The cohomological nature of the Fu-Kane-Mele invariant

In this paper we generalize the definition of the FKMM-invariant introduced in De Nittis and Gomi (2015) for the case of “Quaternionic” vector bundles over involutive base spaces endowed with free involution or with a non-finite fixed-point set. In De Nittis and Gomi (2015) 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 necessarily related to models for topological insulators (e.g. the two-dimensional models of adiabatically perturbed systems discussed in Gat and Robbins, 2017). As a byproduct we provide the complete classification of “Quaternionic” vector bundles over a big class of low dimensional involutive spheres and tori.