$K$-theory of two-dimensional substitution tiling spaces from $AF$-algebras

TL;DR

This paper develops a method to compute the $K$-theory of groupoid $C^*$-algebras associated with two-dimensional substitution tilings, linking it to the $K$-theory of related $AF$-algebras and providing explicit calculations.

Contribution

It generalizes previous calculations for specific tilings to a broader class, introducing a topological exact sequence for $K$-theory and highlighting limitations of using only ordinary $K$-theory.

Findings

Explicit $K$-theory formulas for tiling groupoids

Generalization of chair tiling calculations

Examples computed with Sage software

Abstract

Given a two-dimensional substitution tiling space, we show that, under some reasonable assumptions, the $K$-theory of the groupoid $C^\ast$-algebra of its unstable groupoid can be explicitly reconstructed from the $K$-theory of the $AF$-algebras of the substitution rule and its analogue on the $1$-skeleton. We prove this by generalizing the calculations done for the chair tiling in [JS16] using relative $K$-theory and excision, and packaging the result into an exact sequence purely in topology. From this exact sequence, it appears that one cannot use only ordinary $K$-theory to compute using the dimension-filtration on the unstable groupoid. Several examples are computed using Sage and the results are compiled in a table.