  The primary goal of this paper is to study topological invariants in two
dimensional twofold rotation and time-reversal symmetric spinful systems. In
this paper, firstly we build a new homotopy invariant based on the lifting of
the Wilson loop to the universal covering group of the special orthogonal
group. And we prove the invariant we built agrees with the K theory invariant.
We go beyond the previous understanding of the Wilson loop unwinds in more than
two occupied bands by finding an obstruction of such unwinding. Then, within
this formalism, we classify four occupied bands cases into two categories which
may have the same Wilson loop spectrum but in different topological classes.
Our theory implies that even when the Fu-Kane-Mele invariant vanishes, the
existence of a pair of gapless edge modes is topological protected by this new
topological invariant. Finally, we present a tight binding model realizing the
non-trivial phase.
