking the time derivative of the next equation
\begin{equation}\label{5.3}
\varphi_t^*\alpha_t=\alpha_0.
\end{equation}
So, the desired isotopy is obtained
by integrating $X_t$.

\item
Considering the equations which are solved, this claim is clear.\qedhere
\end{enumerate}
\end{proof}

We continue the proof of Theorem \ref{cont-structures}. In order to derive (\ref{moser_cont-eq})
from Lemma \ref{moser_cont}, we use $x\in C$ as a parameter and, in each fibre $E_x$, we consider
$\alpha_t= t A_{\vert E_x}+(1-t){A_0}_{\vert E_x}$. Due to the formal data, $\alpha_t$ is a contact form
near the origin of $E_x$ for every $t\in [0,1]$.

After this Moser type reduction, we have to state and prove a lemma similar to
Lemma \ref{vertical}. Actually, it is not useful to write it down explicitly
since it is the same:
the holonomy maps are contactomorphisms; thus, in each fibre of a box the vertical isotopy preserves
the contact distribution $\ker A_0\cap E_x$. Then, it is Hamiltonian with respect to $A_0\vert E_x$.\footnote{The Hamiltonian function of a vertical isotopy of contactomorphisms whose infinitesimal generator is $X_t$ is (in our setting) the time dependent function $z\in U \mto A_0(X_t)(z)$.}
Therefore, it extends globally since extending such an isotopy
amounts to extend its
Hamiltonian function; thus, no obstruction is encountered. This finishes the proof of the non-parametric
version.

For the relative parametric version of Theorem \ref{cont-structures}, we have to
consider a family $(\alpha_u,\beta_u)_{u\in \DD^k}$
which underlies a family of $\Gamma_n^\cont$-structures when $u\in \partial\DD^k$. Thanks to the relative parametric version of Lemma \ref{moser_cont}, we may follow word for word the
proof we gave for $\Gamma_n^\symp$.
\end{proof}

\section{Open manifolds}\label{open}

This section is devoted to the proof of Theorem \ref{step_2_thm}. Let us recall the setting: $\Gamma$ is an open subgroupoid of the structural groupoid $\Gamma(X)$
of a model $n$-manifold $X$ (see Section \ref{intro});
$M$ is an open manifold of dimension $n$ and $\xi\in H^1_{\tau M}(M; \Gamma)$ is a $\Gamma$-structure
on $M$ whose normal bundle is $\tau M$, the tangent space to $M$. Let
$TM$ denote its total space and $Z: M\to TM$ denote the $0$-section.
The associated
$\Gamma$-foliation defined near $Z(M)$ in $TM$ is denoted by $\cF=\cF_\xi$.
We recall a topological fact about open manifolds whose proof
is available in \cite[\S4.3]{elias}.

\begin{prop} \label{spine}
Given an open $n$-manifold $M$, there exists an $(n-1)$-polyhedron $K\subset M$, called a
\emph{spine} of $M$, so that the inclusion is a homotopy equivalence. More precisely, for any
regular
neighborhood $V$ of $K$, there exists a \emph{compressing} isotopy of embeddings $f_t: M\to M,\ t\in [0,1],$
from $\Id_M$ to an embedding $f_1:M\to V$, which is stationary on a neighborhood of $K$
and such that $t'>t$ implies $f_{t'}(M)\subset f_t(M)$.
\end{prop}

We are going to prove the next statement from which Theorem \ref{step_2_thm} will be
easily
derived.

\begin{thm}\label{spine_reg} Let $K\subset M$ be an $(n-1)$-dimensional polyhedron in an
$n$\nobreakdash-mani\-fold~$M$ (open or closed).
Let $\xi_s ,\ s\in \DD^k$, be a $k$-parameter family
of $\Gamma$\nobreakdash-structures on~$M$ with normal bundle $\tau M$.
When $s\in \partial \DD^k$, it is assumed that the associated foliation~$\cF_s$ is tangent to $\cF_{\exp}$
along $Z(M)$. Then, there exist an open neighborhood $V$ of $K$ in~$M$ and a
$k$-parameter family $\bar \xi_s$ of $\Gamma$-structures on $M\times[0,1]$
-- that is, concordances of~$\xi_s$ -- such that:
\begin{itemize}
\item $\bar\xi_s\vert V\times\{0\}= \xi_s\vert V$;
\item $\bar\xi_s\vert V\times\{1\}$ is regular and its associated foliation is tangent to $\cF_{\exp}$;
\item $\bar\xi_s= p_1^*\left(\xi_s\vert V\right)$ for every $s\in \partial \DD^k$, where $p_1$ denotes
the projection $M\times[0,1]\to M$.
\end{itemize}
\end{thm}
In other words, $(\bar\xi_s)_{s\in \DD^k}$ is a family of regularization concordances
on a neighborhood of a $K$,
relative to the boundary of the parameter space.

