Pattern equivariant functions and cohomology

TL;DR

This paper introduces a direct method to compute the cohomology of tilings and point patterns by generalizing equivariance, making the process more accessible than traditional hull or groupoid approaches.

Contribution

It presents a new, simplified construction for cohomology of tilings and point patterns based on generalized equivariance, bypassing complex hull or groupoid methods.

Findings

Provides a more direct approach to cohomology calculation.

Generalizes equivariance from lattices to finite local complexity patterns.

Simplifies the understanding and computation of pattern cohomology.

Abstract

The cohomology of a tiling or a point pattern has originally been defined via the construction of the hull or the groupoid associated with the tiling or the pattern. Here we present a construction which is more direct and therefore easier accessible. It is based on generalizing the notion of equivariance from lattices to point patterns of finite local complexity.