论文标题
映射Tori的几何埃利奥特不变和非共同刚性
A geometric Elliott invariant and noncommutative rigidity of mapping tori
论文作者
论文摘要
我们证明了使用非共同几何形状工具与最小拓扑动力学系统相关的绘制托架的刚度属性。更确切地说，我们表明，在轻度的几何假设下，与$ \ Mathbb {z}^d $ actions相关的两个映射Tori的定向叶轮同型等效性可以提升至其叶面$ C^*$ algebras的同构。在叶子空间很单一的叶面空间的背景下，该特性是拓扑刚性的非共同类似物，其中$ c^*$ - 代数代替同构类型的同构类型。我们的技术是为Elliott不变的几何方法开发出依赖于映射圆环的拓扑和索引理论数据的几何方法。我们还讨论了如何将构造扩展到更一般的同质副本商，这是由简单连接的可解决的谎言组的离散共反应亚组产生的，以及如何将理论应用于某些Cantor最小系统的磁性间隙标记问题。
We prove a rigidity property for mapping tori associated to minimal topological dynamical systems using tools from noncommutative geometry. More precisely, we show that under mild geometric assumptions, an orientation-preserving leafwise homotopy equivalence of two mapping tori associated to $\mathbb{Z}^d$-actions on a compact space can be lifted to an isomorphism of their foliation $C^*$-algebras. This property is a noncommutative analogue of topological rigidity in the context of foliated spaces whose space of leaves is singular, where isomorphism type of the $C^*$-algebra replaces homeomorphism type. Our technique is to develop a geometric approach to the Elliott invariant that relies on topological and index-theoretic data from the mapping torus. We also discuss how our construction can be extended to slightly more general homotopy quotients arising from actions of discrete cocompact subgroups of simply connected solvable Lie groups, as well as how the theory can be applied to the magnetic gap-labelling problem for certain Cantor minimal systems.
