On the K-theory of C*-algebras for substitution tilings (a pedestrian version)

TL;DR

This paper explores the K-theory of C*-algebras associated with substitution tilings, providing formulas for computation and revealing torsion properties in specific dimensions.

Contribution

It introduces methods to compute K-theory of stable, unstable, and asymptotic C*-algebras from cohomology and homology, and characterizes torsion in these groups.

Findings

K-theory of S and U can be derived from a single cochain complex.

K-theory groups for 1D tilings are torsion free.

In 2D tilings, only certain K-groups may contain torsion.

Abstract

Under suitable conditions, a substitution tiling gives rise to a Smale space, from which three equivalence relations can be constructed, namely the stable, unstable, and asymptotic equivalence relations. We denote with $S$, $U$, and $A$ their corresponding $C^*$-algebras in the sense of Renault. In this article we show that the $K$-theories of $S$ and $U$ can be computed from the cohomology and homology of a single cochain complex with connecting maps for tilings of the line and of the plane. Moreover, we provide formulas to compute the $K$-theory for these three $C^*$-algebras. Furthermore, we show that the $K$-theory groups for tilings of dimension 1 are always torsion free. For tilings of dimension 2, only $K_0(U)$ and $K_1(S)$ can contain torsion.