论文标题
$ k $ - 二维替代瓷砖空间的理论来自$ af $ - 代理
$K$-theory of two-dimensional substitution tiling spaces from $AF$-algebras
论文作者
论文摘要
鉴于二维替代瓷砖空间，我们表明，在某些合理的假设下，可以从$ k $ a $ k $ - 替代$ af $ - 替换规则的$ af $ a $ k $ theory中明确地重建其不稳定类的$ c $ c^\ ast $ algebra。我们通过使用相对$ k $理论和切除术在[JS16]中为椅子铺平的计算进行概括，并将结果包装到纯粹的拓扑中的精确序列中。从这个确切的序列中，似乎不能仅使用普通的$ k $ - 理论来使用不稳定的groupoid上的尺寸进行计算。使用SAGE计算了几个示例，并将结果编译在表中。
Given a two-dimensional substitution tiling space, we show that, under some reasonable assumptions, the $K$-theory of the groupoid $C^\ast$-algebra of its unstable groupoid can be explicitly reconstructed from the $K$-theory of the $AF$-algebras of the substitution rule and its analogue on the $1$-skeleton. We prove this by generalizing the calculations done for the chair tiling in [JS16] using relative $K$-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 $K$-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.
