  In this note we use techniques in the topology of 2-complexes to recast some
tools that have arisen in the study of planar tiling questions. With spherical
pictures we show that the tile counting group associated to a set $T$ of tiles
and a set of regions tileable by $T$ is isomorphic to a quotient of the second
homology group of a 2-complex built from $T$. In this topological setting we
derive some well-known tile invariants, one of which we apply to the solution
of a tiling question involving modified rectangles.
