  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.
