HK与GL
HK and GL
AI总结：
研究多Z群在Cantor集上自由作用的变换群胚的HK猜想和间隙标记问题，通过比较群胚同调与上同调的长正合序列和Pimsner-Voiculescu正合序列，引入上同调比较映射，对Hirsch长度小的群证明或部分证明HK猜想和间隙标记。
摘要：
我们研究多Z群在Cantor集上自由作用的变换群胚的HK猜想和间隙标记问题。主要工具是比较群胚同调和上同调的长正合序列与Z的交叉积的Pimsner-Voiculescu正合序列。除了典范的同调比较映射μ0和μ1，我们还引入了与作用群的适当K-理论类相关的上同调比较映射。结合庞加莱对偶，这些映射检测了HK猜想中出现的高阶同调项。我们将此方法应用于小Hirsch长度的多Z群的自由作用。对于Z、Z^2和Klein瓶群的作用，我们恢复了HK和间隙标记。对于Hirsch长度为三和四的几类群，我们要么证明了HK，要么得到了用群胚同调和上同调描述K-群的显式正合序列。对于间隙标记，我们结合了de la Harpe-Skandalis行列式、Pimsner-Voiculescu边界映射的迹公式以及拓扑全群中的转置分解。这给出了所有Hirsch长度为三的多Z群自由作用以及某些Hirsch长度为四的群（包括Z^4）的间隙标记（至多相差因子2）。我们还恢复了Z^3作用的间隙标记，并通过使用映射环面的上同调比较映射证明了Z^5作用的间隙标记（至多相差因子2）。
英文摘要：
We study the HK conjecture and the gap-labelling problem for transformation groupoids associated with free actions of poly-$\Z$ groups on Cantor sets. The main tool is a comparison of the long exact sequences in groupoid homology and cohomology with the Pimsner--Voiculescu exact sequence for crossed products by $\Z$. In addition to the canonical homology comparison maps $μ_0$ and $μ_1$, we introduce cohomology comparison maps associated with suitable $K$-theory classes of the acting group. Together with Poincaré duality, these maps detect the higher homology terms occurring in the HK conjecture. We apply this method to free actions of poly-$\Z$ groups of small Hirsch length. For actions of $\Z$, $\Z^2$, and the Klein bottle group, we recover HK and gap-labelling. For several classes of groups of Hirsch length three and four, we either prove HK or obtain explicit exact sequences describing the $K$-groups in terms of groupoid homology and cohomology. For gap-labelling, we combine the de la Harpe--Skandalis determinant, the trace formula for the Pimsner--Voiculescu boundary map, and transposition decompositions in topological full groups. This gives gap-labelling up to a factor of two for all free actions of poly-$\Z$ groups of Hirsch length three and for certain groups of Hirsch length four, including $\Z^4$. We also recover gap-labelling for $\Z^3$-actions and prove gap-labelling up to a factor of two for $\Z^5$-actions by using cohomology comparison maps for mapping tori.
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