  In this paper we generalize the definition of the FKMM-invariant introduced
in [DG2] for the case of "Quaternionic" vector bundles over involutive base
spaces endowed with free involution or with a non-finite fixed-point set. In
[DG2] it has already be shown how the FKMM-invariant provides a cohomological
description of the Fu-Kane-Mele index used to classify topological insulators
in class AII. It follows that the FKMM-invariant described in this paper
provides a cohomological generalization of the Fu-Kane-Mele index which is
applicable to the classification of protected phases for other type of
topological quantum systems (TQS) which are not necesarily related to models
for topological insulators (e.g. the two-dimensional models of adiabatically
perturbed systems discussed in [GR]). As a byproduct we provide the complete
classification of "Quaternionic" vector bundles over a big class of low
dimensional involutive spheres and tori.
