Topological Insulators and the Kane-Mele Invariant: Obstruction and Localisation Theory

TL;DR

This paper provides a homotopy and geometric interpretation of the Kane-Mele invariant in three-dimensional topological insulators, linking it to obstruction theory and deriving a new localisation formula using cohomology.

Contribution

It introduces a new cohomological localisation formula for the Kane-Mele invariant and relates it to obstruction theory and bundle gerbes with $bZ_2$-actions.

Findings

Derived a Mayer-Vietoris theorem for $bZ_2$-actions on manifolds.

Established a localisation formula connecting Pfaffian and geometric index computations.

Provided a unified cohomological framework for topological invariants in time-reversal symmetric systems.

Abstract

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 nonequivariant homotopy. We prove a Mayer-Vietoris Theorem for manifolds with $\mathbb{Z}_2$-actions which intertwines Real and $\mathbb{Z}_2$-equivariant de Rham cohomology groups, and apply it to derive a new localisation 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 $\mathbb{Z}_2$-actions to obtain geometric refinements of this obstruction and localisation 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.