$\mathbb{Z}_2$-Thurston norm in Sol manifolds and embeddability of non-orientable surfaces

For every Sol manifold $M$, we determine the $\mathbb{Z}_2$-Thurston norm of every element in $H_2(M;\mathbb{Z}_2)$. Each Sol manifold is either a torus bundle over the circle or a torus semi-bundle, thus corresponds to a torus map. We discuss the action of this torus map on a curve complex for the torus, whose edges connect curve classes of intersection number 2. For torus bundles over the circle, the $\mathbb{Z}_2$-Thurston norm of any $\mathbb{Z}_2$-homology class equals either zero or the minimum translation distance under the action; and for torus semi-bundles, it equals either zero or the translation distance of a specific curve class. Moreover, we construct incompressible surfaces to realize all the $\mathbb{Z}_2$-homology classes. As a consequence, for any torus bundle over the circle or torus semi-bundle, we determine which non-orientable closed surfaces can be embedded in it.