Achiral Lefschetz fibrations and the moduli space of curves

Symplectic Lefschetz fibrations can be described via classifying maps with values in the Deligne-Mumford compactification of the moduli space of curves, by means of constructions relying on symplectic geometry. In this note we prove the existence of classifying maps for achiral Lefschetz fibrations using Riemannian geometry. We further extend the Smith signature formula to the achiral case, providing a more elementary proof to this statement.