Cohomology of Substitution Tiling Spaces

TL;DR

This paper extends methods for computing the cohomology of substitution tiling spaces to higher dimensions, analyzes the action of rotation groups, and computes the cohomology of the pinwheel tiling space.

Contribution

It generalizes a modified Anderson-Putnam complex to higher dimensions and explores group actions on cohomology, including the pinwheel tiling.

Findings

Extended the modified complex to higher dimensions

Computed cohomology of the pinwheel tiling space

Analyzed rotation group action on cohomology

Abstract

Anderson and Putnam showed that the cohomology of a substitution tiling space may be computed by collaring tiles to obtain a substitution which "forces its border." One can then represent the tiling space as an inverse limit of an inflation and substitution map on a cellular complex formed from the collared tiles; the cohomology of the tiling space is computed as the direct limit of the homomorphism induced by inflation and substitution on the cohomology of the complex. In earlier work, Barge and Diamond described a modification of the Anderson-Putnam complex on collared tiles for one-dimensional substitution tiling spaces that allows for easier computation and provides a means of identifying certain special features of the tiling space with particular elements of the cohomology. In this paper, we extend this modified construction to higher dimensions. We also examine the action of the rotation group on cohomology and compute the cohomology of the pinwheel tiling space.