Cohomology of One-dimensional Mixed Substitution Tiling Spaces

TL;DR

This paper computes the cohomology of one-dimensional tiling spaces generated by multiple substitutions, introducing a universal complex to simplify calculations and establish bounds on cohomology ranks.

Contribution

It introduces a universal Anderson-Putnam complex for multiple substitutions and provides bounds on the first cohomology group of tiling spaces.

Findings

Universal Anderson-Putnam complex simplifies cohomology calculations.

Bounds on the first cohomology rank based on tile count.

Isomorphism between projective limit and tiling space under certain conditions.

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.