Intégration Symplectique des Variétés de Poisson Totalement Asphériques

Abstract

Poisson structures are contravariant structures. Nevertheless there is a good description of regular Poisson manifolds by means of foliated symplectic forms. This point of view makes it easy to lift these structures to the holonomy or homotopy groupoïd of their symplectic (regular) foliation; defining the Poisson realization of the Poisson structure.The second aim of the paper is to construct the universal symplectic integration of totally aspherical Poisson structures that is regular Poisson structures such that: i) the second homotopy group of any symplectic leaf is trivial; ii) any vanishing cycle is trivial. The universal symplectic integration is a symplectic groupoïd with connected and simply connected fibres which realizes the given Poisson structures.This construction generalizes the construction of the simply connected Lie group of a given finite-dimensional Lie algebra.Keywordsfeuilletagestructure de Poissonsymplectiquealgèbroïdegroupoïde