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

Substitutions with Vanishing Rotationally Invariant First Cohomology