\subsection{Proof of Theorem \ref{step_2_thm} from Theorem \ref{spine_reg}}
Here, $M$ is an open manifold. A~spine $K$ of $M$ may be chosen
(Proposition \ref{spine}) and
Theorem \ref{spine_reg} applies to these data: $K\subset M, (\xi_s)_{s\in \DD^k}$.
So, we have a family $\bar\xi_s$ of regularization
concordances on some neighborhood $V$ of $K$ in $M$,
relative to $\partial \DD^k$.
We have to extend this family to a family of regularization concordances
over the whole of $M$,
still relative to $\partial \DD^k$. We may assume there exists $\rho$ close to $1$ so that $\xi_s$
is regular on $M$ when $\Vert s\Vert\in[\rho,1]$.

We first insert the family of concordances
described as follows,
where $t\in [0,1]$ is the parameter of the concordance:
\begin{itemize}
\item for $\Vert s\Vert \leq \rho$, we put the concordance $t\mto f_t^*\xi_s$, where $(f_t)_{t\in[0,1]}$
is the isotopy of embeddings given by Proposition \ref{spine};
\item for $\rho\leq\Vert s\Vert \leq 1$, we put the concordance
$t \mto f_{\left(\frac{1-\Vert s\Vert}{1-\rho}t\right)}^*\xi_s$.
\end{itemize}
When $t=1$ (that is the {\it end} of these concordances depending on $s\in \DD^k$) and when $\Vert s\Vert\geq\rho$,
the structures are regular on $M$.
Then, denoting by $S: [1,2]\to [0,1]$ the shift $t\mto t-1$, we continue, for $t\in [1,2]$ with the concordances $(f_1\times S)^*\bar\xi_s$ when $\Vert s\Vert \leq\rho$; these ones are stationary
when $\Vert s\Vert =\rho$. Thus, we are allowed to extend them by the stationary concordances
when $\rho\leq \Vert s\Vert \leq 1$. Of course, the previous piecewise description can be made smooth
if desired.\qed

\subsection{Proof of Theorem \ref{spine_reg} without parameters ($k=0$)}\label{no_parameter}
We start with a $\Gamma$\nobreakdash-structure $\xi$ on $M$.
Let $ \cF$ be its associated $\Gamma$-foliation
defined in some small neighborhood $U$ of $Z(M)$. Let $\xi^u$
(\resp $\cF^u$) be the underlying $\Gamma_n$-structure (\resp $\Gamma_n$-foliation)
of $\xi$ (\resp $\cF$) where the transverse geometry is forgotten.

The proof will consist of two steps: in the first step, we will make a specific
regularization
of $\xi^u$ by some $\Gamma_n$-concordance
over $M\times [0,3]$;
in the second step, the geometric $\Gamma$-structure of the concordance will be defined
only over a small neighborhood of $K\times [0,3]$.
Finally, we get the $\Gamma$-regularization of $\xi$ near $K$.

\subsubsection*{First step}
Fix a small $\ep>0$.
As in Thurston \cite{thu74}, we consider a one-parameter family $P_t$, $t\in [0,3]$, of
$n$-plane fields on $U$ with the following properties:

\begin{itemize}
\item
$P_t$ is transverse to the fibres for every $t$.
\item
$P_t$ is tangent to $\mathcal F$ when $t\in [0,1+\ep]$.
\item
$P_t$ is tangent to $\mathcal F_{\exp}$ when $t\in [2, 3]$.
\end{itemize}

Such a plane field exists by barycentric combination in the convex set\footnote{\label{convex-note}Take an $n$-plane field $Q$ transverse to the fibres. The above-mentioned convex set is
affinely isomorphic to $\hom (Q,\tau^vTM)$, where $\tau^v$ stands for the sub-bundle of $\tau(TM)$ tangent to the fibres of $TM\to M$.} of the
plane fields transverse to the fibres of $\tau M$.

