Title: Feuilletages en surfaces, cycles \'evanouissants et
vari\'et\'es de Poisson
Author: F. Alcalde Cuesta et G. Hector
Notes: AMS-LaTeX version 1.1, 26 pages, to appear in
{\em Monatshefte f\"ur Mathematik}
\begin{abstract}
A {\em foliated cylinder} of a foliated manifold $(M,{\cal F})$ is a path
$\{c_t\}_{t \in [0,1]}$ of integral loops for $\cal F$, i.e. each loop $c_t$ is
supported by a leaf $F_t \in {\cal F}$. Such a cylinder defines a
{\em non-trivial vanishing cycle} $c_0$ if $c_t$ is null-homotopic in its
support $F_t$ for each $t>0$, but $c_0$ is not null-homotopic in its support
$F_0$. Vanishing cycles were introduced by S. P. Novikov in order to
describe $2$-dimensional foliations on compact $3$-manifolds. Here we
use this concept to study foliations of higher codimension.
Our first aim will be to relate triviality of vanishing
cycles with topological properties of the {\em homotopy groupoid}; indeed,
we show that all vanishing cycles are trivial if and only if the total space
of the homotopy groupoid is Hausdorff.
To do so, we reduce the problem to considering so-called {\em regular}
vanishing cycle (an "orthogonal" version of the classical notion of immersed
vanishing cycle) and {\em coherent} vanishing cycle (for which we require
the existence of a path $\{D_t\}_{t \in ]0,1]}$ of integral discs $D_t$
whose boundary $\partial D_t$ is equal to $c_t$). We show that
triviality of these particular vanishing cycles implies triviality of all
vanishing cycles. For compact foliated manifolds, we obtain the following
criterion: a regular coherent vanishing cycle is non-trivial if and only if the
area of the discs $D_t$ is unbounded.
Finally, we give two applications of the previous results to foliations by
surfaces: we generalize the Reeb stability theorem to higher codimensions
and we solve the problem of the symplectic realization of Poisson
structures supported by a $2$-dimensional foliation.
\end{abstract}
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\noindent
{\em 1991 Mathematics Subject Classification}. Primary 57R30;
Secondary 58F05.
\noindent
{\em Key words}. Feuilletages, cycles \'evanouissants,
vari\'et\'es de Poisson.