Involutions with isolated fixed points on orientable $3$-dimensional flat space forms

In this paper we completely classify (up to conjugacy) all involutions $\iota: M \to M$, where $M$ is an orientable connected flat $3$-dimensional space form, such that $\iota$ has fixed points but only finitely many. If $M_1,\ldots,M_6$ are the $6$ space forms then only $M_1, M_2, M_6$ admit such involutions. Moreover, they are unique up to conjugacy. The main idea behind the proof is to find incompressible tori $T \subseteq M$ so that either $\iota(T) = T$ or $\iota(T) \cap T = \varnothing$ and then cut $M$ into simpler pieces. These results lead to a complete classification of $3$-manifolds containing $\mathbf{Z} \oplus \mathbf{Z} \oplus \mathbf{Z}$ in their fundamental groups.