Substitutions with Vanishing Rotationally Invariant First Cohomology

Abstract

The cohomology groups of tiling spaces with three-fold and nine-fold symmetries are obtained. The substitution tilings are characterized by the fact that they have vanishing first cohomology group in the space of tilings modulo a rotation. The rank of the rational first cohomology, in the tiling space formed by the closure of a translational orbit, equals the Euler totient function evaluated at 𝑁 if the underlying rotation group is 𝐙𝑁. When the symmetries are of crystallographic type, the cohomologies are infinitely generated.