Topological invariants of substitution tilings and rotations

Abstract

To an aperiodic tiling one may assign cohomology groups, topological invariants which retain important combinatorial information about the tiling in question. In this talk I shall review the construction of these cohomology groups and explain some approaches to computing them for substitution tilings. I shall also explain how to visualise them using pattern equivariant (PE) cochains. Via Poincaré duality, one may also visualse them via dual PE chains. When coupled with the action of rotation, this approach can often pick out familiar geometric features of aperiodic patterns. I shall explain recent work in determining the cohomology of the full Euclidean hull of a tiling using the rotation action on the cohomology, allowing us to determine the Čech cohomology of the Euclidean hull of Penrose tilings.