Pattern-Equivariant Homology of Finite Local Complexity Patterns
Pattern-Equivariant Homology of Finite Local Complexity Patterns
复制标题
有限局部复杂性模式的模式等变同调
DOI：
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发表时间：
2014
期刊：
影响因子：
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通讯作者：
James J. Walton
中科院分区：
文献类型：
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作者：
James J. Walton
本文建立了一个统一研究有限局部复杂性(FLC)模式的一般环境。引入了“模式”的抽象概念，它可以被视为保持平铺的等距空间群的类比，但其中，人们考虑的是保持部分等距的部分等距。这些部分变换的逆半群是具有FLC但几乎没有整体对称性的模式的空间群的合适的类似物。类似地，我们引入了模式的环境空间上的等价关系系统Aemph{Colage}的概念，我们证明它能够推广许多适用于研究FLC平铺和Delone集的构造，例如将平铺空间表示为逼近的逆极限。
为我们的抽象模式构造了一个不变量，即所谓的模式等价(PE)同调。这些同调群是使用模式环境空间上的无限奇异链来定义的，尽管我们证明了在适当的条件下可以定义同构的元胞形式。对于FLC瓷砖，这些蜂窝PE链类似于PE蜂窝共链。证明了PE同调和上同调群通过Poincar对偶联系在一起。
提出了一种用于计算分层拼接的PE同调群的有效且高度几何的方法。旋转不变的PE同调群被证明不是相关联的平铺空间的拓扑不变量，并且似乎保留了关于平铺空间中平铺的全局对称性的额外信息。我们展示了如何将PE同调群合并到一个谱序列中，该谱序列收敛到瓦片的刚性壳的v{C}ECh上同调。这些方法允许简单地计算Penrose瓦片的刚性壳的v{C}ECh上同调。
This thesis establishes a generalised setting with which to unify the study of finite local complexity (FLC) patterns. The abstract notion of a "pattern" is introduced, which may be seen as an analogue of the space group of isometries preserving a tiling but where, instead, one considers partial isometries preserving portions of it. These inverse semigroups of partial transformations are the suitable analogue of the space group for patterns with FLC but few global symmetries. In a similar vein we introduce the notion of a \emph{collage}, a system of equivalence relations on the ambient space of a pattern, which we show is capable of generalising many constructions applicable to the study of FLC tilings and Delone sets, such as the expression of the tiling space as an inverse limit of approximants.
An invariant is constructed for our abstract patterns, the so called pattern-equivariant (PE) homology. These homology groups are defined using infinite singular chains on the ambient space of the pattern, although we show that one may define cellular versions which are isomorphic under suitable conditions. For FLC tilings these cellular PE chains are analogous to the PE cellular cochains \cite{Sadun1}. The PE homology and cohomology groups are shown to be related through Poincar\'{e} duality.
An efficient and highly geometric method for the computation of the PE homology groups for hierarchical tilings is presented. The rotationally invariant PE homology groups are shown not to be a topological invariant for the associated tiling space and seem to retain extra information about global symmetries of tilings in the tiling space. We show how the PE homology groups may be incorporated into a spectral sequence converging to the \v{C}ech cohomology of the rigid hull of a tiling. These methods allow for a simple computation of the \v{C}ech cohomology of the rigid hull of the Penrose tilings.
线性扭矩流产生的分离网络上的等价关系
影响因子：
1.8
作者：
Haynes A
通讯作者：
Haynes A