We analyze topological invariants, in particular Z2 invariants, which characterize time reversal invariant topological insulators, in the framework of index theory and K-theory. After giving a careful study of the underlying geometry and K-theory, we formalize topological invariants as elements of KR theory. To be precise, the strong topological invariants lie in the higher KR groups of spheres; KR̃−j−1(SD+1,d). Here j is a KR-cycle index, as well as an index counting off the Altland-Zirnbauer classification of Time Reversal Symmetry (TRS) and Particle Hole Symmetry (PHS)—as we show. In this setting, the computation of the invariants can be seen as the evaluation of the natural pairing between KR-cycles and KR-classes. This fits with topological and analytical index computations as well as with Poincaré Duality and the Baum–Connes isomorphism for free Abelian groups. We provide an introduction starting from the basic objects of real, complex and quaternionic structures which are the mathematical objects corresponding to TRS and PHS. We furthermore detail the relevant bundles and K-theories (Real and Quaternionic) that lead to the classification as well as the topological setting for the base spaces.

中文翻译：

---

拓扑绝缘体和 K 理论

我们在指数理论和 K 理论的框架下分析了拓扑不变量，特别是 Z2 不变量，它表征了时间反转不变的拓扑绝缘体。在仔细研究了基础几何和 K 理论之后，我们将拓扑不变量形式化为 KR 理论的元素。准确地说，强拓扑不变量存在于较高 KR 的球体群中； KR̃−j−1(SD+1,d)。这里 j 是一个 KR 循环指数，也是一个对时间反转对称性 (TRS) 和粒子空穴对称性 (PHS) 的 Altland-Zirnbauer 分类进行计数的指数——正如我们所展示的。在这种情况下，不变量的计算可以看作是对 KR 循环和 KR 类之间自然配对的评估。这符合拓扑和分析索引计算以及庞加莱对偶性和自由阿贝尔群的鲍姆-康纳同构。我们从实数、复数和四元结构的基本对象开始进行介绍，这些对象是TRS和PHS对应的数学对象。我们还详细介绍了导致分类的相关束和 K 理论（实数和四元数）以及基础空间的拓扑设置。