From ball fillings to cube tilings

Which 3-manifolds bound a smooth rational homology 4-ball?  This is a central and open-ended question in low-dimensional topology.  I will describe an obstruction which applies to a certain class of 3-manifolds and which takes the form of a lattice embedding / graph theory condition.  We conjecture that the obstruction is complete, and we prove it in a certain extremal case. The proof utilizes the Hajós-Minkowski theorem that every lattice tiling of Euclidean space by cubes contains a pair of cubes that abut along a facet of each. The talk is based on joint work, part with Slaven Jabuka, part with Brendan Owens.