We present homotopy theoretic and geometric interpretations of the Kane–Mele invariant for gapped fermionic quantum systems in three dimensions with time-reversal symmetry. We show that the invariant is related to a certain 4-equivalence which lends it an interpretation as an obstruction to a block decomposition of the sewing matrix up to non-equivariant homotopy. We prove a Mayer–Vietoris Theorem for manifolds with [Formula: see text]-actions which intertwines Real and [Formula: see text]-equivariant de Rham cohomology groups, and apply it to derive a new localization formula for the Kane–Mele invariant. This provides a unified cohomological explanation for the equivalence between the discrete Pfaffian formula and the known local geometric computations of the index for periodic lattice systems. We build on the relation between the Kane–Mele invariant and the theory of bundle gerbes with [Formula: see text]-actions to obtain geometric refinements of this obstruction and localization technique. In the preliminary part we review the Freed–Moore theory of band insulators on Galilean spacetimes with emphasis on geometric constructions, and present a bottom-up approach to time-reversal symmetric topological phases.

中文翻译：

---

拓扑绝缘体和 Kane-Mele 不变量：阻塞和局部化理论

我们提出了具有时间反转对称性的三个维度的带隙费米子量子系统的 Kane-Mele 不变量的同伦理论和几何解释。我们证明了不变量与某个 4-等价有关，这使其解释为阻碍缝合矩阵的块分解，直至非等变同伦。我们证明了流形的 Mayer-Vietoris 定理，其中 [Formula: see text]-actions which intertwines Real 和 [Formula: see text]-equivariant de Rham 上同调群，并应用它推导出 Kane-Mele 不变量的新定位公式. 这为离散 Pfaffian 公式与已知的周期性晶格系统指数的局部几何计算之间的等价性提供了统一的上同调解释。我们建立在 Kane-Mele 不变量和束 gerbes 理论之间的关系基础上，使用 [公式：参见文本]-动作来获得这种阻碍和定位技术的几何细化。在初步部分，我们回顾了伽利略时空中带绝缘体的 Freed-Moore 理论，重点是几何构造，并提出了一种自下而上的时间反转对称拓扑相位方法。