Let M2nM2n be a Poisson manifold with Poisson bivector field Π. We say that M is b -Poisson if the map Πn:M→Λ2n(TM)Πn:M→Λ2n(TM) intersects the zero section transversally on a codimension one submanifold Z⊂MZ⊂M. This paper will be a systematic investigation of such Poisson manifolds. In particular, we will study in detail the structure of (M,Π)(M,Π) in the neighborhood of Z and using symplectic techniques define topological invariants which determine the structure up to isomorphism. We also investigate a variant of de Rham theory for these manifolds and its connection with Poisson cohomology
