Cohomology of Hierarchical Tilings

This is a chapter in an upcoming book on aperiodic order. We go over different versions of tiling cohomology (\v Cech, pattern-equivariant, PV, quotient) with emphasis on the inverse limit constructions used to compute these cohomologies. We then consider the uses of tiling cohomology to distinguish spaces, to understand deformations, and to help understand maps between tiling spaces. The emphasis of this chapter is on substitution tilings and their generalizations, but the underlying ideas apply equally well to cut-and-project tilings and to tilings defined by local matching rules.