Universe belongs, namely ${\mathcal{G}_3}$, ${\mathcal{G}_4}$ or ${\mathcal{G}_5}$ respectively.[^6] Let us restrict our analysis to the orientable case in order to be specific. The determination of the rotation taking $C_0D_0E_0$ to the parallel transportation of $C_1D_1E_1$ provides also the direction of the axis of rotation of the screw motion linking $C_0$ with $C_1$. If the Universe has a topology of class ${\mathcal{G}_3}$, ${\mathcal{G}_4}$ or ${\mathcal{G}_5}$, the translation vector is parallel to this axis, so elementary geometry can be used to determine the parameter $|a|$ and the position of the axis. Moreover, the isometry linking $C_0$ with $C_2$ has to be a translation, and a parallel transport of the triangle $C_2,D_2,E_2$ to $C_0D_0E_0$ would confirm it. A remarkable fact is that, if the topology of the Universe has been identified to be of class ${\mathcal{G}_3}$, ${\mathcal{G}_4}$ or ${\mathcal{G}_5}$, the vector $c$ is automatically fixed, and observational searches can be performed to find the topological images of $C_0$, $D_0$ and $E_0$ due to the isometries $\gamma$ and $\gamma^{-1}$ for validation of the model.

Let us now consider the case when the angle of rotation taking $C_0D_0E_0$ to the parallel transport of $C_1D_1E_1$ is $\pi$. In this case the topology of the Universe has 