PATTERN-EQUIVARIANT HOMOLOGY
James Walton · 2016
Abstract. Pattern-equivariant (PE) cohomology is a well established tool with which to in-terpret the Čech cohomology groups of a tiling space in a highly geometric way. We consider here homology groups of locally finite but non-compactly supported PE chains. For FLC tilings with respect to translations, we show that these groups are Poincare ́ dual to the PE cohomology groups. For tilings with FLC with respect to rigid motions, the PE chains exhibit a singular behaviour at points of rotational symmetry which often adds extra torsion to the calculated invariants. We present an efficient method for computation of these groups for hierarchical tilings.