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In this paper we generalize the definition of the FKMM-invariant introducedin [DG2] for the case of "Quaternionic" vector bundles over involutive basespaces 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 cohomologicaldescription of the Fu-Kane-Mele index used to classify topological insulatorsin class AII. It follows that the FKMM-invariant described in this paperprovides a cohomological generalization of the Fu-Kane-Mele index which isapplicable to the classification of protected phases for other type oftopological quantum systems (TQS) which are not necesarily related to modelsfor topological insulators (e.g. the two-dimensional models of adiabaticallyperturbed systems discussed in [GR]). As a byproduct we provide the completeclassification of "Quaternionic" vector bundles over a big class of lowdimensional involutive spheres and tori.

The cohomological nature of the Fu-Kane-Mele invariant