  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 \v Cech
cohomology $\check H^1(\Omega)$ gives upper and lower bounds for the size of
the return module. For all sufficiently large patches, the rank of the return
module is at most the same as the rank of the cohomology. 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 sufficiently large patches, the rank of the return module is equal
to the rank of $\check H^1(\Omega)$.
