Cohomology of rotational tiling spaces

TL;DR

This paper introduces a spectral sequence to compute the 7Cech cohomology of the Euclidean hull of planar tilings with finite local complexity, exemplified by Penrose tilings.

Contribution

It develops a new spectral sequence framework linking ePE homology and cohomology to the cohomology of tiling spaces, enabling explicit calculations.

Findings

Computed the 7Cech cohomology of Penrose tilings' Euclidean hull.

Established a simple combinatorial description of the boundary map.

Provided a method applicable to other tilings with finite local complexity.

Abstract

A spectral sequence is defined which converges to the \v{C}ech cohomology of the Euclidean hull of a tiling of the plane with Euclidean finite local complexity. The terms of the second page are determined by the so-called ePE homology and ePE cohomology groups of the tiling, and the only potentially non-trivial boundary map has a simple combinatorial description in terms of its local patches. Using this spectral sequence, we compute the \v{C}ech cohomology of the Euclidean hull of the Penrose tilings.