  The use of topological invariants to describe geometric phases of quantum
matter has become an essential tool in modern solid state physics. The first
instance of this paradigmatic trend can be traced to the study of the quantum
Hall effect, in which the Chern number underlies the quantization of the
transverse Hall conductivity. More recently, in the framework of time-reversal
symmetric topological insulators and quantum spin Hall systems, a new
topological classification has been proposed by Fu, Kane and Mele, where the
label takes value in $\mathbb Z_2$.
  We illustrate how both the Chern number $c \in \mathbb Z$ and the
Fu-Kane-Mele invariant $\delta \in \mathbb Z_2$ of 2-dimensional topological
insulators can be characterized as topological obstructions. Indeed, $c$
quantifies the obstruction to the existence of a frame of Bloch states for the
crystal which is both continuous and periodic with respect to the crystal
momentum. Instead, $\delta$ measures the possibility to impose a further
time-reversal symmetry constraint on the Bloch frame.
