  We provide a classification of type AII topological quantum systems in
dimension d=1,2,3,4. Our analysis is based on the construction of a topological
invariant, the FKMM-invariant, which completely classifies "Quaternionic"
vector bundles (a.k.a. "symplectic" vector bundles) in dimension d<4. This
invariant takes value in a proper equivariant cohomology theory and, in the
case of examples of physical interest, it reproduces the familiar Fu-Kane-Mele
index. In the case d=4 the classification requires a combined use of the
FKMM-invariant and the second Chern class. Among the other things, we prove
that the FKMM-invariant is a bona fide characteristic class for the category of
"Quaternionic" vector bundles in the sense that it can be realized as the
pullback of a universal topological invariant.
