  The homology group of a tiling introduced by M. Reid is studied for certain
topological tilings. As in the planar case, for finite square grids on
topological surfaces, the method of homology groups, namely the non-triviality
of some specific element in the group allows a `coloring proof' of
impossibility of a tiling. Several results about the non-existence of polyomino
tilings on certain square-tiled surfaces are proved in the paper.
