一个解决风车镶嵌间隙标号猜想的指标定理
An index theorem to solve the gap-labeling conjecture for the pinwheel tiling
摘要：
在本文中，我们研究了C？与风车瓷砖相关的代数。我们证明它是由Z + Z^6与一个上同调群的和给出的。C？代数被赋予一个迹，它在它的K 0-群上导出一个线性映射。然后，我们在这个映射下显式地计算被加数Z+Z^6的像，表明Z的像为零，Z^6的像包含在风车镶嵌的补丁频率的模中。最后证明了A. Connes将K 0-群的最后一个被加数的图像与一个更可计算的上同调公式联系起来。这是证明风车镶嵌的间隙标记猜想的第一步。
英文摘要：
In this paper, we study the K0-group of the C?-algebra associated to a pinwheel tiling. We prove that it is given by the sum of Z + Z^6 with a cohomological group. The C?-algebra is endowed with a trace that induces a linear map on its K0-group. We then compute explicitly the image, under this map, of the summand Z+Z^6, showing that the image of Z is zero and the image of Z^6 is included in the module of patch frequencies of the pinwheel tiling. We finally prove that we can apply the measured index theorem due to A. Connes to relate the image of the last summand of the K0-group to a cohomological formula which is more computable. This is the first step in the proof of the gap-labeling conjecture for the pinwheel tiling.
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