Topological meaning of Z₂ numbers in time reversal invariant systems

We show that the Z$_2$ invariant, which classifies the topological properties of time reversal invariant insulators, has deep relationship with the global anomaly. Although the second Chern number is the basic topological invariant characterizing time reversal systems, we show that the relative phase between the Kramers doublet reduces the topological quantum number Z to Z$_2$.