Rotational cohomology and total pattern equivariant cohomology of tiling spaces acted on by infinite groups

In 2003, Johannes Kellendonk and Ian Putnam introduced pattern equivariant cohomology for tilings. In 2006, Betseygail Rand defined a type of pattern equivariant cohomology that incorporates rotational symmetry, using representation of the rotation group. In this doctoral thesis we study the relationship between these two types of pattern equivariant cohomology, showing exactly how to calculate one from the other in the case in which the rotation group is a finitely generated abelian group of free rank 1. We apply our result by calculating the cohomology of the pinwheel tiling.