Wavefunction topology of two-dimensional time-reversal symmetric superconductors

We discuss the topology of the wavefunctions of two-dimensional time-reversal symmetric superconductors. We consider (a) the planar state, (b) a system with broken up-down reflection symmetry, and (c) a system with general spin-orbit interaction. We show explicitly how the relative sign of the order parameter on the two Fermi surfaces affects this topology, and clarify the meaning of the $Z_2$ classification for these topological states.