Let $T$ be a triangulation of $M$ containing a subdivision of $K$ (also called $K$)
as a sub-complex and
fine enough with respect to the open covering
$\{\exp_x(U_x) \mid x\in M\}$ in order that formula (\ref{1-formula}) makes sense. Here, we recall
that formula which holds for~$x$ in any simplex of $T$:
$$
\exp_x(j^r(x)) =\sigma^r(x).
$$

We now consider the Thom jiggling given by formula (\ref{1-formula});
its order $r$ is chosen large enough so that the
$n$-simplices of $j^r(T^r)$ are transverse to
$P_t$ for every $t\in[0,3]$.

The first piece of the concordance, when $t\in [0,1+\ep]$, actually a $\Gamma$-concordance,
consists of moving the zero section from
$Z$ to $j^r$ by traversing any homotopy valued in $U$ and
stationary when $t\in[1,1+\ep]$.
The concordance of $\Gamma$-structure is given
by pulling $\xi$ back by this homotopy of maps $M\to U$
(look at Remark \ref{smoothness}(1) about smoothness).

We now describe the second piece of the concordance, when $t\in [1,2]$.
We consider the codimension $n$-plane field
$\tilde P$ in $U\times [0,3]$ defined by
\begin{equation}
\tilde P(x,t): = P_t(x)\oplus \RR \partial_t .
\end{equation}
It is tangent to $\mathcal F\times [0,1+\ep]$ and to $\mathcal F_{\exp}\times [2,3]$.
The trace of $\tilde P$ on each $(n+1)$-cell of $j^r(M)\times [1,2]$
is one-dimensional. Then, this trace is integrable.
Thus, there is a $C^0$-small smooth approximation of $\tilde P$,
relative to $t\in[0,1]\cup [2, 3]$ and still denoted
by~$\tilde P$, which is integrable near $j^r(M)\times [1,2]$.
Now, the pair $(j^r(M)\times [1,2], \tilde P)$
defines a concordance of $\Gamma_n$-structures. This finishes the second piece.\enlargethispage{-\baselineskip}%

The third piece of the concordance when $ t\in[2,3]$ consists of keeping the foliation
$\cF_{\exp}$ and applying the homotopy
from $j^r$ to the $0$-section $Z$ as provided by Proposition \ref{jig}(2). On the whole, we built a specific regularization concordance
of the underlying $\Gamma_n$-structure $ \xi^u $, which is nearly sufficient for our purpose.

We need more of {\it good position}.
Let $K^r $ denote the $(n-1)$-dimensional complex which is
the $r$-th Thom subdivision of $K$.
Let $\tilde K^r$ be the image of $Z(K^r)\times [0,3]$ along the concordance built above.
This is an $n$-complex
whose $n$-cells are not transverse to $\tilde P$. When $t\in [1,2]$, the only reason
for non-transversality is
that $\tilde K^r$ and $\tilde P$ share the $\partial_t $-direction.
Let $\tilde K^r_{[t,t']}$ (\resp $\tilde K^r_t$) denote the restriction
of $\tilde K^r$ over $M\times[t,t']$ (\resp $M\times\{t\})$.

When the $n$-cells of $\tilde K^r_{[t,t']}$ are prismatic (that is, $\text{simplex}\times\text{interval}$)
which is always the case when $[t,t']\subset [1,2]$,
they will receive the {\it standard subdivision} defined by H.\,Whitney \cite[App.\,II]{whitney};\footnote{This subdivision that W.\,Thurston names {\it crystalline} is clearly explained inside the proof of his famous {\it Jiggling} Lemma.} this latter only depends on an order chosen on the set of vertices of $\tilde K^r_t$.

\begin{claim}
There exist a subdivision $t_1=1,t'_1,\dots,t_i, t'_i,\dots, t_N= 3$
and a small piecewise smooth vertical isotopy, its time-one map being denoted by $\psi$, such that:
\begin{enumeratei}
\item
$\psi\vert \tilde K^r_{t_i}= \Id$ for every $i= 1,\dots,N$;
\item
for every $n$-simplex $\tau$ of the standard subdivision of $\tilde K^r _{[t_i,t'_i]}$
(\resp $\tilde K^r _{[t'_i,t_{i+1}]}$), the image $\psi(\tau)$ is smoothly embedded in $U\times[0,1]$ and
quasi-transverse to $\tilde P$. Here, quasi-transverse means transverse when $\dim\tau\geq n$
and no tangency when $\dim\tau<n$.
\end{enumeratei}
\end{claim}

\begin{proof}[Proof of the claim]
We search for a {\it jiggling in time}. We are going to do it
for $\tilde K^r _{[1,2]}$; the jiggling in time of $\tilde K^r _{[2,3]}$ is similar, but a bit more complicated
due to the fact that the cell decomposition is not purely prismatic. A numbering of the vertices
of~$\tilde K^r _1$ is fixed: $v_1,v_2, \dots$; this numbering propagates to the corresponding
vertices of $\tilde K^r _t$ for every $t\in [1,2]$.

