Geometric obstructions for $\xi$-fillings of 3-manifolds

We consider the realisation problem for normal 1-types of 4-manifolds with a given boundary. More precisely, given a normal 1-type $\xi$ and closed 3-dimensional $\xi$-manifold $Y$, does there exist a compact 4-dimensional $\xi$-manifold with boundary $Y$? We describe a three stage obstruction theory for the existence of such a 4-manifold, with our main contribution being a `tertiary’ obstruction that we describe geometrically via Wall’s quadratic self-intersection form.