Tiling同调与上同调中的挠
Torsion in Tiling Homology and Cohomology
摘要：
第一作者最近意外地发现扭转的积分上同调的蒂宾根三角瓷砖导致重新评估目前的描述和计算方法的拓扑不变量与非周期性瓷砖。挠率的存在使人们对先前假定的上同调不变量和K-理论不变量的等价性以及假定的K-理论不变量不存在挠率提出了质疑。在本文中，我们详细研究了拓扑不变量的规范投影tilings，我们扩展的结果Forrest，Hunton和Kellendonk给出一个完整的治疗扭转的上同调这种tilings在余维至多3，并提出了补充和修正需要以前的结果和计算文献。对于余维为1和2的镶嵌，给出扭转分量的完整处理是简单的，但是余维为3的情况要复杂得多，我们用前面考虑的所有四个二十面体镶嵌的计算来说明我们的方法。转向K-理论不变量，我们表明，上同调和K-理论同意所有规范的投影平铺（物理）维至多3，从而证明存在扭转，例如，K-理论的蒂宾根三角形平铺。高维欧氏空间的上同调和K-理论的等价性问题仍然是开放的。
英文摘要：
The first author's recent unexpected discovery of torsion in the integral cohomology of the Tübingen Triangle Tiling has led to a re-evaluation of current descriptions of and calculational methods for the topological invariants associated with aperiodic tilings. The existence of torsion calls into question the previously assumed equivalence of cohomological and K-theoretic invariants as well as the supposed lack of torsion in the latter. In this paper we examine in detail the topological invariants of canonical projection tilings; we extend results of Forrest, Hunton and Kellendonk to give a full treatment of the torsion in the cohomology of such tilings in codimension at most 3, and present the additions and amendments needed to previous results and calculations in the literature. It is straightforward to give a complete treatment of the torsion components for tilings of codimension 1 and 2, but the case of codimension 3 is a good deal more complicated, and we illustrate our methods with the calculations of all four icosahedral tilings previously considered. Turning to the K-theoretic invariants, we show that cohomology and K-theory agree for all canonical projection tilings in (physical) dimension at most 3, thus proving the existence of torsion in, for example, the K-theory of the Tübingen Triangle Tiling. The question of the equivalence of cohomology and K-theory for tilings of higher dimensional euclidean space remains open.
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