Pattern-Equivariant Homology of Finite Local Complexity Patterns

TL;DR

This thesis introduces a unified framework for studying finite local complexity patterns using pattern-equivariant homology, revealing new insights into tiling symmetries and enabling computations of tiling space invariants.

Contribution

It develops a generalized setting with inverse semigroups and collages for FLC patterns, introduces PE homology, and connects it to tiling invariants and cohomology computations.

Findings

PE homology groups relate to tiling symmetries.

PE homology can be computed via cellular methods.

Spectral sequences link PE homology to Čech cohomology.

Abstract

This thesis establishes a generalised setting with which to unify the study of finite local complexity (FLC) patterns. The abstract notion of a "pattern" is introduced, which may be seen as an analogue of the space group of isometries preserving a tiling but where, instead, one considers partial isometries preserving portions of it. These inverse semigroups of partial transformations are the suitable analogue of the space group for patterns with FLC but few global symmetries. In a similar vein we introduce the notion of a \emph{collage}, a system of equivalence relations on the ambient space of a pattern, which we show is capable of generalising many constructions applicable to the study of FLC tilings and Delone sets, such as the expression of the tiling space as an inverse limit of approximants. An invariant is constructed for our abstract patterns, the so called pattern-equivariant (PE) homology. These homology groups are defined using infinite singular chains on the ambient space of the pattern, although we show that one may define cellular versions which are isomorphic under suitable conditions. For FLC tilings these cellular PE chains are analogous to the PE cellular cochains \cite{Sadun1}. The PE homology and cohomology groups are shown to be related through Poincar\'{e} duality. An efficient and highly geometric method for the computation of the PE homology groups for hierarchical tilings is presented. The rotationally invariant PE homology groups are shown not to be a topological invariant for the associated tiling space and seem to retain extra information about global symmetries of tilings in the tiling space. We show how the PE homology groups may be incorporated into a spectral sequence converging to the \v{C}ech cohomology of the rigid hull of a tiling. These methods allow for a simple computation of the \v{C}ech cohomology of the rigid hull of the Penrose tilings.