Achiral Lefschetz fibrations and the moduli space of curves

TL;DR

This paper demonstrates the existence of classifying maps for achiral Lefschetz fibrations using Riemannian geometry and extends the Smith signature formula to include achiral cases, offering a more elementary proof.

Contribution

It introduces a Riemannian geometric approach to achiral Lefschetz fibrations and generalizes the Smith signature formula to this setting.

Findings

Existence of classifying maps for achiral Lefschetz fibrations.

Extension of Smith signature formula to achiral case.

Elementary proof of the Smith signature formula in this context.

Abstract

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.