"\(K\)-theory of two-dimensional substitution tiling spaces from \(AF\)-algebras"
Liu, JianlongA tiling space carries an action by translations, giving a groupoid \(C^\ast\)-algebra. If the tiling arises from a substitution rule, one obtains an \(AF\)-algebra for each intermediate dimension. For \(d=1\), Putnam (1989) proved that these \(AF\)-algebras are sufficient in constructing the \(K\)-theory of the groupoid \(C^\ast\)-algebra. For \(d=2\), Julien-Savinien (2016) showed a proof-of-concept for the chair tiling. Using the six-term exact sequence in relative \(K\)-theory introduced by Haslehurst (2021), we give a simple proof for \(d=1,2\) that the \(K\)-theory of this groupoid \(C^\ast\)-algebra can always be constructed from that of the attached \(AF\)-algebras, and, incidentally, relate everything back to cohomology.