Cohomology of Hierarchical Tilings

TL;DR

This paper reviews various methods of tiling cohomology, focusing on inverse limit constructions, and discusses their applications in distinguishing tiling spaces, understanding deformations, and analyzing maps, especially for substitution tilings.

Contribution

It provides a comprehensive overview of different tiling cohomology theories and their computational techniques, emphasizing inverse limit methods and their applications to aperiodic tilings.

Findings

Different cohomology theories are interconnected through inverse limit constructions.

Tiling cohomology can distinguish non-equivalent tiling spaces.

Applications include analyzing deformations and maps between tiling spaces.

Abstract

This is a chapter in an upcoming book on aperiodic order. We go over different versions of tiling cohomology (\v Cech, pattern-equivariant, PV, quotient) with emphasis on the inverse limit constructions used to compute these cohomologies. We then consider the uses of tiling cohomology to distinguish spaces, to understand deformations, and to help understand maps between tiling spaces. The emphasis of this chapter is on substitution tilings and their generalizations, but the underlying ideas apply equally well to cut-and-project tilings and to tilings defined by local matching rules.