Poincaré duality for pattern-equivariant (co)homology

Abstract

Two aims in studying the topology of tiling spaces are, firstly, to understand how one may interpret topological invariants of tiling spaces and, secondly, to find ways of actually computing them for specific examples. In the first direction, I will show how one may interpret the Čech cohomology groups of a tiling in a highly geometric way, via a Poincaré duality result, using so called “pattern-equivariant chains” on the tiling. These groups have an analogous definition to the well-known pattern-equivariant cohomology groups. I will present an efficient method for computing these groups for hierarchical tilings. When considering the rotation-invariant versions of these groups, one often finds extra torsion in the calculated invariants to the Čech groups. Above being mere artefacts of the calculations, I will show how one may incorporate these extra torsion groups into a spectral sequence converging to the cohomology of the Euclidean hull of a tiling.