  We discuss means to study topological properties of wavefunctions in a time
reversal invariant crystalline system through $K$-groups. The well-known
methods for calculating $K$-groups of $G$-bundles over spheres are extended
using earlier results in order to deal with wavefunctions defined over toroidal
Brillouin zones, following a method due to Nash. The recently discovered
topological insulator is considered as an illustrative example.
