Abstract:In the spirit of Hjorth’s turbulence theory, we introduce “unbalancedness”: a new dynamical obstruction to classifying orbit equivalence relations by actions of Polish groups which admit a two-sided invariant metric (TSI). Since abelian groups are TSI, unbalancedness can be used for identifying which classification problems cannot be solved by classical homology and cohomology theories. In terms of applications, we show that Morita equivalence of continuous-trace $C^*$-algebras, as well as isomorphism of Hermitian line bundles, are not classifiable by actions of TSI groups. In the process, we show that the Wreath product of any two non-compact subgroups of $S_{\infty }$ admits an action whose orbit equivalence relation is generically ergodic against any action of a TSI group and we deduce that there is an orbit equivalence relation of a CLI group which is not classifiable by actions of TSI groups.

---

中文翻译：

---

通过（共）同源性和其他 TSI 组不变量进行分类的动力学障碍

摘要：本着 Hjorth 湍流理论的精神，我们引入了“不平衡性”：一种新的动力学障碍，用于通过允许双边不变度量 (TSI) 的波兰群的动作对轨道等价关系进行分类。由于阿贝尔群是 TSI，不平衡性可用于识别经典同调和上同调理论无法解决的分类问题。在应用方面，我们证明了连续迹 $C^*$-代数的 Morita 等价，以及 Hermitian 线丛的同构，不能通过 TSI 群的动作进行分类。正在进行中，

---