2017/12/27 by Daniel Gonçalves, Gonçalves, Daniel, Maria Ramirez-Solano +1
Mathematics · #37D15 #46L80 #52C23 #Advanced Algebra and Geometry #Advanced Operator Algebra Research #Advanced Topics in Algebra #Dynamical Systems (math.DS) #FOS: Mathematics #K-Theory and Homology (math.KT) #Operator Algebras (math.OA)
paper · pdf · doi:10.48550/arxiv.1712.09551
openalex publication_date 2017/12/27 · openalex created_date 2018/01/05 · openalex updated_date 2026/07/28
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 K0(U) and K1(S) can contain torsion.