  We describe some applications of group- and bundle-theoretic methods in solid
state physics, showing how symmetries lead to a proof of the localization of
electrons in gapped crystalline solids, as e.g. insulators and semiconductors.
We shortly review the Bloch-Floquet decomposition of periodic operators, and
the related concepts of Bloch frames and composite Wannier functions. We show
that the latter are almost-exponentially localized if and only if there exists
a smooth periodic Bloch frame, and that the obstruction to the latter condition
is the triviality of a Hermitian vector bundle, called the Bloch bundle. The
role of additional $\mathbb{Z}_2$-symmetries, as time-reversal and
space-reflection symmetry, is discussed, showing how time-reversal symmetry
implies the triviality of the Bloch bundle, both in the bosonic and in the
fermionic case. Moreover, the same $\mathbb{Z}_2$-symmetry allows to define a
finer notion of isomorphism and, consequently, to define new topological
invariants, which agree with the indices introduced by Fu, Kane and Mele in the
context of topological insulators.
