How big is a tiling’s return module?

Abstract The rank of a tiling’s return module depends on the geometry of its tiles and is not a topological invariant. However, the rank of the first Čech cohomology $\check H^1(\Omega )$ gives upper and lower bounds for the rank of the return module. For all sufficiently large patches, the rank of the return module is at most the rank of $\check H^1(\Omega )$ . For a generic choice of tile shapes and an arbitrary reference patch, the rank of the return module is at least the rank of $\check H^1(\Omega )$ . Therefore, for generic tile shapes and all sufficiently large patches, the rank of the return module is equal to the rank of $\check H^1(\Omega )$ .