  "Quaternionic" vector bundles are the objects which describe the topological
phases of quantum systems subjected to an odd time-reversal symmetry (class
AII). In this work we prove that the FKMM invariant provides the correct
fundamental characteristic class for the classification of "Quaternionic"
vector bundles in dimension less than, or equal to three (low dimension). The
new insight is provided by the interpretation of the FKMM invariant from the
viewpoint of the Bredon equivariant cohomology. This fact, along with basic
results in equivariant homotopy theory, allows us to achieve the expected
result.
