Cohomology of one-dimensional mixed substitution tiling spaces

Abstract

We compute the Cech cohomology with integer coefficients of one-dimensional tiling spaces arising from not just one, but several different substitutions, all acting on the same set of tiles. These calculations involve the introduction of a universal version of the Anderson–Putnam complex. We show that, under a certain condition on the substitutions, the projective limit of this universal Anderson–Putnam complex is isomorphic to the tiling space, and we introduce a simplified universal Anderson–Putnam complex that can be used to compute Cech cohomology. We then use this simplified complex to place bounds on the rank of the first cohomology group of a one-dimensional substitution tiling space in terms of the number of tiles.