K-theory of crossed products of tiling C*-algebras by rotation groups

Abstract

Let $\Omega$ be a tiling space and let $G$ be the maximal group of rotations which fixes $\Omega$. Then the cohomology of $\Omega$ and $\Omega /G$ are both invariants which give useful geometric information about the tilings in $\Omega$. The noncommutative analog of the cohomology of $\Omega$ is the K-theory of a C*-algebra associated to $\Omega$, and for translationally finite tilings of dimension 2 or less the K-theory is isomorphic to the direct sum of cohomology groups. In this paper we give a prescription for calculating the noncommutative analog of the cohomology of $\Omega /G$, that is, the K-theory of the crossed product of the tiling C*-algebra by $G$. We also provide a table with some calculated K-groups for many common examples, including the Penrose and pinwheel tilings.

14 pages, one table

References in corpus (2)

• Equivariant K-theory of finite dimensional real vector spaces
• C*-algebras of tilings with infinite rotational symmetry

Cited by in corpus (4)

• Fractal spectral triples on Kellendonk's $C^{\ast }$-algebra of a substitution tiling
• Classifying the Dynamics of Architected Materials by Groupoid Methods
• New results on tilings via cup products and Chern characters on tiling spaces
• Aperiodicity, rotational tiling spaces and topological space groups