An uncountable set of tiling spaces with distinct cohomology

TL;DR

This paper extends the Barge-Diamond complex framework to mixed tiling substitution systems, providing a method to compute their cohomology and demonstrating an uncountable diversity of tiling spaces with distinct cohomological properties.

Contribution

It introduces a generalized approach for analyzing tiling spaces via inverse limits and cohomology calculations, revealing uncountably many distinct cohomology classes in specific substitution systems.

Findings

Developed a method to compute Čech cohomology of tiling spaces.

Proved existence of uncountably many tiling spaces with distinct cohomology.

Extended Barge-Diamond complexes to mixed substitution systems.

Abstract

We generalise the notion of a Barge-Diamond complex, in the one-dimensional case, to a mixed system of tiling substitutions. This gives a way of describing the associated tiling space as an inverse limit of Barge-Diamond complexes. We give an effective method for calculating the \v{C}ech cohomology of the tiling space via an exact sequence relating the associated sequence of substitution matrices and certain subcomplexes appearing in the approximants. As an application, we show that there exists a system of three substitutions on two letters which exhibit an uncountable collection of minimal tiling spaces with distinct isomorphism classes of \v{C}ech cohomology.