-theory of two-dimensional substitution tiling spaces from -algebras
arXiv:2211.00580 · doi:10.1016/j.geomphys.2023.105016
Abstract
Given a two-dimensional substitution tiling space, we show that, under some reasonable assumptions, the -theory of the groupoid -algebra of its unstable groupoid can be explicitly reconstructed from the -theory of the -algebras of the substitution rule and its analogue on the -skeleton. We prove this by generalizing the calculations done for the chair tiling in [JS16] using relative -theory and excision, and packaging the result into an exact sequence purely in topology. From this exact sequence, it appears that one cannot use only ordinary -theory to compute using the dimension-filtration on the unstable groupoid. Several examples are computed using Sage and the results are compiled in a table.
Slightly updated to match publication