Fu-Kane-Mele不变量的上同调性质
The cohomological nature of the Fu-Kane-Mele invariant
摘要：
本文将文献[DG 2]中引入的FKMM-不变量的定义推广到对合基空间上的“四元数”向量丛的情形，其中对合基空间具有自由对合或非有限不动点集。在[DG 2]中，已经示出了FKMM-不变量如何提供用于将拓扑绝缘体分类在AII类中的Fu-Kane-Mele指数的上同调描述。由此可见，本文描述的FKMM不变量提供了Fu-Kane-Mele指数的上同调推广，该指数适用于其他类型的拓扑量子系统（TQS）的保护相分类，这些系统不一定与拓扑绝缘体的模型相关（例如[GR]中讨论的绝热扰动系统的二维模型）。作为一个副产品，我们提供了完整的分类“四元数”向量丛上的一大类低维对合球和环面。
英文摘要：
In this paper we generalize the definition of the FKMM-invariant introduced in [DG2] for the case of "Quaternionic" vector bundles over involutive base spaces endowed with free involution or with a non-finite fixed-point set. In [DG2] it has already be shown how the FKMM-invariant provides a cohomological description of the Fu-Kane-Mele index used to classify topological insulators in class AII. It follows that the FKMM-invariant described in this paper provides a cohomological generalization of the Fu-Kane-Mele index which is applicable to the classification of protected phases for other type of topological quantum systems (TQS) which are not necesarily related to models for topological insulators (e.g. the two-dimensional models of adiabatically perturbed systems discussed in [GR]). As a byproduct we provide the complete classification of "Quaternionic" vector bundles over a big class of low dimensional involutive spheres and tori.
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