Comparing different versions of tiling cohomology

TL;DR

This paper establishes isomorphisms between various versions of tiling cohomology, including the Anderson-Putnam-Gähler, PV-cohomology, and pattern equivariant cohomology, providing a unified framework for understanding tiling deformations.

Contribution

It proves that different tiling cohomology theories are isomorphic and extends these isomorphisms to their weak versions, offering a new perspective on tiling deformations.

Findings

Isomorphisms between three tiling cohomology versions.

Extension of isomorphisms to weak cohomology groups.

New formulation of the pattern equivariant mixed quotient group.

Abstract

We establish direct isomorphisms between different versions of tiling cohomology. The first version is the direct limit of the cohomologies of the approximants in the Anderson-Putnam-G\"ahler complex, the second is the recently introduced PV-cohomolgy of Bellissard and Savinien and the third is pattern equivariant cohomology. For the last two versions one can define weak cohomology groups. We show that the isomorphisms extend to the weak versions. This leads to an alternative formulation of the pattern equivariant mixed quotient group which describes deformations of the tiling modulo topological conjugacy.