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Given a Lie algebroid we discuss the existence of a smooth abelian integration of its abelianization. We show that the obstructions are related to the extended monodromy groups introduced recently in [9]. We also show that this groupoid can be obtained by a path-space construction, similar to the Weinstein groupoid of [6], but where the underlying homotopies are now supported in surfaces with arbitrary genus. As an application, we show that the prequantization condition for a (possibly non-simply connected) manifold is equivalent to the smoothness of an abelian integration. Our results can be interpreted as a generalization of the classical Hurewicz theorem.

 

We survey recent results on the local and global integrability of a Lie algebroid, as well as the integrability of infinitesimal multiplicative geometric structures on it.

 

Abstract This is the first in a series of papers dedicated to the study of Poisson manifolds of compact types (PMCTs). This notion encompassesseveral classes of Poisson manifolds defined via properties of their symplectic integrations. In this first paper we establish some fundamentalproperties and constructions of PMCTs. For instance, we show that their Poisson cohomology behaves very much like thede Rham cohomology of a compact manifold (Hodge decomposition, non-degenerate Poincaré duality pairing, etc.)and that the Moser trick can be adapted to PMCTs. More important, we find unexpected connections between PMCTs and symplectic topology: PMCTsare related with the theory of Lagrangian fibrations and we exhibit a construction of a non-trivialPMCT related to a classical question on the topology of the orbits of a free symplectic circle action.In subsequent papers, we will establish deep connections between PMCTs and integral affine geometry,Hamiltonian G -spaces, foliation theory, orbifolds, Lie theory and symplectic gerbes.