The time subdivision is chosen so that, for every $(n-1)$-simplex $c\subset \tilde K^r_t$ and every $x\in c$,
the hyperplane $H_t(x,c):= T_x(c)+P_t(x)$
varies very little in $TU$ when $t$ traverses $[t_i,t_{i+1}]$, uniformly when $x$ runs in any star.
Let $t'_i$ be the middle of this interval. Each~$\tilde K^r_{[t_i,t'_i]}$ (\resp $\tilde K^r_{[t'_i,t_{i+1}]}$)
receives the {\it standard subdivision} of the prismatic cells. On $\tilde K^r_{[t_i,t'_i]}$, the desired embedding $\psi$ and
its isotopy
from $\Id$ are constructed recursively on the numbered stars of vertices $\Star(v_1), \Star(v_2),\dots$.
Precisely, there is a locally finite family of isotopies $\chi_1, \chi_2,\dots $
(that is, only finitely many supports intersect) and $\psi$ will be the composition of their time-one map:
$\cdots\circ\chi_2^1\circ \chi_1^1$.
The reversed isotopies are used over the interval $[t'_i,t_{i+1}]$.

Let $v_1$ be the {\it first} vertex of $\tilde K^r_{t_i}$. Let $X_1$ be a small vertical vector in $T_{v_1}U$
which is chosen linearly independent form all above-mentioned hyperplanes
$H_{t_i}(x,c)$, where~$c$ is any $n$-simplex of $\tilde K^r_{t_i}$ passing through $v_1$.
Let $v'_1$ be the corresponding vertex in~$\tilde K^r_{t'_i}$. By definition of the standard subdivision
of $\tilde K^r_{[t_i,t'_i]}$, the vertex
$v'_1$ is joined to $\Star(v_1, \tilde K^r_{t_i})$, the star of $v_1$ in $\tilde K^r_{t_i}$. In
affine notation $\chi_1^1$ is obtained by replacing~in~$\tilde K^r_{[t_i,t'_i]}$
\begin{equation}
\Star(v'_1, \tilde K^r_{[t_i,t'_i]})\quad \text{with}\quad \bigl(v'_1+X_1\bigr)*\lk(v'_1, \tilde K^r_{[t_i,t'_i]}), \end{equation}
where $\lk(a,-)$ stands for the {\it link} of $a$, that is, the union of simplices in
$\Star(a,-)$ which do not contain $a$.
After this step, we get the property that every $n$-simplex in
$B_1:= \chi_1^1\bigl(\tilde K^r_{[t_i,t'_i]}\bigr)$ passing through $v'_1+X_1$ is transversal to $\tilde P$.

The second step consists of a similar construction related to the star of
$v'_2$ in $B_1$ by using a small vertical vector $X_2$ which is linearly independent of the
hyperplanes
made with the $n$-simplices of $B_1$ passing through $v'_2$. This yields $\chi_2^1$, the time-one map
of the second isotopy which allows us to gain the transversality of new $n$-simplices to~$\tilde P$.
When $\Star(v'_2,B_1)$ meets
$\Star(v'_1+X_1, B_1)$, the vector $X_2$ is chosen so small that the property gained in the first step
is preserved. And so on.
\end{proof}

In what follows, we still denote by $\tilde K^r$ the outcome of the previous jiggling. As a result,
every $n$-cell of $\tilde K^r_{[1, 3]}$ is transverse to $\tilde P$.
This simplicial complex $\tilde K^r$
collapses\footnote{A simplicial complex $L$
collapses to $K$ if there is a sequence of elementary collapses $L_q\searrow L_{q+1}$
starting with $L$ and ending with $K$. An elementary collapse means that $L_q$ is the union of
$L_{q+1}$ and a simplex $\sigma$ so that $\sigma\cap L_{q+1}$ is made of the boundary of $\sigma$ with an
open facet removed.}
successively to $\tilde K^r_{[0,2]}$ and then to $\tilde K^r_{[0,1]}$.

