Pattern-equivariant cohomology of tiling spaces with rotations

This paper develops a new cohomology theory on generalized tiling spaces. This theory incorporates both the rotational geometry of the tiling space and the local pattern geometry into the structure of the cohomology groups. Our use of the local pattern geometry is a generalization of pattern-equivariant cohomology, a theory developed by Ian Putnam and Johannes Kellendonk in 2003. It was defined for tilings whose tiles appear as translates. The most general setting in tiling theory is to work with tiling spaces, with an action of a subgroup of the Euclidean group. This paper defines a new, general pattern-equivariant cohomology for tiling spaces with finite rotation groups, and proves that it is preserved under homeomorphisms which commute with the action of the group. It is conjectured here that this theory is not a topological invariant for tiling spaces with infinite rotation group.