The Fu–Kane–Mele \(\mathbb {Z}_2\) index characterizes two-dimensional time-reversal symmetric topological phases of matter. We shed some light on some features of this index by investigating projection-valued maps endowed with a fermionic time-reversal symmetry. Our main contributions are threefold. First, we establish a splitting theorem, proving that any such projection-valued map admits a splitting into two projection-valued maps that are related to each other via time-reversal symmetry. Second, we provide a complete homotopy classification theorem for these maps, thereby clarifying their topological structure. Third, by means of the previous analysis, we connect the Fu–Kane–Mele index to the Chern number of one of the factors in the previously-mentioned decomposition, which in turn allows to exhibit how the \(\mathbb {Z}_2\) -valued topological obstruction to constructing a periodic and smooth Bloch frame for the projection-valued map, measured by the Fu–Kane–Mele index, can be concentrated in a single pseudo-periodic Kramers pair.
