The Milnor–Wood inequality for Euler numbers of hyperbolic 4-manifold bundles

Let $M=M(e,g)$ be homeomorphic to an $\mathbb R^2$-bundle over a closed orientable surface of genus $g>1$, with Euler number $e=\langle[F],[F]\rangle$ given by the self-intersection of the zero section $F$. Milnor–Wood inequality. The inequality

$$0\leq |e|\leq 2g-2$$

is necessary for $M$ to admit a complete hyperbolic structure. This gives a necessary topological restriction on the Euler number of such bundles; the supplied text does not establish whether the bound is sufficient.