Pattern-Equivariant Homology

Pattern-equivariant (PE) cohomology is a well-established tool with which to interpret the \v{C}ech cohomology groups of a tiling space in a highly geometric way. In this paper we consider homology groups of PE infinite chains. We establish Poincar\'{e} duality between the PE cohomology and PE homology. The Penrose kite and dart tilings are taken as our central running example, we show how through this formalism one may give highly approachable geometric descriptions of the generators of the \v{C}ech cohomology of their tiling space. These invariants are also considered in the context of rotational symmetry. Poincar\'{e} duality fails over integer coefficients for the `ePE homology groups' based upon chains which are PE with respect to orientation-preserving Euclidean motions between patches. As a result we construct a new invariant, which is of relevance to the cohomology of rotational tiling spaces. We present an efficient method of computation of the PE and ePE (co)homology groups for hierarchical tilings.