Three dimensional two-band Floquet topological insulator with $Z_2$ index

We present a class of three dimensional (3D) two-band Floquet topological insulators constructed from two-dimensional Floquet topological insulators with a $Z$ topological index. It is shown that the 3D two-band Floquet topological insulator has a $Z_2$ topological index, whose value can be obtained by numerical calculations or by using a relation to the winding number. The classification of the 3D $Z_2$ Floquet topological insulator, however, cannot be attributed to the stable homotopy groups. Thus, it is an example outside the proposed K-theory classifications of Floquet topological insulators. We also analyze the edge modes of the 3D $Z_2$ Floquet topological insulator and find the parity of the number of edge modes reflects the bulk $Z_2$ index.