The geometry of $b^k$ manifolds

Let $Z$ be a hypersurface of a manifold $M$. The $b$-tangent bundle of $(M, Z)$, whose sections are vector fields tangent to $Z$, is used to study pseudodifferential operators and stable Poisson structures on $M$. In this paper we introduce the $b^k$-tangent bundle, whose sections are vector fields with "order $k$ tangency" to $Z$. We describe the geometry of this bundle and its dual, generalize the celebrated Mazzeo-Melrose theorem of the de Rham theory of $b$-manifolds, and apply these tools to classify certain Poisson structures on compact oriented surfaces.