THE TOPOLOGY OF TILE INVARIANTS

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.