\subsubsection*{Second step}
We now focus on $\tilde K^r_{[0,1]}$ on which $\tilde K^r$ collapses.
Since the cells of $\tilde K^r_{[1,3]}$ of positive dimension are quasi-transverse to the foliation $\tilde P$,
the collapse $\tilde K^r \searrow \tilde K^r_{[0,1]}$
extends to a collapse of pair
\begin{equation}\label{collapse}
(\tilde K^r, \tilde P)\searrow(\tilde K^r_{[0,1]}, \tilde P).
\end{equation}
Let $N(\tilde K^r)$ denote a small neighborhood of $\tilde K^r$ in $U\times [0,3]$.
From the sequence of elementary collapses, one derives step by step
an embedding of pair
\begin{equation}\label{emb}
\Phi: \bigl(N(\tilde K^r), \tilde P\bigr)\to \bigl(U\times [0,1+\ep), \,\mathcal F\times [0,1+ \ep)\bigr),
\end{equation}
which induces the inclusion $N(\tilde K^r_{[0,1]}) \hto U\times[0,1+\ep)$.
Since $\mathcal F\times [0,1+\ep]$
is a $\Gamma$\nobreakdash-foliation, $\tilde P_{\vert N(\tilde K^r)}$ is so
by pulling back through $\Phi$. Therefore,
$\Phi^*\bigl(\mathcal F\times[0,1+\ep)\bigr)$ is a regularization concordance
of the $\Gamma$-structure which is induced near $K$.\qed

\smallskip
This last process associated with collapses is named {\it inflation} in W.\,Thurston's article \cite{thu74}.

\subsection{Relative parametric version of Theorem \ref{spine_reg}}
Here, the data consist of a family $(\xi_s)_{s\in\DD^k}$ of $\Gamma$-structures whose normal bundle is
the tangent bundle $\tau M$. It is assumed that,
for every $s\in \partial \DD^k$, the associated foliation $\cF_s$ is tangent to $\cF_{\exp}$
along the $0$-section $Z(M)$,
hence
$\xi_s$ is regular. Without loss of generality, we may assume~$\xi_s$ is regular
when $1\geq \Vert s\Vert\geq \rho $ for some $\rho$ close to $1$.
The proof just consists of two remarks.

\begin{enumerate}
\item
The previous proof (see Subsection \ref{no_parameter})
works directly for our $k$-parameter family of data if we do not take care of the boundary condition.
Indeed,
observe that a common order $r$ of Thom jiggling $j^r$ can
be chosen for all $s\in \DD^k$ since the family of $n$-plane fields $P_{s,t}$ we have to look at
is compact. Thus, if the jiggling is vertical enough, that is, for $r$ large enough, its $n$-simplices are transverse to $P_{s,t}$
for every $(s,t)\in \DD^k\times [0,3]$. The $0$-parameter process applies for every $s\in \DD^k$
and yields a regularization in a fixed neighborhood $V$ of $K$ in $M$. Precisely, we have formulas
(\ref{collapse}) and (\ref{emb}) depending on the parameter $s\in \DD^k$, yielding
regularization concordances $\Phi_s^*\bigl(\mathcal F_s\times[0,1+\ep)\bigr)$.

\item
We may assume that $P_{s,t}$ is tangent to $\cF_s$ for every
$s\in\{1\geq \Vert s\Vert\geq \rho\}$ and $ t\in [0,3]$. For $\Vert s\Vert\in [\rho,1]$,
set $\mu(s):= \frac{1- \Vert s\Vert}{1-\rho}$. Recall that $\Phi_s$ is the time-one map of an isotopy
of embeddings $\Phi_s^w: N(\tilde K^r)\to U\times [0,3] $, $w\in [0,1]$, relative to
$N(\tilde K^r_{[0,1]})$ and such that
$\Phi_s^0= \Id$ and $\Phi_s^1\bigl(N(\tilde K^r)\bigl) \subset N(\tilde K^r_{[0,1+\ep]})$.
\end{enumerate}

We finish, for $s\in\{1\geq \Vert s\Vert\geq \rho\}$, with the regularization concordance
$(\Phi_s^{\mu(s)})^*\bigl(\mathcal F_s\times[0,3]\bigr)$. When $\Vert s\Vert =1$, this is the trivial
concordance $\mathcal F_s\times [0,3]$. Then, the relative version is proved.\qed

\backmatter
\bibliographystyle{jepplain}
\bibliography{laudenbach-meignez}
\end{document}