Symplectic and Poisson geometry on b-manifolds

Let M 2 n be a Poisson manifold with Poisson bivector field Π . We say that M is b -Poisson if the map Π n : M → Λ 2 n ( T M ) intersects the zero section transversally on a codimension one submanifold Z ⊂ M . This paper will be a systematic investigation of such Poisson manifolds. In particular, we will study in detail the structure of ( 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.