  This work concerns with the description of the topological phases of band
insulators of class DIII by using the equivariant cohomology. The main result
is the definition of a cohomology class for general systems of class DIII which
generalizes the well known $\mathbb{Z}_2$-invariant given by the Teo-Kane
formula in the one-dimension case. In the two-dimensional case this cohomology
invariant allows a complete description of the strong and weak phases. The
relation with the KR-theory, the Noether-Fredholm index and the classification
of "Real" gerbes are also discussed.
