关于AF-离散群胚的指标映射

中文摘要 AI 辅助

我们给出了AF-离散群胚的指标映射的显式几何与代数描述。该指标记录了无限瓦片之间的输运以及芽群中的阿贝尔化回返，并受限于相关Bratteli图的维数群中的一个缺陷。我们确定了指标映射的核，并构造了在单点轨道上具有平凡芽的零指标元素的换位分解。特别地，在每一个极小情形下，该核等于由动力学换位生成的子群。我们还刻画了由固定AF核逼近的性质，并证明了紧生成的极小系统的几乎有限性。我们以芽群的术语显式地给出了更高同调。我们通过提升曲面关系和由平衡边界数据构造的扩张类来表达次级奇偶微分。在极小性和比较性条件下，提升有限关系给出一个分裂的AH序列。显式的边界和矩阵计算，以及一个具有非零奇偶微分的非极小例子，说明了该理论。

英文摘要

We give an explicit geometric and algebraic description of the index map for AF-by-discrete groupoids. The index records transport between infinite tiles and abelianized returns in the groups of germs, subject to a defect in the dimension group of the associated Bratteli diagram. We determine the kernel of the index map and construct transposition factorizations of zero-index elements with trivial germs at singleton orbits. In particular, in every minimal case, this kernel equals the subgroup generated by dynamical transpositions. We also characterize approximation by the fixed AF core and prove almost finiteness for minimal systems that are compactly generated. We make higher homology explicit in terms of the groups of germs. We express the secondary parity differential by lifted surface relations and by an extension class constructed from balanced boundary data. Under minimality and comparison, lifting finite relations gives a split AH sequence. Explicit boundary and matrix calculations, together with a nonminimal example with nonzero parity differential, illustrate the theory.

发表机构

• School of Mathematics, South China University of Technology（华南理工大学数学学院）

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