Cohomology of substitution tiling spaces
Cohomology of substitution tiling spaces
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替代平铺空间的上同调
DOI：
10.1017/s0143385709000777
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发表时间：
2008
影响因子：
0.9
通讯作者：
L. Sadun
中科院分区：
文献类型：
--
作者：
M. Barge;B. Diamond;J. Hunton;L. Sadun
Anderson和Putnam证明了替换平铺空间的上同调性可以通过将平铺空间的平铺空间围合来计算，从而得到一个“强制其边界”的替换。然后，可以将瓷砖空间表示为由带圈瓷砖形成的细胞复合体上的膨胀和替代映射的逆极限；平铺空间的上同调被计算为复上同调上由膨胀和取代引起的同态的直接极限。在早期的工作中，Barge和Diamond描述了一种在一维替代平铺空间中对有圈瓷砖的Anderson-Putnam复合体的修改，这种修改允许更容易的计算，并提供了一种方法来识别平铺空间的某些特殊特征与上同调的特定元素。在本文中，我们将这种改进的构造扩展到更高的维度。我们还研究了旋转群对上同调的作用，并计算了风车平铺空间的上同调。
Abstract Anderson and Putnam showed that the cohomology of a substitution tiling space may be computed by collaring tiles to obtain a substitution which ‘forces its border’. One can then represent the tiling space as an inverse limit of an inflation and substitution map on a cellular complex formed from the collared tiles; the cohomology of the tiling space is computed as the direct limit of the homomorphism induced by inflation and substitution on the cohomology of the complex. In earlier work, Barge and Diamond described a modification of the Anderson–Putnam complex on collared tiles for one-dimensional substitution tiling spaces that allows for easier computation and provides a means of identifying certain special features of the tiling space with particular elements of the cohomology. In this paper, we extend this modified construction to higher dimensions. We also examine the action of the rotation group on cohomology and compute the cohomology of the pinwheel tiling space.