Let Ω be a tiling space and let G be the maximal group of rotations which fixes Ω. Then the cohomology of Ω and Ω/G are both invariants which give useful geometric information about the tilings in Ω. The noncommutative analog of the cohomology of Ω is the K-theory of a C*-algebra associated to Ω, 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 Ω/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.

Communications in Mathematical Physics

Starling, C. (2014). K-Theory of Crossed Products of Tiling C*-Algebras by Rotation Groups. Communications in Mathematical Physics, 334(1), 301–311. doi:10.1007/s00220-014-2070-5