Feuilletages en surfaces, cycles évanouissants et variétés de Poisson

Abstract

Surface foliations, vanishing cycles and Poisson manifolds. Afoliated cylinder of a foliated manifold (M,F) is a path of integral loopsc _t forF. Such a cylinder defines anon-trivial vanishing cycle c ₀ ifc _t is null-homotopic in its supportF _t for eacht>0, butc ₀ is not null-homotopic in its supportF ₀. Vanishing cycles were introduced by S. P. Novikov to study qualitative aspects of codimension one foliations. In this paper we apply this notion to the study of higher codimensional foliations.

The first aim is to show the influence that the triviality of vanishing cycles exerts on the topology of thehomotopy groupoid ofF. It is natural to try to reduce the study of triviality to more regular vanishing cycles, as well as to obtain a nice criterion of triviality. In this way, we introduce the notion ofregular vanishing cycle as the “orthogonal” version (for a riemannian metric onM) of the classical notion of immersed vanishing cycle and the notion ofcoherent vanishing cycle, i.e. an integral discD ₁ with boundaryc ₁ extends to a global foliated homotopyD _t such thatc _t is the boundary ofD _t for eacht>0. We also prove that the triviality of these vanishing cycles implies the 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 discsD _t converges to infinity.

Finally, we give two applications of these results to surface foliations: we generalize the Reeb stability theorem to higher codimensions and we resolve the problem of the symplectic realization of Poisson structures supported by surface foliations.