Facultad de Ciencias Matemáticas y Físicas, Universidad de Oviedo, 33007 Oviedo, Spain
Copyright © 2012 Juan García Escudero. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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 if the underlying rotation group is . When the symmetries
are of crystallographic type, the cohomologies are infinitely generated.