Substitutions with vanishing rotationally invariant first cohomology (Q764590)

For any tiling \(X\) we can consider a tiling space \(\Omega_X\) formed by the closure of the translational orbit of one tiling \(X\). If \(X\) has \(N\)-fold rotational symmetry, we can also consider the tiling space \(\overline{\Omega_X}=\Omega_X/Z_N\). Many properties of tiling \(X\) can be described in the terms of cohomologies of the tiling spaces \(\Omega_X\) and \(\overline{\Omega_X}\). In the paper Čech cohomologies of \(\Omega_X\) and \(\overline{\Omega_X}\) for some substitution tilings \(X\) with 3-fold and 9-fold rotational symmetry are computed.