  We study the topological band theory of time reversal invariant topological
insulators and interpret the topological $\mathbb{Z}_2$ invariant as an
obstruction in terms of Stiefel--Whitney classes. The band structure of a
topological insulator defines a Pfaffian line bundle over the momentum space,
whose structure group can be reduced to $\mathbb{Z}_2$. So the topological
$\mathbb{Z}_2$ invariant will be understood by the Stiefel--Whitney theory,
which detects the orientability of a principal $\mathbb{Z}_2$-bundle. Moreover,
the relation between weak and strong topological insulators will be understood
based on cobordism theory. Finally, the topological $\mathbb{Z}_2$ invariant
gives rise to a fully extended topological quantum field theory (TQFT).
