Weinstein manifolds as cotangent buildings

We introduce the framework of cotangent buildings to complement and refine that of Weinstein handlebodies. While Weinstein handlebodies are suitable for a bottom-up" analysis of the Weinstein structure, cotangent buildings also enable a top down" analysis. Further, cotangent buildings include a precise control over the interaction of any subcollection of the various building blocks, each of which is modeled on the cotangent bundle of a manifold with corners. Our main result is that any Weinstein manifold is Weinstein homotopic to one admitting the structure of a cotangent building.